Research ArticleSimilarity Solution for Free Convection Flow ofa Micropolar Fluid under Convective Boundary Condition viaLie Scaling Group Transformations

Ch RamReddy T Pradeepa and D Srinivasacharya

Department of Mathematics National Institute of Technology Warangal 506004 India

Correspondence should be addressed to Ch RamReddy chittetiramgmailcom

Received 19 January 2015 Revised 26 April 2015 Accepted 27 April 2015

Academic Editor Ming Liu

Copyright copy 2015 Ch RamReddy et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

The free convective flow of an incompressible micropolar fluid along permeable vertical plate under the convective boundarycondition is investigated The Lie scaling group of transformations is applied to get the similarity representation for the systemof partial differential equations and then the resulting systems of equations are solved using spectral quasi-linearisation methodA quantitative comparison of the numerical results is made with previously published results for special cases and the results arefound to be in good agreement The results of the physical parameters on the developments of flow temperature concentrationskinfriction wall couple stress heat transfer and mass transfer characteristics along vertical plate are given and the salient featuresare discussed

1 Introduction

In the past few decades most of the researchers consideredconvective heat transfer problems with either constant walltemperature (CWT) constant heat flux (CHF) or Newtonianheating (NH) in a Newtonian andor non-Newtonian fluidRecently a novel mechanism for the heating process hasdrawn the involvement of many researchers namely convec-tive boundary condition (CBC) where the heat is supplied tothe convecting fluid through a bounding surface with a finiteheat capacity Further this results in the heat transfer ratethrough the surface being proportional to the local differencein temperature with the ambient conditions (Merkin [1])Besides it ismore general and realistic particularly in varioustechnologies and industrial operations such as transpirationcooling process textile drying and laser pulse heating Aziz[2] reported similarity solution for thermal boundary layerflow over a flat plate in a uniform stream of fluid withthe convective boundary condition and he concluded that asimilarity solution is possible if the convective heat transferrelated to hot fluid on the lower surface of the plate is

proportional to the inverse square root of the axial lengthIn the presence of an internal heat generation local similaritysolution for free convection heat transfer from a movingvertical plate with the convective boundary condition isdiscussed by Makinde [3] The laminar natural convectionflow over a semi-infinite moving vertical plate under theconvective boundary condition is examined by Ibrahim andBhashar Reddy [4] RamReddy et al [5] investigated theinfluence of the prominent Soret effect on mixed convectionin a nanofluid under the convective boundary conditionsThenonsimilar result has been presented for the free convectionboundary layer flow along a solid sphere under the convectiveboundary conditions by Alkasasbeh et al [6] More recentlya note on the natural convection along convectively heatedvertical plate is given by Pantokratoras [7]

One of the best established theories of fluids withmicrostructure is the theory of micropolar fluids and thistheory can be found in the books by Lukaszewicz [8] andEremeyev et al [9] It has gathered a good deal of attentiondue to the obvious reasons that theNavier Stokes equation forNewtonian fluids cannot successfully explain the attributes of

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015 Article ID 650813 16 pageshttpdxdoiorg1011552015650813

2 Advances in High Energy Physics

fluids with a substructure Physically the micropolar fluidsmay be treated as non-Newtonian fluids consisting of dumb-bell molecules or rigid cylindrical element polymer fluidsfluid suspension animal blood and so forth Further thetheory of micropolar fluids includes microrotation as wellas microinertia effects This theory studies viscous fluids inwhich microconstituents are rigid and spherical or randomlyoriented as well The subject of free convection boundarylayer flow in a micropolar fluid has been keyed out byseveral investigators due to its immense applications in theengineering problems such as solar energy collecting devicesair conditioning of a room material processing and passivecooling of nuclear reactors The boundary layer flow overa semi-infinite flat plate is considered for analyzing theoryof micropolar fluid and its application to low concentrationsuspension flow by Ahmadi [10] Rees and Pop [11] discussedthe free convection boundary layer flow of a micropolar fluidfrom a vertical flat plate The nonsimilarity transformationsare used to analyze the effects of double stratification onfreemixed convective transport in a micropolar fluid bySrinivasacharya and RamReddy [12ndash14] (also see the ref-erences cited therein) The problems of a steady laminarstagnation point flow towards a stretchingshrinking sheetin an incompressible micropolar fluid under the convectivesurface boundary condition are discussed by Yacob and Ishak[15] and Zaimi and Ishak [16] Merely from the literature itis noted that the majority of the researchers have found thelocal similarity or nonsimilarity solutions for the problemsinvolving convective boundary conditions since most of theresearchers have taken a convective heat transfer coefficientas a function 119909 for getting the similarity solutions in theirproblems Nevertheless the assumption of a heat transfercoefficient varying along the plate as a function of 119909 is notrealistic and very difficult to be obtained in practice For thatcause it could be supposed that the above works have onlytheoretical value

In the recent past several researchers are focused onobtaining the similarity solutions of the convective transportphenomena problems arising in fluid dynamics aerodynam-ics plasma physics meteorology and some branches of engi-neering by using different procedures One such procedureis Lie group analysis The concept of Lie group analysis alsocalled symmetry analysis is developed by Sophius Lie todetermine transformations which map a given differentialequation to itself and it unifies almost all known exactintegration techniques (see [17ndash19]) It provides a potentsophisticated and systematic tool for generating the invariantsolutions of the system of nonlinear partial differential equa-tions (PDEs) with relevant initial or boundary conditionsA special form of Lie group transformations known as thescaling group has been suggested by various researchers tostudy convection flows of different flow phenomena (seeTapanidis et al [20] Hassanien and Hamad [21] Kandasamyet al [22] Aziz et al [23] Mutlag et al [24] etc they areworth observing)

From the literature survey it seems that the problem ofthe free convective heat and mass transport along permeablevertical plate in a micropolar fluid under the convective

boundary condition has not been investigated so far Moti-vated by all these works this paper attempts to present thenew similarity transformations and corresponding similaritysolution to investigate the free convection flow of a micropo-lar fluid under the convective boundary condition using theLie group transformations The mathematical model involv-ing the convective boundary conditions becomes slightlymore complicated leading to the complex interactions of theflow heat and mass transfer mechanism Further the analyt-ical solution is out of scope in the present set-up and hence anumerical solution is obtained for the current problem Alsothe influence of important parameters namely micropolarsuctioninjection and convective heat transfer parameterson the physical quantities of the flow heat and mass transferrates is analyzed in different flow situations

2 Mathematical Formulation

Consider the steady laminar and free convective flow ofan incompressible micropolar fluid with the free streamtemperature and concentration 119879

infinand 119862

infin respectively

Choose the coordinate system such that the 119909-axis is alongthe vertical plate and 119910-axis normal to the plate as shownin Figure 1 The suctioninjection velocity distribution isassumed to be V

119908 The plate is either heated or cooled from

left by convection from a fluid of temperature 119879119891with 119879

119891gt

119879infin

corresponding to a heated surface (assisting flow) and119879119891lt 119879infin

corresponding to a cooled surface (opposing flow)respectively On thewall concentration is taken to be constantand is given by 119862

119908

By employing Boussinesq approximation and makinguse of the standard boundary layer approximations thegoverning equations for the micropolar fluid [10] are givenby

120597119906

120597119909

+

120597V120597119910

= 0 (1)

120588(119906

120597119906

120597119909

+ V120597119906

120597119910

)

= (120583 + 120581)

1205972119906

1205971199102 + 120581

120597120596

120597119910

+120588119892lowast

(120573119879(119909) (119879 minus119879

infin) + 120573119862(119909) (119862minus119862

infin))

(2)

120588119895 (119906

120597120596

120597119909

+ V120597120596

120597119910

) = 120574

1205972120596

1205971199102 minus 120581(2120596+

120597119906

120597119910

) (3)

119906

120597119879

120597119909

+ V120597119879

120597119910

= 120572

1205972119879

1205971199102 (4)

119906

120597119862

120597119909

+ V120597119862

120597119910

= 119863

1205972119862

1205971199102 (5)

where 119906 and V are the velocity components in 119909 and 119910

directions respectively 120596 is the component of microrotationwhose direction of rotation lies in the 119909 119910-plane 119879 is thetemperature 119862 is the concentration 119892lowast is the accelerationdue to gravity 120588 is the density 120583 is the dynamic coefficient

Advances in High Energy Physics 3

x

y

glowast

u = 0

= w

u

C rarr Cinfin

T rarr Tinfinminusk

120655T

120655y= hf(Tf minus T)

C = Cw

Figure 1 Physical model and coordinate system

of viscosity 120573119879(119909) is the volumetric coefficient of thermal

expansion 120573119862(119909) is the volumetric coefficient of solutal

expansions 120581 is the vortex viscosity 119895 is the microinertiadensity 120574 is the spin-gradient viscosity 120572 is the thermaldiffusivity and119863 is the solutal diffusivity of the medium

The boundary conditions are

119906 = 0

V = V119908

120596 = minus 119899

120597119906

120597119910

minus 119896

120597119879

120597119910

= ℎ119891(119879119891minus119879)

119862 = 119862119908

at 119910 = 0

(6a)

119906 = 0

120596 = 0

119879 = 119879infin

119862 = 119862infin

as 119910 997888rarr infin

(6b)

where subscripts119908 andinfin indicate the conditions at the walland at the outer edge of the boundary layer respectively ℎ

119891

is the convective heat transfer coefficient 119896 is the thermalconductivity of the fluid and 119899 is amaterial constant Furtherwe follow the work of many recent authors by assuming that120574 = (120583 + 1205812)119895 This assumption is invoked to allow the fieldof equations to predict the correct behavior in the limitingcase when the microstructure effects become negligible andthe total spin 120596 reduces to the angular velocity [10]

3 Nondimensionalization ofthe Governing Equations

Introduce the following dimensionless variables

119909 =

119909

119871

119910 =

119910

119871

Gr14

119906 =

119871

]Gr12119906

V =119871

]Gr14V

120596 =

1198712

]Gr34120596

120579 =

119879 minus 119879infin

119879119891minus 119879infin

120601 =

119862 minus 119862infin

119862119908minus 119862infin

(7)

where Gr = 119892lowast

1205731198790(119879119891minus 119879infin)119871

3]2 is the Grashof number

In view of the continuity equation (1) we introduce thestream function 120595 by

119906 =

120597120595

120597119910

V = minus

120597120595

120597119909

(8)

Using (7) and (8) into (2)ndash(5) we get the following momen-tum angular momentum energy and concentration equa-tions

Δ 1 =120597120595

120597119910

1205972120595

120597119909120597119910

minus

120597120595

120597119909

1205972120595

1205971199102 minus(

11 minus 119873

)[

1205973120595

1205971199103 minus119873

120597120596

120597119910

]

minus

119892lowast

120573119879(119909) (119879

119891minus 119879infin) 119871

3

]2Gr120579

minus

119892lowast

120573119862(119909) (119862

119908minus 119862infin) 119871

3

]2Gr120601 = 0

Δ 2 =120597120595

120597119910

120597120596

120597119909

minus

120597120595

120597119909

120597120596

120597119910

minus(

2 minus 119873

2 minus 2119873)

1205972120596

1205971199102

+(

119873

1 minus 119873

)[2120596+

1205972120595

1205971199102 ] = 0

Δ 3 =120597120595

120597119910

120597120579

120597119909

minus

120597120595

120597119909

120597120579

120597119910

minus

1Pr

1205972120579

1205971199102 = 0

Δ 4 =120597120595

120597119910

120597120601

120597119909

minus

120597120595

120597119909

120597120601

120597119910

minus

1Sc

1205972120601

1205971199102 = 0

(9)

In usual definitions ] is the kinematic viscosity Pr = ]120572 isthe Prandtl number Sc = ]119863 is the Schmidt number 119873 =

120581(120583 + 120581) (0 le 119873 lt 1) is the coupling number [25] and themicroinertia density is taken to be 119895 = 119871

2Gr12

4 Advances in High Energy Physics

Now boundary conditions (6a) and (6b) become

120597120595

120597119910

= 0

120597120595

120597119909

= 119891119908

120596 = minus 119899

1205972120595

1205971199102

120597120579

120597119910

= minusBi (1minus 120579)

120601 = 1

at 119910 = 0

(10a)

120597120595

120597119910

= 0

120596 = 0

120579 = 0

120601 = 0

as 119910 997888rarr infin

(10b)

where 119891119908= minus(119871]Gr14)V

119908is the suctioninjection parame-

ter It is worth mentioning that 119891119908determines the transpira-

tion rate at the surface with119891119908gt 0 for suction and119891

119908lt 0 for

injection and119891119908= 0 corresponds to an impermeable surface

Further Bi = ℎ119891119871119896Gr14 is the Biot number It is a ratio of

the internal thermal resistance of the plate to the boundarylayer thermal resistance of the hot fluid at the bottom of thesurface

4 Similarity Equations via Lie Scaling GroupTransformations

Aone-parameter Lie scaling group of transformations whichis a simplified form of Lie group transformation is selected as(for more see [26ndash31])

Γ 119909lowast

= 1199091198901205761205721

119910lowast

= 1199101198901205761205722

120595lowast

= 1205951198901205761205723

120596lowast

= 1205961198901205761205724

120579lowast

= 1205791198901205761205725

120601lowast

= 1206011198901205761205726

120573lowast

119879= 1205731198791198901205761205727

120573lowast

119862= 1205731198621198901205761205728

(11)

Here 120576 = 0 is the parameter of the group and 120572119894(where

119894 = 1 2 3 8) are arbitrary real numbers whose interrela-tionship will be determined by our analysis Transformations

in (11) may be treated as a point transformation transformingthe coordinates

(119909 119910 120595 120596 120579 120601 120573119879 120573119862)

= (119909lowast

119910lowast

120595lowast

120596lowast

120579lowast

120601lowast

120573lowast

119879 120573lowast

119862)

(12)

We now investigate the relationship among the exponents 120572119894

(where 119894 = 1 2 3 8) such that

Δ119895[119909lowast

119910lowast

119906lowast

Vlowast 1205973120595lowast

120597119910lowast3 ]

= 119867119895[119909 119910 119906 V

1205973120595

1205971199103 119886]

sdot Δ119895[119909 119910 119906 V

1205973120595

1205971199103 ] (119895 = 1 2 3 4)

(13)

This is the requirement that the differential formsΔ 1Δ 2Δ 3and Δ 4 are conformally invariant under transformation (11)Substituting transformations (11) in (9) we have

Δ 1 = 119890120576(1205721+21205722minus21205723)

(

120597120595lowast

120597119910lowast

1205972120595lowast

120597119909lowast120597119910lowastminus

120597120595lowast

120597119909lowast

1205972120595lowast

120597119910lowast2 )

minus(

11 minus 119873

) 119890120576(31205722minus1205723) 120597

3120595lowast

120597119910lowast3

minus(

119873

1 minus 119873

) 119890120576(1205722minus1205724) 120597120596

lowast

120597119910lowast

minus

119892lowast

120573lowast

119879(119879119891minus 119879infin) 119871

3

]2Gr119890minus120576(1205725+1205727)

120579lowast

minus

119892lowast

120573lowast

119862(119862119908minus 119862infin) 119871

3

]2Gr119890minus120576(1205726+1205728)

120601lowast

= 0

(14a)

Δ 2 = 119890120576(1205721+1205722minus1205723minus1205724)

(

120597120595lowast

120597119910lowast

120597120596lowast

120597119909lowastminus

120597120595lowast

120597119909lowast

120597120596lowast

120597119910lowast)

minus(

2 minus 119873

2 minus 2119873) 119890120576(21205722minus1205724) 120597

2120596lowast

120597119910lowast2

+(

119873

1 minus 119873

)(2120596lowast119890minus1205761205724 + 119890120576(21205722minus1205723) 120597

2120595lowast

120597119910lowast2 )

= 0

(14b)

Δ 3 = 119890120576(1205721+1205722minus1205723minus1205725)

(

120597120595lowast

120597119910lowast

120597120579lowast

120597119909lowastminus

120597120595lowast

120597119909lowast

120597120579lowast

120597119910lowast)

minus

1Pr

119890120576(21205722minus1205725)

(

1205972120579lowast

120597119910lowast2) = 0

(14c)

Δ 4 = 119890120576(1205721+1205722minus1205723minus1205726)

(

120597120595lowast

120597119910lowast

120597120601lowast

120597119909lowastminus

120597120595lowast

120597119909lowast

120597120601lowast

120597119910lowast)

minus

1Sc

119890120576(21205722minus1205726)

(

1205972120601lowast

120597119910lowast2) = 0

(14d)

Advances in High Energy Physics 5

Now boundary conditions (10a) and (10b) become

119890120576(1205722minus1205723)

120597120595lowast

120597119910lowast= 0

119890120576(1205721minus1205723)

120597120595lowast

120597119909lowast= 119891119908

119890minus1205761205724

120596lowast

= minus 119899119890120576(21205722minus1205723) 120597

2120595lowast

120597119910lowast2

119890120576(1205722minus1205725) 120597120579

lowast

120597119910lowast= minusBi (1minus 119890

minus1205761205725120579lowast

)

119890minus1205761205726

120601lowast

= 1at 119910lowast = 0

(15a)

119890120576(1205722minus1205723)

120597120595lowast

120597119910lowast= 0

119890minus1205761205724

120596lowast

= 0119890minus1205761205725

120579lowast

= 0119890minus1205761205726

120601lowast

= 0as 119910lowast 997888rarr infin

(15b)

The system remains invariant under the group transforma-tion Γ We then have the following relationships for theparameters

1205721 + 21205722 minus 21205723 = 31205722 minus1205723 = 1205722 minus1205724 = minus1205725 minus1205727= minus1205726 minus1205728

1205721 +1205722 minus1205723 minus1205724 = 21205722 minus1205724 = minus1205724 = 21205722 minus1205723

1205721 +1205722 minus1205723 minus1205725 = 21205722 minus1205725

1205721 +1205722 minus1205723 minus1205726 = 21205722 minus1205726

1205721 minus1205723 = 0

minus 1205724 = 21205722 minus1205723

1205722 minus1205725 = 0 = minus1205725

1205726 = 0

(16)

Solving linear system (16) we have the following relationshipamong the exponents

1205721 = 1205723 = 1205724 = 1205727 = 1205728

1205722 = 1205725 = 1205726 = 0(17)

The set of transformations Γ reduces to

119909lowast

= 1199091198901205761205721

119910lowast

= 119910

120595lowast

= 1205951198901205761205721

120596lowast

= 1205961198901205761205721

120579lowast

= 120579

120601lowast

= 120601

120573lowast

119879= 1205731198791198901205761205721

120573lowast

119862= 1205731198621198901205761205721

(18)

Expanding by theTaylor series in power of 120576 keeping the termup to the first degree (neglecting higher power of 120576) we get

119909lowast

minus119909 = 1205761205721119909

119910lowast

= 119910

120595lowast

minus120595 = 1205761205721120595

120596lowast

minus120596 = 1205761205721120596

120579lowast

= 120579

120601lowast

= 120601

120573lowast

119879minus120573119879= 1205761205721120573119879

120573lowast

119862minus120573119862= 1205761205721120573119862

(19)

The characteristic equations are

119889119909

1205721119909=

119889119910

0=

119889120595

1205721120595=

119889120596

1205721120596=

119889120579

0=

119889120601

0=

119889120573119879

1205721120573119879

=

119889120573119862

1205721120573119862

(20)

Solving the above characteristic equations we have thefollowing similarity transformations

120578 = 119910

120595 = 119909119891 (120578)

120596 = 119909119892 (120578)

120573119879= 1205731198790119909

120573119862= 1205731198620119909

120579 = 120579 (120578)

120601 = 120601 (120578)

(21)

where 1205731198790and 120573

1198620are constant thermal and mass coefficient

of expansionUsing (21) into (9) we get the following similarity

equations

(

11 minus 119873

)119891101584010158401015840

+11989111989110158401015840

minus11989110158402+(

119873

1 minus 119873

)1198921015840

+ 120579+B120601

= 0

(

2 minus 119873

2 minus 2119873)11989210158401015840

+1198911198921015840

minus1198911015840

119892minus(

119873

1 minus 119873

) (2119892+11989110158401015840

)

= 0

1Pr

12057910158401015840

+1198911205791015840

= 0

1Sc

12060110158401015840

+1198911206011015840

= 0

(22)

where the primes indicate differentiation with respect to 120578

alone andB = 1205731198620(119862119908minus 119862infin)1205731198790(119879119891minus 119879infin) is the buoyancy

ratio

6 Advances in High Energy Physics

Boundary conditions (10a) and (10b) in terms of 119891 119892 120579and 120601 become

120578 = 0 119891 (0) = 119891119908

1198911015840

(0) = 0

119892 (0) = minus 11989911989110158401015840

(0)

1205791015840

(0) = minusBi [1minus 120579 (0)]

120601 (0) = 1

(23a)

120578 997888rarr infin 1198911015840

(infin) = 0

119892 (infin) = 0

120579 (infin) = 0

120601 (infin) = 0

(23b)

5 Skin Friction Wall Couple Stress and Heatand Mass Transfer Coefficients

The wall shear stress and the wall couple stress are

120591119908= [(120583 + 120581)

120597119906

120597119910

+ 120581120596]

119910=0

119898119908= 120574 [

120597120596

120597119910

]

119910=0

(24a)

and the heat and mass transfers from the plate respectivelyare given by

119902119908= minus 119896 [

120597119879

120597119910

]

119910=0

119902119898= minus119863[

120597119862

120597119910

]

119910=0

(24b)

The nondimensional skin friction 119862119891

= 2120591119908120588119906

2lowast wall

couple stress 119872119908

= 119898119908120588119906

2lowast119909 the local Nusselt number

119873119906119909= 119902119908119909119896(119879

119891minus 119879infin) and local Sherwood number Sh

119909=

119902119898119909119863(119862

119908minus 119862infin) are given by

119862119891Gr14119909

= 2(1 minus 119899119873

1 minus 119873

)11989110158401015840

(0)

119872119908Gr12119909

= (

2 minus 119873

2 minus 2119873)1198921015840

(0)

119873119906119909

Gr14119909

= minus 1205791015840

(0)

Sh119909

Gr14119909

= minus1206011015840

(0)

(25)

where 1199062lowastis the characteristic velocity and Gr

119909= 119892lowast

1205731198790(119879119891minus

119879infin)119909

3]2 is the local Grashof number

6 Numerical Solution Using the SpectralQuasi-Linearization Method (SQLM)

In this section we describe the quasi-linearization method(QLM) for solving the governing system of (22) alongwith boundary conditions (23a) and (23b) This QLM isa generalization of the Newton-Raphson method and wasproposed by Bellman and Kalaba [32] for solving nonlinearboundary value problems

Assume that the solutions 119891119903 119892119903 120579119903 and 120601

119903of (22) at the

(119903+1)th iteration are119891119903+1 119892119903+1 120579119903+1 and 120601119903+1 If the solutions

at the previous iteration are sufficiently close to the solutionsat the present iteration the nonlinear components of (22)can be linearised using one-term Taylor series of multiplevariables so that (22) give the following iterative sequence oflinear differential equations

(

11 minus 119873

)119891101584010158401015840

119903+1 + 119886111990311989110158401015840

119903+1 + 11988621199031198911015840

119903+1 + 1198863119903119891119903+1

+(

119873

1 minus 119873

)1198921015840

119903+1 + 120579119903+1 +B120601

119903+1 = 1198771119903

(

2 minus 119873

2 minus 2119873)11989210158401015840

119903+1 + 11988731199031198921015840

119903+1 + 1198874119903119892119903+1 + 11988711199031198911015840

119903+1

+ 1198872119903119891119903+1 minus(

119873

1 minus 119873

)11989110158401015840

119903+1 = 1198772119903

1198881119903119891119903+1 +1Pr

12057910158401015840

119903+1 + 11988821199031205791015840

119903+1 = 1198773119903

1198891119903119891119903+1 +1Sc

12060110158401015840

119903+1 +11988921199031206011015840

119903+1 = 1198774119903

(26)

where the coefficients 1198861199041 119903

(1199041 = 1 2 3) 1198871199042 119903

(1199042 = 1 2 4) 1198881199043 119903

(1199043 = 1 2) 1198891199044 119903

(1199044 = 1 2) and 1198771199045 119903

(1199045 = 1 2 4) are known functions (from previous iterations) and aredefined as

1198861119903 = 119891119903

1198862119903 = minus 21198911015840119903

1198863119903 = 11989110158401015840

119903

1198771119903 = 11989111990311989110158401015840

119903minus (1198911015840

119903)

2

1198871119903 = minus119892119903

1198872119903 = 1198921015840

119903

1198873119903 = 119891119903

1198874119903 = minus1198911015840

119903minus(

21198731 minus 119873

)

1198772119903 = 1198911199031198921015840

119903minus1198911015840

119903119892119903

1198881119903 = 1205791015840

119903

1198882119903 = 119891119903

1198773119903 = 1198911199031205791015840

119903

1198891119903 = 1206011015840

119903

1198892119903 = 119891119903

1198774119903 = 1198911199031206011015840

119903

(27)

Advances in High Energy Physics 7

subject to boundary conditions

119891119903+1 (0) = 119891

119908

1198911015840

119903+1 = 0

1198911015840

119903+1 (infin) = 0

119892119903+1 = minus 119899119891

10158401015840

119903+1 (0)

119892119903+1 (infin) = 0

1205791015840

119903+1 (0) = minusBi (1minus 120579 (0))

120579119903+1 (infin) = 0

120601119903+1 (0) = 1

120601119903+1 (infin) = 0

(28)

System (26) constitutes a linear system of coupled differen-tial equations with variable coefficients and can be solvediteratively using any numerical method for 119903 = 1 2 3 In this work as will be discussed below the Chebyshevpseudospectral method was used to solve the QLM scheme(26) (for more details refer to the works of Motsa et al[33 34])

1198910 (120578) = 119891119908+ 1minus 119890

minus120578

1198920 (120578) = minus 119899119890minus120578

1205790 = 119890minus120578

BiBi + 1

1206010 = 119890minus120578

(29)

and starting from these sets of initial approximations 1198910 11989201205790 and 1206010 the iteration schemes (26) can be solved iterativelyfor 119891119903+1(120578) 119892119903+1(120578) 120579119903+1(120578) and120601119903+1(120578) when 119903 = 0 1 2

For this we discretise the equation using the Chebyshevspectral collocation method The unknown functions areapproximated by the Chebyshev interpolating polynomialsin such way that they are collocated at the Gauss-Lobattocollocation points defined as

120591119895= cos

120587119895

119873

119895 = 0 1 2 119873 (30)

where 119873 is the number of collocation points The physicalregion [0infin) is transformed into the region [minus1 1] using thedomain truncation technique in which the problem is solvedon the interval [0 120578

infin] instead of [0infin) This leads to the

mapping

120578

120578infin

=

120591 + 12

minus1 le 120591 le 1 (31)

where 120578infin

is the scaling parameter used to invoke theboundary condition at infinity The functions 119891 119892 120579 and 120601

are approximated at the collocation points by

119891 (120591) =

119873

sum

119896=0119891 (120591119896) 119879119896(120591119895)

119892 (120591) =

119873

sum

119896=0119892 (120591119896) 119879119896(120591119895)

120579 (120591) =

119873

sum

119896=0120579 (120591119896) 119879119896(120591119895)

120601 (120591) =

119873

sum

119896=0120601 (120591119896) 119879119896(120591119895)

119895 = 0 1 2 119873

(32)

where 119879119896is the 119896th Chebyshev polynomial defined as

119879119896(120591) = cos [119896 cosminus1 (120591)] (33)

The derivatives of the variables at the collocation points arerepresented as

119889119901

119891

119889120578119901=

119873

sum

119896=0D119901119897119896119891 (120591119896)

119889119901

119892

119889120578119901=

119873

sum

119896=0D119901119897119896119892 (120591119896)

119889119901

120579

119889120578119901=

119873

sum

119896=0D119901119897119896120579 (120591119896)

119889119901

120601

119889120578119901=

119873

sum

119896=0D119901119897119896120601 (120591119896)

119897 = 0 1 119873

(34)

where 119901 is the order of the derivative and D = 2D120578infin

is theChebyshev spectral differentiation matrix and its entries areclearly defined in Canuto et al [35]

Substituting (31)ndash(34) into (26) leads to the matrix equa-tion

119860119883 = 119877 (35)

subject to the boundary conditions

119891119903+1 (120591119873) = 119891

119908

119873

sum

119896=0D119873119896

119891 (120591119896) = 0

119873

sum

119896=0D0119896119891 (120591

119896) = 0

119892119903+1 (120591119873) = minus 119899

119873

sum

119896=0D2119873119896

119891 (120591119896)

119892119903+1 (1205910) = 0

8 Advances in High Energy Physics

119873

sum

119896=0D119873119896

120579119903+1 (120591119896) minusBi 120579

119903+1 (120591119873) = minusBi

120579119903+1 (1205910) = 0

120601119903+1 (120591119873) = 1120601119903+1 (1205910) = 0

(36)

In (35) 119860 is a (4119873+ 4) times (4119873+ 4) square matrix and119883 and 119877are (4119873 + 1) times 1 column vectors defined by

119860 =

[

[

[

[

[

[

11986011 11986012 11986013 11986014

11986021 11986022 11986023 11986024

11986031 11986032 11986033 11986034

11986041 11986042 11986043 11986044

]

]

]

]

]

]

119883 =

[

[

[

[

[

[

F119903+1

G119903+1

Θ119903+1

Φ119903+1

]

]

]

]

]

]

119877 =

[

[

[

[

[

[

R1

R2

R3

R4

]

]

]

]

]

]

(37)

whereF = [119891

119903+1 (1205910) 119891119903+1 (1205911) 119891119903+1 (120591119873)]119879

G = [119892119903+1 (1205910) 119892119903+1 (1205911) 119892119903+1 (120591119873)]

119879

Θ = [120579119903+1 (1205910) 120579119903+1 (1205911) 120579119903+1 (120591119873)]

119879

Φ = [120601119903+1 (1205910) 120601119903+1 (1205911) 120601119903+1 (120591119873)]

119879

11986011 = (

11 minus 119873

)D3+ diag [1198861119903]D

2+ diag [1198862119903]D

+ diag [1198863119903]

11986012 = (

119873

1 minus 119873

)D

11986013 = I

11986014 = BI

11986021 = minus(

119873

1 minus 119873

)D2+ diag [1198871119903]D+ diag [1198872119903]

11986022 = (

2 minus 119873

2 minus 2119873)D2

+ diag [1198873119903]D+ diag [1198874119903]

11986023 = 0

11986024 = 0

11986031 = diag [1198881119903]

11986032 = 0

11986033 =1Pr

D2+ diag [1198882119903]D

Table 1 Comparison of minus1205791015840(0) for free convection along a verticalflat plate in Newtonian fluid when 119873 = 0 119899 = 0 B = 0 Pr = 1Bi rarr infin and 119891

119908= 0

Merkin [36] Nazar et al [37] Molla et al [38] Present04214 04214 04214 04214313

11986034 = 0

11986041 = diag [1198891119903]

11986042 = 0

11986043 = 0

11986044 =1Sc

D2+ diag [1198892119903]D

R1 = 1198771119903

R2 = 1198772119903

R3 = 1198773119903

R4 = 1198774119903

(38)

and here I is an identity matrix the size of the matrix 0 is(119873+1)times1 and diag[ ] is a diagonalmatrix of size (119873+1)times(119873+

1) Subscript r denotes the iteration number After modifyingmatrix system (35) to incorporate boundary condition (36)the solution is obtained as

119883 = 119860minus1119877 (39)

7 Results and Discussions

It is noticed that the present problem reduces to free convec-tion heat transfer along an impermeable vertical plate in amicropolar fluid without the convective boundary conditionwhen119891

119908= 0 Bi rarr infin andB = 0 Also in the limit as119873 rarr

0 governing equations (2)ndash(5) reduce to the correspondingequations for a free convection heat and mass transfer ina viscous fluid In order to validate the code generated theresults of the present problem have been compared with theresults obtained by Merkin [36] Nazar et al [37] and Mollaet al [38] as a special case by taking 119873 = 0 119899 = 0 B = 0Pr = 1 Bi rarr infin and 119891

119908= 0 and it was found that

they are in good agreement as presented in Table 1 Also thecomparison of heat transfer coefficient has been made withthe results obtained by Nazar et al [37] as shown in Table 2when 119899 = 05 B = 0 Pr = 1 Bi rarr infin and 119891

119908= 0

To study the effects of coupling number119873 suctioninjectionparameter 119891

119908 Biot number Bi and material parameter 119899

computations were carried out in the cases of B = 10Pr = 071 and Sc = 022

The effects of the coupling number119873 on the dimension-less velocity microrotation temperature and concentrationare illustrated in Figures 2(a)ndash2(d) with fixed values ofother parameters The coupling number 119873 characterizes thecoupling of linear and rotational motion arising from the

Advances in High Energy Physics 9

00

01

02

03

04

05

06f998400

0 4 8 12 16

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(a)

g

minus06

minus05

minus04

minus03

minus02

minus01

00

01

0 2 4 6 8 10 12

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(b)

000

004

008

012

016

120579

0 1 2 3 4 5 6

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(c)

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

120601

00

02

04

06

08

10

120578

0 2 4 6 8 10 12

Bi = 01 fw = 05

(d)

Figure 2 Effect of119873 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Table 2 Comparison of minus1205791015840

(0) for free convection flow in amicropolar fluid obtained by Nazar et al [37] when 119899 = 05B = 0Pr = 1 Bi rarr infin and 119891

119908= 0

119873 Nazar et al [37] Present000 04214 04214033 03991 03990050 03834 03834060 03709 03709066 03608 03608071 03522 03522075 03447 03447

fluid particles In the case of 119873 = 0 (ie as 120581 tends tozero) the micropolarity is absent and fluid becomes nonpolarfluid With a large value of 119873 effect of microstructurebecomes significant whereas with a diminished value of 119873the individuality of the substructure is much less articulatedAs 119873 increases it is found from Figure 2(a) that the max-imum velocity decreases in amplitude and the location ofthe maximum velocity moves farther away from the wallSince 119873 rarr 0 corresponds to viscous fluid the velocity incase of a micropolar fluid has been less compared to thatof viscous fluid case It can be observed from Figure 2(b)that as 119873 increases initially microrotation profiles tend tobecome flatter and then approach their free stream valuesfar away from the wall This happens due to the vanishing

10 Advances in High Energy Physics

00

01

02

03

04

05

06

f998400

120578

0 2 4 6 8 10 12

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(a)

minus06

minus04

minus03

minus02

minus05

minus01

00

01

g

0 2 4 6 8 10

120578

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(b)

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

120578

0 1 2 3 4 5

120579

00

02

04

06

08

10

(c)

120601

00

02

04

06

08

10

fw = 05 n = 05N = 05

0 2 4 6 8 10 12

120578

Bi = 01

Bi = 10

Bi = 50Bi = 200

(d)

Figure 3 Effect of Bi on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

of antisymmetric part of the stress on the boundary thatcorresponds to a weak concentration of microelements Thisis because an increment in the value of 119873 implies a highervortex viscosity of fluid which promotes the microrotationof micropolar fluids It is seen from Figures 2(c) and 2(d)that thermal and concentration boundary layers of thefluid increase with increase in coupling number 119873 Hencetemperature and concentration in case of micropolar fluidsare more than those of the viscous fluid case

The Biot number Bi is the ratio of internal thermalresistance of a solid to boundary layer thermal resistanceWhen Bi = 0 the plate is totally insulated internal thermalresistance of the plate is extremely high and no convectiveheat transfer to the cold fluid on the upper part of the

plate takes place Figure 3(a) depicts fluid velocity profilesfor different values of the Biot number with 119873 = 05 119891

119908=

05 and 119899 = 05 Generally fluid velocity is zero at platesurface and increases gradually away fromplate to free streamvalue satisfying boundary conditions It is interesting to notethat an increase in the intensity of convective surface heattransfer Bi produces significant enhancement in fluid velocitywithin the momentum boundary layer In Figure 3(b) webring out the behavior of microrotation with different valuesof Biot number Bi for fixed values of other parameters As theparameter value Bi increases microrotation showing reverserotation near the two boundaries Hence the condition ofvanishing of antisymmetric part of the stress on the boundaryresults in a drastic change of the microrotation profiles

Advances in High Energy Physics 11

07

06

05

04

03

02

01

00

f998400

fw = 00

fw = 10

fw = 20

0 3 6 9 12 15

120578

N = 05 Bi = 01 n = 05

fw = minus05

(a)

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

120578

0 2 4 6 8 10 12

g

minus04

minus03

minus02

minus01

00

01

fw = minus05

(b)

0 1 2 3 4 5 6

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

000

005

010

015

020

025

030

120579

fw = minus05

(c)

0 3 6 9 12

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

00

02

04

06

08

10

120601

fw = minus05

(d)

Figure 4 Effect of 119891119908on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Given that convective heating increases with Biot numberBi rarr infin simulates the isothermal surface which is clearlyseen from Figure 3(c) where 120579(0) = 1 as Bi rarr infin Infact a high Biot number indicates higher internal thermalresistance of the plate than boundary layer thermal resistanceIn this fluid temperature is maximum at the plate surfaceand decreases exponentially to zero value far out from theplate satisfying boundary conditions As a consequence anincrement in the Biot number leads to increase of fluidtemperature efficiency Figure 3(d) illustrates the variation ofdimensionless concentration for different values of Bi It isclear that the concentration of fluid decreases with increaseof Bi

The effect of 119891119908

on velocity profile is depicted inFigure 4(a) Here 119891

119908gt 0 represents suction and 119891

119908lt 0

denotes injectionThe lower velocity is noticed in case of suc-tion when compared to case of injection From Figure 4(b)we note that microrotation is showing reverse rotation neartwo boundaries with both suction and injection parame-ter The dimensionless temperature for different values ofsuctioninjection parameters is drawn in Figure 4(c) It isreadable that the temperature of the fluid is more in case ofinjection whereas it is less in case of suction in comparisonwith the impermeable surface case (119891

119908= 0) Figure 4(d)

demonstrates dimensionless concentration for different val-ues of suctioninjection parameters It is determined that the

12 Advances in High Energy Physics

00

01

02

03

04

05

06f998400

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(a)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10

120578

minus10

minus08

minus06

minus04

minus02

00

g

(b)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

120578

0 1 2 3 4 5

120579

000

003

006

009

012

015

(c)

00

02

04

06

08

10

120601

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(d)

Figure 5 Effect of material parameter 119899 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

concentration of fluid ismore with injection whereas it is lesswith suctionwhen compared to the impermeable surface case(119891119908= 0) As a finale the thermal and solutal boundary layer

thicknesses increase in case of injection compared to case ofsuction as displayed in Figures 4(c) and 4(d)

In Figures 5(a)ndash5(d) the effect of material parameter119899 on dimensionless velocity microrotation temperatureand concentration is presented for fixed values of otherparameters since the material parameter 119899 signifies themicrorotation effects (ie for 119899 = 0 particles are not freeto revolve near the surface whereas as 119899 increases from 0 to1 the microrotation term gets augmented and induces flowenhancement) As 119899 increases it is found from Figure 5(a)that the minimum velocity increases in amplitude and the

location of the minimum velocity moves farther away fromthe wall From Figure 5(b) we observe that themicrorotationis decreasing with increasing value of material parameter119899 within the boundary layer It is clear from Figures 5(c)-5(d) that the thermal and solutal boundary layer thicknessesdecrease with increase of material parameter 119899

The variations of minus1205791015840

(0) and minus1206011015840

(0) versus couplingnumber 119873 are shown in Figures 6ndash8 It can be noticed fromthese figures that the heat and mass transfer coefficientsare less in case of micropolar fluids when compared to theviscous fluids This is because as 119873 increases the thermaland solutal boundary layer thicknesses become larger thusgiving rise to a small value of local heat and mass transferrates The effect of the Biot number Bi with fixed 119891

119908= 05

Advances in High Energy Physics 13

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

fw = 05 n = 05

01

00

02

03

04

05

06

07Nu x

Grminus

14

x

(a)

fw = 05 n = 05

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

024

026

028

030

032

ShxGrminus

14

x

(b)

Figure 6 Effect of Bi on (a) heat transfer rate and (b) mass transfer rate

005

006

007

008

009

010

Nu x

Grminus

14

x

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

(a)

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

01

02

03

04

05

06

ShxGrminus

14

x

(b)

Figure 7 Effect of 119891119908on (a) heat transfer rate and (b) mass transfer rate

and 119899 = 05 on local heat transfer coefficient is exhibitedin Figure 6 It is found from Figure 6 that local heat transferrate increases nonlinearly with the increase in Biot numberBiThe influence of the Biot number Bi on local mass transfercoefficient is shown in Figure 6 Figure 6 reveals that the localmass transfer coefficient is enhanced by the growth in theBiot number Bi In Figure 7 the effect of the suctioninjectionparameter 119891

119908with fixed Bi = 01 and 119899 = 05 on local heat

and mass transfer coefficients is displayed It is found fromFigure 7 that the local heat and mass transfer coefficients areless in the case of injection 119891

119908lt 0 in comparison with the

case of suction 119891119908

gt 0 Figure 8 is prepared to analyze theeffect of the material parameter 119899 with fixed Bi = 01 and

119891119908= 05 on local heat andmass transfer coefficients Figure 8

reveals that the local heat and mass transfer coefficients areenhanced by the increase in material parameter 119899 This isbecause when 119899 increases from 0 to 1 the microrotation termgets augmented and induces flow enhancement

The variations of 119862119891Gr14119909

and 119872119908Gr12119909

which areproportional to the coefficients of skin friction and wallcouple stress are shown in Table 3 with different values ofthe coupling number 119873 for fixed 119899 = 05 119891

119908= 05 and

Bi = 01 It indicates that the skin friction factor is higherfor micropolar fluid than the viscous fluids (119873 = 0) Sincemicropolar fluids offer a heavy resistance (resulting fromvortex viscosity) to fluid movement and cause larger skin

14 Advances in High Energy Physics

0084

0085

0086

0087Nu x

Grminus

14

x

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

fw = 05 Bi = 01

(a)

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

024

026

028

030

ShxGrminus

14

x

fw = 05 Bi = 01

(b)

Figure 8 Effect of material parameter 119899 on (a) heat transfer rate and (b) mass transfer rate

Table 3 Effects of skin friction and wall couple stress for varyingvalues of Biot numbers Bi micropolar parameter 119873 materialparameter 119899 and suctioninjection parameter 119891

119908

119873 Bi 119891119908

119899 119862119891Gr14119909

119872119908Gr12119909

01 01 05 05 2424227 06327903 01 05 05 2502716 063996605 01 05 05 2647987 063086307 01 05 05 2943561 060387909 01 05 05 3874631 055089105 01 05 05 2647987 063086305 10 05 05 3211755 080587905 50 05 05 3533141 090807805 200 05 05 3629791 093913705 01 minus05 05 2426265 033621905 01 00 05 2530014 046542905 01 10 05 2737553 082089905 01 20 05 2721244 119359605 01 05 00 2916655 minus028933905 01 05 05 2647987 063086305 01 05 10 2232556 2066191

friction factor compared to viscous fluid the results as wellsuggest larger values of coupling number 119873 and lower wallcouple stresses Since the skin friction coefficient 11989110158401015840(0) andwall couple stress coefficient as well as high temperature andmass transport rates are more depressed in the micropolarfluid comparing to the viscous fluid which may be beneficialin flow temperature and concentration control of polymerprocessing thus the presence of microscopic effects arisingfrom the local structure and of the fluid elements reduces thehigh temperature and mass transfer coefficients The effectof Bi on 119862

119891Gr14119909

and 119872119908Gr12119909

for 119891119908

= 05 119899 = 05 and

119873 = 05 is illustrated in Table 3 It can be noticed that theskin friction and wall couple stress coefficients are increasingwith increase of Bi for fixed values of other parametersThis notice is consistent with physical profiles presented inFigure 3 The effects of suctioninjection parameter on theskin friction andwall couple stress coefficients are also shownin Table 3 It is noted that the skin friction and wall couplestress coefficients are less with injection case whereas theyare more with suction case when compared to the case ofimpermeable surface Finally the detailed behavior of thematerial parameter 119899 is given in Table 3 The skin frictiondecreases and wall couple stress increases with increase ofmaterial parameter 119899

8 Conclusions

In this composition the similarity solution of the free convec-tion flow on a permeable vertical plate of a micropolar fluidunder the convective boundary condition is obtained usingLie group transformations Using the similarity variables thegoverning equations are transformed into a set of nondi-mensional parabolic equations These equations are solvednumerically using spectral quasi-linearisation method Thenumerical computation is carried out for various values ofnondimensional physical parameters The main findings aresummarized as follows

(i) The numerical results indicate that velocity distribu-tion is less near the plate but it is more far away fromthe plate the wall couple stress coefficient and rate ofheat andmass transfers are lower but the temperatureand concentration distributions and the skin frictioncoefficient are higher for the micropolar fluids incomparison with those of viscous fluids Also thereverse rotation ofmicrorotation near two boundariesis found with the increasing value of119873

Advances in High Energy Physics 15

(ii) An increase in Biot number Bi decreases concen-tration distribution whereas it causes an increasein temperature distribution skin friction and wallcouple stress coefficients and heat and mass transferrates within the boundary layer Further enhancingin Biot number Bi enhances velocity distributionnear the plate but shows the reverse behavior faraway from the plate We observe reverse rotationof microrotation near two boundaries within theboundary layer in the presence of Biot number

(iii) Less velocity temperature and concentration distri-butions are observed more skin friction and wallcouple stress coefficients and heat and mass transferrates in the case of suction compared to the case ofinjection Further microrotation decreases near thewall and depicts the opposite trend far away from thewall

(iv) It is found that microrotation temperature and con-centration distributions and skin friction coefficientsare more in the case of a micropolar fluid with strongconcentration (ie 119899 = 0) when compared to the caseof a micropolar fluid with weak concentration (ie119899 = 12) Further velocity is less in the case of 119899 = 0when compared to the case of 119899 = 12 near the walland shows the opposite trend far away from the wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their valuablesuggestions and comments

References

[1] J H Merkin ldquoNatural-convection boundary-layer flow on avertical surface with Newtonian heatingrdquo International Journalof Heat and Fluid Flow vol 15 no 5 pp 392ndash398 1994

[2] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009

[3] O DMakinde ldquoSimilarity solution for natural convection froma moving vertical plate with internal heat generation and aconvective boundary conditionrdquo Thermal Science vol 15 ppS137ndashS143 2011

[4] S M Ibrahim and N Bhashar Reddy ldquoSimilarity solution ofheat and mass transfer for natural convection over a movingvertical plate with internal heat generation and a convectiveboundary condition in the presence of thermal radiationviscous dissipation and chemical reactionrdquo ISRN Thermody-namics vol 2013 Article ID 790604 10 pages 2013

[5] C RamReddy P V S N Murthy A J Chamkha and A MRashad ldquoSoret effect on mixed convection flow in a nanofluidunder convective boundary conditionrdquo International Journal ofHeat and Mass Transfer vol 64 pp 384ndash392 2013

[6] H T Alkasasbeh M Z Salleh R M Tahar and R NazarldquoNumerical solutions of free convection boundary layer flow ona solid sphere with convective boundary conditionsrdquo Journal ofPhysics Conference Series vol 495 Article ID 012025 2014

[7] A Pantokratoras ldquoA note on natural convection along a con-vectively heated vertical platerdquo International Journal of ThermalSciences vol 76 pp 221ndash224 2014

[8] G Lukaszewicz Micropolar FluidsmdashTheory and ApplicationsBirkhaauser Basel Switzerland 1999

[9] V A Eremeyev L P Lebedev and H Altenbach Foundationsof Micropolar Mechanics Springer New York NY USA 2013

[10] G Ahmadi ldquoSelf-similar solution of imcompressible micropo-lar boundary layer flow over a semi-infinite platerdquo InternationalJournal of Engineering Science vol 14 no 7 pp 639ndash646 1976

[11] D A Rees and I Pop ldquoFree convection boundary-layer flowof a micropolar fluid from a vertical flat platerdquo IMA Journal ofApplied Mathematics vol 61 no 2 pp 179ndash197 1998

[12] D Srinivasacharya and C RamReddy ldquoEffect of double stratifi-cation onmixed convection in a micropolar fluidrdquoMatematikavol 28 no 2 pp 133ndash149 2012

[13] D Srinivasacharya and C Ramreddy ldquoMixed convection heatand mass transfer in a doubly stratified micropolar fluidrdquoComputationalThermal Sciences vol 5 no 4 pp 273ndash287 2013

[14] D Srinivasacharya and C RamReddy ldquoNatural convection heatand mass transfer in a micropolar fluid with thermal and massstratificationrdquo International Journal for Computational Methodsin Engineering Science andMechanics vol 14 no 5 pp 401ndash4132013

[15] N A Yacob and A Ishak ldquoStagnation point flow towardsa stretchingshrinking sheet in a micropolar fluid with aconvective surface boundary conditionrdquoThe Canadian Journalof Chemical Engineering vol 90 no 3 pp 621ndash626 2012

[16] K Zaimi and A Ishak ldquoStagnation-point flow and heat transferover a nonlinearly stretchingshrinking sheet in a micropolarfluidrdquo Abstract and Applied Analysis vol 2014 Article ID261630 6 pages 2014

[17] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[18] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[19] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer 2009

[20] T Tapanidis G Tsagas and H P Mazumdar ldquoApplication ofscaling group of transformations to viscoelastic second-gradefluid flowrdquo Nonlinear Functional Analysis and Applications vol8 no 3 pp 345ndash350 2003

[21] I A A Hassanien andM A Hamad ldquoGroup theoretic methodfor unsteady free convection flow of a micropolar fluid along avertical plate in a thermally stratified mediumrdquo Applied Mathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 32 no 6 pp 1099ndash1114 2008

[22] R Kandasamy K Gunasekaran and S B H Hasan ldquoScalinggroup transformation on fluid flow with variable stream con-ditionsrdquo International Journal of Non-Linear Mechanics vol 46no 7 pp 976ndash985 2011

[23] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat TransfermdashAsian Research vol41 no 3 pp 241ndash259 2012

16 Advances in High Energy Physics

[24] A A Mutlag M J Uddin M A A Hamad and A I M IsmailldquoHeat transfer analysis for falkner-skan boundary layer flowpast a stationary wedge with slips boundary conditions con-sidering temperature-dependent thermal conductivityrdquo SainsMalaysiana vol 42 no 6 pp 855ndash862 2013

[25] S C Cowin ldquoPolar fluidsrdquo Physics of Fluids vol 11 no 9 pp1919ndash1927 1968

[26] M A Seddeek M Y Akl and A M Al-Hanaya ldquoThermalradiation effects on mixed convection and mass transfer flowon vertical porous plate with heat generation and chemicalreaction by using scaling grouprdquo Journal of Natural Sciences andMathematics vol 4 pp 41ndash60 2010

[27] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[28] M J Uddin W A Khan and A I M Ismail ldquoMHD freeconvective boundary layer flow of a nanofluid past a flat verticalplate with Newtonian heating boundary conditionrdquo PLoS ONEvol 7 no 11 Article ID e49499 2012

[29] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012

[30] M J Uddin W A Khan and A I M Ismail ldquoScalinggroup transformation for MHD boundary layer slip flow of ananofluid over a convectively heated stretching sheet with heatgenerationrdquo Mathematical Problems in Engineering vol 2012Article ID 934964 20 pages 2012

[31] MM RashidiM Ferdows A B Parsa and S Abelman ldquoMHDnatural convection with convective surface boundary conditionover a at platerdquo Abstract and Applied Analysis vol 2014 ArticleID 923487 10 pages 2014

[32] R E Bellman and R E Kalaba Quasilinearisation and Non-Linear Boundary-Value Problems Elsevier New York NY USA1965

[33] S SMotsa P GDlamini andMKhumalo ldquoSpectral relaxationmethod and spectral quasilinearization method for solvingunsteady boundary layer flow problemsrdquo Advances in Mathe-matical Physics vol 2014 Article ID 341964 12 pages 2014

[34] S S Motsa P Sibanda J M Ngnotchouye and G T MarewoldquoA spectral relaxation approach for unsteady boundary-layerflow and heat transfer of a nanofluid over a permeable stretch-ingshrinking sheetrdquo Advances in Mathematical Physics vol2014 Article ID 564942 10 pages 2014

[35] C Canuto M Y Hussaini A Quarteroni and T ZangSpectral Methods Fundamentals in Single Domains ScientificComputation Springer Berlin Germany 2006

[36] J HMerkin ldquoFree convection boundary layer on an isothermalhorizontal cylinderrdquo in Proceedings of the American Societyof Mechanical Engineers and American Institute of ChemicalEngineersHeat Transfer Conference p 911 ASME St LouisMoUSA August 1976

[37] R Nazar N Amin and I Pop ldquoFree convection boundary layeron an isothermal horizontal circular cylinder in a micropolarfluidrdquo in Proceedings of the 12th International Heat TransferConference vol 2 pp 525ndash530 Elsevier Paris France August2002

[38] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

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AstronomyAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

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 Computational  Methods in Physics

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Soft MatterJournal of

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PhotonicsJournal of

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ThermodynamicsJournal of

2 Advances in High Energy Physics

fluids with a substructure Physically the micropolar fluidsmay be treated as non-Newtonian fluids consisting of dumb-bell molecules or rigid cylindrical element polymer fluidsfluid suspension animal blood and so forth Further thetheory of micropolar fluids includes microrotation as wellas microinertia effects This theory studies viscous fluids inwhich microconstituents are rigid and spherical or randomlyoriented as well The subject of free convection boundarylayer flow in a micropolar fluid has been keyed out byseveral investigators due to its immense applications in theengineering problems such as solar energy collecting devicesair conditioning of a room material processing and passivecooling of nuclear reactors The boundary layer flow overa semi-infinite flat plate is considered for analyzing theoryof micropolar fluid and its application to low concentrationsuspension flow by Ahmadi [10] Rees and Pop [11] discussedthe free convection boundary layer flow of a micropolar fluidfrom a vertical flat plate The nonsimilarity transformationsare used to analyze the effects of double stratification onfreemixed convective transport in a micropolar fluid bySrinivasacharya and RamReddy [12ndash14] (also see the ref-erences cited therein) The problems of a steady laminarstagnation point flow towards a stretchingshrinking sheetin an incompressible micropolar fluid under the convectivesurface boundary condition are discussed by Yacob and Ishak[15] and Zaimi and Ishak [16] Merely from the literature itis noted that the majority of the researchers have found thelocal similarity or nonsimilarity solutions for the problemsinvolving convective boundary conditions since most of theresearchers have taken a convective heat transfer coefficientas a function 119909 for getting the similarity solutions in theirproblems Nevertheless the assumption of a heat transfercoefficient varying along the plate as a function of 119909 is notrealistic and very difficult to be obtained in practice For thatcause it could be supposed that the above works have onlytheoretical value

In the recent past several researchers are focused onobtaining the similarity solutions of the convective transportphenomena problems arising in fluid dynamics aerodynam-ics plasma physics meteorology and some branches of engi-neering by using different procedures One such procedureis Lie group analysis The concept of Lie group analysis alsocalled symmetry analysis is developed by Sophius Lie todetermine transformations which map a given differentialequation to itself and it unifies almost all known exactintegration techniques (see [17ndash19]) It provides a potentsophisticated and systematic tool for generating the invariantsolutions of the system of nonlinear partial differential equa-tions (PDEs) with relevant initial or boundary conditionsA special form of Lie group transformations known as thescaling group has been suggested by various researchers tostudy convection flows of different flow phenomena (seeTapanidis et al [20] Hassanien and Hamad [21] Kandasamyet al [22] Aziz et al [23] Mutlag et al [24] etc they areworth observing)

From the literature survey it seems that the problem ofthe free convective heat and mass transport along permeablevertical plate in a micropolar fluid under the convective

boundary condition has not been investigated so far Moti-vated by all these works this paper attempts to present thenew similarity transformations and corresponding similaritysolution to investigate the free convection flow of a micropo-lar fluid under the convective boundary condition using theLie group transformations The mathematical model involv-ing the convective boundary conditions becomes slightlymore complicated leading to the complex interactions of theflow heat and mass transfer mechanism Further the analyt-ical solution is out of scope in the present set-up and hence anumerical solution is obtained for the current problem Alsothe influence of important parameters namely micropolarsuctioninjection and convective heat transfer parameterson the physical quantities of the flow heat and mass transferrates is analyzed in different flow situations

2 Mathematical Formulation

Consider the steady laminar and free convective flow ofan incompressible micropolar fluid with the free streamtemperature and concentration 119879

infinand 119862

infin respectively

Choose the coordinate system such that the 119909-axis is alongthe vertical plate and 119910-axis normal to the plate as shownin Figure 1 The suctioninjection velocity distribution isassumed to be V

119908 The plate is either heated or cooled from

left by convection from a fluid of temperature 119879119891with 119879

119891gt

119879infin

corresponding to a heated surface (assisting flow) and119879119891lt 119879infin

corresponding to a cooled surface (opposing flow)respectively On thewall concentration is taken to be constantand is given by 119862

119908

By employing Boussinesq approximation and makinguse of the standard boundary layer approximations thegoverning equations for the micropolar fluid [10] are givenby

120597119906

120597119909

+

120597V120597119910

= 0 (1)

120588(119906

120597119906

120597119909

+ V120597119906

120597119910

)

= (120583 + 120581)

1205972119906

1205971199102 + 120581

120597120596

120597119910

+120588119892lowast

(120573119879(119909) (119879 minus119879

infin) + 120573119862(119909) (119862minus119862

infin))

(2)

120588119895 (119906

120597120596

120597119909

+ V120597120596

120597119910

) = 120574

1205972120596

1205971199102 minus 120581(2120596+

120597119906

120597119910

) (3)

119906

120597119879

120597119909

+ V120597119879

120597119910

= 120572

1205972119879

1205971199102 (4)

119906

120597119862

120597119909

+ V120597119862

120597119910

= 119863

1205972119862

1205971199102 (5)

where 119906 and V are the velocity components in 119909 and 119910

directions respectively 120596 is the component of microrotationwhose direction of rotation lies in the 119909 119910-plane 119879 is thetemperature 119862 is the concentration 119892lowast is the accelerationdue to gravity 120588 is the density 120583 is the dynamic coefficient

Advances in High Energy Physics 3

x

y

glowast

u = 0

= w

u

C rarr Cinfin

T rarr Tinfinminusk

120655T

120655y= hf(Tf minus T)

C = Cw

Figure 1 Physical model and coordinate system

of viscosity 120573119879(119909) is the volumetric coefficient of thermal

expansion 120573119862(119909) is the volumetric coefficient of solutal

expansions 120581 is the vortex viscosity 119895 is the microinertiadensity 120574 is the spin-gradient viscosity 120572 is the thermaldiffusivity and119863 is the solutal diffusivity of the medium

The boundary conditions are

119906 = 0

V = V119908

120596 = minus 119899

120597119906

120597119910

minus 119896

120597119879

120597119910

= ℎ119891(119879119891minus119879)

119862 = 119862119908

at 119910 = 0

(6a)

119906 = 0

120596 = 0

119879 = 119879infin

119862 = 119862infin

as 119910 997888rarr infin

(6b)

where subscripts119908 andinfin indicate the conditions at the walland at the outer edge of the boundary layer respectively ℎ

119891

is the convective heat transfer coefficient 119896 is the thermalconductivity of the fluid and 119899 is amaterial constant Furtherwe follow the work of many recent authors by assuming that120574 = (120583 + 1205812)119895 This assumption is invoked to allow the fieldof equations to predict the correct behavior in the limitingcase when the microstructure effects become negligible andthe total spin 120596 reduces to the angular velocity [10]

3 Nondimensionalization ofthe Governing Equations

Introduce the following dimensionless variables

119909 =

119909

119871

119910 =

119910

119871

Gr14

119906 =

119871

]Gr12119906

V =119871

]Gr14V

120596 =

1198712

]Gr34120596

120579 =

119879 minus 119879infin

119879119891minus 119879infin

120601 =

119862 minus 119862infin

119862119908minus 119862infin

(7)

where Gr = 119892lowast

1205731198790(119879119891minus 119879infin)119871

3]2 is the Grashof number

In view of the continuity equation (1) we introduce thestream function 120595 by

119906 =

120597120595

120597119910

V = minus

120597120595

120597119909

(8)

Using (7) and (8) into (2)ndash(5) we get the following momen-tum angular momentum energy and concentration equa-tions

Δ 1 =120597120595

120597119910

1205972120595

120597119909120597119910

minus

120597120595

120597119909

1205972120595

1205971199102 minus(

11 minus 119873

)[

1205973120595

1205971199103 minus119873

120597120596

120597119910

]

minus

119892lowast

120573119879(119909) (119879

119891minus 119879infin) 119871

3

]2Gr120579

minus

119892lowast

120573119862(119909) (119862

119908minus 119862infin) 119871

3

]2Gr120601 = 0

Δ 2 =120597120595

120597119910

120597120596

120597119909

minus

120597120595

120597119909

120597120596

120597119910

minus(

2 minus 119873

2 minus 2119873)

1205972120596

1205971199102

+(

119873

1 minus 119873

)[2120596+

1205972120595

1205971199102 ] = 0

Δ 3 =120597120595

120597119910

120597120579

120597119909

minus

120597120595

120597119909

120597120579

120597119910

minus

1Pr

1205972120579

1205971199102 = 0

Δ 4 =120597120595

120597119910

120597120601

120597119909

minus

120597120595

120597119909

120597120601

120597119910

minus

1Sc

1205972120601

1205971199102 = 0

(9)

In usual definitions ] is the kinematic viscosity Pr = ]120572 isthe Prandtl number Sc = ]119863 is the Schmidt number 119873 =

120581(120583 + 120581) (0 le 119873 lt 1) is the coupling number [25] and themicroinertia density is taken to be 119895 = 119871

2Gr12

4 Advances in High Energy Physics

Now boundary conditions (6a) and (6b) become

120597120595

120597119910

= 0

120597120595

120597119909

= 119891119908

120596 = minus 119899

1205972120595

1205971199102

120597120579

120597119910

= minusBi (1minus 120579)

120601 = 1

at 119910 = 0

(10a)

120597120595

120597119910

= 0

120596 = 0

120579 = 0

120601 = 0

as 119910 997888rarr infin

(10b)

where 119891119908= minus(119871]Gr14)V

119908is the suctioninjection parame-

ter It is worth mentioning that 119891119908determines the transpira-

tion rate at the surface with119891119908gt 0 for suction and119891

119908lt 0 for

injection and119891119908= 0 corresponds to an impermeable surface

Further Bi = ℎ119891119871119896Gr14 is the Biot number It is a ratio of

the internal thermal resistance of the plate to the boundarylayer thermal resistance of the hot fluid at the bottom of thesurface

4 Similarity Equations via Lie Scaling GroupTransformations

Aone-parameter Lie scaling group of transformations whichis a simplified form of Lie group transformation is selected as(for more see [26ndash31])

Γ 119909lowast

= 1199091198901205761205721

119910lowast

= 1199101198901205761205722

120595lowast

= 1205951198901205761205723

120596lowast

= 1205961198901205761205724

120579lowast

= 1205791198901205761205725

120601lowast

= 1206011198901205761205726

120573lowast

119879= 1205731198791198901205761205727

120573lowast

119862= 1205731198621198901205761205728

(11)

Here 120576 = 0 is the parameter of the group and 120572119894(where

119894 = 1 2 3 8) are arbitrary real numbers whose interrela-tionship will be determined by our analysis Transformations

in (11) may be treated as a point transformation transformingthe coordinates

(119909 119910 120595 120596 120579 120601 120573119879 120573119862)

= (119909lowast

119910lowast

120595lowast

120596lowast

120579lowast

120601lowast

120573lowast

119879 120573lowast

119862)

(12)

We now investigate the relationship among the exponents 120572119894

(where 119894 = 1 2 3 8) such that

Δ119895[119909lowast

119910lowast

119906lowast

Vlowast 1205973120595lowast

120597119910lowast3 ]

= 119867119895[119909 119910 119906 V

1205973120595

1205971199103 119886]

sdot Δ119895[119909 119910 119906 V

1205973120595

1205971199103 ] (119895 = 1 2 3 4)

(13)

This is the requirement that the differential formsΔ 1Δ 2Δ 3and Δ 4 are conformally invariant under transformation (11)Substituting transformations (11) in (9) we have

Δ 1 = 119890120576(1205721+21205722minus21205723)

(

120597120595lowast

120597119910lowast

1205972120595lowast

120597119909lowast120597119910lowastminus

120597120595lowast

120597119909lowast

1205972120595lowast

120597119910lowast2 )

minus(

11 minus 119873

) 119890120576(31205722minus1205723) 120597

3120595lowast

120597119910lowast3

minus(

119873

1 minus 119873

) 119890120576(1205722minus1205724) 120597120596

lowast

120597119910lowast

minus

119892lowast

120573lowast

119879(119879119891minus 119879infin) 119871

3

]2Gr119890minus120576(1205725+1205727)

120579lowast

minus

119892lowast

120573lowast

119862(119862119908minus 119862infin) 119871

3

]2Gr119890minus120576(1205726+1205728)

120601lowast

= 0

(14a)

Δ 2 = 119890120576(1205721+1205722minus1205723minus1205724)

(

120597120595lowast

120597119910lowast

120597120596lowast

120597119909lowastminus

120597120595lowast

120597119909lowast

120597120596lowast

120597119910lowast)

minus(

2 minus 119873

2 minus 2119873) 119890120576(21205722minus1205724) 120597

2120596lowast

120597119910lowast2

+(

119873

1 minus 119873

)(2120596lowast119890minus1205761205724 + 119890120576(21205722minus1205723) 120597

2120595lowast

120597119910lowast2 )

= 0

(14b)

Δ 3 = 119890120576(1205721+1205722minus1205723minus1205725)

(

120597120595lowast

120597119910lowast

120597120579lowast

120597119909lowastminus

120597120595lowast

120597119909lowast

120597120579lowast

120597119910lowast)

minus

1Pr

119890120576(21205722minus1205725)

(

1205972120579lowast

120597119910lowast2) = 0

(14c)

Δ 4 = 119890120576(1205721+1205722minus1205723minus1205726)

(

120597120595lowast

120597119910lowast

120597120601lowast

120597119909lowastminus

120597120595lowast

120597119909lowast

120597120601lowast

120597119910lowast)

minus

1Sc

119890120576(21205722minus1205726)

(

1205972120601lowast

120597119910lowast2) = 0

(14d)

Advances in High Energy Physics 5

Now boundary conditions (10a) and (10b) become

119890120576(1205722minus1205723)

120597120595lowast

120597119910lowast= 0

119890120576(1205721minus1205723)

120597120595lowast

120597119909lowast= 119891119908

119890minus1205761205724

120596lowast

= minus 119899119890120576(21205722minus1205723) 120597

2120595lowast

120597119910lowast2

119890120576(1205722minus1205725) 120597120579

lowast

120597119910lowast= minusBi (1minus 119890

minus1205761205725120579lowast

)

119890minus1205761205726

120601lowast

= 1at 119910lowast = 0

(15a)

119890120576(1205722minus1205723)

120597120595lowast

120597119910lowast= 0

119890minus1205761205724

120596lowast

= 0119890minus1205761205725

120579lowast

= 0119890minus1205761205726

120601lowast

= 0as 119910lowast 997888rarr infin

(15b)

The system remains invariant under the group transforma-tion Γ We then have the following relationships for theparameters

1205721 + 21205722 minus 21205723 = 31205722 minus1205723 = 1205722 minus1205724 = minus1205725 minus1205727= minus1205726 minus1205728

1205721 +1205722 minus1205723 minus1205724 = 21205722 minus1205724 = minus1205724 = 21205722 minus1205723

1205721 +1205722 minus1205723 minus1205725 = 21205722 minus1205725

1205721 +1205722 minus1205723 minus1205726 = 21205722 minus1205726

1205721 minus1205723 = 0

minus 1205724 = 21205722 minus1205723

1205722 minus1205725 = 0 = minus1205725

1205726 = 0

(16)

Solving linear system (16) we have the following relationshipamong the exponents

1205721 = 1205723 = 1205724 = 1205727 = 1205728

1205722 = 1205725 = 1205726 = 0(17)

The set of transformations Γ reduces to

119909lowast

= 1199091198901205761205721

119910lowast

= 119910

120595lowast

= 1205951198901205761205721

120596lowast

= 1205961198901205761205721

120579lowast

= 120579

120601lowast

= 120601

120573lowast

119879= 1205731198791198901205761205721

120573lowast

119862= 1205731198621198901205761205721

(18)

Expanding by theTaylor series in power of 120576 keeping the termup to the first degree (neglecting higher power of 120576) we get

119909lowast

minus119909 = 1205761205721119909

119910lowast

= 119910

120595lowast

minus120595 = 1205761205721120595

120596lowast

minus120596 = 1205761205721120596

120579lowast

= 120579

120601lowast

= 120601

120573lowast

119879minus120573119879= 1205761205721120573119879

120573lowast

119862minus120573119862= 1205761205721120573119862

(19)

The characteristic equations are

119889119909

1205721119909=

119889119910

0=

119889120595

1205721120595=

119889120596

1205721120596=

119889120579

0=

119889120601

0=

119889120573119879

1205721120573119879

=

119889120573119862

1205721120573119862

(20)

Solving the above characteristic equations we have thefollowing similarity transformations

120578 = 119910

120595 = 119909119891 (120578)

120596 = 119909119892 (120578)

120573119879= 1205731198790119909

120573119862= 1205731198620119909

120579 = 120579 (120578)

120601 = 120601 (120578)

(21)

where 1205731198790and 120573

1198620are constant thermal and mass coefficient

of expansionUsing (21) into (9) we get the following similarity

equations

(

11 minus 119873

)119891101584010158401015840

+11989111989110158401015840

minus11989110158402+(

119873

1 minus 119873

)1198921015840

+ 120579+B120601

= 0

(

2 minus 119873

2 minus 2119873)11989210158401015840

+1198911198921015840

minus1198911015840

119892minus(

119873

1 minus 119873

) (2119892+11989110158401015840

)

= 0

1Pr

12057910158401015840

+1198911205791015840

= 0

1Sc

12060110158401015840

+1198911206011015840

= 0

(22)

where the primes indicate differentiation with respect to 120578

alone andB = 1205731198620(119862119908minus 119862infin)1205731198790(119879119891minus 119879infin) is the buoyancy

ratio

6 Advances in High Energy Physics

Boundary conditions (10a) and (10b) in terms of 119891 119892 120579and 120601 become

120578 = 0 119891 (0) = 119891119908

1198911015840

(0) = 0

119892 (0) = minus 11989911989110158401015840

(0)

1205791015840

(0) = minusBi [1minus 120579 (0)]

120601 (0) = 1

(23a)

120578 997888rarr infin 1198911015840

(infin) = 0

119892 (infin) = 0

120579 (infin) = 0

120601 (infin) = 0

(23b)

5 Skin Friction Wall Couple Stress and Heatand Mass Transfer Coefficients

The wall shear stress and the wall couple stress are

120591119908= [(120583 + 120581)

120597119906

120597119910

+ 120581120596]

119910=0

119898119908= 120574 [

120597120596

120597119910

]

119910=0

(24a)

and the heat and mass transfers from the plate respectivelyare given by

119902119908= minus 119896 [

120597119879

120597119910

]

119910=0

119902119898= minus119863[

120597119862

120597119910

]

119910=0

(24b)

The nondimensional skin friction 119862119891

= 2120591119908120588119906

2lowast wall

couple stress 119872119908

= 119898119908120588119906

2lowast119909 the local Nusselt number

119873119906119909= 119902119908119909119896(119879

119891minus 119879infin) and local Sherwood number Sh

119909=

119902119898119909119863(119862

119908minus 119862infin) are given by

119862119891Gr14119909

= 2(1 minus 119899119873

1 minus 119873

)11989110158401015840

(0)

119872119908Gr12119909

= (

2 minus 119873

2 minus 2119873)1198921015840

(0)

119873119906119909

Gr14119909

= minus 1205791015840

(0)

Sh119909

Gr14119909

= minus1206011015840

(0)

(25)

where 1199062lowastis the characteristic velocity and Gr

119909= 119892lowast

1205731198790(119879119891minus

119879infin)119909

3]2 is the local Grashof number

6 Numerical Solution Using the SpectralQuasi-Linearization Method (SQLM)

In this section we describe the quasi-linearization method(QLM) for solving the governing system of (22) alongwith boundary conditions (23a) and (23b) This QLM isa generalization of the Newton-Raphson method and wasproposed by Bellman and Kalaba [32] for solving nonlinearboundary value problems

Assume that the solutions 119891119903 119892119903 120579119903 and 120601

119903of (22) at the

(119903+1)th iteration are119891119903+1 119892119903+1 120579119903+1 and 120601119903+1 If the solutions

at the previous iteration are sufficiently close to the solutionsat the present iteration the nonlinear components of (22)can be linearised using one-term Taylor series of multiplevariables so that (22) give the following iterative sequence oflinear differential equations

(

11 minus 119873

)119891101584010158401015840

119903+1 + 119886111990311989110158401015840

119903+1 + 11988621199031198911015840

119903+1 + 1198863119903119891119903+1

+(

119873

1 minus 119873

)1198921015840

119903+1 + 120579119903+1 +B120601

119903+1 = 1198771119903

(

2 minus 119873

2 minus 2119873)11989210158401015840

119903+1 + 11988731199031198921015840

119903+1 + 1198874119903119892119903+1 + 11988711199031198911015840

119903+1

+ 1198872119903119891119903+1 minus(

119873

1 minus 119873

)11989110158401015840

119903+1 = 1198772119903

1198881119903119891119903+1 +1Pr

12057910158401015840

119903+1 + 11988821199031205791015840

119903+1 = 1198773119903

1198891119903119891119903+1 +1Sc

12060110158401015840

119903+1 +11988921199031206011015840

119903+1 = 1198774119903

(26)

where the coefficients 1198861199041 119903

(1199041 = 1 2 3) 1198871199042 119903

(1199042 = 1 2 4) 1198881199043 119903

(1199043 = 1 2) 1198891199044 119903

(1199044 = 1 2) and 1198771199045 119903

(1199045 = 1 2 4) are known functions (from previous iterations) and aredefined as

1198861119903 = 119891119903

1198862119903 = minus 21198911015840119903

1198863119903 = 11989110158401015840

119903

1198771119903 = 11989111990311989110158401015840

119903minus (1198911015840

119903)

2

1198871119903 = minus119892119903

1198872119903 = 1198921015840

119903

1198873119903 = 119891119903

1198874119903 = minus1198911015840

119903minus(

21198731 minus 119873

)

1198772119903 = 1198911199031198921015840

119903minus1198911015840

119903119892119903

1198881119903 = 1205791015840

119903

1198882119903 = 119891119903

1198773119903 = 1198911199031205791015840

119903

1198891119903 = 1206011015840

119903

1198892119903 = 119891119903

1198774119903 = 1198911199031206011015840

119903

(27)

Advances in High Energy Physics 7

subject to boundary conditions

119891119903+1 (0) = 119891

119908

1198911015840

119903+1 = 0

1198911015840

119903+1 (infin) = 0

119892119903+1 = minus 119899119891

10158401015840

119903+1 (0)

119892119903+1 (infin) = 0

1205791015840

119903+1 (0) = minusBi (1minus 120579 (0))

120579119903+1 (infin) = 0

120601119903+1 (0) = 1

120601119903+1 (infin) = 0

(28)

System (26) constitutes a linear system of coupled differen-tial equations with variable coefficients and can be solvediteratively using any numerical method for 119903 = 1 2 3 In this work as will be discussed below the Chebyshevpseudospectral method was used to solve the QLM scheme(26) (for more details refer to the works of Motsa et al[33 34])

1198910 (120578) = 119891119908+ 1minus 119890

minus120578

1198920 (120578) = minus 119899119890minus120578

1205790 = 119890minus120578

BiBi + 1

1206010 = 119890minus120578

(29)

and starting from these sets of initial approximations 1198910 11989201205790 and 1206010 the iteration schemes (26) can be solved iterativelyfor 119891119903+1(120578) 119892119903+1(120578) 120579119903+1(120578) and120601119903+1(120578) when 119903 = 0 1 2

For this we discretise the equation using the Chebyshevspectral collocation method The unknown functions areapproximated by the Chebyshev interpolating polynomialsin such way that they are collocated at the Gauss-Lobattocollocation points defined as

120591119895= cos

120587119895

119873

119895 = 0 1 2 119873 (30)

where 119873 is the number of collocation points The physicalregion [0infin) is transformed into the region [minus1 1] using thedomain truncation technique in which the problem is solvedon the interval [0 120578

infin] instead of [0infin) This leads to the

mapping

120578

120578infin

=

120591 + 12

minus1 le 120591 le 1 (31)

where 120578infin

is the scaling parameter used to invoke theboundary condition at infinity The functions 119891 119892 120579 and 120601

are approximated at the collocation points by

119891 (120591) =

119873

sum

119896=0119891 (120591119896) 119879119896(120591119895)

119892 (120591) =

119873

sum

119896=0119892 (120591119896) 119879119896(120591119895)

120579 (120591) =

119873

sum

119896=0120579 (120591119896) 119879119896(120591119895)

120601 (120591) =

119873

sum

119896=0120601 (120591119896) 119879119896(120591119895)

119895 = 0 1 2 119873

(32)

where 119879119896is the 119896th Chebyshev polynomial defined as

119879119896(120591) = cos [119896 cosminus1 (120591)] (33)

The derivatives of the variables at the collocation points arerepresented as

119889119901

119891

119889120578119901=

119873

sum

119896=0D119901119897119896119891 (120591119896)

119889119901

119892

119889120578119901=

119873

sum

119896=0D119901119897119896119892 (120591119896)

119889119901

120579

119889120578119901=

119873

sum

119896=0D119901119897119896120579 (120591119896)

119889119901

120601

119889120578119901=

119873

sum

119896=0D119901119897119896120601 (120591119896)

119897 = 0 1 119873

(34)

where 119901 is the order of the derivative and D = 2D120578infin

is theChebyshev spectral differentiation matrix and its entries areclearly defined in Canuto et al [35]

Substituting (31)ndash(34) into (26) leads to the matrix equa-tion

119860119883 = 119877 (35)

subject to the boundary conditions

119891119903+1 (120591119873) = 119891

119908

119873

sum

119896=0D119873119896

119891 (120591119896) = 0

119873

sum

119896=0D0119896119891 (120591

119896) = 0

119892119903+1 (120591119873) = minus 119899

119873

sum

119896=0D2119873119896

119891 (120591119896)

119892119903+1 (1205910) = 0

8 Advances in High Energy Physics

119873

sum

119896=0D119873119896

120579119903+1 (120591119896) minusBi 120579

119903+1 (120591119873) = minusBi

120579119903+1 (1205910) = 0

120601119903+1 (120591119873) = 1120601119903+1 (1205910) = 0

(36)

In (35) 119860 is a (4119873+ 4) times (4119873+ 4) square matrix and119883 and 119877are (4119873 + 1) times 1 column vectors defined by

119860 =

[

[

[

[

[

[

11986011 11986012 11986013 11986014

11986021 11986022 11986023 11986024

11986031 11986032 11986033 11986034

11986041 11986042 11986043 11986044

]

]

]

]

]

]

119883 =

[

[

[

[

[

[

F119903+1

G119903+1

Θ119903+1

Φ119903+1

]

]

]

]

]

]

119877 =

[

[

[

[

[

[

R1

R2

R3

R4

]

]

]

]

]

]

(37)

whereF = [119891

119903+1 (1205910) 119891119903+1 (1205911) 119891119903+1 (120591119873)]119879

G = [119892119903+1 (1205910) 119892119903+1 (1205911) 119892119903+1 (120591119873)]

119879

Θ = [120579119903+1 (1205910) 120579119903+1 (1205911) 120579119903+1 (120591119873)]

119879

Φ = [120601119903+1 (1205910) 120601119903+1 (1205911) 120601119903+1 (120591119873)]

119879

11986011 = (

11 minus 119873

)D3+ diag [1198861119903]D

2+ diag [1198862119903]D

+ diag [1198863119903]

11986012 = (

119873

1 minus 119873

)D

11986013 = I

11986014 = BI

11986021 = minus(

119873

1 minus 119873

)D2+ diag [1198871119903]D+ diag [1198872119903]

11986022 = (

2 minus 119873

2 minus 2119873)D2

+ diag [1198873119903]D+ diag [1198874119903]

11986023 = 0

11986024 = 0

11986031 = diag [1198881119903]

11986032 = 0

11986033 =1Pr

D2+ diag [1198882119903]D

Table 1 Comparison of minus1205791015840(0) for free convection along a verticalflat plate in Newtonian fluid when 119873 = 0 119899 = 0 B = 0 Pr = 1Bi rarr infin and 119891

119908= 0

Merkin [36] Nazar et al [37] Molla et al [38] Present04214 04214 04214 04214313

11986034 = 0

11986041 = diag [1198891119903]

11986042 = 0

11986043 = 0

11986044 =1Sc

D2+ diag [1198892119903]D

R1 = 1198771119903

R2 = 1198772119903

R3 = 1198773119903

R4 = 1198774119903

(38)

and here I is an identity matrix the size of the matrix 0 is(119873+1)times1 and diag[ ] is a diagonalmatrix of size (119873+1)times(119873+

1) Subscript r denotes the iteration number After modifyingmatrix system (35) to incorporate boundary condition (36)the solution is obtained as

119883 = 119860minus1119877 (39)

7 Results and Discussions

It is noticed that the present problem reduces to free convec-tion heat transfer along an impermeable vertical plate in amicropolar fluid without the convective boundary conditionwhen119891

119908= 0 Bi rarr infin andB = 0 Also in the limit as119873 rarr

0 governing equations (2)ndash(5) reduce to the correspondingequations for a free convection heat and mass transfer ina viscous fluid In order to validate the code generated theresults of the present problem have been compared with theresults obtained by Merkin [36] Nazar et al [37] and Mollaet al [38] as a special case by taking 119873 = 0 119899 = 0 B = 0Pr = 1 Bi rarr infin and 119891

119908= 0 and it was found that

they are in good agreement as presented in Table 1 Also thecomparison of heat transfer coefficient has been made withthe results obtained by Nazar et al [37] as shown in Table 2when 119899 = 05 B = 0 Pr = 1 Bi rarr infin and 119891

119908= 0

To study the effects of coupling number119873 suctioninjectionparameter 119891

119908 Biot number Bi and material parameter 119899

computations were carried out in the cases of B = 10Pr = 071 and Sc = 022

The effects of the coupling number119873 on the dimension-less velocity microrotation temperature and concentrationare illustrated in Figures 2(a)ndash2(d) with fixed values ofother parameters The coupling number 119873 characterizes thecoupling of linear and rotational motion arising from the

Advances in High Energy Physics 9

00

01

02

03

04

05

06f998400

0 4 8 12 16

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(a)

g

minus06

minus05

minus04

minus03

minus02

minus01

00

01

0 2 4 6 8 10 12

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(b)

000

004

008

012

016

120579

0 1 2 3 4 5 6

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(c)

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

120601

00

02

04

06

08

10

120578

0 2 4 6 8 10 12

Bi = 01 fw = 05

(d)

Figure 2 Effect of119873 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Table 2 Comparison of minus1205791015840

(0) for free convection flow in amicropolar fluid obtained by Nazar et al [37] when 119899 = 05B = 0Pr = 1 Bi rarr infin and 119891

119908= 0

119873 Nazar et al [37] Present000 04214 04214033 03991 03990050 03834 03834060 03709 03709066 03608 03608071 03522 03522075 03447 03447

fluid particles In the case of 119873 = 0 (ie as 120581 tends tozero) the micropolarity is absent and fluid becomes nonpolarfluid With a large value of 119873 effect of microstructurebecomes significant whereas with a diminished value of 119873the individuality of the substructure is much less articulatedAs 119873 increases it is found from Figure 2(a) that the max-imum velocity decreases in amplitude and the location ofthe maximum velocity moves farther away from the wallSince 119873 rarr 0 corresponds to viscous fluid the velocity incase of a micropolar fluid has been less compared to thatof viscous fluid case It can be observed from Figure 2(b)that as 119873 increases initially microrotation profiles tend tobecome flatter and then approach their free stream valuesfar away from the wall This happens due to the vanishing

10 Advances in High Energy Physics

00

01

02

03

04

05

06

f998400

120578

0 2 4 6 8 10 12

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(a)

minus06

minus04

minus03

minus02

minus05

minus01

00

01

g

0 2 4 6 8 10

120578

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(b)

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

120578

0 1 2 3 4 5

120579

00

02

04

06

08

10

(c)

120601

00

02

04

06

08

10

fw = 05 n = 05N = 05

0 2 4 6 8 10 12

120578

Bi = 01

Bi = 10

Bi = 50Bi = 200

(d)

Figure 3 Effect of Bi on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

of antisymmetric part of the stress on the boundary thatcorresponds to a weak concentration of microelements Thisis because an increment in the value of 119873 implies a highervortex viscosity of fluid which promotes the microrotationof micropolar fluids It is seen from Figures 2(c) and 2(d)that thermal and concentration boundary layers of thefluid increase with increase in coupling number 119873 Hencetemperature and concentration in case of micropolar fluidsare more than those of the viscous fluid case

The Biot number Bi is the ratio of internal thermalresistance of a solid to boundary layer thermal resistanceWhen Bi = 0 the plate is totally insulated internal thermalresistance of the plate is extremely high and no convectiveheat transfer to the cold fluid on the upper part of the

plate takes place Figure 3(a) depicts fluid velocity profilesfor different values of the Biot number with 119873 = 05 119891

119908=

05 and 119899 = 05 Generally fluid velocity is zero at platesurface and increases gradually away fromplate to free streamvalue satisfying boundary conditions It is interesting to notethat an increase in the intensity of convective surface heattransfer Bi produces significant enhancement in fluid velocitywithin the momentum boundary layer In Figure 3(b) webring out the behavior of microrotation with different valuesof Biot number Bi for fixed values of other parameters As theparameter value Bi increases microrotation showing reverserotation near the two boundaries Hence the condition ofvanishing of antisymmetric part of the stress on the boundaryresults in a drastic change of the microrotation profiles

Advances in High Energy Physics 11

07

06

05

04

03

02

01

00

f998400

fw = 00

fw = 10

fw = 20

0 3 6 9 12 15

120578

N = 05 Bi = 01 n = 05

fw = minus05

(a)

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

120578

0 2 4 6 8 10 12

g

minus04

minus03

minus02

minus01

00

01

fw = minus05

(b)

0 1 2 3 4 5 6

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

000

005

010

015

020

025

030

120579

fw = minus05

(c)

0 3 6 9 12

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

00

02

04

06

08

10

120601

fw = minus05

(d)

Figure 4 Effect of 119891119908on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Given that convective heating increases with Biot numberBi rarr infin simulates the isothermal surface which is clearlyseen from Figure 3(c) where 120579(0) = 1 as Bi rarr infin Infact a high Biot number indicates higher internal thermalresistance of the plate than boundary layer thermal resistanceIn this fluid temperature is maximum at the plate surfaceand decreases exponentially to zero value far out from theplate satisfying boundary conditions As a consequence anincrement in the Biot number leads to increase of fluidtemperature efficiency Figure 3(d) illustrates the variation ofdimensionless concentration for different values of Bi It isclear that the concentration of fluid decreases with increaseof Bi

The effect of 119891119908

on velocity profile is depicted inFigure 4(a) Here 119891

119908gt 0 represents suction and 119891

119908lt 0

denotes injectionThe lower velocity is noticed in case of suc-tion when compared to case of injection From Figure 4(b)we note that microrotation is showing reverse rotation neartwo boundaries with both suction and injection parame-ter The dimensionless temperature for different values ofsuctioninjection parameters is drawn in Figure 4(c) It isreadable that the temperature of the fluid is more in case ofinjection whereas it is less in case of suction in comparisonwith the impermeable surface case (119891

119908= 0) Figure 4(d)

demonstrates dimensionless concentration for different val-ues of suctioninjection parameters It is determined that the

12 Advances in High Energy Physics

00

01

02

03

04

05

06f998400

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(a)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10

120578

minus10

minus08

minus06

minus04

minus02

00

g

(b)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

120578

0 1 2 3 4 5

120579

000

003

006

009

012

015

(c)

00

02

04

06

08

10

120601

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(d)

Figure 5 Effect of material parameter 119899 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

concentration of fluid ismore with injection whereas it is lesswith suctionwhen compared to the impermeable surface case(119891119908= 0) As a finale the thermal and solutal boundary layer

thicknesses increase in case of injection compared to case ofsuction as displayed in Figures 4(c) and 4(d)

In Figures 5(a)ndash5(d) the effect of material parameter119899 on dimensionless velocity microrotation temperatureand concentration is presented for fixed values of otherparameters since the material parameter 119899 signifies themicrorotation effects (ie for 119899 = 0 particles are not freeto revolve near the surface whereas as 119899 increases from 0 to1 the microrotation term gets augmented and induces flowenhancement) As 119899 increases it is found from Figure 5(a)that the minimum velocity increases in amplitude and the

location of the minimum velocity moves farther away fromthe wall From Figure 5(b) we observe that themicrorotationis decreasing with increasing value of material parameter119899 within the boundary layer It is clear from Figures 5(c)-5(d) that the thermal and solutal boundary layer thicknessesdecrease with increase of material parameter 119899

The variations of minus1205791015840

(0) and minus1206011015840

(0) versus couplingnumber 119873 are shown in Figures 6ndash8 It can be noticed fromthese figures that the heat and mass transfer coefficientsare less in case of micropolar fluids when compared to theviscous fluids This is because as 119873 increases the thermaland solutal boundary layer thicknesses become larger thusgiving rise to a small value of local heat and mass transferrates The effect of the Biot number Bi with fixed 119891

119908= 05

Advances in High Energy Physics 13

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

fw = 05 n = 05

01

00

02

03

04

05

06

07Nu x

Grminus

14

x

(a)

fw = 05 n = 05

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

024

026

028

030

032

ShxGrminus

14

x

(b)

Figure 6 Effect of Bi on (a) heat transfer rate and (b) mass transfer rate

005

006

007

008

009

010

Nu x

Grminus

14

x

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

(a)

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

01

02

03

04

05

06

ShxGrminus

14

x

(b)

Figure 7 Effect of 119891119908on (a) heat transfer rate and (b) mass transfer rate

and 119899 = 05 on local heat transfer coefficient is exhibitedin Figure 6 It is found from Figure 6 that local heat transferrate increases nonlinearly with the increase in Biot numberBiThe influence of the Biot number Bi on local mass transfercoefficient is shown in Figure 6 Figure 6 reveals that the localmass transfer coefficient is enhanced by the growth in theBiot number Bi In Figure 7 the effect of the suctioninjectionparameter 119891

119908with fixed Bi = 01 and 119899 = 05 on local heat

and mass transfer coefficients is displayed It is found fromFigure 7 that the local heat and mass transfer coefficients areless in the case of injection 119891

119908lt 0 in comparison with the

case of suction 119891119908

gt 0 Figure 8 is prepared to analyze theeffect of the material parameter 119899 with fixed Bi = 01 and

119891119908= 05 on local heat andmass transfer coefficients Figure 8

reveals that the local heat and mass transfer coefficients areenhanced by the increase in material parameter 119899 This isbecause when 119899 increases from 0 to 1 the microrotation termgets augmented and induces flow enhancement

The variations of 119862119891Gr14119909

and 119872119908Gr12119909

which areproportional to the coefficients of skin friction and wallcouple stress are shown in Table 3 with different values ofthe coupling number 119873 for fixed 119899 = 05 119891

119908= 05 and

Bi = 01 It indicates that the skin friction factor is higherfor micropolar fluid than the viscous fluids (119873 = 0) Sincemicropolar fluids offer a heavy resistance (resulting fromvortex viscosity) to fluid movement and cause larger skin

14 Advances in High Energy Physics

0084

0085

0086

0087Nu x

Grminus

14

x

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

fw = 05 Bi = 01

(a)

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

024

026

028

030

ShxGrminus

14

x

fw = 05 Bi = 01

(b)

Figure 8 Effect of material parameter 119899 on (a) heat transfer rate and (b) mass transfer rate

Table 3 Effects of skin friction and wall couple stress for varyingvalues of Biot numbers Bi micropolar parameter 119873 materialparameter 119899 and suctioninjection parameter 119891

119908

119873 Bi 119891119908

119899 119862119891Gr14119909

119872119908Gr12119909

01 01 05 05 2424227 06327903 01 05 05 2502716 063996605 01 05 05 2647987 063086307 01 05 05 2943561 060387909 01 05 05 3874631 055089105 01 05 05 2647987 063086305 10 05 05 3211755 080587905 50 05 05 3533141 090807805 200 05 05 3629791 093913705 01 minus05 05 2426265 033621905 01 00 05 2530014 046542905 01 10 05 2737553 082089905 01 20 05 2721244 119359605 01 05 00 2916655 minus028933905 01 05 05 2647987 063086305 01 05 10 2232556 2066191

friction factor compared to viscous fluid the results as wellsuggest larger values of coupling number 119873 and lower wallcouple stresses Since the skin friction coefficient 11989110158401015840(0) andwall couple stress coefficient as well as high temperature andmass transport rates are more depressed in the micropolarfluid comparing to the viscous fluid which may be beneficialin flow temperature and concentration control of polymerprocessing thus the presence of microscopic effects arisingfrom the local structure and of the fluid elements reduces thehigh temperature and mass transfer coefficients The effectof Bi on 119862

119891Gr14119909

and 119872119908Gr12119909

for 119891119908

= 05 119899 = 05 and

119873 = 05 is illustrated in Table 3 It can be noticed that theskin friction and wall couple stress coefficients are increasingwith increase of Bi for fixed values of other parametersThis notice is consistent with physical profiles presented inFigure 3 The effects of suctioninjection parameter on theskin friction andwall couple stress coefficients are also shownin Table 3 It is noted that the skin friction and wall couplestress coefficients are less with injection case whereas theyare more with suction case when compared to the case ofimpermeable surface Finally the detailed behavior of thematerial parameter 119899 is given in Table 3 The skin frictiondecreases and wall couple stress increases with increase ofmaterial parameter 119899

8 Conclusions

In this composition the similarity solution of the free convec-tion flow on a permeable vertical plate of a micropolar fluidunder the convective boundary condition is obtained usingLie group transformations Using the similarity variables thegoverning equations are transformed into a set of nondi-mensional parabolic equations These equations are solvednumerically using spectral quasi-linearisation method Thenumerical computation is carried out for various values ofnondimensional physical parameters The main findings aresummarized as follows

(i) The numerical results indicate that velocity distribu-tion is less near the plate but it is more far away fromthe plate the wall couple stress coefficient and rate ofheat andmass transfers are lower but the temperatureand concentration distributions and the skin frictioncoefficient are higher for the micropolar fluids incomparison with those of viscous fluids Also thereverse rotation ofmicrorotation near two boundariesis found with the increasing value of119873

Advances in High Energy Physics 15

(ii) An increase in Biot number Bi decreases concen-tration distribution whereas it causes an increasein temperature distribution skin friction and wallcouple stress coefficients and heat and mass transferrates within the boundary layer Further enhancingin Biot number Bi enhances velocity distributionnear the plate but shows the reverse behavior faraway from the plate We observe reverse rotationof microrotation near two boundaries within theboundary layer in the presence of Biot number

(iii) Less velocity temperature and concentration distri-butions are observed more skin friction and wallcouple stress coefficients and heat and mass transferrates in the case of suction compared to the case ofinjection Further microrotation decreases near thewall and depicts the opposite trend far away from thewall

(iv) It is found that microrotation temperature and con-centration distributions and skin friction coefficientsare more in the case of a micropolar fluid with strongconcentration (ie 119899 = 0) when compared to the caseof a micropolar fluid with weak concentration (ie119899 = 12) Further velocity is less in the case of 119899 = 0when compared to the case of 119899 = 12 near the walland shows the opposite trend far away from the wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their valuablesuggestions and comments

References

[1] J H Merkin ldquoNatural-convection boundary-layer flow on avertical surface with Newtonian heatingrdquo International Journalof Heat and Fluid Flow vol 15 no 5 pp 392ndash398 1994

[2] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009

[3] O DMakinde ldquoSimilarity solution for natural convection froma moving vertical plate with internal heat generation and aconvective boundary conditionrdquo Thermal Science vol 15 ppS137ndashS143 2011

[4] S M Ibrahim and N Bhashar Reddy ldquoSimilarity solution ofheat and mass transfer for natural convection over a movingvertical plate with internal heat generation and a convectiveboundary condition in the presence of thermal radiationviscous dissipation and chemical reactionrdquo ISRN Thermody-namics vol 2013 Article ID 790604 10 pages 2013

[5] C RamReddy P V S N Murthy A J Chamkha and A MRashad ldquoSoret effect on mixed convection flow in a nanofluidunder convective boundary conditionrdquo International Journal ofHeat and Mass Transfer vol 64 pp 384ndash392 2013

[6] H T Alkasasbeh M Z Salleh R M Tahar and R NazarldquoNumerical solutions of free convection boundary layer flow ona solid sphere with convective boundary conditionsrdquo Journal ofPhysics Conference Series vol 495 Article ID 012025 2014

[7] A Pantokratoras ldquoA note on natural convection along a con-vectively heated vertical platerdquo International Journal of ThermalSciences vol 76 pp 221ndash224 2014

[8] G Lukaszewicz Micropolar FluidsmdashTheory and ApplicationsBirkhaauser Basel Switzerland 1999

[9] V A Eremeyev L P Lebedev and H Altenbach Foundationsof Micropolar Mechanics Springer New York NY USA 2013

[10] G Ahmadi ldquoSelf-similar solution of imcompressible micropo-lar boundary layer flow over a semi-infinite platerdquo InternationalJournal of Engineering Science vol 14 no 7 pp 639ndash646 1976

[11] D A Rees and I Pop ldquoFree convection boundary-layer flowof a micropolar fluid from a vertical flat platerdquo IMA Journal ofApplied Mathematics vol 61 no 2 pp 179ndash197 1998

[12] D Srinivasacharya and C RamReddy ldquoEffect of double stratifi-cation onmixed convection in a micropolar fluidrdquoMatematikavol 28 no 2 pp 133ndash149 2012

[13] D Srinivasacharya and C Ramreddy ldquoMixed convection heatand mass transfer in a doubly stratified micropolar fluidrdquoComputationalThermal Sciences vol 5 no 4 pp 273ndash287 2013

[14] D Srinivasacharya and C RamReddy ldquoNatural convection heatand mass transfer in a micropolar fluid with thermal and massstratificationrdquo International Journal for Computational Methodsin Engineering Science andMechanics vol 14 no 5 pp 401ndash4132013

[15] N A Yacob and A Ishak ldquoStagnation point flow towardsa stretchingshrinking sheet in a micropolar fluid with aconvective surface boundary conditionrdquoThe Canadian Journalof Chemical Engineering vol 90 no 3 pp 621ndash626 2012

[16] K Zaimi and A Ishak ldquoStagnation-point flow and heat transferover a nonlinearly stretchingshrinking sheet in a micropolarfluidrdquo Abstract and Applied Analysis vol 2014 Article ID261630 6 pages 2014

[17] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[18] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[19] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer 2009

[20] T Tapanidis G Tsagas and H P Mazumdar ldquoApplication ofscaling group of transformations to viscoelastic second-gradefluid flowrdquo Nonlinear Functional Analysis and Applications vol8 no 3 pp 345ndash350 2003

[21] I A A Hassanien andM A Hamad ldquoGroup theoretic methodfor unsteady free convection flow of a micropolar fluid along avertical plate in a thermally stratified mediumrdquo Applied Mathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 32 no 6 pp 1099ndash1114 2008

[22] R Kandasamy K Gunasekaran and S B H Hasan ldquoScalinggroup transformation on fluid flow with variable stream con-ditionsrdquo International Journal of Non-Linear Mechanics vol 46no 7 pp 976ndash985 2011

[23] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat TransfermdashAsian Research vol41 no 3 pp 241ndash259 2012

16 Advances in High Energy Physics

[24] A A Mutlag M J Uddin M A A Hamad and A I M IsmailldquoHeat transfer analysis for falkner-skan boundary layer flowpast a stationary wedge with slips boundary conditions con-sidering temperature-dependent thermal conductivityrdquo SainsMalaysiana vol 42 no 6 pp 855ndash862 2013

[25] S C Cowin ldquoPolar fluidsrdquo Physics of Fluids vol 11 no 9 pp1919ndash1927 1968

[26] M A Seddeek M Y Akl and A M Al-Hanaya ldquoThermalradiation effects on mixed convection and mass transfer flowon vertical porous plate with heat generation and chemicalreaction by using scaling grouprdquo Journal of Natural Sciences andMathematics vol 4 pp 41ndash60 2010

[27] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[28] M J Uddin W A Khan and A I M Ismail ldquoMHD freeconvective boundary layer flow of a nanofluid past a flat verticalplate with Newtonian heating boundary conditionrdquo PLoS ONEvol 7 no 11 Article ID e49499 2012

[29] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012

[30] M J Uddin W A Khan and A I M Ismail ldquoScalinggroup transformation for MHD boundary layer slip flow of ananofluid over a convectively heated stretching sheet with heatgenerationrdquo Mathematical Problems in Engineering vol 2012Article ID 934964 20 pages 2012

[31] MM RashidiM Ferdows A B Parsa and S Abelman ldquoMHDnatural convection with convective surface boundary conditionover a at platerdquo Abstract and Applied Analysis vol 2014 ArticleID 923487 10 pages 2014

[32] R E Bellman and R E Kalaba Quasilinearisation and Non-Linear Boundary-Value Problems Elsevier New York NY USA1965

[33] S SMotsa P GDlamini andMKhumalo ldquoSpectral relaxationmethod and spectral quasilinearization method for solvingunsteady boundary layer flow problemsrdquo Advances in Mathe-matical Physics vol 2014 Article ID 341964 12 pages 2014

[34] S S Motsa P Sibanda J M Ngnotchouye and G T MarewoldquoA spectral relaxation approach for unsteady boundary-layerflow and heat transfer of a nanofluid over a permeable stretch-ingshrinking sheetrdquo Advances in Mathematical Physics vol2014 Article ID 564942 10 pages 2014

[35] C Canuto M Y Hussaini A Quarteroni and T ZangSpectral Methods Fundamentals in Single Domains ScientificComputation Springer Berlin Germany 2006

[36] J HMerkin ldquoFree convection boundary layer on an isothermalhorizontal cylinderrdquo in Proceedings of the American Societyof Mechanical Engineers and American Institute of ChemicalEngineersHeat Transfer Conference p 911 ASME St LouisMoUSA August 1976

[37] R Nazar N Amin and I Pop ldquoFree convection boundary layeron an isothermal horizontal circular cylinder in a micropolarfluidrdquo in Proceedings of the 12th International Heat TransferConference vol 2 pp 525ndash530 Elsevier Paris France August2002

[38] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

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AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 3

x

y

glowast

u = 0

= w

u

C rarr Cinfin

T rarr Tinfinminusk

120655T

120655y= hf(Tf minus T)

C = Cw

Figure 1 Physical model and coordinate system

of viscosity 120573119879(119909) is the volumetric coefficient of thermal

expansion 120573119862(119909) is the volumetric coefficient of solutal

expansions 120581 is the vortex viscosity 119895 is the microinertiadensity 120574 is the spin-gradient viscosity 120572 is the thermaldiffusivity and119863 is the solutal diffusivity of the medium

The boundary conditions are

119906 = 0

V = V119908

120596 = minus 119899

120597119906

120597119910

minus 119896

120597119879

120597119910

= ℎ119891(119879119891minus119879)

119862 = 119862119908

at 119910 = 0

(6a)

119906 = 0

120596 = 0

119879 = 119879infin

119862 = 119862infin

as 119910 997888rarr infin

(6b)

where subscripts119908 andinfin indicate the conditions at the walland at the outer edge of the boundary layer respectively ℎ

119891

is the convective heat transfer coefficient 119896 is the thermalconductivity of the fluid and 119899 is amaterial constant Furtherwe follow the work of many recent authors by assuming that120574 = (120583 + 1205812)119895 This assumption is invoked to allow the fieldof equations to predict the correct behavior in the limitingcase when the microstructure effects become negligible andthe total spin 120596 reduces to the angular velocity [10]

3 Nondimensionalization ofthe Governing Equations

Introduce the following dimensionless variables

119909 =

119909

119871

119910 =

119910

119871

Gr14

119906 =

119871

]Gr12119906

V =119871

]Gr14V

120596 =

1198712

]Gr34120596

120579 =

119879 minus 119879infin

119879119891minus 119879infin

120601 =

119862 minus 119862infin

119862119908minus 119862infin

(7)

where Gr = 119892lowast

1205731198790(119879119891minus 119879infin)119871

3]2 is the Grashof number

In view of the continuity equation (1) we introduce thestream function 120595 by

119906 =

120597120595

120597119910

V = minus

120597120595

120597119909

(8)

Using (7) and (8) into (2)ndash(5) we get the following momen-tum angular momentum energy and concentration equa-tions

Δ 1 =120597120595

120597119910

1205972120595

120597119909120597119910

minus

120597120595

120597119909

1205972120595

1205971199102 minus(

11 minus 119873

)[

1205973120595

1205971199103 minus119873

120597120596

120597119910

]

minus

119892lowast

120573119879(119909) (119879

119891minus 119879infin) 119871

3

]2Gr120579

minus

119892lowast

120573119862(119909) (119862

119908minus 119862infin) 119871

3

]2Gr120601 = 0

Δ 2 =120597120595

120597119910

120597120596

120597119909

minus

120597120595

120597119909

120597120596

120597119910

minus(

2 minus 119873

2 minus 2119873)

1205972120596

1205971199102

+(

119873

1 minus 119873

)[2120596+

1205972120595

1205971199102 ] = 0

Δ 3 =120597120595

120597119910

120597120579

120597119909

minus

120597120595

120597119909

120597120579

120597119910

minus

1Pr

1205972120579

1205971199102 = 0

Δ 4 =120597120595

120597119910

120597120601

120597119909

minus

120597120595

120597119909

120597120601

120597119910

minus

1Sc

1205972120601

1205971199102 = 0

(9)

In usual definitions ] is the kinematic viscosity Pr = ]120572 isthe Prandtl number Sc = ]119863 is the Schmidt number 119873 =

120581(120583 + 120581) (0 le 119873 lt 1) is the coupling number [25] and themicroinertia density is taken to be 119895 = 119871

2Gr12

4 Advances in High Energy Physics

Now boundary conditions (6a) and (6b) become

120597120595

120597119910

= 0

120597120595

120597119909

= 119891119908

120596 = minus 119899

1205972120595

1205971199102

120597120579

120597119910

= minusBi (1minus 120579)

120601 = 1

at 119910 = 0

(10a)

120597120595

120597119910

= 0

120596 = 0

120579 = 0

120601 = 0

as 119910 997888rarr infin

(10b)

where 119891119908= minus(119871]Gr14)V

119908is the suctioninjection parame-

ter It is worth mentioning that 119891119908determines the transpira-

tion rate at the surface with119891119908gt 0 for suction and119891

119908lt 0 for

injection and119891119908= 0 corresponds to an impermeable surface

Further Bi = ℎ119891119871119896Gr14 is the Biot number It is a ratio of

the internal thermal resistance of the plate to the boundarylayer thermal resistance of the hot fluid at the bottom of thesurface

4 Similarity Equations via Lie Scaling GroupTransformations

Aone-parameter Lie scaling group of transformations whichis a simplified form of Lie group transformation is selected as(for more see [26ndash31])

Γ 119909lowast

= 1199091198901205761205721

119910lowast

= 1199101198901205761205722

120595lowast

= 1205951198901205761205723

120596lowast

= 1205961198901205761205724

120579lowast

= 1205791198901205761205725

120601lowast

= 1206011198901205761205726

120573lowast

119879= 1205731198791198901205761205727

120573lowast

119862= 1205731198621198901205761205728

(11)

Here 120576 = 0 is the parameter of the group and 120572119894(where

119894 = 1 2 3 8) are arbitrary real numbers whose interrela-tionship will be determined by our analysis Transformations

in (11) may be treated as a point transformation transformingthe coordinates

(119909 119910 120595 120596 120579 120601 120573119879 120573119862)

= (119909lowast

119910lowast

120595lowast

120596lowast

120579lowast

120601lowast

120573lowast

119879 120573lowast

119862)

(12)

We now investigate the relationship among the exponents 120572119894

(where 119894 = 1 2 3 8) such that

Δ119895[119909lowast

119910lowast

119906lowast

Vlowast 1205973120595lowast

120597119910lowast3 ]

= 119867119895[119909 119910 119906 V

1205973120595

1205971199103 119886]

sdot Δ119895[119909 119910 119906 V

1205973120595

1205971199103 ] (119895 = 1 2 3 4)

(13)

This is the requirement that the differential formsΔ 1Δ 2Δ 3and Δ 4 are conformally invariant under transformation (11)Substituting transformations (11) in (9) we have

Δ 1 = 119890120576(1205721+21205722minus21205723)

(

120597120595lowast

120597119910lowast

1205972120595lowast

120597119909lowast120597119910lowastminus

120597120595lowast

120597119909lowast

1205972120595lowast

120597119910lowast2 )

minus(

11 minus 119873

) 119890120576(31205722minus1205723) 120597

3120595lowast

120597119910lowast3

minus(

119873

1 minus 119873

) 119890120576(1205722minus1205724) 120597120596

lowast

120597119910lowast

minus

119892lowast

120573lowast

119879(119879119891minus 119879infin) 119871

3

]2Gr119890minus120576(1205725+1205727)

120579lowast

minus

119892lowast

120573lowast

119862(119862119908minus 119862infin) 119871

3

]2Gr119890minus120576(1205726+1205728)

120601lowast

= 0

(14a)

Δ 2 = 119890120576(1205721+1205722minus1205723minus1205724)

(

120597120595lowast

120597119910lowast

120597120596lowast

120597119909lowastminus

120597120595lowast

120597119909lowast

120597120596lowast

120597119910lowast)

minus(

2 minus 119873

2 minus 2119873) 119890120576(21205722minus1205724) 120597

2120596lowast

120597119910lowast2

+(

119873

1 minus 119873

)(2120596lowast119890minus1205761205724 + 119890120576(21205722minus1205723) 120597

2120595lowast

120597119910lowast2 )

= 0

(14b)

Δ 3 = 119890120576(1205721+1205722minus1205723minus1205725)

(

120597120595lowast

120597119910lowast

120597120579lowast

120597119909lowastminus

120597120595lowast

120597119909lowast

120597120579lowast

120597119910lowast)

minus

1Pr

119890120576(21205722minus1205725)

(

1205972120579lowast

120597119910lowast2) = 0

(14c)

Δ 4 = 119890120576(1205721+1205722minus1205723minus1205726)

(

120597120595lowast

120597119910lowast

120597120601lowast

120597119909lowastminus

120597120595lowast

120597119909lowast

120597120601lowast

120597119910lowast)

minus

1Sc

119890120576(21205722minus1205726)

(

1205972120601lowast

120597119910lowast2) = 0

(14d)

Advances in High Energy Physics 5

Now boundary conditions (10a) and (10b) become

119890120576(1205722minus1205723)

120597120595lowast

120597119910lowast= 0

119890120576(1205721minus1205723)

120597120595lowast

120597119909lowast= 119891119908

119890minus1205761205724

120596lowast

= minus 119899119890120576(21205722minus1205723) 120597

2120595lowast

120597119910lowast2

119890120576(1205722minus1205725) 120597120579

lowast

120597119910lowast= minusBi (1minus 119890

minus1205761205725120579lowast

)

119890minus1205761205726

120601lowast

= 1at 119910lowast = 0

(15a)

119890120576(1205722minus1205723)

120597120595lowast

120597119910lowast= 0

119890minus1205761205724

120596lowast

= 0119890minus1205761205725

120579lowast

= 0119890minus1205761205726

120601lowast

= 0as 119910lowast 997888rarr infin

(15b)

The system remains invariant under the group transforma-tion Γ We then have the following relationships for theparameters

1205721 + 21205722 minus 21205723 = 31205722 minus1205723 = 1205722 minus1205724 = minus1205725 minus1205727= minus1205726 minus1205728

1205721 +1205722 minus1205723 minus1205724 = 21205722 minus1205724 = minus1205724 = 21205722 minus1205723

1205721 +1205722 minus1205723 minus1205725 = 21205722 minus1205725

1205721 +1205722 minus1205723 minus1205726 = 21205722 minus1205726

1205721 minus1205723 = 0

minus 1205724 = 21205722 minus1205723

1205722 minus1205725 = 0 = minus1205725

1205726 = 0

(16)

Solving linear system (16) we have the following relationshipamong the exponents

1205721 = 1205723 = 1205724 = 1205727 = 1205728

1205722 = 1205725 = 1205726 = 0(17)

The set of transformations Γ reduces to

119909lowast

= 1199091198901205761205721

119910lowast

= 119910

120595lowast

= 1205951198901205761205721

120596lowast

= 1205961198901205761205721

120579lowast

= 120579

120601lowast

= 120601

120573lowast

119879= 1205731198791198901205761205721

120573lowast

119862= 1205731198621198901205761205721

(18)

Expanding by theTaylor series in power of 120576 keeping the termup to the first degree (neglecting higher power of 120576) we get

119909lowast

minus119909 = 1205761205721119909

119910lowast

= 119910

120595lowast

minus120595 = 1205761205721120595

120596lowast

minus120596 = 1205761205721120596

120579lowast

= 120579

120601lowast

= 120601

120573lowast

119879minus120573119879= 1205761205721120573119879

120573lowast

119862minus120573119862= 1205761205721120573119862

(19)

The characteristic equations are

119889119909

1205721119909=

119889119910

0=

119889120595

1205721120595=

119889120596

1205721120596=

119889120579

0=

119889120601

0=

119889120573119879

1205721120573119879

=

119889120573119862

1205721120573119862

(20)

Solving the above characteristic equations we have thefollowing similarity transformations

120578 = 119910

120595 = 119909119891 (120578)

120596 = 119909119892 (120578)

120573119879= 1205731198790119909

120573119862= 1205731198620119909

120579 = 120579 (120578)

120601 = 120601 (120578)

(21)

where 1205731198790and 120573

1198620are constant thermal and mass coefficient

of expansionUsing (21) into (9) we get the following similarity

equations

(

11 minus 119873

)119891101584010158401015840

+11989111989110158401015840

minus11989110158402+(

119873

1 minus 119873

)1198921015840

+ 120579+B120601

= 0

(

2 minus 119873

2 minus 2119873)11989210158401015840

+1198911198921015840

minus1198911015840

119892minus(

119873

1 minus 119873

) (2119892+11989110158401015840

)

= 0

1Pr

12057910158401015840

+1198911205791015840

= 0

1Sc

12060110158401015840

+1198911206011015840

= 0

(22)

where the primes indicate differentiation with respect to 120578

alone andB = 1205731198620(119862119908minus 119862infin)1205731198790(119879119891minus 119879infin) is the buoyancy

ratio

6 Advances in High Energy Physics

Boundary conditions (10a) and (10b) in terms of 119891 119892 120579and 120601 become

120578 = 0 119891 (0) = 119891119908

1198911015840

(0) = 0

119892 (0) = minus 11989911989110158401015840

(0)

1205791015840

(0) = minusBi [1minus 120579 (0)]

120601 (0) = 1

(23a)

120578 997888rarr infin 1198911015840

(infin) = 0

119892 (infin) = 0

120579 (infin) = 0

120601 (infin) = 0

(23b)

5 Skin Friction Wall Couple Stress and Heatand Mass Transfer Coefficients

The wall shear stress and the wall couple stress are

120591119908= [(120583 + 120581)

120597119906

120597119910

+ 120581120596]

119910=0

119898119908= 120574 [

120597120596

120597119910

]

119910=0

(24a)

and the heat and mass transfers from the plate respectivelyare given by

119902119908= minus 119896 [

120597119879

120597119910

]

119910=0

119902119898= minus119863[

120597119862

120597119910

]

119910=0

(24b)

The nondimensional skin friction 119862119891

= 2120591119908120588119906

2lowast wall

couple stress 119872119908

= 119898119908120588119906

2lowast119909 the local Nusselt number

119873119906119909= 119902119908119909119896(119879

119891minus 119879infin) and local Sherwood number Sh

119909=

119902119898119909119863(119862

119908minus 119862infin) are given by

119862119891Gr14119909

= 2(1 minus 119899119873

1 minus 119873

)11989110158401015840

(0)

119872119908Gr12119909

= (

2 minus 119873

2 minus 2119873)1198921015840

(0)

119873119906119909

Gr14119909

= minus 1205791015840

(0)

Sh119909

Gr14119909

= minus1206011015840

(0)

(25)

where 1199062lowastis the characteristic velocity and Gr

119909= 119892lowast

1205731198790(119879119891minus

119879infin)119909

3]2 is the local Grashof number

6 Numerical Solution Using the SpectralQuasi-Linearization Method (SQLM)

In this section we describe the quasi-linearization method(QLM) for solving the governing system of (22) alongwith boundary conditions (23a) and (23b) This QLM isa generalization of the Newton-Raphson method and wasproposed by Bellman and Kalaba [32] for solving nonlinearboundary value problems

Assume that the solutions 119891119903 119892119903 120579119903 and 120601

119903of (22) at the

(119903+1)th iteration are119891119903+1 119892119903+1 120579119903+1 and 120601119903+1 If the solutions

at the previous iteration are sufficiently close to the solutionsat the present iteration the nonlinear components of (22)can be linearised using one-term Taylor series of multiplevariables so that (22) give the following iterative sequence oflinear differential equations

(

11 minus 119873

)119891101584010158401015840

119903+1 + 119886111990311989110158401015840

119903+1 + 11988621199031198911015840

119903+1 + 1198863119903119891119903+1

+(

119873

1 minus 119873

)1198921015840

119903+1 + 120579119903+1 +B120601

119903+1 = 1198771119903

(

2 minus 119873

2 minus 2119873)11989210158401015840

119903+1 + 11988731199031198921015840

119903+1 + 1198874119903119892119903+1 + 11988711199031198911015840

119903+1

+ 1198872119903119891119903+1 minus(

119873

1 minus 119873

)11989110158401015840

119903+1 = 1198772119903

1198881119903119891119903+1 +1Pr

12057910158401015840

119903+1 + 11988821199031205791015840

119903+1 = 1198773119903

1198891119903119891119903+1 +1Sc

12060110158401015840

119903+1 +11988921199031206011015840

119903+1 = 1198774119903

(26)

where the coefficients 1198861199041 119903

(1199041 = 1 2 3) 1198871199042 119903

(1199042 = 1 2 4) 1198881199043 119903

(1199043 = 1 2) 1198891199044 119903

(1199044 = 1 2) and 1198771199045 119903

(1199045 = 1 2 4) are known functions (from previous iterations) and aredefined as

1198861119903 = 119891119903

1198862119903 = minus 21198911015840119903

1198863119903 = 11989110158401015840

119903

1198771119903 = 11989111990311989110158401015840

119903minus (1198911015840

119903)

2

1198871119903 = minus119892119903

1198872119903 = 1198921015840

119903

1198873119903 = 119891119903

1198874119903 = minus1198911015840

119903minus(

21198731 minus 119873

)

1198772119903 = 1198911199031198921015840

119903minus1198911015840

119903119892119903

1198881119903 = 1205791015840

119903

1198882119903 = 119891119903

1198773119903 = 1198911199031205791015840

119903

1198891119903 = 1206011015840

119903

1198892119903 = 119891119903

1198774119903 = 1198911199031206011015840

119903

(27)

Advances in High Energy Physics 7

subject to boundary conditions

119891119903+1 (0) = 119891

119908

1198911015840

119903+1 = 0

1198911015840

119903+1 (infin) = 0

119892119903+1 = minus 119899119891

10158401015840

119903+1 (0)

119892119903+1 (infin) = 0

1205791015840

119903+1 (0) = minusBi (1minus 120579 (0))

120579119903+1 (infin) = 0

120601119903+1 (0) = 1

120601119903+1 (infin) = 0

(28)

System (26) constitutes a linear system of coupled differen-tial equations with variable coefficients and can be solvediteratively using any numerical method for 119903 = 1 2 3 In this work as will be discussed below the Chebyshevpseudospectral method was used to solve the QLM scheme(26) (for more details refer to the works of Motsa et al[33 34])

1198910 (120578) = 119891119908+ 1minus 119890

minus120578

1198920 (120578) = minus 119899119890minus120578

1205790 = 119890minus120578

BiBi + 1

1206010 = 119890minus120578

(29)

and starting from these sets of initial approximations 1198910 11989201205790 and 1206010 the iteration schemes (26) can be solved iterativelyfor 119891119903+1(120578) 119892119903+1(120578) 120579119903+1(120578) and120601119903+1(120578) when 119903 = 0 1 2

For this we discretise the equation using the Chebyshevspectral collocation method The unknown functions areapproximated by the Chebyshev interpolating polynomialsin such way that they are collocated at the Gauss-Lobattocollocation points defined as

120591119895= cos

120587119895

119873

119895 = 0 1 2 119873 (30)

where 119873 is the number of collocation points The physicalregion [0infin) is transformed into the region [minus1 1] using thedomain truncation technique in which the problem is solvedon the interval [0 120578

infin] instead of [0infin) This leads to the

mapping

120578

120578infin

=

120591 + 12

minus1 le 120591 le 1 (31)

where 120578infin

is the scaling parameter used to invoke theboundary condition at infinity The functions 119891 119892 120579 and 120601

are approximated at the collocation points by

119891 (120591) =

119873

sum

119896=0119891 (120591119896) 119879119896(120591119895)

119892 (120591) =

119873

sum

119896=0119892 (120591119896) 119879119896(120591119895)

120579 (120591) =

119873

sum

119896=0120579 (120591119896) 119879119896(120591119895)

120601 (120591) =

119873

sum

119896=0120601 (120591119896) 119879119896(120591119895)

119895 = 0 1 2 119873

(32)

where 119879119896is the 119896th Chebyshev polynomial defined as

119879119896(120591) = cos [119896 cosminus1 (120591)] (33)

The derivatives of the variables at the collocation points arerepresented as

119889119901

119891

119889120578119901=

119873

sum

119896=0D119901119897119896119891 (120591119896)

119889119901

119892

119889120578119901=

119873

sum

119896=0D119901119897119896119892 (120591119896)

119889119901

120579

119889120578119901=

119873

sum

119896=0D119901119897119896120579 (120591119896)

119889119901

120601

119889120578119901=

119873

sum

119896=0D119901119897119896120601 (120591119896)

119897 = 0 1 119873

(34)

where 119901 is the order of the derivative and D = 2D120578infin

is theChebyshev spectral differentiation matrix and its entries areclearly defined in Canuto et al [35]

Substituting (31)ndash(34) into (26) leads to the matrix equa-tion

119860119883 = 119877 (35)

subject to the boundary conditions

119891119903+1 (120591119873) = 119891

119908

119873

sum

119896=0D119873119896

119891 (120591119896) = 0

119873

sum

119896=0D0119896119891 (120591

119896) = 0

119892119903+1 (120591119873) = minus 119899

119873

sum

119896=0D2119873119896

119891 (120591119896)

119892119903+1 (1205910) = 0

8 Advances in High Energy Physics

119873

sum

119896=0D119873119896

120579119903+1 (120591119896) minusBi 120579

119903+1 (120591119873) = minusBi

120579119903+1 (1205910) = 0

120601119903+1 (120591119873) = 1120601119903+1 (1205910) = 0

(36)

In (35) 119860 is a (4119873+ 4) times (4119873+ 4) square matrix and119883 and 119877are (4119873 + 1) times 1 column vectors defined by

119860 =

[

[

[

[

[

[

11986011 11986012 11986013 11986014

11986021 11986022 11986023 11986024

11986031 11986032 11986033 11986034

11986041 11986042 11986043 11986044

]

]

]

]

]

]

119883 =

[

[

[

[

[

[

F119903+1

G119903+1

Θ119903+1

Φ119903+1

]

]

]

]

]

]

119877 =

[

[

[

[

[

[

R1

R2

R3

R4

]

]

]

]

]

]

(37)

whereF = [119891

119903+1 (1205910) 119891119903+1 (1205911) 119891119903+1 (120591119873)]119879

G = [119892119903+1 (1205910) 119892119903+1 (1205911) 119892119903+1 (120591119873)]

119879

Θ = [120579119903+1 (1205910) 120579119903+1 (1205911) 120579119903+1 (120591119873)]

119879

Φ = [120601119903+1 (1205910) 120601119903+1 (1205911) 120601119903+1 (120591119873)]

119879

11986011 = (

11 minus 119873

)D3+ diag [1198861119903]D

2+ diag [1198862119903]D

+ diag [1198863119903]

11986012 = (

119873

1 minus 119873

)D

11986013 = I

11986014 = BI

11986021 = minus(

119873

1 minus 119873

)D2+ diag [1198871119903]D+ diag [1198872119903]

11986022 = (

2 minus 119873

2 minus 2119873)D2

+ diag [1198873119903]D+ diag [1198874119903]

11986023 = 0

11986024 = 0

11986031 = diag [1198881119903]

11986032 = 0

11986033 =1Pr

D2+ diag [1198882119903]D

Table 1 Comparison of minus1205791015840(0) for free convection along a verticalflat plate in Newtonian fluid when 119873 = 0 119899 = 0 B = 0 Pr = 1Bi rarr infin and 119891

119908= 0

Merkin [36] Nazar et al [37] Molla et al [38] Present04214 04214 04214 04214313

11986034 = 0

11986041 = diag [1198891119903]

11986042 = 0

11986043 = 0

11986044 =1Sc

D2+ diag [1198892119903]D

R1 = 1198771119903

R2 = 1198772119903

R3 = 1198773119903

R4 = 1198774119903

(38)

and here I is an identity matrix the size of the matrix 0 is(119873+1)times1 and diag[ ] is a diagonalmatrix of size (119873+1)times(119873+

1) Subscript r denotes the iteration number After modifyingmatrix system (35) to incorporate boundary condition (36)the solution is obtained as

119883 = 119860minus1119877 (39)

7 Results and Discussions

It is noticed that the present problem reduces to free convec-tion heat transfer along an impermeable vertical plate in amicropolar fluid without the convective boundary conditionwhen119891

119908= 0 Bi rarr infin andB = 0 Also in the limit as119873 rarr

0 governing equations (2)ndash(5) reduce to the correspondingequations for a free convection heat and mass transfer ina viscous fluid In order to validate the code generated theresults of the present problem have been compared with theresults obtained by Merkin [36] Nazar et al [37] and Mollaet al [38] as a special case by taking 119873 = 0 119899 = 0 B = 0Pr = 1 Bi rarr infin and 119891

119908= 0 and it was found that

they are in good agreement as presented in Table 1 Also thecomparison of heat transfer coefficient has been made withthe results obtained by Nazar et al [37] as shown in Table 2when 119899 = 05 B = 0 Pr = 1 Bi rarr infin and 119891

119908= 0

To study the effects of coupling number119873 suctioninjectionparameter 119891

119908 Biot number Bi and material parameter 119899

computations were carried out in the cases of B = 10Pr = 071 and Sc = 022

The effects of the coupling number119873 on the dimension-less velocity microrotation temperature and concentrationare illustrated in Figures 2(a)ndash2(d) with fixed values ofother parameters The coupling number 119873 characterizes thecoupling of linear and rotational motion arising from the

Advances in High Energy Physics 9

00

01

02

03

04

05

06f998400

0 4 8 12 16

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(a)

g

minus06

minus05

minus04

minus03

minus02

minus01

00

01

0 2 4 6 8 10 12

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(b)

000

004

008

012

016

120579

0 1 2 3 4 5 6

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(c)

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

120601

00

02

04

06

08

10

120578

0 2 4 6 8 10 12

Bi = 01 fw = 05

(d)

Figure 2 Effect of119873 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Table 2 Comparison of minus1205791015840

(0) for free convection flow in amicropolar fluid obtained by Nazar et al [37] when 119899 = 05B = 0Pr = 1 Bi rarr infin and 119891

119908= 0

119873 Nazar et al [37] Present000 04214 04214033 03991 03990050 03834 03834060 03709 03709066 03608 03608071 03522 03522075 03447 03447

fluid particles In the case of 119873 = 0 (ie as 120581 tends tozero) the micropolarity is absent and fluid becomes nonpolarfluid With a large value of 119873 effect of microstructurebecomes significant whereas with a diminished value of 119873the individuality of the substructure is much less articulatedAs 119873 increases it is found from Figure 2(a) that the max-imum velocity decreases in amplitude and the location ofthe maximum velocity moves farther away from the wallSince 119873 rarr 0 corresponds to viscous fluid the velocity incase of a micropolar fluid has been less compared to thatof viscous fluid case It can be observed from Figure 2(b)that as 119873 increases initially microrotation profiles tend tobecome flatter and then approach their free stream valuesfar away from the wall This happens due to the vanishing

10 Advances in High Energy Physics

00

01

02

03

04

05

06

f998400

120578

0 2 4 6 8 10 12

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(a)

minus06

minus04

minus03

minus02

minus05

minus01

00

01

g

0 2 4 6 8 10

120578

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(b)

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

120578

0 1 2 3 4 5

120579

00

02

04

06

08

10

(c)

120601

00

02

04

06

08

10

fw = 05 n = 05N = 05

0 2 4 6 8 10 12

120578

Bi = 01

Bi = 10

Bi = 50Bi = 200

(d)

Figure 3 Effect of Bi on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

of antisymmetric part of the stress on the boundary thatcorresponds to a weak concentration of microelements Thisis because an increment in the value of 119873 implies a highervortex viscosity of fluid which promotes the microrotationof micropolar fluids It is seen from Figures 2(c) and 2(d)that thermal and concentration boundary layers of thefluid increase with increase in coupling number 119873 Hencetemperature and concentration in case of micropolar fluidsare more than those of the viscous fluid case

The Biot number Bi is the ratio of internal thermalresistance of a solid to boundary layer thermal resistanceWhen Bi = 0 the plate is totally insulated internal thermalresistance of the plate is extremely high and no convectiveheat transfer to the cold fluid on the upper part of the

plate takes place Figure 3(a) depicts fluid velocity profilesfor different values of the Biot number with 119873 = 05 119891

119908=

05 and 119899 = 05 Generally fluid velocity is zero at platesurface and increases gradually away fromplate to free streamvalue satisfying boundary conditions It is interesting to notethat an increase in the intensity of convective surface heattransfer Bi produces significant enhancement in fluid velocitywithin the momentum boundary layer In Figure 3(b) webring out the behavior of microrotation with different valuesof Biot number Bi for fixed values of other parameters As theparameter value Bi increases microrotation showing reverserotation near the two boundaries Hence the condition ofvanishing of antisymmetric part of the stress on the boundaryresults in a drastic change of the microrotation profiles

Advances in High Energy Physics 11

07

06

05

04

03

02

01

00

f998400

fw = 00

fw = 10

fw = 20

0 3 6 9 12 15

120578

N = 05 Bi = 01 n = 05

fw = minus05

(a)

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

120578

0 2 4 6 8 10 12

g

minus04

minus03

minus02

minus01

00

01

fw = minus05

(b)

0 1 2 3 4 5 6

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

000

005

010

015

020

025

030

120579

fw = minus05

(c)

0 3 6 9 12

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

00

02

04

06

08

10

120601

fw = minus05

(d)

Figure 4 Effect of 119891119908on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Given that convective heating increases with Biot numberBi rarr infin simulates the isothermal surface which is clearlyseen from Figure 3(c) where 120579(0) = 1 as Bi rarr infin Infact a high Biot number indicates higher internal thermalresistance of the plate than boundary layer thermal resistanceIn this fluid temperature is maximum at the plate surfaceand decreases exponentially to zero value far out from theplate satisfying boundary conditions As a consequence anincrement in the Biot number leads to increase of fluidtemperature efficiency Figure 3(d) illustrates the variation ofdimensionless concentration for different values of Bi It isclear that the concentration of fluid decreases with increaseof Bi

The effect of 119891119908

on velocity profile is depicted inFigure 4(a) Here 119891

119908gt 0 represents suction and 119891

119908lt 0

denotes injectionThe lower velocity is noticed in case of suc-tion when compared to case of injection From Figure 4(b)we note that microrotation is showing reverse rotation neartwo boundaries with both suction and injection parame-ter The dimensionless temperature for different values ofsuctioninjection parameters is drawn in Figure 4(c) It isreadable that the temperature of the fluid is more in case ofinjection whereas it is less in case of suction in comparisonwith the impermeable surface case (119891

119908= 0) Figure 4(d)

demonstrates dimensionless concentration for different val-ues of suctioninjection parameters It is determined that the

12 Advances in High Energy Physics

00

01

02

03

04

05

06f998400

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(a)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10

120578

minus10

minus08

minus06

minus04

minus02

00

g

(b)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

120578

0 1 2 3 4 5

120579

000

003

006

009

012

015

(c)

00

02

04

06

08

10

120601

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(d)

Figure 5 Effect of material parameter 119899 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

concentration of fluid ismore with injection whereas it is lesswith suctionwhen compared to the impermeable surface case(119891119908= 0) As a finale the thermal and solutal boundary layer

thicknesses increase in case of injection compared to case ofsuction as displayed in Figures 4(c) and 4(d)

In Figures 5(a)ndash5(d) the effect of material parameter119899 on dimensionless velocity microrotation temperatureand concentration is presented for fixed values of otherparameters since the material parameter 119899 signifies themicrorotation effects (ie for 119899 = 0 particles are not freeto revolve near the surface whereas as 119899 increases from 0 to1 the microrotation term gets augmented and induces flowenhancement) As 119899 increases it is found from Figure 5(a)that the minimum velocity increases in amplitude and the

location of the minimum velocity moves farther away fromthe wall From Figure 5(b) we observe that themicrorotationis decreasing with increasing value of material parameter119899 within the boundary layer It is clear from Figures 5(c)-5(d) that the thermal and solutal boundary layer thicknessesdecrease with increase of material parameter 119899

The variations of minus1205791015840

(0) and minus1206011015840

(0) versus couplingnumber 119873 are shown in Figures 6ndash8 It can be noticed fromthese figures that the heat and mass transfer coefficientsare less in case of micropolar fluids when compared to theviscous fluids This is because as 119873 increases the thermaland solutal boundary layer thicknesses become larger thusgiving rise to a small value of local heat and mass transferrates The effect of the Biot number Bi with fixed 119891

119908= 05

Advances in High Energy Physics 13

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

fw = 05 n = 05

01

00

02

03

04

05

06

07Nu x

Grminus

14

x

(a)

fw = 05 n = 05

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

024

026

028

030

032

ShxGrminus

14

x

(b)

Figure 6 Effect of Bi on (a) heat transfer rate and (b) mass transfer rate

005

006

007

008

009

010

Nu x

Grminus

14

x

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

(a)

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

01

02

03

04

05

06

ShxGrminus

14

x

(b)

Figure 7 Effect of 119891119908on (a) heat transfer rate and (b) mass transfer rate

and 119899 = 05 on local heat transfer coefficient is exhibitedin Figure 6 It is found from Figure 6 that local heat transferrate increases nonlinearly with the increase in Biot numberBiThe influence of the Biot number Bi on local mass transfercoefficient is shown in Figure 6 Figure 6 reveals that the localmass transfer coefficient is enhanced by the growth in theBiot number Bi In Figure 7 the effect of the suctioninjectionparameter 119891

119908with fixed Bi = 01 and 119899 = 05 on local heat

and mass transfer coefficients is displayed It is found fromFigure 7 that the local heat and mass transfer coefficients areless in the case of injection 119891

119908lt 0 in comparison with the

case of suction 119891119908

gt 0 Figure 8 is prepared to analyze theeffect of the material parameter 119899 with fixed Bi = 01 and

119891119908= 05 on local heat andmass transfer coefficients Figure 8

reveals that the local heat and mass transfer coefficients areenhanced by the increase in material parameter 119899 This isbecause when 119899 increases from 0 to 1 the microrotation termgets augmented and induces flow enhancement

The variations of 119862119891Gr14119909

and 119872119908Gr12119909

which areproportional to the coefficients of skin friction and wallcouple stress are shown in Table 3 with different values ofthe coupling number 119873 for fixed 119899 = 05 119891

119908= 05 and

Bi = 01 It indicates that the skin friction factor is higherfor micropolar fluid than the viscous fluids (119873 = 0) Sincemicropolar fluids offer a heavy resistance (resulting fromvortex viscosity) to fluid movement and cause larger skin

14 Advances in High Energy Physics

0084

0085

0086

0087Nu x

Grminus

14

x

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

fw = 05 Bi = 01

(a)

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

024

026

028

030

ShxGrminus

14

x

fw = 05 Bi = 01

(b)

Figure 8 Effect of material parameter 119899 on (a) heat transfer rate and (b) mass transfer rate

Table 3 Effects of skin friction and wall couple stress for varyingvalues of Biot numbers Bi micropolar parameter 119873 materialparameter 119899 and suctioninjection parameter 119891

119908

119873 Bi 119891119908

119899 119862119891Gr14119909

119872119908Gr12119909

01 01 05 05 2424227 06327903 01 05 05 2502716 063996605 01 05 05 2647987 063086307 01 05 05 2943561 060387909 01 05 05 3874631 055089105 01 05 05 2647987 063086305 10 05 05 3211755 080587905 50 05 05 3533141 090807805 200 05 05 3629791 093913705 01 minus05 05 2426265 033621905 01 00 05 2530014 046542905 01 10 05 2737553 082089905 01 20 05 2721244 119359605 01 05 00 2916655 minus028933905 01 05 05 2647987 063086305 01 05 10 2232556 2066191

friction factor compared to viscous fluid the results as wellsuggest larger values of coupling number 119873 and lower wallcouple stresses Since the skin friction coefficient 11989110158401015840(0) andwall couple stress coefficient as well as high temperature andmass transport rates are more depressed in the micropolarfluid comparing to the viscous fluid which may be beneficialin flow temperature and concentration control of polymerprocessing thus the presence of microscopic effects arisingfrom the local structure and of the fluid elements reduces thehigh temperature and mass transfer coefficients The effectof Bi on 119862

119891Gr14119909

and 119872119908Gr12119909

for 119891119908

= 05 119899 = 05 and

119873 = 05 is illustrated in Table 3 It can be noticed that theskin friction and wall couple stress coefficients are increasingwith increase of Bi for fixed values of other parametersThis notice is consistent with physical profiles presented inFigure 3 The effects of suctioninjection parameter on theskin friction andwall couple stress coefficients are also shownin Table 3 It is noted that the skin friction and wall couplestress coefficients are less with injection case whereas theyare more with suction case when compared to the case ofimpermeable surface Finally the detailed behavior of thematerial parameter 119899 is given in Table 3 The skin frictiondecreases and wall couple stress increases with increase ofmaterial parameter 119899

8 Conclusions

In this composition the similarity solution of the free convec-tion flow on a permeable vertical plate of a micropolar fluidunder the convective boundary condition is obtained usingLie group transformations Using the similarity variables thegoverning equations are transformed into a set of nondi-mensional parabolic equations These equations are solvednumerically using spectral quasi-linearisation method Thenumerical computation is carried out for various values ofnondimensional physical parameters The main findings aresummarized as follows

(i) The numerical results indicate that velocity distribu-tion is less near the plate but it is more far away fromthe plate the wall couple stress coefficient and rate ofheat andmass transfers are lower but the temperatureand concentration distributions and the skin frictioncoefficient are higher for the micropolar fluids incomparison with those of viscous fluids Also thereverse rotation ofmicrorotation near two boundariesis found with the increasing value of119873

Advances in High Energy Physics 15

(ii) An increase in Biot number Bi decreases concen-tration distribution whereas it causes an increasein temperature distribution skin friction and wallcouple stress coefficients and heat and mass transferrates within the boundary layer Further enhancingin Biot number Bi enhances velocity distributionnear the plate but shows the reverse behavior faraway from the plate We observe reverse rotationof microrotation near two boundaries within theboundary layer in the presence of Biot number

(iii) Less velocity temperature and concentration distri-butions are observed more skin friction and wallcouple stress coefficients and heat and mass transferrates in the case of suction compared to the case ofinjection Further microrotation decreases near thewall and depicts the opposite trend far away from thewall

(iv) It is found that microrotation temperature and con-centration distributions and skin friction coefficientsare more in the case of a micropolar fluid with strongconcentration (ie 119899 = 0) when compared to the caseof a micropolar fluid with weak concentration (ie119899 = 12) Further velocity is less in the case of 119899 = 0when compared to the case of 119899 = 12 near the walland shows the opposite trend far away from the wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their valuablesuggestions and comments

References

[1] J H Merkin ldquoNatural-convection boundary-layer flow on avertical surface with Newtonian heatingrdquo International Journalof Heat and Fluid Flow vol 15 no 5 pp 392ndash398 1994

[2] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009

[3] O DMakinde ldquoSimilarity solution for natural convection froma moving vertical plate with internal heat generation and aconvective boundary conditionrdquo Thermal Science vol 15 ppS137ndashS143 2011

[4] S M Ibrahim and N Bhashar Reddy ldquoSimilarity solution ofheat and mass transfer for natural convection over a movingvertical plate with internal heat generation and a convectiveboundary condition in the presence of thermal radiationviscous dissipation and chemical reactionrdquo ISRN Thermody-namics vol 2013 Article ID 790604 10 pages 2013

[5] C RamReddy P V S N Murthy A J Chamkha and A MRashad ldquoSoret effect on mixed convection flow in a nanofluidunder convective boundary conditionrdquo International Journal ofHeat and Mass Transfer vol 64 pp 384ndash392 2013

[6] H T Alkasasbeh M Z Salleh R M Tahar and R NazarldquoNumerical solutions of free convection boundary layer flow ona solid sphere with convective boundary conditionsrdquo Journal ofPhysics Conference Series vol 495 Article ID 012025 2014

[7] A Pantokratoras ldquoA note on natural convection along a con-vectively heated vertical platerdquo International Journal of ThermalSciences vol 76 pp 221ndash224 2014

[8] G Lukaszewicz Micropolar FluidsmdashTheory and ApplicationsBirkhaauser Basel Switzerland 1999

[9] V A Eremeyev L P Lebedev and H Altenbach Foundationsof Micropolar Mechanics Springer New York NY USA 2013

[10] G Ahmadi ldquoSelf-similar solution of imcompressible micropo-lar boundary layer flow over a semi-infinite platerdquo InternationalJournal of Engineering Science vol 14 no 7 pp 639ndash646 1976

[11] D A Rees and I Pop ldquoFree convection boundary-layer flowof a micropolar fluid from a vertical flat platerdquo IMA Journal ofApplied Mathematics vol 61 no 2 pp 179ndash197 1998

[12] D Srinivasacharya and C RamReddy ldquoEffect of double stratifi-cation onmixed convection in a micropolar fluidrdquoMatematikavol 28 no 2 pp 133ndash149 2012

[13] D Srinivasacharya and C Ramreddy ldquoMixed convection heatand mass transfer in a doubly stratified micropolar fluidrdquoComputationalThermal Sciences vol 5 no 4 pp 273ndash287 2013

[14] D Srinivasacharya and C RamReddy ldquoNatural convection heatand mass transfer in a micropolar fluid with thermal and massstratificationrdquo International Journal for Computational Methodsin Engineering Science andMechanics vol 14 no 5 pp 401ndash4132013

[15] N A Yacob and A Ishak ldquoStagnation point flow towardsa stretchingshrinking sheet in a micropolar fluid with aconvective surface boundary conditionrdquoThe Canadian Journalof Chemical Engineering vol 90 no 3 pp 621ndash626 2012

[16] K Zaimi and A Ishak ldquoStagnation-point flow and heat transferover a nonlinearly stretchingshrinking sheet in a micropolarfluidrdquo Abstract and Applied Analysis vol 2014 Article ID261630 6 pages 2014

[17] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[18] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[19] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer 2009

[20] T Tapanidis G Tsagas and H P Mazumdar ldquoApplication ofscaling group of transformations to viscoelastic second-gradefluid flowrdquo Nonlinear Functional Analysis and Applications vol8 no 3 pp 345ndash350 2003

[21] I A A Hassanien andM A Hamad ldquoGroup theoretic methodfor unsteady free convection flow of a micropolar fluid along avertical plate in a thermally stratified mediumrdquo Applied Mathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 32 no 6 pp 1099ndash1114 2008

[22] R Kandasamy K Gunasekaran and S B H Hasan ldquoScalinggroup transformation on fluid flow with variable stream con-ditionsrdquo International Journal of Non-Linear Mechanics vol 46no 7 pp 976ndash985 2011

[23] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat TransfermdashAsian Research vol41 no 3 pp 241ndash259 2012

16 Advances in High Energy Physics

[24] A A Mutlag M J Uddin M A A Hamad and A I M IsmailldquoHeat transfer analysis for falkner-skan boundary layer flowpast a stationary wedge with slips boundary conditions con-sidering temperature-dependent thermal conductivityrdquo SainsMalaysiana vol 42 no 6 pp 855ndash862 2013

[25] S C Cowin ldquoPolar fluidsrdquo Physics of Fluids vol 11 no 9 pp1919ndash1927 1968

[26] M A Seddeek M Y Akl and A M Al-Hanaya ldquoThermalradiation effects on mixed convection and mass transfer flowon vertical porous plate with heat generation and chemicalreaction by using scaling grouprdquo Journal of Natural Sciences andMathematics vol 4 pp 41ndash60 2010

[27] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[28] M J Uddin W A Khan and A I M Ismail ldquoMHD freeconvective boundary layer flow of a nanofluid past a flat verticalplate with Newtonian heating boundary conditionrdquo PLoS ONEvol 7 no 11 Article ID e49499 2012

[29] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012

[30] M J Uddin W A Khan and A I M Ismail ldquoScalinggroup transformation for MHD boundary layer slip flow of ananofluid over a convectively heated stretching sheet with heatgenerationrdquo Mathematical Problems in Engineering vol 2012Article ID 934964 20 pages 2012

[31] MM RashidiM Ferdows A B Parsa and S Abelman ldquoMHDnatural convection with convective surface boundary conditionover a at platerdquo Abstract and Applied Analysis vol 2014 ArticleID 923487 10 pages 2014

[32] R E Bellman and R E Kalaba Quasilinearisation and Non-Linear Boundary-Value Problems Elsevier New York NY USA1965

[33] S SMotsa P GDlamini andMKhumalo ldquoSpectral relaxationmethod and spectral quasilinearization method for solvingunsteady boundary layer flow problemsrdquo Advances in Mathe-matical Physics vol 2014 Article ID 341964 12 pages 2014

[34] S S Motsa P Sibanda J M Ngnotchouye and G T MarewoldquoA spectral relaxation approach for unsteady boundary-layerflow and heat transfer of a nanofluid over a permeable stretch-ingshrinking sheetrdquo Advances in Mathematical Physics vol2014 Article ID 564942 10 pages 2014

[35] C Canuto M Y Hussaini A Quarteroni and T ZangSpectral Methods Fundamentals in Single Domains ScientificComputation Springer Berlin Germany 2006

[36] J HMerkin ldquoFree convection boundary layer on an isothermalhorizontal cylinderrdquo in Proceedings of the American Societyof Mechanical Engineers and American Institute of ChemicalEngineersHeat Transfer Conference p 911 ASME St LouisMoUSA August 1976

[37] R Nazar N Amin and I Pop ldquoFree convection boundary layeron an isothermal horizontal circular cylinder in a micropolarfluidrdquo in Proceedings of the 12th International Heat TransferConference vol 2 pp 525ndash530 Elsevier Paris France August2002

[38] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

4 Advances in High Energy Physics

Now boundary conditions (6a) and (6b) become

120597120595

120597119910

= 0

120597120595

120597119909

= 119891119908

120596 = minus 119899

1205972120595

1205971199102

120597120579

120597119910

= minusBi (1minus 120579)

120601 = 1

at 119910 = 0

(10a)

120597120595

120597119910

= 0

120596 = 0

120579 = 0

120601 = 0

as 119910 997888rarr infin

(10b)

where 119891119908= minus(119871]Gr14)V

119908is the suctioninjection parame-

ter It is worth mentioning that 119891119908determines the transpira-

tion rate at the surface with119891119908gt 0 for suction and119891

119908lt 0 for

injection and119891119908= 0 corresponds to an impermeable surface

Further Bi = ℎ119891119871119896Gr14 is the Biot number It is a ratio of

the internal thermal resistance of the plate to the boundarylayer thermal resistance of the hot fluid at the bottom of thesurface

4 Similarity Equations via Lie Scaling GroupTransformations

Aone-parameter Lie scaling group of transformations whichis a simplified form of Lie group transformation is selected as(for more see [26ndash31])

Γ 119909lowast

= 1199091198901205761205721

119910lowast

= 1199101198901205761205722

120595lowast

= 1205951198901205761205723

120596lowast

= 1205961198901205761205724

120579lowast

= 1205791198901205761205725

120601lowast

= 1206011198901205761205726

120573lowast

119879= 1205731198791198901205761205727

120573lowast

119862= 1205731198621198901205761205728

(11)

Here 120576 = 0 is the parameter of the group and 120572119894(where

119894 = 1 2 3 8) are arbitrary real numbers whose interrela-tionship will be determined by our analysis Transformations

in (11) may be treated as a point transformation transformingthe coordinates

(119909 119910 120595 120596 120579 120601 120573119879 120573119862)

= (119909lowast

119910lowast

120595lowast

120596lowast

120579lowast

120601lowast

120573lowast

119879 120573lowast

119862)

(12)

We now investigate the relationship among the exponents 120572119894

(where 119894 = 1 2 3 8) such that

Δ119895[119909lowast

119910lowast

119906lowast

Vlowast 1205973120595lowast

120597119910lowast3 ]

= 119867119895[119909 119910 119906 V

1205973120595

1205971199103 119886]

sdot Δ119895[119909 119910 119906 V

1205973120595

1205971199103 ] (119895 = 1 2 3 4)

(13)

This is the requirement that the differential formsΔ 1Δ 2Δ 3and Δ 4 are conformally invariant under transformation (11)Substituting transformations (11) in (9) we have

Δ 1 = 119890120576(1205721+21205722minus21205723)

(

120597120595lowast

120597119910lowast

1205972120595lowast

120597119909lowast120597119910lowastminus

120597120595lowast

120597119909lowast

1205972120595lowast

120597119910lowast2 )

minus(

11 minus 119873

) 119890120576(31205722minus1205723) 120597

3120595lowast

120597119910lowast3

minus(

119873

1 minus 119873

) 119890120576(1205722minus1205724) 120597120596

lowast

120597119910lowast

minus

119892lowast

120573lowast

119879(119879119891minus 119879infin) 119871

3

]2Gr119890minus120576(1205725+1205727)

120579lowast

minus

119892lowast

120573lowast

119862(119862119908minus 119862infin) 119871

3

]2Gr119890minus120576(1205726+1205728)

120601lowast

= 0

(14a)

Δ 2 = 119890120576(1205721+1205722minus1205723minus1205724)

(

120597120595lowast

120597119910lowast

120597120596lowast

120597119909lowastminus

120597120595lowast

120597119909lowast

120597120596lowast

120597119910lowast)

minus(

2 minus 119873

2 minus 2119873) 119890120576(21205722minus1205724) 120597

2120596lowast

120597119910lowast2

+(

119873

1 minus 119873

)(2120596lowast119890minus1205761205724 + 119890120576(21205722minus1205723) 120597

2120595lowast

120597119910lowast2 )

= 0

(14b)

Δ 3 = 119890120576(1205721+1205722minus1205723minus1205725)

(

120597120595lowast

120597119910lowast

120597120579lowast

120597119909lowastminus

120597120595lowast

120597119909lowast

120597120579lowast

120597119910lowast)

minus

1Pr

119890120576(21205722minus1205725)

(

1205972120579lowast

120597119910lowast2) = 0

(14c)

Δ 4 = 119890120576(1205721+1205722minus1205723minus1205726)

(

120597120595lowast

120597119910lowast

120597120601lowast

120597119909lowastminus

120597120595lowast

120597119909lowast

120597120601lowast

120597119910lowast)

minus

1Sc

119890120576(21205722minus1205726)

(

1205972120601lowast

120597119910lowast2) = 0

(14d)

Advances in High Energy Physics 5

Now boundary conditions (10a) and (10b) become

119890120576(1205722minus1205723)

120597120595lowast

120597119910lowast= 0

119890120576(1205721minus1205723)

120597120595lowast

120597119909lowast= 119891119908

119890minus1205761205724

120596lowast

= minus 119899119890120576(21205722minus1205723) 120597

2120595lowast

120597119910lowast2

119890120576(1205722minus1205725) 120597120579

lowast

120597119910lowast= minusBi (1minus 119890

minus1205761205725120579lowast

)

119890minus1205761205726

120601lowast

= 1at 119910lowast = 0

(15a)

119890120576(1205722minus1205723)

120597120595lowast

120597119910lowast= 0

119890minus1205761205724

120596lowast

= 0119890minus1205761205725

120579lowast

= 0119890minus1205761205726

120601lowast

= 0as 119910lowast 997888rarr infin

(15b)

The system remains invariant under the group transforma-tion Γ We then have the following relationships for theparameters

1205721 + 21205722 minus 21205723 = 31205722 minus1205723 = 1205722 minus1205724 = minus1205725 minus1205727= minus1205726 minus1205728

1205721 +1205722 minus1205723 minus1205724 = 21205722 minus1205724 = minus1205724 = 21205722 minus1205723

1205721 +1205722 minus1205723 minus1205725 = 21205722 minus1205725

1205721 +1205722 minus1205723 minus1205726 = 21205722 minus1205726

1205721 minus1205723 = 0

minus 1205724 = 21205722 minus1205723

1205722 minus1205725 = 0 = minus1205725

1205726 = 0

(16)

Solving linear system (16) we have the following relationshipamong the exponents

1205721 = 1205723 = 1205724 = 1205727 = 1205728

1205722 = 1205725 = 1205726 = 0(17)

The set of transformations Γ reduces to

119909lowast

= 1199091198901205761205721

119910lowast

= 119910

120595lowast

= 1205951198901205761205721

120596lowast

= 1205961198901205761205721

120579lowast

= 120579

120601lowast

= 120601

120573lowast

119879= 1205731198791198901205761205721

120573lowast

119862= 1205731198621198901205761205721

(18)

Expanding by theTaylor series in power of 120576 keeping the termup to the first degree (neglecting higher power of 120576) we get

119909lowast

minus119909 = 1205761205721119909

119910lowast

= 119910

120595lowast

minus120595 = 1205761205721120595

120596lowast

minus120596 = 1205761205721120596

120579lowast

= 120579

120601lowast

= 120601

120573lowast

119879minus120573119879= 1205761205721120573119879

120573lowast

119862minus120573119862= 1205761205721120573119862

(19)

The characteristic equations are

119889119909

1205721119909=

119889119910

0=

119889120595

1205721120595=

119889120596

1205721120596=

119889120579

0=

119889120601

0=

119889120573119879

1205721120573119879

=

119889120573119862

1205721120573119862

(20)

Solving the above characteristic equations we have thefollowing similarity transformations

120578 = 119910

120595 = 119909119891 (120578)

120596 = 119909119892 (120578)

120573119879= 1205731198790119909

120573119862= 1205731198620119909

120579 = 120579 (120578)

120601 = 120601 (120578)

(21)

where 1205731198790and 120573

1198620are constant thermal and mass coefficient

of expansionUsing (21) into (9) we get the following similarity

equations

(

11 minus 119873

)119891101584010158401015840

+11989111989110158401015840

minus11989110158402+(

119873

1 minus 119873

)1198921015840

+ 120579+B120601

= 0

(

2 minus 119873

2 minus 2119873)11989210158401015840

+1198911198921015840

minus1198911015840

119892minus(

119873

1 minus 119873

) (2119892+11989110158401015840

)

= 0

1Pr

12057910158401015840

+1198911205791015840

= 0

1Sc

12060110158401015840

+1198911206011015840

= 0

(22)

where the primes indicate differentiation with respect to 120578

alone andB = 1205731198620(119862119908minus 119862infin)1205731198790(119879119891minus 119879infin) is the buoyancy

ratio

6 Advances in High Energy Physics

Boundary conditions (10a) and (10b) in terms of 119891 119892 120579and 120601 become

120578 = 0 119891 (0) = 119891119908

1198911015840

(0) = 0

119892 (0) = minus 11989911989110158401015840

(0)

1205791015840

(0) = minusBi [1minus 120579 (0)]

120601 (0) = 1

(23a)

120578 997888rarr infin 1198911015840

(infin) = 0

119892 (infin) = 0

120579 (infin) = 0

120601 (infin) = 0

(23b)

5 Skin Friction Wall Couple Stress and Heatand Mass Transfer Coefficients

The wall shear stress and the wall couple stress are

120591119908= [(120583 + 120581)

120597119906

120597119910

+ 120581120596]

119910=0

119898119908= 120574 [

120597120596

120597119910

]

119910=0

(24a)

and the heat and mass transfers from the plate respectivelyare given by

119902119908= minus 119896 [

120597119879

120597119910

]

119910=0

119902119898= minus119863[

120597119862

120597119910

]

119910=0

(24b)

The nondimensional skin friction 119862119891

= 2120591119908120588119906

2lowast wall

couple stress 119872119908

= 119898119908120588119906

2lowast119909 the local Nusselt number

119873119906119909= 119902119908119909119896(119879

119891minus 119879infin) and local Sherwood number Sh

119909=

119902119898119909119863(119862

119908minus 119862infin) are given by

119862119891Gr14119909

= 2(1 minus 119899119873

1 minus 119873

)11989110158401015840

(0)

119872119908Gr12119909

= (

2 minus 119873

2 minus 2119873)1198921015840

(0)

119873119906119909

Gr14119909

= minus 1205791015840

(0)

Sh119909

Gr14119909

= minus1206011015840

(0)

(25)

where 1199062lowastis the characteristic velocity and Gr

119909= 119892lowast

1205731198790(119879119891minus

119879infin)119909

3]2 is the local Grashof number

6 Numerical Solution Using the SpectralQuasi-Linearization Method (SQLM)

In this section we describe the quasi-linearization method(QLM) for solving the governing system of (22) alongwith boundary conditions (23a) and (23b) This QLM isa generalization of the Newton-Raphson method and wasproposed by Bellman and Kalaba [32] for solving nonlinearboundary value problems

Assume that the solutions 119891119903 119892119903 120579119903 and 120601

119903of (22) at the

(119903+1)th iteration are119891119903+1 119892119903+1 120579119903+1 and 120601119903+1 If the solutions

at the previous iteration are sufficiently close to the solutionsat the present iteration the nonlinear components of (22)can be linearised using one-term Taylor series of multiplevariables so that (22) give the following iterative sequence oflinear differential equations

(

11 minus 119873

)119891101584010158401015840

119903+1 + 119886111990311989110158401015840

119903+1 + 11988621199031198911015840

119903+1 + 1198863119903119891119903+1

+(

119873

1 minus 119873

)1198921015840

119903+1 + 120579119903+1 +B120601

119903+1 = 1198771119903

(

2 minus 119873

2 minus 2119873)11989210158401015840

119903+1 + 11988731199031198921015840

119903+1 + 1198874119903119892119903+1 + 11988711199031198911015840

119903+1

+ 1198872119903119891119903+1 minus(

119873

1 minus 119873

)11989110158401015840

119903+1 = 1198772119903

1198881119903119891119903+1 +1Pr

12057910158401015840

119903+1 + 11988821199031205791015840

119903+1 = 1198773119903

1198891119903119891119903+1 +1Sc

12060110158401015840

119903+1 +11988921199031206011015840

119903+1 = 1198774119903

(26)

where the coefficients 1198861199041 119903

(1199041 = 1 2 3) 1198871199042 119903

(1199042 = 1 2 4) 1198881199043 119903

(1199043 = 1 2) 1198891199044 119903

(1199044 = 1 2) and 1198771199045 119903

(1199045 = 1 2 4) are known functions (from previous iterations) and aredefined as

1198861119903 = 119891119903

1198862119903 = minus 21198911015840119903

1198863119903 = 11989110158401015840

119903

1198771119903 = 11989111990311989110158401015840

119903minus (1198911015840

119903)

2

1198871119903 = minus119892119903

1198872119903 = 1198921015840

119903

1198873119903 = 119891119903

1198874119903 = minus1198911015840

119903minus(

21198731 minus 119873

)

1198772119903 = 1198911199031198921015840

119903minus1198911015840

119903119892119903

1198881119903 = 1205791015840

119903

1198882119903 = 119891119903

1198773119903 = 1198911199031205791015840

119903

1198891119903 = 1206011015840

119903

1198892119903 = 119891119903

1198774119903 = 1198911199031206011015840

119903

(27)

Advances in High Energy Physics 7

subject to boundary conditions

119891119903+1 (0) = 119891

119908

1198911015840

119903+1 = 0

1198911015840

119903+1 (infin) = 0

119892119903+1 = minus 119899119891

10158401015840

119903+1 (0)

119892119903+1 (infin) = 0

1205791015840

119903+1 (0) = minusBi (1minus 120579 (0))

120579119903+1 (infin) = 0

120601119903+1 (0) = 1

120601119903+1 (infin) = 0

(28)

System (26) constitutes a linear system of coupled differen-tial equations with variable coefficients and can be solvediteratively using any numerical method for 119903 = 1 2 3 In this work as will be discussed below the Chebyshevpseudospectral method was used to solve the QLM scheme(26) (for more details refer to the works of Motsa et al[33 34])

1198910 (120578) = 119891119908+ 1minus 119890

minus120578

1198920 (120578) = minus 119899119890minus120578

1205790 = 119890minus120578

BiBi + 1

1206010 = 119890minus120578

(29)

and starting from these sets of initial approximations 1198910 11989201205790 and 1206010 the iteration schemes (26) can be solved iterativelyfor 119891119903+1(120578) 119892119903+1(120578) 120579119903+1(120578) and120601119903+1(120578) when 119903 = 0 1 2

For this we discretise the equation using the Chebyshevspectral collocation method The unknown functions areapproximated by the Chebyshev interpolating polynomialsin such way that they are collocated at the Gauss-Lobattocollocation points defined as

120591119895= cos

120587119895

119873

119895 = 0 1 2 119873 (30)

where 119873 is the number of collocation points The physicalregion [0infin) is transformed into the region [minus1 1] using thedomain truncation technique in which the problem is solvedon the interval [0 120578

infin] instead of [0infin) This leads to the

mapping

120578

120578infin

=

120591 + 12

minus1 le 120591 le 1 (31)

where 120578infin

is the scaling parameter used to invoke theboundary condition at infinity The functions 119891 119892 120579 and 120601

are approximated at the collocation points by

119891 (120591) =

119873

sum

119896=0119891 (120591119896) 119879119896(120591119895)

119892 (120591) =

119873

sum

119896=0119892 (120591119896) 119879119896(120591119895)

120579 (120591) =

119873

sum

119896=0120579 (120591119896) 119879119896(120591119895)

120601 (120591) =

119873

sum

119896=0120601 (120591119896) 119879119896(120591119895)

119895 = 0 1 2 119873

(32)

where 119879119896is the 119896th Chebyshev polynomial defined as

119879119896(120591) = cos [119896 cosminus1 (120591)] (33)

The derivatives of the variables at the collocation points arerepresented as

119889119901

119891

119889120578119901=

119873

sum

119896=0D119901119897119896119891 (120591119896)

119889119901

119892

119889120578119901=

119873

sum

119896=0D119901119897119896119892 (120591119896)

119889119901

120579

119889120578119901=

119873

sum

119896=0D119901119897119896120579 (120591119896)

119889119901

120601

119889120578119901=

119873

sum

119896=0D119901119897119896120601 (120591119896)

119897 = 0 1 119873

(34)

where 119901 is the order of the derivative and D = 2D120578infin

is theChebyshev spectral differentiation matrix and its entries areclearly defined in Canuto et al [35]

Substituting (31)ndash(34) into (26) leads to the matrix equa-tion

119860119883 = 119877 (35)

subject to the boundary conditions

119891119903+1 (120591119873) = 119891

119908

119873

sum

119896=0D119873119896

119891 (120591119896) = 0

119873

sum

119896=0D0119896119891 (120591

119896) = 0

119892119903+1 (120591119873) = minus 119899

119873

sum

119896=0D2119873119896

119891 (120591119896)

119892119903+1 (1205910) = 0

8 Advances in High Energy Physics

119873

sum

119896=0D119873119896

120579119903+1 (120591119896) minusBi 120579

119903+1 (120591119873) = minusBi

120579119903+1 (1205910) = 0

120601119903+1 (120591119873) = 1120601119903+1 (1205910) = 0

(36)

In (35) 119860 is a (4119873+ 4) times (4119873+ 4) square matrix and119883 and 119877are (4119873 + 1) times 1 column vectors defined by

119860 =

[

[

[

[

[

[

11986011 11986012 11986013 11986014

11986021 11986022 11986023 11986024

11986031 11986032 11986033 11986034

11986041 11986042 11986043 11986044

]

]

]

]

]

]

119883 =

[

[

[

[

[

[

F119903+1

G119903+1

Θ119903+1

Φ119903+1

]

]

]

]

]

]

119877 =

[

[

[

[

[

[

R1

R2

R3

R4

]

]

]

]

]

]

(37)

whereF = [119891

119903+1 (1205910) 119891119903+1 (1205911) 119891119903+1 (120591119873)]119879

G = [119892119903+1 (1205910) 119892119903+1 (1205911) 119892119903+1 (120591119873)]

119879

Θ = [120579119903+1 (1205910) 120579119903+1 (1205911) 120579119903+1 (120591119873)]

119879

Φ = [120601119903+1 (1205910) 120601119903+1 (1205911) 120601119903+1 (120591119873)]

119879

11986011 = (

11 minus 119873

)D3+ diag [1198861119903]D

2+ diag [1198862119903]D

+ diag [1198863119903]

11986012 = (

119873

1 minus 119873

)D

11986013 = I

11986014 = BI

11986021 = minus(

119873

1 minus 119873

)D2+ diag [1198871119903]D+ diag [1198872119903]

11986022 = (

2 minus 119873

2 minus 2119873)D2

+ diag [1198873119903]D+ diag [1198874119903]

11986023 = 0

11986024 = 0

11986031 = diag [1198881119903]

11986032 = 0

11986033 =1Pr

D2+ diag [1198882119903]D

Table 1 Comparison of minus1205791015840(0) for free convection along a verticalflat plate in Newtonian fluid when 119873 = 0 119899 = 0 B = 0 Pr = 1Bi rarr infin and 119891

119908= 0

Merkin [36] Nazar et al [37] Molla et al [38] Present04214 04214 04214 04214313

11986034 = 0

11986041 = diag [1198891119903]

11986042 = 0

11986043 = 0

11986044 =1Sc

D2+ diag [1198892119903]D

R1 = 1198771119903

R2 = 1198772119903

R3 = 1198773119903

R4 = 1198774119903

(38)

and here I is an identity matrix the size of the matrix 0 is(119873+1)times1 and diag[ ] is a diagonalmatrix of size (119873+1)times(119873+

1) Subscript r denotes the iteration number After modifyingmatrix system (35) to incorporate boundary condition (36)the solution is obtained as

119883 = 119860minus1119877 (39)

7 Results and Discussions

It is noticed that the present problem reduces to free convec-tion heat transfer along an impermeable vertical plate in amicropolar fluid without the convective boundary conditionwhen119891

119908= 0 Bi rarr infin andB = 0 Also in the limit as119873 rarr

0 governing equations (2)ndash(5) reduce to the correspondingequations for a free convection heat and mass transfer ina viscous fluid In order to validate the code generated theresults of the present problem have been compared with theresults obtained by Merkin [36] Nazar et al [37] and Mollaet al [38] as a special case by taking 119873 = 0 119899 = 0 B = 0Pr = 1 Bi rarr infin and 119891

119908= 0 and it was found that

they are in good agreement as presented in Table 1 Also thecomparison of heat transfer coefficient has been made withthe results obtained by Nazar et al [37] as shown in Table 2when 119899 = 05 B = 0 Pr = 1 Bi rarr infin and 119891

119908= 0

To study the effects of coupling number119873 suctioninjectionparameter 119891

119908 Biot number Bi and material parameter 119899

computations were carried out in the cases of B = 10Pr = 071 and Sc = 022

The effects of the coupling number119873 on the dimension-less velocity microrotation temperature and concentrationare illustrated in Figures 2(a)ndash2(d) with fixed values ofother parameters The coupling number 119873 characterizes thecoupling of linear and rotational motion arising from the

Advances in High Energy Physics 9

00

01

02

03

04

05

06f998400

0 4 8 12 16

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(a)

g

minus06

minus05

minus04

minus03

minus02

minus01

00

01

0 2 4 6 8 10 12

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(b)

000

004

008

012

016

120579

0 1 2 3 4 5 6

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(c)

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

120601

00

02

04

06

08

10

120578

0 2 4 6 8 10 12

Bi = 01 fw = 05

(d)

Figure 2 Effect of119873 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Table 2 Comparison of minus1205791015840

(0) for free convection flow in amicropolar fluid obtained by Nazar et al [37] when 119899 = 05B = 0Pr = 1 Bi rarr infin and 119891

119908= 0

119873 Nazar et al [37] Present000 04214 04214033 03991 03990050 03834 03834060 03709 03709066 03608 03608071 03522 03522075 03447 03447

fluid particles In the case of 119873 = 0 (ie as 120581 tends tozero) the micropolarity is absent and fluid becomes nonpolarfluid With a large value of 119873 effect of microstructurebecomes significant whereas with a diminished value of 119873the individuality of the substructure is much less articulatedAs 119873 increases it is found from Figure 2(a) that the max-imum velocity decreases in amplitude and the location ofthe maximum velocity moves farther away from the wallSince 119873 rarr 0 corresponds to viscous fluid the velocity incase of a micropolar fluid has been less compared to thatof viscous fluid case It can be observed from Figure 2(b)that as 119873 increases initially microrotation profiles tend tobecome flatter and then approach their free stream valuesfar away from the wall This happens due to the vanishing

10 Advances in High Energy Physics

00

01

02

03

04

05

06

f998400

120578

0 2 4 6 8 10 12

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(a)

minus06

minus04

minus03

minus02

minus05

minus01

00

01

g

0 2 4 6 8 10

120578

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(b)

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

120578

0 1 2 3 4 5

120579

00

02

04

06

08

10

(c)

120601

00

02

04

06

08

10

fw = 05 n = 05N = 05

0 2 4 6 8 10 12

120578

Bi = 01

Bi = 10

Bi = 50Bi = 200

(d)

Figure 3 Effect of Bi on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

of antisymmetric part of the stress on the boundary thatcorresponds to a weak concentration of microelements Thisis because an increment in the value of 119873 implies a highervortex viscosity of fluid which promotes the microrotationof micropolar fluids It is seen from Figures 2(c) and 2(d)that thermal and concentration boundary layers of thefluid increase with increase in coupling number 119873 Hencetemperature and concentration in case of micropolar fluidsare more than those of the viscous fluid case

The Biot number Bi is the ratio of internal thermalresistance of a solid to boundary layer thermal resistanceWhen Bi = 0 the plate is totally insulated internal thermalresistance of the plate is extremely high and no convectiveheat transfer to the cold fluid on the upper part of the

plate takes place Figure 3(a) depicts fluid velocity profilesfor different values of the Biot number with 119873 = 05 119891

119908=

05 and 119899 = 05 Generally fluid velocity is zero at platesurface and increases gradually away fromplate to free streamvalue satisfying boundary conditions It is interesting to notethat an increase in the intensity of convective surface heattransfer Bi produces significant enhancement in fluid velocitywithin the momentum boundary layer In Figure 3(b) webring out the behavior of microrotation with different valuesof Biot number Bi for fixed values of other parameters As theparameter value Bi increases microrotation showing reverserotation near the two boundaries Hence the condition ofvanishing of antisymmetric part of the stress on the boundaryresults in a drastic change of the microrotation profiles

Advances in High Energy Physics 11

07

06

05

04

03

02

01

00

f998400

fw = 00

fw = 10

fw = 20

0 3 6 9 12 15

120578

N = 05 Bi = 01 n = 05

fw = minus05

(a)

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

120578

0 2 4 6 8 10 12

g

minus04

minus03

minus02

minus01

00

01

fw = minus05

(b)

0 1 2 3 4 5 6

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

000

005

010

015

020

025

030

120579

fw = minus05

(c)

0 3 6 9 12

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

00

02

04

06

08

10

120601

fw = minus05

(d)

Figure 4 Effect of 119891119908on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Given that convective heating increases with Biot numberBi rarr infin simulates the isothermal surface which is clearlyseen from Figure 3(c) where 120579(0) = 1 as Bi rarr infin Infact a high Biot number indicates higher internal thermalresistance of the plate than boundary layer thermal resistanceIn this fluid temperature is maximum at the plate surfaceand decreases exponentially to zero value far out from theplate satisfying boundary conditions As a consequence anincrement in the Biot number leads to increase of fluidtemperature efficiency Figure 3(d) illustrates the variation ofdimensionless concentration for different values of Bi It isclear that the concentration of fluid decreases with increaseof Bi

The effect of 119891119908

on velocity profile is depicted inFigure 4(a) Here 119891

119908gt 0 represents suction and 119891

119908lt 0

denotes injectionThe lower velocity is noticed in case of suc-tion when compared to case of injection From Figure 4(b)we note that microrotation is showing reverse rotation neartwo boundaries with both suction and injection parame-ter The dimensionless temperature for different values ofsuctioninjection parameters is drawn in Figure 4(c) It isreadable that the temperature of the fluid is more in case ofinjection whereas it is less in case of suction in comparisonwith the impermeable surface case (119891

119908= 0) Figure 4(d)

demonstrates dimensionless concentration for different val-ues of suctioninjection parameters It is determined that the

12 Advances in High Energy Physics

00

01

02

03

04

05

06f998400

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(a)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10

120578

minus10

minus08

minus06

minus04

minus02

00

g

(b)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

120578

0 1 2 3 4 5

120579

000

003

006

009

012

015

(c)

00

02

04

06

08

10

120601

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(d)

Figure 5 Effect of material parameter 119899 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

concentration of fluid ismore with injection whereas it is lesswith suctionwhen compared to the impermeable surface case(119891119908= 0) As a finale the thermal and solutal boundary layer

thicknesses increase in case of injection compared to case ofsuction as displayed in Figures 4(c) and 4(d)

In Figures 5(a)ndash5(d) the effect of material parameter119899 on dimensionless velocity microrotation temperatureand concentration is presented for fixed values of otherparameters since the material parameter 119899 signifies themicrorotation effects (ie for 119899 = 0 particles are not freeto revolve near the surface whereas as 119899 increases from 0 to1 the microrotation term gets augmented and induces flowenhancement) As 119899 increases it is found from Figure 5(a)that the minimum velocity increases in amplitude and the

location of the minimum velocity moves farther away fromthe wall From Figure 5(b) we observe that themicrorotationis decreasing with increasing value of material parameter119899 within the boundary layer It is clear from Figures 5(c)-5(d) that the thermal and solutal boundary layer thicknessesdecrease with increase of material parameter 119899

The variations of minus1205791015840

(0) and minus1206011015840

(0) versus couplingnumber 119873 are shown in Figures 6ndash8 It can be noticed fromthese figures that the heat and mass transfer coefficientsare less in case of micropolar fluids when compared to theviscous fluids This is because as 119873 increases the thermaland solutal boundary layer thicknesses become larger thusgiving rise to a small value of local heat and mass transferrates The effect of the Biot number Bi with fixed 119891

119908= 05

Advances in High Energy Physics 13

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

fw = 05 n = 05

01

00

02

03

04

05

06

07Nu x

Grminus

14

x

(a)

fw = 05 n = 05

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

024

026

028

030

032

ShxGrminus

14

x

(b)

Figure 6 Effect of Bi on (a) heat transfer rate and (b) mass transfer rate

005

006

007

008

009

010

Nu x

Grminus

14

x

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

(a)

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

01

02

03

04

05

06

ShxGrminus

14

x

(b)

Figure 7 Effect of 119891119908on (a) heat transfer rate and (b) mass transfer rate

and 119899 = 05 on local heat transfer coefficient is exhibitedin Figure 6 It is found from Figure 6 that local heat transferrate increases nonlinearly with the increase in Biot numberBiThe influence of the Biot number Bi on local mass transfercoefficient is shown in Figure 6 Figure 6 reveals that the localmass transfer coefficient is enhanced by the growth in theBiot number Bi In Figure 7 the effect of the suctioninjectionparameter 119891

119908with fixed Bi = 01 and 119899 = 05 on local heat

and mass transfer coefficients is displayed It is found fromFigure 7 that the local heat and mass transfer coefficients areless in the case of injection 119891

119908lt 0 in comparison with the

case of suction 119891119908

gt 0 Figure 8 is prepared to analyze theeffect of the material parameter 119899 with fixed Bi = 01 and

119891119908= 05 on local heat andmass transfer coefficients Figure 8

reveals that the local heat and mass transfer coefficients areenhanced by the increase in material parameter 119899 This isbecause when 119899 increases from 0 to 1 the microrotation termgets augmented and induces flow enhancement

The variations of 119862119891Gr14119909

and 119872119908Gr12119909

which areproportional to the coefficients of skin friction and wallcouple stress are shown in Table 3 with different values ofthe coupling number 119873 for fixed 119899 = 05 119891

119908= 05 and

Bi = 01 It indicates that the skin friction factor is higherfor micropolar fluid than the viscous fluids (119873 = 0) Sincemicropolar fluids offer a heavy resistance (resulting fromvortex viscosity) to fluid movement and cause larger skin

14 Advances in High Energy Physics

0084

0085

0086

0087Nu x

Grminus

14

x

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

fw = 05 Bi = 01

(a)

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

024

026

028

030

ShxGrminus

14

x

fw = 05 Bi = 01

(b)

Figure 8 Effect of material parameter 119899 on (a) heat transfer rate and (b) mass transfer rate

Table 3 Effects of skin friction and wall couple stress for varyingvalues of Biot numbers Bi micropolar parameter 119873 materialparameter 119899 and suctioninjection parameter 119891

119908

119873 Bi 119891119908

119899 119862119891Gr14119909

119872119908Gr12119909

01 01 05 05 2424227 06327903 01 05 05 2502716 063996605 01 05 05 2647987 063086307 01 05 05 2943561 060387909 01 05 05 3874631 055089105 01 05 05 2647987 063086305 10 05 05 3211755 080587905 50 05 05 3533141 090807805 200 05 05 3629791 093913705 01 minus05 05 2426265 033621905 01 00 05 2530014 046542905 01 10 05 2737553 082089905 01 20 05 2721244 119359605 01 05 00 2916655 minus028933905 01 05 05 2647987 063086305 01 05 10 2232556 2066191

friction factor compared to viscous fluid the results as wellsuggest larger values of coupling number 119873 and lower wallcouple stresses Since the skin friction coefficient 11989110158401015840(0) andwall couple stress coefficient as well as high temperature andmass transport rates are more depressed in the micropolarfluid comparing to the viscous fluid which may be beneficialin flow temperature and concentration control of polymerprocessing thus the presence of microscopic effects arisingfrom the local structure and of the fluid elements reduces thehigh temperature and mass transfer coefficients The effectof Bi on 119862

119891Gr14119909

and 119872119908Gr12119909

for 119891119908

= 05 119899 = 05 and

119873 = 05 is illustrated in Table 3 It can be noticed that theskin friction and wall couple stress coefficients are increasingwith increase of Bi for fixed values of other parametersThis notice is consistent with physical profiles presented inFigure 3 The effects of suctioninjection parameter on theskin friction andwall couple stress coefficients are also shownin Table 3 It is noted that the skin friction and wall couplestress coefficients are less with injection case whereas theyare more with suction case when compared to the case ofimpermeable surface Finally the detailed behavior of thematerial parameter 119899 is given in Table 3 The skin frictiondecreases and wall couple stress increases with increase ofmaterial parameter 119899

8 Conclusions

In this composition the similarity solution of the free convec-tion flow on a permeable vertical plate of a micropolar fluidunder the convective boundary condition is obtained usingLie group transformations Using the similarity variables thegoverning equations are transformed into a set of nondi-mensional parabolic equations These equations are solvednumerically using spectral quasi-linearisation method Thenumerical computation is carried out for various values ofnondimensional physical parameters The main findings aresummarized as follows

(i) The numerical results indicate that velocity distribu-tion is less near the plate but it is more far away fromthe plate the wall couple stress coefficient and rate ofheat andmass transfers are lower but the temperatureand concentration distributions and the skin frictioncoefficient are higher for the micropolar fluids incomparison with those of viscous fluids Also thereverse rotation ofmicrorotation near two boundariesis found with the increasing value of119873

Advances in High Energy Physics 15

(ii) An increase in Biot number Bi decreases concen-tration distribution whereas it causes an increasein temperature distribution skin friction and wallcouple stress coefficients and heat and mass transferrates within the boundary layer Further enhancingin Biot number Bi enhances velocity distributionnear the plate but shows the reverse behavior faraway from the plate We observe reverse rotationof microrotation near two boundaries within theboundary layer in the presence of Biot number

(iii) Less velocity temperature and concentration distri-butions are observed more skin friction and wallcouple stress coefficients and heat and mass transferrates in the case of suction compared to the case ofinjection Further microrotation decreases near thewall and depicts the opposite trend far away from thewall

(iv) It is found that microrotation temperature and con-centration distributions and skin friction coefficientsare more in the case of a micropolar fluid with strongconcentration (ie 119899 = 0) when compared to the caseof a micropolar fluid with weak concentration (ie119899 = 12) Further velocity is less in the case of 119899 = 0when compared to the case of 119899 = 12 near the walland shows the opposite trend far away from the wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their valuablesuggestions and comments

References

[1] J H Merkin ldquoNatural-convection boundary-layer flow on avertical surface with Newtonian heatingrdquo International Journalof Heat and Fluid Flow vol 15 no 5 pp 392ndash398 1994

[2] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009

[3] O DMakinde ldquoSimilarity solution for natural convection froma moving vertical plate with internal heat generation and aconvective boundary conditionrdquo Thermal Science vol 15 ppS137ndashS143 2011

[4] S M Ibrahim and N Bhashar Reddy ldquoSimilarity solution ofheat and mass transfer for natural convection over a movingvertical plate with internal heat generation and a convectiveboundary condition in the presence of thermal radiationviscous dissipation and chemical reactionrdquo ISRN Thermody-namics vol 2013 Article ID 790604 10 pages 2013

[5] C RamReddy P V S N Murthy A J Chamkha and A MRashad ldquoSoret effect on mixed convection flow in a nanofluidunder convective boundary conditionrdquo International Journal ofHeat and Mass Transfer vol 64 pp 384ndash392 2013

[6] H T Alkasasbeh M Z Salleh R M Tahar and R NazarldquoNumerical solutions of free convection boundary layer flow ona solid sphere with convective boundary conditionsrdquo Journal ofPhysics Conference Series vol 495 Article ID 012025 2014

[7] A Pantokratoras ldquoA note on natural convection along a con-vectively heated vertical platerdquo International Journal of ThermalSciences vol 76 pp 221ndash224 2014

[8] G Lukaszewicz Micropolar FluidsmdashTheory and ApplicationsBirkhaauser Basel Switzerland 1999

[9] V A Eremeyev L P Lebedev and H Altenbach Foundationsof Micropolar Mechanics Springer New York NY USA 2013

[10] G Ahmadi ldquoSelf-similar solution of imcompressible micropo-lar boundary layer flow over a semi-infinite platerdquo InternationalJournal of Engineering Science vol 14 no 7 pp 639ndash646 1976

[11] D A Rees and I Pop ldquoFree convection boundary-layer flowof a micropolar fluid from a vertical flat platerdquo IMA Journal ofApplied Mathematics vol 61 no 2 pp 179ndash197 1998

[12] D Srinivasacharya and C RamReddy ldquoEffect of double stratifi-cation onmixed convection in a micropolar fluidrdquoMatematikavol 28 no 2 pp 133ndash149 2012

[13] D Srinivasacharya and C Ramreddy ldquoMixed convection heatand mass transfer in a doubly stratified micropolar fluidrdquoComputationalThermal Sciences vol 5 no 4 pp 273ndash287 2013

[14] D Srinivasacharya and C RamReddy ldquoNatural convection heatand mass transfer in a micropolar fluid with thermal and massstratificationrdquo International Journal for Computational Methodsin Engineering Science andMechanics vol 14 no 5 pp 401ndash4132013

[15] N A Yacob and A Ishak ldquoStagnation point flow towardsa stretchingshrinking sheet in a micropolar fluid with aconvective surface boundary conditionrdquoThe Canadian Journalof Chemical Engineering vol 90 no 3 pp 621ndash626 2012

[16] K Zaimi and A Ishak ldquoStagnation-point flow and heat transferover a nonlinearly stretchingshrinking sheet in a micropolarfluidrdquo Abstract and Applied Analysis vol 2014 Article ID261630 6 pages 2014

[17] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[18] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[19] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer 2009

[20] T Tapanidis G Tsagas and H P Mazumdar ldquoApplication ofscaling group of transformations to viscoelastic second-gradefluid flowrdquo Nonlinear Functional Analysis and Applications vol8 no 3 pp 345ndash350 2003

[21] I A A Hassanien andM A Hamad ldquoGroup theoretic methodfor unsteady free convection flow of a micropolar fluid along avertical plate in a thermally stratified mediumrdquo Applied Mathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 32 no 6 pp 1099ndash1114 2008

[22] R Kandasamy K Gunasekaran and S B H Hasan ldquoScalinggroup transformation on fluid flow with variable stream con-ditionsrdquo International Journal of Non-Linear Mechanics vol 46no 7 pp 976ndash985 2011

[23] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat TransfermdashAsian Research vol41 no 3 pp 241ndash259 2012

16 Advances in High Energy Physics

[24] A A Mutlag M J Uddin M A A Hamad and A I M IsmailldquoHeat transfer analysis for falkner-skan boundary layer flowpast a stationary wedge with slips boundary conditions con-sidering temperature-dependent thermal conductivityrdquo SainsMalaysiana vol 42 no 6 pp 855ndash862 2013

[25] S C Cowin ldquoPolar fluidsrdquo Physics of Fluids vol 11 no 9 pp1919ndash1927 1968

[26] M A Seddeek M Y Akl and A M Al-Hanaya ldquoThermalradiation effects on mixed convection and mass transfer flowon vertical porous plate with heat generation and chemicalreaction by using scaling grouprdquo Journal of Natural Sciences andMathematics vol 4 pp 41ndash60 2010

[27] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[28] M J Uddin W A Khan and A I M Ismail ldquoMHD freeconvective boundary layer flow of a nanofluid past a flat verticalplate with Newtonian heating boundary conditionrdquo PLoS ONEvol 7 no 11 Article ID e49499 2012

[29] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012

[30] M J Uddin W A Khan and A I M Ismail ldquoScalinggroup transformation for MHD boundary layer slip flow of ananofluid over a convectively heated stretching sheet with heatgenerationrdquo Mathematical Problems in Engineering vol 2012Article ID 934964 20 pages 2012

[31] MM RashidiM Ferdows A B Parsa and S Abelman ldquoMHDnatural convection with convective surface boundary conditionover a at platerdquo Abstract and Applied Analysis vol 2014 ArticleID 923487 10 pages 2014

[32] R E Bellman and R E Kalaba Quasilinearisation and Non-Linear Boundary-Value Problems Elsevier New York NY USA1965

[33] S SMotsa P GDlamini andMKhumalo ldquoSpectral relaxationmethod and spectral quasilinearization method for solvingunsteady boundary layer flow problemsrdquo Advances in Mathe-matical Physics vol 2014 Article ID 341964 12 pages 2014

[34] S S Motsa P Sibanda J M Ngnotchouye and G T MarewoldquoA spectral relaxation approach for unsteady boundary-layerflow and heat transfer of a nanofluid over a permeable stretch-ingshrinking sheetrdquo Advances in Mathematical Physics vol2014 Article ID 564942 10 pages 2014

[35] C Canuto M Y Hussaini A Quarteroni and T ZangSpectral Methods Fundamentals in Single Domains ScientificComputation Springer Berlin Germany 2006

[36] J HMerkin ldquoFree convection boundary layer on an isothermalhorizontal cylinderrdquo in Proceedings of the American Societyof Mechanical Engineers and American Institute of ChemicalEngineersHeat Transfer Conference p 911 ASME St LouisMoUSA August 1976

[37] R Nazar N Amin and I Pop ldquoFree convection boundary layeron an isothermal horizontal circular cylinder in a micropolarfluidrdquo in Proceedings of the 12th International Heat TransferConference vol 2 pp 525ndash530 Elsevier Paris France August2002

[38] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 5

Now boundary conditions (10a) and (10b) become

119890120576(1205722minus1205723)

120597120595lowast

120597119910lowast= 0

119890120576(1205721minus1205723)

120597120595lowast

120597119909lowast= 119891119908

119890minus1205761205724

120596lowast

= minus 119899119890120576(21205722minus1205723) 120597

2120595lowast

120597119910lowast2

119890120576(1205722minus1205725) 120597120579

lowast

120597119910lowast= minusBi (1minus 119890

minus1205761205725120579lowast

)

119890minus1205761205726

120601lowast

= 1at 119910lowast = 0

(15a)

119890120576(1205722minus1205723)

120597120595lowast

120597119910lowast= 0

119890minus1205761205724

120596lowast

= 0119890minus1205761205725

120579lowast

= 0119890minus1205761205726

120601lowast

= 0as 119910lowast 997888rarr infin

(15b)

The system remains invariant under the group transforma-tion Γ We then have the following relationships for theparameters

1205721 + 21205722 minus 21205723 = 31205722 minus1205723 = 1205722 minus1205724 = minus1205725 minus1205727= minus1205726 minus1205728

1205721 +1205722 minus1205723 minus1205724 = 21205722 minus1205724 = minus1205724 = 21205722 minus1205723

1205721 +1205722 minus1205723 minus1205725 = 21205722 minus1205725

1205721 +1205722 minus1205723 minus1205726 = 21205722 minus1205726

1205721 minus1205723 = 0

minus 1205724 = 21205722 minus1205723

1205722 minus1205725 = 0 = minus1205725

1205726 = 0

(16)

Solving linear system (16) we have the following relationshipamong the exponents

1205721 = 1205723 = 1205724 = 1205727 = 1205728

1205722 = 1205725 = 1205726 = 0(17)

The set of transformations Γ reduces to

119909lowast

= 1199091198901205761205721

119910lowast

= 119910

120595lowast

= 1205951198901205761205721

120596lowast

= 1205961198901205761205721

120579lowast

= 120579

120601lowast

= 120601

120573lowast

119879= 1205731198791198901205761205721

120573lowast

119862= 1205731198621198901205761205721

(18)

Expanding by theTaylor series in power of 120576 keeping the termup to the first degree (neglecting higher power of 120576) we get

119909lowast

minus119909 = 1205761205721119909

119910lowast

= 119910

120595lowast

minus120595 = 1205761205721120595

120596lowast

minus120596 = 1205761205721120596

120579lowast

= 120579

120601lowast

= 120601

120573lowast

119879minus120573119879= 1205761205721120573119879

120573lowast

119862minus120573119862= 1205761205721120573119862

(19)

The characteristic equations are

119889119909

1205721119909=

119889119910

0=

119889120595

1205721120595=

119889120596

1205721120596=

119889120579

0=

119889120601

0=

119889120573119879

1205721120573119879

=

119889120573119862

1205721120573119862

(20)

Solving the above characteristic equations we have thefollowing similarity transformations

120578 = 119910

120595 = 119909119891 (120578)

120596 = 119909119892 (120578)

120573119879= 1205731198790119909

120573119862= 1205731198620119909

120579 = 120579 (120578)

120601 = 120601 (120578)

(21)

where 1205731198790and 120573

1198620are constant thermal and mass coefficient

of expansionUsing (21) into (9) we get the following similarity

equations

(

11 minus 119873

)119891101584010158401015840

+11989111989110158401015840

minus11989110158402+(

119873

1 minus 119873

)1198921015840

+ 120579+B120601

= 0

(

2 minus 119873

2 minus 2119873)11989210158401015840

+1198911198921015840

minus1198911015840

119892minus(

119873

1 minus 119873

) (2119892+11989110158401015840

)

= 0

1Pr

12057910158401015840

+1198911205791015840

= 0

1Sc

12060110158401015840

+1198911206011015840

= 0

(22)

where the primes indicate differentiation with respect to 120578

alone andB = 1205731198620(119862119908minus 119862infin)1205731198790(119879119891minus 119879infin) is the buoyancy

ratio

6 Advances in High Energy Physics

Boundary conditions (10a) and (10b) in terms of 119891 119892 120579and 120601 become

120578 = 0 119891 (0) = 119891119908

1198911015840

(0) = 0

119892 (0) = minus 11989911989110158401015840

(0)

1205791015840

(0) = minusBi [1minus 120579 (0)]

120601 (0) = 1

(23a)

120578 997888rarr infin 1198911015840

(infin) = 0

119892 (infin) = 0

120579 (infin) = 0

120601 (infin) = 0

(23b)

5 Skin Friction Wall Couple Stress and Heatand Mass Transfer Coefficients

The wall shear stress and the wall couple stress are

120591119908= [(120583 + 120581)

120597119906

120597119910

+ 120581120596]

119910=0

119898119908= 120574 [

120597120596

120597119910

]

119910=0

(24a)

and the heat and mass transfers from the plate respectivelyare given by

119902119908= minus 119896 [

120597119879

120597119910

]

119910=0

119902119898= minus119863[

120597119862

120597119910

]

119910=0

(24b)

The nondimensional skin friction 119862119891

= 2120591119908120588119906

2lowast wall

couple stress 119872119908

= 119898119908120588119906

2lowast119909 the local Nusselt number

119873119906119909= 119902119908119909119896(119879

119891minus 119879infin) and local Sherwood number Sh

119909=

119902119898119909119863(119862

119908minus 119862infin) are given by

119862119891Gr14119909

= 2(1 minus 119899119873

1 minus 119873

)11989110158401015840

(0)

119872119908Gr12119909

= (

2 minus 119873

2 minus 2119873)1198921015840

(0)

119873119906119909

Gr14119909

= minus 1205791015840

(0)

Sh119909

Gr14119909

= minus1206011015840

(0)

(25)

where 1199062lowastis the characteristic velocity and Gr

119909= 119892lowast

1205731198790(119879119891minus

119879infin)119909

3]2 is the local Grashof number

6 Numerical Solution Using the SpectralQuasi-Linearization Method (SQLM)

In this section we describe the quasi-linearization method(QLM) for solving the governing system of (22) alongwith boundary conditions (23a) and (23b) This QLM isa generalization of the Newton-Raphson method and wasproposed by Bellman and Kalaba [32] for solving nonlinearboundary value problems

Assume that the solutions 119891119903 119892119903 120579119903 and 120601

119903of (22) at the

(119903+1)th iteration are119891119903+1 119892119903+1 120579119903+1 and 120601119903+1 If the solutions

at the previous iteration are sufficiently close to the solutionsat the present iteration the nonlinear components of (22)can be linearised using one-term Taylor series of multiplevariables so that (22) give the following iterative sequence oflinear differential equations

(

11 minus 119873

)119891101584010158401015840

119903+1 + 119886111990311989110158401015840

119903+1 + 11988621199031198911015840

119903+1 + 1198863119903119891119903+1

+(

119873

1 minus 119873

)1198921015840

119903+1 + 120579119903+1 +B120601

119903+1 = 1198771119903

(

2 minus 119873

2 minus 2119873)11989210158401015840

119903+1 + 11988731199031198921015840

119903+1 + 1198874119903119892119903+1 + 11988711199031198911015840

119903+1

+ 1198872119903119891119903+1 minus(

119873

1 minus 119873

)11989110158401015840

119903+1 = 1198772119903

1198881119903119891119903+1 +1Pr

12057910158401015840

119903+1 + 11988821199031205791015840

119903+1 = 1198773119903

1198891119903119891119903+1 +1Sc

12060110158401015840

119903+1 +11988921199031206011015840

119903+1 = 1198774119903

(26)

where the coefficients 1198861199041 119903

(1199041 = 1 2 3) 1198871199042 119903

(1199042 = 1 2 4) 1198881199043 119903

(1199043 = 1 2) 1198891199044 119903

(1199044 = 1 2) and 1198771199045 119903

(1199045 = 1 2 4) are known functions (from previous iterations) and aredefined as

1198861119903 = 119891119903

1198862119903 = minus 21198911015840119903

1198863119903 = 11989110158401015840

119903

1198771119903 = 11989111990311989110158401015840

119903minus (1198911015840

119903)

2

1198871119903 = minus119892119903

1198872119903 = 1198921015840

119903

1198873119903 = 119891119903

1198874119903 = minus1198911015840

119903minus(

21198731 minus 119873

)

1198772119903 = 1198911199031198921015840

119903minus1198911015840

119903119892119903

1198881119903 = 1205791015840

119903

1198882119903 = 119891119903

1198773119903 = 1198911199031205791015840

119903

1198891119903 = 1206011015840

119903

1198892119903 = 119891119903

1198774119903 = 1198911199031206011015840

119903

(27)

Advances in High Energy Physics 7

subject to boundary conditions

119891119903+1 (0) = 119891

119908

1198911015840

119903+1 = 0

1198911015840

119903+1 (infin) = 0

119892119903+1 = minus 119899119891

10158401015840

119903+1 (0)

119892119903+1 (infin) = 0

1205791015840

119903+1 (0) = minusBi (1minus 120579 (0))

120579119903+1 (infin) = 0

120601119903+1 (0) = 1

120601119903+1 (infin) = 0

(28)

System (26) constitutes a linear system of coupled differen-tial equations with variable coefficients and can be solvediteratively using any numerical method for 119903 = 1 2 3 In this work as will be discussed below the Chebyshevpseudospectral method was used to solve the QLM scheme(26) (for more details refer to the works of Motsa et al[33 34])

1198910 (120578) = 119891119908+ 1minus 119890

minus120578

1198920 (120578) = minus 119899119890minus120578

1205790 = 119890minus120578

BiBi + 1

1206010 = 119890minus120578

(29)

and starting from these sets of initial approximations 1198910 11989201205790 and 1206010 the iteration schemes (26) can be solved iterativelyfor 119891119903+1(120578) 119892119903+1(120578) 120579119903+1(120578) and120601119903+1(120578) when 119903 = 0 1 2

For this we discretise the equation using the Chebyshevspectral collocation method The unknown functions areapproximated by the Chebyshev interpolating polynomialsin such way that they are collocated at the Gauss-Lobattocollocation points defined as

120591119895= cos

120587119895

119873

119895 = 0 1 2 119873 (30)

where 119873 is the number of collocation points The physicalregion [0infin) is transformed into the region [minus1 1] using thedomain truncation technique in which the problem is solvedon the interval [0 120578

infin] instead of [0infin) This leads to the

mapping

120578

120578infin

=

120591 + 12

minus1 le 120591 le 1 (31)

where 120578infin

is the scaling parameter used to invoke theboundary condition at infinity The functions 119891 119892 120579 and 120601

are approximated at the collocation points by

119891 (120591) =

119873

sum

119896=0119891 (120591119896) 119879119896(120591119895)

119892 (120591) =

119873

sum

119896=0119892 (120591119896) 119879119896(120591119895)

120579 (120591) =

119873

sum

119896=0120579 (120591119896) 119879119896(120591119895)

120601 (120591) =

119873

sum

119896=0120601 (120591119896) 119879119896(120591119895)

119895 = 0 1 2 119873

(32)

where 119879119896is the 119896th Chebyshev polynomial defined as

119879119896(120591) = cos [119896 cosminus1 (120591)] (33)

The derivatives of the variables at the collocation points arerepresented as

119889119901

119891

119889120578119901=

119873

sum

119896=0D119901119897119896119891 (120591119896)

119889119901

119892

119889120578119901=

119873

sum

119896=0D119901119897119896119892 (120591119896)

119889119901

120579

119889120578119901=

119873

sum

119896=0D119901119897119896120579 (120591119896)

119889119901

120601

119889120578119901=

119873

sum

119896=0D119901119897119896120601 (120591119896)

119897 = 0 1 119873

(34)

where 119901 is the order of the derivative and D = 2D120578infin

is theChebyshev spectral differentiation matrix and its entries areclearly defined in Canuto et al [35]

Substituting (31)ndash(34) into (26) leads to the matrix equa-tion

119860119883 = 119877 (35)

subject to the boundary conditions

119891119903+1 (120591119873) = 119891

119908

119873

sum

119896=0D119873119896

119891 (120591119896) = 0

119873

sum

119896=0D0119896119891 (120591

119896) = 0

119892119903+1 (120591119873) = minus 119899

119873

sum

119896=0D2119873119896

119891 (120591119896)

119892119903+1 (1205910) = 0

8 Advances in High Energy Physics

119873

sum

119896=0D119873119896

120579119903+1 (120591119896) minusBi 120579

119903+1 (120591119873) = minusBi

120579119903+1 (1205910) = 0

120601119903+1 (120591119873) = 1120601119903+1 (1205910) = 0

(36)

In (35) 119860 is a (4119873+ 4) times (4119873+ 4) square matrix and119883 and 119877are (4119873 + 1) times 1 column vectors defined by

119860 =

[

[

[

[

[

[

11986011 11986012 11986013 11986014

11986021 11986022 11986023 11986024

11986031 11986032 11986033 11986034

11986041 11986042 11986043 11986044

]

]

]

]

]

]

119883 =

[

[

[

[

[

[

F119903+1

G119903+1

Θ119903+1

Φ119903+1

]

]

]

]

]

]

119877 =

[

[

[

[

[

[

R1

R2

R3

R4

]

]

]

]

]

]

(37)

whereF = [119891

119903+1 (1205910) 119891119903+1 (1205911) 119891119903+1 (120591119873)]119879

G = [119892119903+1 (1205910) 119892119903+1 (1205911) 119892119903+1 (120591119873)]

119879

Θ = [120579119903+1 (1205910) 120579119903+1 (1205911) 120579119903+1 (120591119873)]

119879

Φ = [120601119903+1 (1205910) 120601119903+1 (1205911) 120601119903+1 (120591119873)]

119879

11986011 = (

11 minus 119873

)D3+ diag [1198861119903]D

2+ diag [1198862119903]D

+ diag [1198863119903]

11986012 = (

119873

1 minus 119873

)D

11986013 = I

11986014 = BI

11986021 = minus(

119873

1 minus 119873

)D2+ diag [1198871119903]D+ diag [1198872119903]

11986022 = (

2 minus 119873

2 minus 2119873)D2

+ diag [1198873119903]D+ diag [1198874119903]

11986023 = 0

11986024 = 0

11986031 = diag [1198881119903]

11986032 = 0

11986033 =1Pr

D2+ diag [1198882119903]D

Table 1 Comparison of minus1205791015840(0) for free convection along a verticalflat plate in Newtonian fluid when 119873 = 0 119899 = 0 B = 0 Pr = 1Bi rarr infin and 119891

119908= 0

Merkin [36] Nazar et al [37] Molla et al [38] Present04214 04214 04214 04214313

11986034 = 0

11986041 = diag [1198891119903]

11986042 = 0

11986043 = 0

11986044 =1Sc

D2+ diag [1198892119903]D

R1 = 1198771119903

R2 = 1198772119903

R3 = 1198773119903

R4 = 1198774119903

(38)

and here I is an identity matrix the size of the matrix 0 is(119873+1)times1 and diag[ ] is a diagonalmatrix of size (119873+1)times(119873+

1) Subscript r denotes the iteration number After modifyingmatrix system (35) to incorporate boundary condition (36)the solution is obtained as

119883 = 119860minus1119877 (39)

7 Results and Discussions

It is noticed that the present problem reduces to free convec-tion heat transfer along an impermeable vertical plate in amicropolar fluid without the convective boundary conditionwhen119891

119908= 0 Bi rarr infin andB = 0 Also in the limit as119873 rarr

0 governing equations (2)ndash(5) reduce to the correspondingequations for a free convection heat and mass transfer ina viscous fluid In order to validate the code generated theresults of the present problem have been compared with theresults obtained by Merkin [36] Nazar et al [37] and Mollaet al [38] as a special case by taking 119873 = 0 119899 = 0 B = 0Pr = 1 Bi rarr infin and 119891

119908= 0 and it was found that

they are in good agreement as presented in Table 1 Also thecomparison of heat transfer coefficient has been made withthe results obtained by Nazar et al [37] as shown in Table 2when 119899 = 05 B = 0 Pr = 1 Bi rarr infin and 119891

119908= 0

To study the effects of coupling number119873 suctioninjectionparameter 119891

119908 Biot number Bi and material parameter 119899

computations were carried out in the cases of B = 10Pr = 071 and Sc = 022

The effects of the coupling number119873 on the dimension-less velocity microrotation temperature and concentrationare illustrated in Figures 2(a)ndash2(d) with fixed values ofother parameters The coupling number 119873 characterizes thecoupling of linear and rotational motion arising from the

Advances in High Energy Physics 9

00

01

02

03

04

05

06f998400

0 4 8 12 16

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(a)

g

minus06

minus05

minus04

minus03

minus02

minus01

00

01

0 2 4 6 8 10 12

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(b)

000

004

008

012

016

120579

0 1 2 3 4 5 6

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(c)

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

120601

00

02

04

06

08

10

120578

0 2 4 6 8 10 12

Bi = 01 fw = 05

(d)

Figure 2 Effect of119873 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Table 2 Comparison of minus1205791015840

(0) for free convection flow in amicropolar fluid obtained by Nazar et al [37] when 119899 = 05B = 0Pr = 1 Bi rarr infin and 119891

119908= 0

119873 Nazar et al [37] Present000 04214 04214033 03991 03990050 03834 03834060 03709 03709066 03608 03608071 03522 03522075 03447 03447

fluid particles In the case of 119873 = 0 (ie as 120581 tends tozero) the micropolarity is absent and fluid becomes nonpolarfluid With a large value of 119873 effect of microstructurebecomes significant whereas with a diminished value of 119873the individuality of the substructure is much less articulatedAs 119873 increases it is found from Figure 2(a) that the max-imum velocity decreases in amplitude and the location ofthe maximum velocity moves farther away from the wallSince 119873 rarr 0 corresponds to viscous fluid the velocity incase of a micropolar fluid has been less compared to thatof viscous fluid case It can be observed from Figure 2(b)that as 119873 increases initially microrotation profiles tend tobecome flatter and then approach their free stream valuesfar away from the wall This happens due to the vanishing

10 Advances in High Energy Physics

00

01

02

03

04

05

06

f998400

120578

0 2 4 6 8 10 12

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(a)

minus06

minus04

minus03

minus02

minus05

minus01

00

01

g

0 2 4 6 8 10

120578

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(b)

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

120578

0 1 2 3 4 5

120579

00

02

04

06

08

10

(c)

120601

00

02

04

06

08

10

fw = 05 n = 05N = 05

0 2 4 6 8 10 12

120578

Bi = 01

Bi = 10

Bi = 50Bi = 200

(d)

Figure 3 Effect of Bi on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

of antisymmetric part of the stress on the boundary thatcorresponds to a weak concentration of microelements Thisis because an increment in the value of 119873 implies a highervortex viscosity of fluid which promotes the microrotationof micropolar fluids It is seen from Figures 2(c) and 2(d)that thermal and concentration boundary layers of thefluid increase with increase in coupling number 119873 Hencetemperature and concentration in case of micropolar fluidsare more than those of the viscous fluid case

The Biot number Bi is the ratio of internal thermalresistance of a solid to boundary layer thermal resistanceWhen Bi = 0 the plate is totally insulated internal thermalresistance of the plate is extremely high and no convectiveheat transfer to the cold fluid on the upper part of the

plate takes place Figure 3(a) depicts fluid velocity profilesfor different values of the Biot number with 119873 = 05 119891

119908=

05 and 119899 = 05 Generally fluid velocity is zero at platesurface and increases gradually away fromplate to free streamvalue satisfying boundary conditions It is interesting to notethat an increase in the intensity of convective surface heattransfer Bi produces significant enhancement in fluid velocitywithin the momentum boundary layer In Figure 3(b) webring out the behavior of microrotation with different valuesof Biot number Bi for fixed values of other parameters As theparameter value Bi increases microrotation showing reverserotation near the two boundaries Hence the condition ofvanishing of antisymmetric part of the stress on the boundaryresults in a drastic change of the microrotation profiles

Advances in High Energy Physics 11

07

06

05

04

03

02

01

00

f998400

fw = 00

fw = 10

fw = 20

0 3 6 9 12 15

120578

N = 05 Bi = 01 n = 05

fw = minus05

(a)

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

120578

0 2 4 6 8 10 12

g

minus04

minus03

minus02

minus01

00

01

fw = minus05

(b)

0 1 2 3 4 5 6

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

000

005

010

015

020

025

030

120579

fw = minus05

(c)

0 3 6 9 12

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

00

02

04

06

08

10

120601

fw = minus05

(d)

Figure 4 Effect of 119891119908on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Given that convective heating increases with Biot numberBi rarr infin simulates the isothermal surface which is clearlyseen from Figure 3(c) where 120579(0) = 1 as Bi rarr infin Infact a high Biot number indicates higher internal thermalresistance of the plate than boundary layer thermal resistanceIn this fluid temperature is maximum at the plate surfaceand decreases exponentially to zero value far out from theplate satisfying boundary conditions As a consequence anincrement in the Biot number leads to increase of fluidtemperature efficiency Figure 3(d) illustrates the variation ofdimensionless concentration for different values of Bi It isclear that the concentration of fluid decreases with increaseof Bi

The effect of 119891119908

on velocity profile is depicted inFigure 4(a) Here 119891

119908gt 0 represents suction and 119891

119908lt 0

denotes injectionThe lower velocity is noticed in case of suc-tion when compared to case of injection From Figure 4(b)we note that microrotation is showing reverse rotation neartwo boundaries with both suction and injection parame-ter The dimensionless temperature for different values ofsuctioninjection parameters is drawn in Figure 4(c) It isreadable that the temperature of the fluid is more in case ofinjection whereas it is less in case of suction in comparisonwith the impermeable surface case (119891

119908= 0) Figure 4(d)

demonstrates dimensionless concentration for different val-ues of suctioninjection parameters It is determined that the

12 Advances in High Energy Physics

00

01

02

03

04

05

06f998400

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(a)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10

120578

minus10

minus08

minus06

minus04

minus02

00

g

(b)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

120578

0 1 2 3 4 5

120579

000

003

006

009

012

015

(c)

00

02

04

06

08

10

120601

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(d)

Figure 5 Effect of material parameter 119899 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

concentration of fluid ismore with injection whereas it is lesswith suctionwhen compared to the impermeable surface case(119891119908= 0) As a finale the thermal and solutal boundary layer

thicknesses increase in case of injection compared to case ofsuction as displayed in Figures 4(c) and 4(d)

In Figures 5(a)ndash5(d) the effect of material parameter119899 on dimensionless velocity microrotation temperatureand concentration is presented for fixed values of otherparameters since the material parameter 119899 signifies themicrorotation effects (ie for 119899 = 0 particles are not freeto revolve near the surface whereas as 119899 increases from 0 to1 the microrotation term gets augmented and induces flowenhancement) As 119899 increases it is found from Figure 5(a)that the minimum velocity increases in amplitude and the

location of the minimum velocity moves farther away fromthe wall From Figure 5(b) we observe that themicrorotationis decreasing with increasing value of material parameter119899 within the boundary layer It is clear from Figures 5(c)-5(d) that the thermal and solutal boundary layer thicknessesdecrease with increase of material parameter 119899

The variations of minus1205791015840

(0) and minus1206011015840

(0) versus couplingnumber 119873 are shown in Figures 6ndash8 It can be noticed fromthese figures that the heat and mass transfer coefficientsare less in case of micropolar fluids when compared to theviscous fluids This is because as 119873 increases the thermaland solutal boundary layer thicknesses become larger thusgiving rise to a small value of local heat and mass transferrates The effect of the Biot number Bi with fixed 119891

119908= 05

Advances in High Energy Physics 13

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

fw = 05 n = 05

01

00

02

03

04

05

06

07Nu x

Grminus

14

x

(a)

fw = 05 n = 05

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

024

026

028

030

032

ShxGrminus

14

x

(b)

Figure 6 Effect of Bi on (a) heat transfer rate and (b) mass transfer rate

005

006

007

008

009

010

Nu x

Grminus

14

x

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

(a)

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

01

02

03

04

05

06

ShxGrminus

14

x

(b)

Figure 7 Effect of 119891119908on (a) heat transfer rate and (b) mass transfer rate

and 119899 = 05 on local heat transfer coefficient is exhibitedin Figure 6 It is found from Figure 6 that local heat transferrate increases nonlinearly with the increase in Biot numberBiThe influence of the Biot number Bi on local mass transfercoefficient is shown in Figure 6 Figure 6 reveals that the localmass transfer coefficient is enhanced by the growth in theBiot number Bi In Figure 7 the effect of the suctioninjectionparameter 119891

119908with fixed Bi = 01 and 119899 = 05 on local heat

and mass transfer coefficients is displayed It is found fromFigure 7 that the local heat and mass transfer coefficients areless in the case of injection 119891

119908lt 0 in comparison with the

case of suction 119891119908

gt 0 Figure 8 is prepared to analyze theeffect of the material parameter 119899 with fixed Bi = 01 and

119891119908= 05 on local heat andmass transfer coefficients Figure 8

reveals that the local heat and mass transfer coefficients areenhanced by the increase in material parameter 119899 This isbecause when 119899 increases from 0 to 1 the microrotation termgets augmented and induces flow enhancement

The variations of 119862119891Gr14119909

and 119872119908Gr12119909

which areproportional to the coefficients of skin friction and wallcouple stress are shown in Table 3 with different values ofthe coupling number 119873 for fixed 119899 = 05 119891

119908= 05 and

Bi = 01 It indicates that the skin friction factor is higherfor micropolar fluid than the viscous fluids (119873 = 0) Sincemicropolar fluids offer a heavy resistance (resulting fromvortex viscosity) to fluid movement and cause larger skin

14 Advances in High Energy Physics

0084

0085

0086

0087Nu x

Grminus

14

x

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

fw = 05 Bi = 01

(a)

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

024

026

028

030

ShxGrminus

14

x

fw = 05 Bi = 01

(b)

Figure 8 Effect of material parameter 119899 on (a) heat transfer rate and (b) mass transfer rate

Table 3 Effects of skin friction and wall couple stress for varyingvalues of Biot numbers Bi micropolar parameter 119873 materialparameter 119899 and suctioninjection parameter 119891

119908

119873 Bi 119891119908

119899 119862119891Gr14119909

119872119908Gr12119909

01 01 05 05 2424227 06327903 01 05 05 2502716 063996605 01 05 05 2647987 063086307 01 05 05 2943561 060387909 01 05 05 3874631 055089105 01 05 05 2647987 063086305 10 05 05 3211755 080587905 50 05 05 3533141 090807805 200 05 05 3629791 093913705 01 minus05 05 2426265 033621905 01 00 05 2530014 046542905 01 10 05 2737553 082089905 01 20 05 2721244 119359605 01 05 00 2916655 minus028933905 01 05 05 2647987 063086305 01 05 10 2232556 2066191

friction factor compared to viscous fluid the results as wellsuggest larger values of coupling number 119873 and lower wallcouple stresses Since the skin friction coefficient 11989110158401015840(0) andwall couple stress coefficient as well as high temperature andmass transport rates are more depressed in the micropolarfluid comparing to the viscous fluid which may be beneficialin flow temperature and concentration control of polymerprocessing thus the presence of microscopic effects arisingfrom the local structure and of the fluid elements reduces thehigh temperature and mass transfer coefficients The effectof Bi on 119862

119891Gr14119909

and 119872119908Gr12119909

for 119891119908

= 05 119899 = 05 and

119873 = 05 is illustrated in Table 3 It can be noticed that theskin friction and wall couple stress coefficients are increasingwith increase of Bi for fixed values of other parametersThis notice is consistent with physical profiles presented inFigure 3 The effects of suctioninjection parameter on theskin friction andwall couple stress coefficients are also shownin Table 3 It is noted that the skin friction and wall couplestress coefficients are less with injection case whereas theyare more with suction case when compared to the case ofimpermeable surface Finally the detailed behavior of thematerial parameter 119899 is given in Table 3 The skin frictiondecreases and wall couple stress increases with increase ofmaterial parameter 119899

8 Conclusions

In this composition the similarity solution of the free convec-tion flow on a permeable vertical plate of a micropolar fluidunder the convective boundary condition is obtained usingLie group transformations Using the similarity variables thegoverning equations are transformed into a set of nondi-mensional parabolic equations These equations are solvednumerically using spectral quasi-linearisation method Thenumerical computation is carried out for various values ofnondimensional physical parameters The main findings aresummarized as follows

(i) The numerical results indicate that velocity distribu-tion is less near the plate but it is more far away fromthe plate the wall couple stress coefficient and rate ofheat andmass transfers are lower but the temperatureand concentration distributions and the skin frictioncoefficient are higher for the micropolar fluids incomparison with those of viscous fluids Also thereverse rotation ofmicrorotation near two boundariesis found with the increasing value of119873

Advances in High Energy Physics 15

(ii) An increase in Biot number Bi decreases concen-tration distribution whereas it causes an increasein temperature distribution skin friction and wallcouple stress coefficients and heat and mass transferrates within the boundary layer Further enhancingin Biot number Bi enhances velocity distributionnear the plate but shows the reverse behavior faraway from the plate We observe reverse rotationof microrotation near two boundaries within theboundary layer in the presence of Biot number

(iii) Less velocity temperature and concentration distri-butions are observed more skin friction and wallcouple stress coefficients and heat and mass transferrates in the case of suction compared to the case ofinjection Further microrotation decreases near thewall and depicts the opposite trend far away from thewall

(iv) It is found that microrotation temperature and con-centration distributions and skin friction coefficientsare more in the case of a micropolar fluid with strongconcentration (ie 119899 = 0) when compared to the caseof a micropolar fluid with weak concentration (ie119899 = 12) Further velocity is less in the case of 119899 = 0when compared to the case of 119899 = 12 near the walland shows the opposite trend far away from the wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their valuablesuggestions and comments

References

[1] J H Merkin ldquoNatural-convection boundary-layer flow on avertical surface with Newtonian heatingrdquo International Journalof Heat and Fluid Flow vol 15 no 5 pp 392ndash398 1994

[2] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009

[3] O DMakinde ldquoSimilarity solution for natural convection froma moving vertical plate with internal heat generation and aconvective boundary conditionrdquo Thermal Science vol 15 ppS137ndashS143 2011

[4] S M Ibrahim and N Bhashar Reddy ldquoSimilarity solution ofheat and mass transfer for natural convection over a movingvertical plate with internal heat generation and a convectiveboundary condition in the presence of thermal radiationviscous dissipation and chemical reactionrdquo ISRN Thermody-namics vol 2013 Article ID 790604 10 pages 2013

[5] C RamReddy P V S N Murthy A J Chamkha and A MRashad ldquoSoret effect on mixed convection flow in a nanofluidunder convective boundary conditionrdquo International Journal ofHeat and Mass Transfer vol 64 pp 384ndash392 2013

[6] H T Alkasasbeh M Z Salleh R M Tahar and R NazarldquoNumerical solutions of free convection boundary layer flow ona solid sphere with convective boundary conditionsrdquo Journal ofPhysics Conference Series vol 495 Article ID 012025 2014

[7] A Pantokratoras ldquoA note on natural convection along a con-vectively heated vertical platerdquo International Journal of ThermalSciences vol 76 pp 221ndash224 2014

[8] G Lukaszewicz Micropolar FluidsmdashTheory and ApplicationsBirkhaauser Basel Switzerland 1999

[9] V A Eremeyev L P Lebedev and H Altenbach Foundationsof Micropolar Mechanics Springer New York NY USA 2013

[10] G Ahmadi ldquoSelf-similar solution of imcompressible micropo-lar boundary layer flow over a semi-infinite platerdquo InternationalJournal of Engineering Science vol 14 no 7 pp 639ndash646 1976

[11] D A Rees and I Pop ldquoFree convection boundary-layer flowof a micropolar fluid from a vertical flat platerdquo IMA Journal ofApplied Mathematics vol 61 no 2 pp 179ndash197 1998

[12] D Srinivasacharya and C RamReddy ldquoEffect of double stratifi-cation onmixed convection in a micropolar fluidrdquoMatematikavol 28 no 2 pp 133ndash149 2012

[13] D Srinivasacharya and C Ramreddy ldquoMixed convection heatand mass transfer in a doubly stratified micropolar fluidrdquoComputationalThermal Sciences vol 5 no 4 pp 273ndash287 2013

[14] D Srinivasacharya and C RamReddy ldquoNatural convection heatand mass transfer in a micropolar fluid with thermal and massstratificationrdquo International Journal for Computational Methodsin Engineering Science andMechanics vol 14 no 5 pp 401ndash4132013

[15] N A Yacob and A Ishak ldquoStagnation point flow towardsa stretchingshrinking sheet in a micropolar fluid with aconvective surface boundary conditionrdquoThe Canadian Journalof Chemical Engineering vol 90 no 3 pp 621ndash626 2012

[16] K Zaimi and A Ishak ldquoStagnation-point flow and heat transferover a nonlinearly stretchingshrinking sheet in a micropolarfluidrdquo Abstract and Applied Analysis vol 2014 Article ID261630 6 pages 2014

[17] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[18] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[19] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer 2009

[20] T Tapanidis G Tsagas and H P Mazumdar ldquoApplication ofscaling group of transformations to viscoelastic second-gradefluid flowrdquo Nonlinear Functional Analysis and Applications vol8 no 3 pp 345ndash350 2003

[21] I A A Hassanien andM A Hamad ldquoGroup theoretic methodfor unsteady free convection flow of a micropolar fluid along avertical plate in a thermally stratified mediumrdquo Applied Mathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 32 no 6 pp 1099ndash1114 2008

[22] R Kandasamy K Gunasekaran and S B H Hasan ldquoScalinggroup transformation on fluid flow with variable stream con-ditionsrdquo International Journal of Non-Linear Mechanics vol 46no 7 pp 976ndash985 2011

[23] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat TransfermdashAsian Research vol41 no 3 pp 241ndash259 2012

16 Advances in High Energy Physics

[24] A A Mutlag M J Uddin M A A Hamad and A I M IsmailldquoHeat transfer analysis for falkner-skan boundary layer flowpast a stationary wedge with slips boundary conditions con-sidering temperature-dependent thermal conductivityrdquo SainsMalaysiana vol 42 no 6 pp 855ndash862 2013

[25] S C Cowin ldquoPolar fluidsrdquo Physics of Fluids vol 11 no 9 pp1919ndash1927 1968

[26] M A Seddeek M Y Akl and A M Al-Hanaya ldquoThermalradiation effects on mixed convection and mass transfer flowon vertical porous plate with heat generation and chemicalreaction by using scaling grouprdquo Journal of Natural Sciences andMathematics vol 4 pp 41ndash60 2010

[27] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[28] M J Uddin W A Khan and A I M Ismail ldquoMHD freeconvective boundary layer flow of a nanofluid past a flat verticalplate with Newtonian heating boundary conditionrdquo PLoS ONEvol 7 no 11 Article ID e49499 2012

[29] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012

[30] M J Uddin W A Khan and A I M Ismail ldquoScalinggroup transformation for MHD boundary layer slip flow of ananofluid over a convectively heated stretching sheet with heatgenerationrdquo Mathematical Problems in Engineering vol 2012Article ID 934964 20 pages 2012

[31] MM RashidiM Ferdows A B Parsa and S Abelman ldquoMHDnatural convection with convective surface boundary conditionover a at platerdquo Abstract and Applied Analysis vol 2014 ArticleID 923487 10 pages 2014

[32] R E Bellman and R E Kalaba Quasilinearisation and Non-Linear Boundary-Value Problems Elsevier New York NY USA1965

[33] S SMotsa P GDlamini andMKhumalo ldquoSpectral relaxationmethod and spectral quasilinearization method for solvingunsteady boundary layer flow problemsrdquo Advances in Mathe-matical Physics vol 2014 Article ID 341964 12 pages 2014

[34] S S Motsa P Sibanda J M Ngnotchouye and G T MarewoldquoA spectral relaxation approach for unsteady boundary-layerflow and heat transfer of a nanofluid over a permeable stretch-ingshrinking sheetrdquo Advances in Mathematical Physics vol2014 Article ID 564942 10 pages 2014

[35] C Canuto M Y Hussaini A Quarteroni and T ZangSpectral Methods Fundamentals in Single Domains ScientificComputation Springer Berlin Germany 2006

[36] J HMerkin ldquoFree convection boundary layer on an isothermalhorizontal cylinderrdquo in Proceedings of the American Societyof Mechanical Engineers and American Institute of ChemicalEngineersHeat Transfer Conference p 911 ASME St LouisMoUSA August 1976

[37] R Nazar N Amin and I Pop ldquoFree convection boundary layeron an isothermal horizontal circular cylinder in a micropolarfluidrdquo in Proceedings of the 12th International Heat TransferConference vol 2 pp 525ndash530 Elsevier Paris France August2002

[38] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

Submit your manuscripts athttpwwwhindawicom

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6 Advances in High Energy Physics

Boundary conditions (10a) and (10b) in terms of 119891 119892 120579and 120601 become

120578 = 0 119891 (0) = 119891119908

1198911015840

(0) = 0

119892 (0) = minus 11989911989110158401015840

(0)

1205791015840

(0) = minusBi [1minus 120579 (0)]

120601 (0) = 1

(23a)

120578 997888rarr infin 1198911015840

(infin) = 0

119892 (infin) = 0

120579 (infin) = 0

120601 (infin) = 0

(23b)

5 Skin Friction Wall Couple Stress and Heatand Mass Transfer Coefficients

The wall shear stress and the wall couple stress are

120591119908= [(120583 + 120581)

120597119906

120597119910

+ 120581120596]

119910=0

119898119908= 120574 [

120597120596

120597119910

]

119910=0

(24a)

and the heat and mass transfers from the plate respectivelyare given by

119902119908= minus 119896 [

120597119879

120597119910

]

119910=0

119902119898= minus119863[

120597119862

120597119910

]

119910=0

(24b)

The nondimensional skin friction 119862119891

= 2120591119908120588119906

2lowast wall

couple stress 119872119908

= 119898119908120588119906

2lowast119909 the local Nusselt number

119873119906119909= 119902119908119909119896(119879

119891minus 119879infin) and local Sherwood number Sh

119909=

119902119898119909119863(119862

119908minus 119862infin) are given by

119862119891Gr14119909

= 2(1 minus 119899119873

1 minus 119873

)11989110158401015840

(0)

119872119908Gr12119909

= (

2 minus 119873

2 minus 2119873)1198921015840

(0)

119873119906119909

Gr14119909

= minus 1205791015840

(0)

Sh119909

Gr14119909

= minus1206011015840

(0)

(25)

where 1199062lowastis the characteristic velocity and Gr

119909= 119892lowast

1205731198790(119879119891minus

119879infin)119909

3]2 is the local Grashof number

6 Numerical Solution Using the SpectralQuasi-Linearization Method (SQLM)

In this section we describe the quasi-linearization method(QLM) for solving the governing system of (22) alongwith boundary conditions (23a) and (23b) This QLM isa generalization of the Newton-Raphson method and wasproposed by Bellman and Kalaba [32] for solving nonlinearboundary value problems

Assume that the solutions 119891119903 119892119903 120579119903 and 120601

119903of (22) at the

(119903+1)th iteration are119891119903+1 119892119903+1 120579119903+1 and 120601119903+1 If the solutions

at the previous iteration are sufficiently close to the solutionsat the present iteration the nonlinear components of (22)can be linearised using one-term Taylor series of multiplevariables so that (22) give the following iterative sequence oflinear differential equations

(

11 minus 119873

)119891101584010158401015840

119903+1 + 119886111990311989110158401015840

119903+1 + 11988621199031198911015840

119903+1 + 1198863119903119891119903+1

+(

119873

1 minus 119873

)1198921015840

119903+1 + 120579119903+1 +B120601

119903+1 = 1198771119903

(

2 minus 119873

2 minus 2119873)11989210158401015840

119903+1 + 11988731199031198921015840

119903+1 + 1198874119903119892119903+1 + 11988711199031198911015840

119903+1

+ 1198872119903119891119903+1 minus(

119873

1 minus 119873

)11989110158401015840

119903+1 = 1198772119903

1198881119903119891119903+1 +1Pr

12057910158401015840

119903+1 + 11988821199031205791015840

119903+1 = 1198773119903

1198891119903119891119903+1 +1Sc

12060110158401015840

119903+1 +11988921199031206011015840

119903+1 = 1198774119903

(26)

where the coefficients 1198861199041 119903

(1199041 = 1 2 3) 1198871199042 119903

(1199042 = 1 2 4) 1198881199043 119903

(1199043 = 1 2) 1198891199044 119903

(1199044 = 1 2) and 1198771199045 119903

(1199045 = 1 2 4) are known functions (from previous iterations) and aredefined as

1198861119903 = 119891119903

1198862119903 = minus 21198911015840119903

1198863119903 = 11989110158401015840

119903

1198771119903 = 11989111990311989110158401015840

119903minus (1198911015840

119903)

2

1198871119903 = minus119892119903

1198872119903 = 1198921015840

119903

1198873119903 = 119891119903

1198874119903 = minus1198911015840

119903minus(

21198731 minus 119873

)

1198772119903 = 1198911199031198921015840

119903minus1198911015840

119903119892119903

1198881119903 = 1205791015840

119903

1198882119903 = 119891119903

1198773119903 = 1198911199031205791015840

119903

1198891119903 = 1206011015840

119903

1198892119903 = 119891119903

1198774119903 = 1198911199031206011015840

119903

(27)

Advances in High Energy Physics 7

subject to boundary conditions

119891119903+1 (0) = 119891

119908

1198911015840

119903+1 = 0

1198911015840

119903+1 (infin) = 0

119892119903+1 = minus 119899119891

10158401015840

119903+1 (0)

119892119903+1 (infin) = 0

1205791015840

119903+1 (0) = minusBi (1minus 120579 (0))

120579119903+1 (infin) = 0

120601119903+1 (0) = 1

120601119903+1 (infin) = 0

(28)

System (26) constitutes a linear system of coupled differen-tial equations with variable coefficients and can be solvediteratively using any numerical method for 119903 = 1 2 3 In this work as will be discussed below the Chebyshevpseudospectral method was used to solve the QLM scheme(26) (for more details refer to the works of Motsa et al[33 34])

1198910 (120578) = 119891119908+ 1minus 119890

minus120578

1198920 (120578) = minus 119899119890minus120578

1205790 = 119890minus120578

BiBi + 1

1206010 = 119890minus120578

(29)

and starting from these sets of initial approximations 1198910 11989201205790 and 1206010 the iteration schemes (26) can be solved iterativelyfor 119891119903+1(120578) 119892119903+1(120578) 120579119903+1(120578) and120601119903+1(120578) when 119903 = 0 1 2

For this we discretise the equation using the Chebyshevspectral collocation method The unknown functions areapproximated by the Chebyshev interpolating polynomialsin such way that they are collocated at the Gauss-Lobattocollocation points defined as

120591119895= cos

120587119895

119873

119895 = 0 1 2 119873 (30)

where 119873 is the number of collocation points The physicalregion [0infin) is transformed into the region [minus1 1] using thedomain truncation technique in which the problem is solvedon the interval [0 120578

infin] instead of [0infin) This leads to the

mapping

120578

120578infin

=

120591 + 12

minus1 le 120591 le 1 (31)

where 120578infin

is the scaling parameter used to invoke theboundary condition at infinity The functions 119891 119892 120579 and 120601

are approximated at the collocation points by

119891 (120591) =

119873

sum

119896=0119891 (120591119896) 119879119896(120591119895)

119892 (120591) =

119873

sum

119896=0119892 (120591119896) 119879119896(120591119895)

120579 (120591) =

119873

sum

119896=0120579 (120591119896) 119879119896(120591119895)

120601 (120591) =

119873

sum

119896=0120601 (120591119896) 119879119896(120591119895)

119895 = 0 1 2 119873

(32)

where 119879119896is the 119896th Chebyshev polynomial defined as

119879119896(120591) = cos [119896 cosminus1 (120591)] (33)

The derivatives of the variables at the collocation points arerepresented as

119889119901

119891

119889120578119901=

119873

sum

119896=0D119901119897119896119891 (120591119896)

119889119901

119892

119889120578119901=

119873

sum

119896=0D119901119897119896119892 (120591119896)

119889119901

120579

119889120578119901=

119873

sum

119896=0D119901119897119896120579 (120591119896)

119889119901

120601

119889120578119901=

119873

sum

119896=0D119901119897119896120601 (120591119896)

119897 = 0 1 119873

(34)

where 119901 is the order of the derivative and D = 2D120578infin

is theChebyshev spectral differentiation matrix and its entries areclearly defined in Canuto et al [35]

Substituting (31)ndash(34) into (26) leads to the matrix equa-tion

119860119883 = 119877 (35)

subject to the boundary conditions

119891119903+1 (120591119873) = 119891

119908

119873

sum

119896=0D119873119896

119891 (120591119896) = 0

119873

sum

119896=0D0119896119891 (120591

119896) = 0

119892119903+1 (120591119873) = minus 119899

119873

sum

119896=0D2119873119896

119891 (120591119896)

119892119903+1 (1205910) = 0

8 Advances in High Energy Physics

119873

sum

119896=0D119873119896

120579119903+1 (120591119896) minusBi 120579

119903+1 (120591119873) = minusBi

120579119903+1 (1205910) = 0

120601119903+1 (120591119873) = 1120601119903+1 (1205910) = 0

(36)

In (35) 119860 is a (4119873+ 4) times (4119873+ 4) square matrix and119883 and 119877are (4119873 + 1) times 1 column vectors defined by

119860 =

[

[

[

[

[

[

11986011 11986012 11986013 11986014

11986021 11986022 11986023 11986024

11986031 11986032 11986033 11986034

11986041 11986042 11986043 11986044

]

]

]

]

]

]

119883 =

[

[

[

[

[

[

F119903+1

G119903+1

Θ119903+1

Φ119903+1

]

]

]

]

]

]

119877 =

[

[

[

[

[

[

R1

R2

R3

R4

]

]

]

]

]

]

(37)

whereF = [119891

119903+1 (1205910) 119891119903+1 (1205911) 119891119903+1 (120591119873)]119879

G = [119892119903+1 (1205910) 119892119903+1 (1205911) 119892119903+1 (120591119873)]

119879

Θ = [120579119903+1 (1205910) 120579119903+1 (1205911) 120579119903+1 (120591119873)]

119879

Φ = [120601119903+1 (1205910) 120601119903+1 (1205911) 120601119903+1 (120591119873)]

119879

11986011 = (

11 minus 119873

)D3+ diag [1198861119903]D

2+ diag [1198862119903]D

+ diag [1198863119903]

11986012 = (

119873

1 minus 119873

)D

11986013 = I

11986014 = BI

11986021 = minus(

119873

1 minus 119873

)D2+ diag [1198871119903]D+ diag [1198872119903]

11986022 = (

2 minus 119873

2 minus 2119873)D2

+ diag [1198873119903]D+ diag [1198874119903]

11986023 = 0

11986024 = 0

11986031 = diag [1198881119903]

11986032 = 0

11986033 =1Pr

D2+ diag [1198882119903]D

Table 1 Comparison of minus1205791015840(0) for free convection along a verticalflat plate in Newtonian fluid when 119873 = 0 119899 = 0 B = 0 Pr = 1Bi rarr infin and 119891

119908= 0

Merkin [36] Nazar et al [37] Molla et al [38] Present04214 04214 04214 04214313

11986034 = 0

11986041 = diag [1198891119903]

11986042 = 0

11986043 = 0

11986044 =1Sc

D2+ diag [1198892119903]D

R1 = 1198771119903

R2 = 1198772119903

R3 = 1198773119903

R4 = 1198774119903

(38)

and here I is an identity matrix the size of the matrix 0 is(119873+1)times1 and diag[ ] is a diagonalmatrix of size (119873+1)times(119873+

1) Subscript r denotes the iteration number After modifyingmatrix system (35) to incorporate boundary condition (36)the solution is obtained as

119883 = 119860minus1119877 (39)

7 Results and Discussions

It is noticed that the present problem reduces to free convec-tion heat transfer along an impermeable vertical plate in amicropolar fluid without the convective boundary conditionwhen119891

119908= 0 Bi rarr infin andB = 0 Also in the limit as119873 rarr

0 governing equations (2)ndash(5) reduce to the correspondingequations for a free convection heat and mass transfer ina viscous fluid In order to validate the code generated theresults of the present problem have been compared with theresults obtained by Merkin [36] Nazar et al [37] and Mollaet al [38] as a special case by taking 119873 = 0 119899 = 0 B = 0Pr = 1 Bi rarr infin and 119891

119908= 0 and it was found that

they are in good agreement as presented in Table 1 Also thecomparison of heat transfer coefficient has been made withthe results obtained by Nazar et al [37] as shown in Table 2when 119899 = 05 B = 0 Pr = 1 Bi rarr infin and 119891

119908= 0

To study the effects of coupling number119873 suctioninjectionparameter 119891

119908 Biot number Bi and material parameter 119899

computations were carried out in the cases of B = 10Pr = 071 and Sc = 022

The effects of the coupling number119873 on the dimension-less velocity microrotation temperature and concentrationare illustrated in Figures 2(a)ndash2(d) with fixed values ofother parameters The coupling number 119873 characterizes thecoupling of linear and rotational motion arising from the

Advances in High Energy Physics 9

00

01

02

03

04

05

06f998400

0 4 8 12 16

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(a)

g

minus06

minus05

minus04

minus03

minus02

minus01

00

01

0 2 4 6 8 10 12

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(b)

000

004

008

012

016

120579

0 1 2 3 4 5 6

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(c)

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

120601

00

02

04

06

08

10

120578

0 2 4 6 8 10 12

Bi = 01 fw = 05

(d)

Figure 2 Effect of119873 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Table 2 Comparison of minus1205791015840

(0) for free convection flow in amicropolar fluid obtained by Nazar et al [37] when 119899 = 05B = 0Pr = 1 Bi rarr infin and 119891

119908= 0

119873 Nazar et al [37] Present000 04214 04214033 03991 03990050 03834 03834060 03709 03709066 03608 03608071 03522 03522075 03447 03447

fluid particles In the case of 119873 = 0 (ie as 120581 tends tozero) the micropolarity is absent and fluid becomes nonpolarfluid With a large value of 119873 effect of microstructurebecomes significant whereas with a diminished value of 119873the individuality of the substructure is much less articulatedAs 119873 increases it is found from Figure 2(a) that the max-imum velocity decreases in amplitude and the location ofthe maximum velocity moves farther away from the wallSince 119873 rarr 0 corresponds to viscous fluid the velocity incase of a micropolar fluid has been less compared to thatof viscous fluid case It can be observed from Figure 2(b)that as 119873 increases initially microrotation profiles tend tobecome flatter and then approach their free stream valuesfar away from the wall This happens due to the vanishing

10 Advances in High Energy Physics

00

01

02

03

04

05

06

f998400

120578

0 2 4 6 8 10 12

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(a)

minus06

minus04

minus03

minus02

minus05

minus01

00

01

g

0 2 4 6 8 10

120578

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(b)

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

120578

0 1 2 3 4 5

120579

00

02

04

06

08

10

(c)

120601

00

02

04

06

08

10

fw = 05 n = 05N = 05

0 2 4 6 8 10 12

120578

Bi = 01

Bi = 10

Bi = 50Bi = 200

(d)

Figure 3 Effect of Bi on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

of antisymmetric part of the stress on the boundary thatcorresponds to a weak concentration of microelements Thisis because an increment in the value of 119873 implies a highervortex viscosity of fluid which promotes the microrotationof micropolar fluids It is seen from Figures 2(c) and 2(d)that thermal and concentration boundary layers of thefluid increase with increase in coupling number 119873 Hencetemperature and concentration in case of micropolar fluidsare more than those of the viscous fluid case

The Biot number Bi is the ratio of internal thermalresistance of a solid to boundary layer thermal resistanceWhen Bi = 0 the plate is totally insulated internal thermalresistance of the plate is extremely high and no convectiveheat transfer to the cold fluid on the upper part of the

plate takes place Figure 3(a) depicts fluid velocity profilesfor different values of the Biot number with 119873 = 05 119891

119908=

05 and 119899 = 05 Generally fluid velocity is zero at platesurface and increases gradually away fromplate to free streamvalue satisfying boundary conditions It is interesting to notethat an increase in the intensity of convective surface heattransfer Bi produces significant enhancement in fluid velocitywithin the momentum boundary layer In Figure 3(b) webring out the behavior of microrotation with different valuesof Biot number Bi for fixed values of other parameters As theparameter value Bi increases microrotation showing reverserotation near the two boundaries Hence the condition ofvanishing of antisymmetric part of the stress on the boundaryresults in a drastic change of the microrotation profiles

Advances in High Energy Physics 11

07

06

05

04

03

02

01

00

f998400

fw = 00

fw = 10

fw = 20

0 3 6 9 12 15

120578

N = 05 Bi = 01 n = 05

fw = minus05

(a)

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

120578

0 2 4 6 8 10 12

g

minus04

minus03

minus02

minus01

00

01

fw = minus05

(b)

0 1 2 3 4 5 6

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

000

005

010

015

020

025

030

120579

fw = minus05

(c)

0 3 6 9 12

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

00

02

04

06

08

10

120601

fw = minus05

(d)

Figure 4 Effect of 119891119908on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Given that convective heating increases with Biot numberBi rarr infin simulates the isothermal surface which is clearlyseen from Figure 3(c) where 120579(0) = 1 as Bi rarr infin Infact a high Biot number indicates higher internal thermalresistance of the plate than boundary layer thermal resistanceIn this fluid temperature is maximum at the plate surfaceand decreases exponentially to zero value far out from theplate satisfying boundary conditions As a consequence anincrement in the Biot number leads to increase of fluidtemperature efficiency Figure 3(d) illustrates the variation ofdimensionless concentration for different values of Bi It isclear that the concentration of fluid decreases with increaseof Bi

The effect of 119891119908

on velocity profile is depicted inFigure 4(a) Here 119891

119908gt 0 represents suction and 119891

119908lt 0

denotes injectionThe lower velocity is noticed in case of suc-tion when compared to case of injection From Figure 4(b)we note that microrotation is showing reverse rotation neartwo boundaries with both suction and injection parame-ter The dimensionless temperature for different values ofsuctioninjection parameters is drawn in Figure 4(c) It isreadable that the temperature of the fluid is more in case ofinjection whereas it is less in case of suction in comparisonwith the impermeable surface case (119891

119908= 0) Figure 4(d)

demonstrates dimensionless concentration for different val-ues of suctioninjection parameters It is determined that the

12 Advances in High Energy Physics

00

01

02

03

04

05

06f998400

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(a)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10

120578

minus10

minus08

minus06

minus04

minus02

00

g

(b)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

120578

0 1 2 3 4 5

120579

000

003

006

009

012

015

(c)

00

02

04

06

08

10

120601

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(d)

Figure 5 Effect of material parameter 119899 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

concentration of fluid ismore with injection whereas it is lesswith suctionwhen compared to the impermeable surface case(119891119908= 0) As a finale the thermal and solutal boundary layer

thicknesses increase in case of injection compared to case ofsuction as displayed in Figures 4(c) and 4(d)

In Figures 5(a)ndash5(d) the effect of material parameter119899 on dimensionless velocity microrotation temperatureand concentration is presented for fixed values of otherparameters since the material parameter 119899 signifies themicrorotation effects (ie for 119899 = 0 particles are not freeto revolve near the surface whereas as 119899 increases from 0 to1 the microrotation term gets augmented and induces flowenhancement) As 119899 increases it is found from Figure 5(a)that the minimum velocity increases in amplitude and the

location of the minimum velocity moves farther away fromthe wall From Figure 5(b) we observe that themicrorotationis decreasing with increasing value of material parameter119899 within the boundary layer It is clear from Figures 5(c)-5(d) that the thermal and solutal boundary layer thicknessesdecrease with increase of material parameter 119899

The variations of minus1205791015840

(0) and minus1206011015840

(0) versus couplingnumber 119873 are shown in Figures 6ndash8 It can be noticed fromthese figures that the heat and mass transfer coefficientsare less in case of micropolar fluids when compared to theviscous fluids This is because as 119873 increases the thermaland solutal boundary layer thicknesses become larger thusgiving rise to a small value of local heat and mass transferrates The effect of the Biot number Bi with fixed 119891

119908= 05

Advances in High Energy Physics 13

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

fw = 05 n = 05

01

00

02

03

04

05

06

07Nu x

Grminus

14

x

(a)

fw = 05 n = 05

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

024

026

028

030

032

ShxGrminus

14

x

(b)

Figure 6 Effect of Bi on (a) heat transfer rate and (b) mass transfer rate

005

006

007

008

009

010

Nu x

Grminus

14

x

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

(a)

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

01

02

03

04

05

06

ShxGrminus

14

x

(b)

Figure 7 Effect of 119891119908on (a) heat transfer rate and (b) mass transfer rate

and 119899 = 05 on local heat transfer coefficient is exhibitedin Figure 6 It is found from Figure 6 that local heat transferrate increases nonlinearly with the increase in Biot numberBiThe influence of the Biot number Bi on local mass transfercoefficient is shown in Figure 6 Figure 6 reveals that the localmass transfer coefficient is enhanced by the growth in theBiot number Bi In Figure 7 the effect of the suctioninjectionparameter 119891

119908with fixed Bi = 01 and 119899 = 05 on local heat

and mass transfer coefficients is displayed It is found fromFigure 7 that the local heat and mass transfer coefficients areless in the case of injection 119891

119908lt 0 in comparison with the

case of suction 119891119908

gt 0 Figure 8 is prepared to analyze theeffect of the material parameter 119899 with fixed Bi = 01 and

119891119908= 05 on local heat andmass transfer coefficients Figure 8

reveals that the local heat and mass transfer coefficients areenhanced by the increase in material parameter 119899 This isbecause when 119899 increases from 0 to 1 the microrotation termgets augmented and induces flow enhancement

The variations of 119862119891Gr14119909

and 119872119908Gr12119909

which areproportional to the coefficients of skin friction and wallcouple stress are shown in Table 3 with different values ofthe coupling number 119873 for fixed 119899 = 05 119891

119908= 05 and

Bi = 01 It indicates that the skin friction factor is higherfor micropolar fluid than the viscous fluids (119873 = 0) Sincemicropolar fluids offer a heavy resistance (resulting fromvortex viscosity) to fluid movement and cause larger skin

14 Advances in High Energy Physics

0084

0085

0086

0087Nu x

Grminus

14

x

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

fw = 05 Bi = 01

(a)

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

024

026

028

030

ShxGrminus

14

x

fw = 05 Bi = 01

(b)

Figure 8 Effect of material parameter 119899 on (a) heat transfer rate and (b) mass transfer rate

Table 3 Effects of skin friction and wall couple stress for varyingvalues of Biot numbers Bi micropolar parameter 119873 materialparameter 119899 and suctioninjection parameter 119891

119908

119873 Bi 119891119908

119899 119862119891Gr14119909

119872119908Gr12119909

01 01 05 05 2424227 06327903 01 05 05 2502716 063996605 01 05 05 2647987 063086307 01 05 05 2943561 060387909 01 05 05 3874631 055089105 01 05 05 2647987 063086305 10 05 05 3211755 080587905 50 05 05 3533141 090807805 200 05 05 3629791 093913705 01 minus05 05 2426265 033621905 01 00 05 2530014 046542905 01 10 05 2737553 082089905 01 20 05 2721244 119359605 01 05 00 2916655 minus028933905 01 05 05 2647987 063086305 01 05 10 2232556 2066191

friction factor compared to viscous fluid the results as wellsuggest larger values of coupling number 119873 and lower wallcouple stresses Since the skin friction coefficient 11989110158401015840(0) andwall couple stress coefficient as well as high temperature andmass transport rates are more depressed in the micropolarfluid comparing to the viscous fluid which may be beneficialin flow temperature and concentration control of polymerprocessing thus the presence of microscopic effects arisingfrom the local structure and of the fluid elements reduces thehigh temperature and mass transfer coefficients The effectof Bi on 119862

119891Gr14119909

and 119872119908Gr12119909

for 119891119908

= 05 119899 = 05 and

119873 = 05 is illustrated in Table 3 It can be noticed that theskin friction and wall couple stress coefficients are increasingwith increase of Bi for fixed values of other parametersThis notice is consistent with physical profiles presented inFigure 3 The effects of suctioninjection parameter on theskin friction andwall couple stress coefficients are also shownin Table 3 It is noted that the skin friction and wall couplestress coefficients are less with injection case whereas theyare more with suction case when compared to the case ofimpermeable surface Finally the detailed behavior of thematerial parameter 119899 is given in Table 3 The skin frictiondecreases and wall couple stress increases with increase ofmaterial parameter 119899

8 Conclusions

In this composition the similarity solution of the free convec-tion flow on a permeable vertical plate of a micropolar fluidunder the convective boundary condition is obtained usingLie group transformations Using the similarity variables thegoverning equations are transformed into a set of nondi-mensional parabolic equations These equations are solvednumerically using spectral quasi-linearisation method Thenumerical computation is carried out for various values ofnondimensional physical parameters The main findings aresummarized as follows

(i) The numerical results indicate that velocity distribu-tion is less near the plate but it is more far away fromthe plate the wall couple stress coefficient and rate ofheat andmass transfers are lower but the temperatureand concentration distributions and the skin frictioncoefficient are higher for the micropolar fluids incomparison with those of viscous fluids Also thereverse rotation ofmicrorotation near two boundariesis found with the increasing value of119873

Advances in High Energy Physics 15

(ii) An increase in Biot number Bi decreases concen-tration distribution whereas it causes an increasein temperature distribution skin friction and wallcouple stress coefficients and heat and mass transferrates within the boundary layer Further enhancingin Biot number Bi enhances velocity distributionnear the plate but shows the reverse behavior faraway from the plate We observe reverse rotationof microrotation near two boundaries within theboundary layer in the presence of Biot number

(iii) Less velocity temperature and concentration distri-butions are observed more skin friction and wallcouple stress coefficients and heat and mass transferrates in the case of suction compared to the case ofinjection Further microrotation decreases near thewall and depicts the opposite trend far away from thewall

(iv) It is found that microrotation temperature and con-centration distributions and skin friction coefficientsare more in the case of a micropolar fluid with strongconcentration (ie 119899 = 0) when compared to the caseof a micropolar fluid with weak concentration (ie119899 = 12) Further velocity is less in the case of 119899 = 0when compared to the case of 119899 = 12 near the walland shows the opposite trend far away from the wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their valuablesuggestions and comments

References

[1] J H Merkin ldquoNatural-convection boundary-layer flow on avertical surface with Newtonian heatingrdquo International Journalof Heat and Fluid Flow vol 15 no 5 pp 392ndash398 1994

[2] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009

[3] O DMakinde ldquoSimilarity solution for natural convection froma moving vertical plate with internal heat generation and aconvective boundary conditionrdquo Thermal Science vol 15 ppS137ndashS143 2011

[4] S M Ibrahim and N Bhashar Reddy ldquoSimilarity solution ofheat and mass transfer for natural convection over a movingvertical plate with internal heat generation and a convectiveboundary condition in the presence of thermal radiationviscous dissipation and chemical reactionrdquo ISRN Thermody-namics vol 2013 Article ID 790604 10 pages 2013

[5] C RamReddy P V S N Murthy A J Chamkha and A MRashad ldquoSoret effect on mixed convection flow in a nanofluidunder convective boundary conditionrdquo International Journal ofHeat and Mass Transfer vol 64 pp 384ndash392 2013

[6] H T Alkasasbeh M Z Salleh R M Tahar and R NazarldquoNumerical solutions of free convection boundary layer flow ona solid sphere with convective boundary conditionsrdquo Journal ofPhysics Conference Series vol 495 Article ID 012025 2014

[7] A Pantokratoras ldquoA note on natural convection along a con-vectively heated vertical platerdquo International Journal of ThermalSciences vol 76 pp 221ndash224 2014

[8] G Lukaszewicz Micropolar FluidsmdashTheory and ApplicationsBirkhaauser Basel Switzerland 1999

[9] V A Eremeyev L P Lebedev and H Altenbach Foundationsof Micropolar Mechanics Springer New York NY USA 2013

[10] G Ahmadi ldquoSelf-similar solution of imcompressible micropo-lar boundary layer flow over a semi-infinite platerdquo InternationalJournal of Engineering Science vol 14 no 7 pp 639ndash646 1976

[11] D A Rees and I Pop ldquoFree convection boundary-layer flowof a micropolar fluid from a vertical flat platerdquo IMA Journal ofApplied Mathematics vol 61 no 2 pp 179ndash197 1998

[12] D Srinivasacharya and C RamReddy ldquoEffect of double stratifi-cation onmixed convection in a micropolar fluidrdquoMatematikavol 28 no 2 pp 133ndash149 2012

[13] D Srinivasacharya and C Ramreddy ldquoMixed convection heatand mass transfer in a doubly stratified micropolar fluidrdquoComputationalThermal Sciences vol 5 no 4 pp 273ndash287 2013

[14] D Srinivasacharya and C RamReddy ldquoNatural convection heatand mass transfer in a micropolar fluid with thermal and massstratificationrdquo International Journal for Computational Methodsin Engineering Science andMechanics vol 14 no 5 pp 401ndash4132013

[15] N A Yacob and A Ishak ldquoStagnation point flow towardsa stretchingshrinking sheet in a micropolar fluid with aconvective surface boundary conditionrdquoThe Canadian Journalof Chemical Engineering vol 90 no 3 pp 621ndash626 2012

[16] K Zaimi and A Ishak ldquoStagnation-point flow and heat transferover a nonlinearly stretchingshrinking sheet in a micropolarfluidrdquo Abstract and Applied Analysis vol 2014 Article ID261630 6 pages 2014

[17] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[18] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[19] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer 2009

[20] T Tapanidis G Tsagas and H P Mazumdar ldquoApplication ofscaling group of transformations to viscoelastic second-gradefluid flowrdquo Nonlinear Functional Analysis and Applications vol8 no 3 pp 345ndash350 2003

[21] I A A Hassanien andM A Hamad ldquoGroup theoretic methodfor unsteady free convection flow of a micropolar fluid along avertical plate in a thermally stratified mediumrdquo Applied Mathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 32 no 6 pp 1099ndash1114 2008

[22] R Kandasamy K Gunasekaran and S B H Hasan ldquoScalinggroup transformation on fluid flow with variable stream con-ditionsrdquo International Journal of Non-Linear Mechanics vol 46no 7 pp 976ndash985 2011

[23] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat TransfermdashAsian Research vol41 no 3 pp 241ndash259 2012

16 Advances in High Energy Physics

[24] A A Mutlag M J Uddin M A A Hamad and A I M IsmailldquoHeat transfer analysis for falkner-skan boundary layer flowpast a stationary wedge with slips boundary conditions con-sidering temperature-dependent thermal conductivityrdquo SainsMalaysiana vol 42 no 6 pp 855ndash862 2013

[25] S C Cowin ldquoPolar fluidsrdquo Physics of Fluids vol 11 no 9 pp1919ndash1927 1968

[26] M A Seddeek M Y Akl and A M Al-Hanaya ldquoThermalradiation effects on mixed convection and mass transfer flowon vertical porous plate with heat generation and chemicalreaction by using scaling grouprdquo Journal of Natural Sciences andMathematics vol 4 pp 41ndash60 2010

[27] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[28] M J Uddin W A Khan and A I M Ismail ldquoMHD freeconvective boundary layer flow of a nanofluid past a flat verticalplate with Newtonian heating boundary conditionrdquo PLoS ONEvol 7 no 11 Article ID e49499 2012

[29] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012

[30] M J Uddin W A Khan and A I M Ismail ldquoScalinggroup transformation for MHD boundary layer slip flow of ananofluid over a convectively heated stretching sheet with heatgenerationrdquo Mathematical Problems in Engineering vol 2012Article ID 934964 20 pages 2012

[31] MM RashidiM Ferdows A B Parsa and S Abelman ldquoMHDnatural convection with convective surface boundary conditionover a at platerdquo Abstract and Applied Analysis vol 2014 ArticleID 923487 10 pages 2014

[32] R E Bellman and R E Kalaba Quasilinearisation and Non-Linear Boundary-Value Problems Elsevier New York NY USA1965

[33] S SMotsa P GDlamini andMKhumalo ldquoSpectral relaxationmethod and spectral quasilinearization method for solvingunsteady boundary layer flow problemsrdquo Advances in Mathe-matical Physics vol 2014 Article ID 341964 12 pages 2014

[34] S S Motsa P Sibanda J M Ngnotchouye and G T MarewoldquoA spectral relaxation approach for unsteady boundary-layerflow and heat transfer of a nanofluid over a permeable stretch-ingshrinking sheetrdquo Advances in Mathematical Physics vol2014 Article ID 564942 10 pages 2014

[35] C Canuto M Y Hussaini A Quarteroni and T ZangSpectral Methods Fundamentals in Single Domains ScientificComputation Springer Berlin Germany 2006

[36] J HMerkin ldquoFree convection boundary layer on an isothermalhorizontal cylinderrdquo in Proceedings of the American Societyof Mechanical Engineers and American Institute of ChemicalEngineersHeat Transfer Conference p 911 ASME St LouisMoUSA August 1976

[37] R Nazar N Amin and I Pop ldquoFree convection boundary layeron an isothermal horizontal circular cylinder in a micropolarfluidrdquo in Proceedings of the 12th International Heat TransferConference vol 2 pp 525ndash530 Elsevier Paris France August2002

[38] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

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AstrophysicsJournal of

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Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 7

subject to boundary conditions

119891119903+1 (0) = 119891

119908

1198911015840

119903+1 = 0

1198911015840

119903+1 (infin) = 0

119892119903+1 = minus 119899119891

10158401015840

119903+1 (0)

119892119903+1 (infin) = 0

1205791015840

119903+1 (0) = minusBi (1minus 120579 (0))

120579119903+1 (infin) = 0

120601119903+1 (0) = 1

120601119903+1 (infin) = 0

(28)

System (26) constitutes a linear system of coupled differen-tial equations with variable coefficients and can be solvediteratively using any numerical method for 119903 = 1 2 3 In this work as will be discussed below the Chebyshevpseudospectral method was used to solve the QLM scheme(26) (for more details refer to the works of Motsa et al[33 34])

1198910 (120578) = 119891119908+ 1minus 119890

minus120578

1198920 (120578) = minus 119899119890minus120578

1205790 = 119890minus120578

BiBi + 1

1206010 = 119890minus120578

(29)

and starting from these sets of initial approximations 1198910 11989201205790 and 1206010 the iteration schemes (26) can be solved iterativelyfor 119891119903+1(120578) 119892119903+1(120578) 120579119903+1(120578) and120601119903+1(120578) when 119903 = 0 1 2

For this we discretise the equation using the Chebyshevspectral collocation method The unknown functions areapproximated by the Chebyshev interpolating polynomialsin such way that they are collocated at the Gauss-Lobattocollocation points defined as

120591119895= cos

120587119895

119873

119895 = 0 1 2 119873 (30)

where 119873 is the number of collocation points The physicalregion [0infin) is transformed into the region [minus1 1] using thedomain truncation technique in which the problem is solvedon the interval [0 120578

infin] instead of [0infin) This leads to the

mapping

120578

120578infin

=

120591 + 12

minus1 le 120591 le 1 (31)

where 120578infin

is the scaling parameter used to invoke theboundary condition at infinity The functions 119891 119892 120579 and 120601

are approximated at the collocation points by

119891 (120591) =

119873

sum

119896=0119891 (120591119896) 119879119896(120591119895)

119892 (120591) =

119873

sum

119896=0119892 (120591119896) 119879119896(120591119895)

120579 (120591) =

119873

sum

119896=0120579 (120591119896) 119879119896(120591119895)

120601 (120591) =

119873

sum

119896=0120601 (120591119896) 119879119896(120591119895)

119895 = 0 1 2 119873

(32)

where 119879119896is the 119896th Chebyshev polynomial defined as

119879119896(120591) = cos [119896 cosminus1 (120591)] (33)

The derivatives of the variables at the collocation points arerepresented as

119889119901

119891

119889120578119901=

119873

sum

119896=0D119901119897119896119891 (120591119896)

119889119901

119892

119889120578119901=

119873

sum

119896=0D119901119897119896119892 (120591119896)

119889119901

120579

119889120578119901=

119873

sum

119896=0D119901119897119896120579 (120591119896)

119889119901

120601

119889120578119901=

119873

sum

119896=0D119901119897119896120601 (120591119896)

119897 = 0 1 119873

(34)

where 119901 is the order of the derivative and D = 2D120578infin

is theChebyshev spectral differentiation matrix and its entries areclearly defined in Canuto et al [35]

Substituting (31)ndash(34) into (26) leads to the matrix equa-tion

119860119883 = 119877 (35)

subject to the boundary conditions

119891119903+1 (120591119873) = 119891

119908

119873

sum

119896=0D119873119896

119891 (120591119896) = 0

119873

sum

119896=0D0119896119891 (120591

119896) = 0

119892119903+1 (120591119873) = minus 119899

119873

sum

119896=0D2119873119896

119891 (120591119896)

119892119903+1 (1205910) = 0

8 Advances in High Energy Physics

119873

sum

119896=0D119873119896

120579119903+1 (120591119896) minusBi 120579

119903+1 (120591119873) = minusBi

120579119903+1 (1205910) = 0

120601119903+1 (120591119873) = 1120601119903+1 (1205910) = 0

(36)

In (35) 119860 is a (4119873+ 4) times (4119873+ 4) square matrix and119883 and 119877are (4119873 + 1) times 1 column vectors defined by

119860 =

[

[

[

[

[

[

11986011 11986012 11986013 11986014

11986021 11986022 11986023 11986024

11986031 11986032 11986033 11986034

11986041 11986042 11986043 11986044

]

]

]

]

]

]

119883 =

[

[

[

[

[

[

F119903+1

G119903+1

Θ119903+1

Φ119903+1

]

]

]

]

]

]

119877 =

[

[

[

[

[

[

R1

R2

R3

R4

]

]

]

]

]

]

(37)

whereF = [119891

119903+1 (1205910) 119891119903+1 (1205911) 119891119903+1 (120591119873)]119879

G = [119892119903+1 (1205910) 119892119903+1 (1205911) 119892119903+1 (120591119873)]

119879

Θ = [120579119903+1 (1205910) 120579119903+1 (1205911) 120579119903+1 (120591119873)]

119879

Φ = [120601119903+1 (1205910) 120601119903+1 (1205911) 120601119903+1 (120591119873)]

119879

11986011 = (

11 minus 119873

)D3+ diag [1198861119903]D

2+ diag [1198862119903]D

+ diag [1198863119903]

11986012 = (

119873

1 minus 119873

)D

11986013 = I

11986014 = BI

11986021 = minus(

119873

1 minus 119873

)D2+ diag [1198871119903]D+ diag [1198872119903]

11986022 = (

2 minus 119873

2 minus 2119873)D2

+ diag [1198873119903]D+ diag [1198874119903]

11986023 = 0

11986024 = 0

11986031 = diag [1198881119903]

11986032 = 0

11986033 =1Pr

D2+ diag [1198882119903]D

Table 1 Comparison of minus1205791015840(0) for free convection along a verticalflat plate in Newtonian fluid when 119873 = 0 119899 = 0 B = 0 Pr = 1Bi rarr infin and 119891

119908= 0

Merkin [36] Nazar et al [37] Molla et al [38] Present04214 04214 04214 04214313

11986034 = 0

11986041 = diag [1198891119903]

11986042 = 0

11986043 = 0

11986044 =1Sc

D2+ diag [1198892119903]D

R1 = 1198771119903

R2 = 1198772119903

R3 = 1198773119903

R4 = 1198774119903

(38)

and here I is an identity matrix the size of the matrix 0 is(119873+1)times1 and diag[ ] is a diagonalmatrix of size (119873+1)times(119873+

1) Subscript r denotes the iteration number After modifyingmatrix system (35) to incorporate boundary condition (36)the solution is obtained as

119883 = 119860minus1119877 (39)

7 Results and Discussions

It is noticed that the present problem reduces to free convec-tion heat transfer along an impermeable vertical plate in amicropolar fluid without the convective boundary conditionwhen119891

119908= 0 Bi rarr infin andB = 0 Also in the limit as119873 rarr

0 governing equations (2)ndash(5) reduce to the correspondingequations for a free convection heat and mass transfer ina viscous fluid In order to validate the code generated theresults of the present problem have been compared with theresults obtained by Merkin [36] Nazar et al [37] and Mollaet al [38] as a special case by taking 119873 = 0 119899 = 0 B = 0Pr = 1 Bi rarr infin and 119891

119908= 0 and it was found that

they are in good agreement as presented in Table 1 Also thecomparison of heat transfer coefficient has been made withthe results obtained by Nazar et al [37] as shown in Table 2when 119899 = 05 B = 0 Pr = 1 Bi rarr infin and 119891

119908= 0

To study the effects of coupling number119873 suctioninjectionparameter 119891

119908 Biot number Bi and material parameter 119899

computations were carried out in the cases of B = 10Pr = 071 and Sc = 022

The effects of the coupling number119873 on the dimension-less velocity microrotation temperature and concentrationare illustrated in Figures 2(a)ndash2(d) with fixed values ofother parameters The coupling number 119873 characterizes thecoupling of linear and rotational motion arising from the

Advances in High Energy Physics 9

00

01

02

03

04

05

06f998400

0 4 8 12 16

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(a)

g

minus06

minus05

minus04

minus03

minus02

minus01

00

01

0 2 4 6 8 10 12

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(b)

000

004

008

012

016

120579

0 1 2 3 4 5 6

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(c)

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

120601

00

02

04

06

08

10

120578

0 2 4 6 8 10 12

Bi = 01 fw = 05

(d)

Figure 2 Effect of119873 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Table 2 Comparison of minus1205791015840

(0) for free convection flow in amicropolar fluid obtained by Nazar et al [37] when 119899 = 05B = 0Pr = 1 Bi rarr infin and 119891

119908= 0

119873 Nazar et al [37] Present000 04214 04214033 03991 03990050 03834 03834060 03709 03709066 03608 03608071 03522 03522075 03447 03447

fluid particles In the case of 119873 = 0 (ie as 120581 tends tozero) the micropolarity is absent and fluid becomes nonpolarfluid With a large value of 119873 effect of microstructurebecomes significant whereas with a diminished value of 119873the individuality of the substructure is much less articulatedAs 119873 increases it is found from Figure 2(a) that the max-imum velocity decreases in amplitude and the location ofthe maximum velocity moves farther away from the wallSince 119873 rarr 0 corresponds to viscous fluid the velocity incase of a micropolar fluid has been less compared to thatof viscous fluid case It can be observed from Figure 2(b)that as 119873 increases initially microrotation profiles tend tobecome flatter and then approach their free stream valuesfar away from the wall This happens due to the vanishing

10 Advances in High Energy Physics

00

01

02

03

04

05

06

f998400

120578

0 2 4 6 8 10 12

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(a)

minus06

minus04

minus03

minus02

minus05

minus01

00

01

g

0 2 4 6 8 10

120578

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(b)

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

120578

0 1 2 3 4 5

120579

00

02

04

06

08

10

(c)

120601

00

02

04

06

08

10

fw = 05 n = 05N = 05

0 2 4 6 8 10 12

120578

Bi = 01

Bi = 10

Bi = 50Bi = 200

(d)

Figure 3 Effect of Bi on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

of antisymmetric part of the stress on the boundary thatcorresponds to a weak concentration of microelements Thisis because an increment in the value of 119873 implies a highervortex viscosity of fluid which promotes the microrotationof micropolar fluids It is seen from Figures 2(c) and 2(d)that thermal and concentration boundary layers of thefluid increase with increase in coupling number 119873 Hencetemperature and concentration in case of micropolar fluidsare more than those of the viscous fluid case

The Biot number Bi is the ratio of internal thermalresistance of a solid to boundary layer thermal resistanceWhen Bi = 0 the plate is totally insulated internal thermalresistance of the plate is extremely high and no convectiveheat transfer to the cold fluid on the upper part of the

plate takes place Figure 3(a) depicts fluid velocity profilesfor different values of the Biot number with 119873 = 05 119891

119908=

05 and 119899 = 05 Generally fluid velocity is zero at platesurface and increases gradually away fromplate to free streamvalue satisfying boundary conditions It is interesting to notethat an increase in the intensity of convective surface heattransfer Bi produces significant enhancement in fluid velocitywithin the momentum boundary layer In Figure 3(b) webring out the behavior of microrotation with different valuesof Biot number Bi for fixed values of other parameters As theparameter value Bi increases microrotation showing reverserotation near the two boundaries Hence the condition ofvanishing of antisymmetric part of the stress on the boundaryresults in a drastic change of the microrotation profiles

Advances in High Energy Physics 11

07

06

05

04

03

02

01

00

f998400

fw = 00

fw = 10

fw = 20

0 3 6 9 12 15

120578

N = 05 Bi = 01 n = 05

fw = minus05

(a)

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

120578

0 2 4 6 8 10 12

g

minus04

minus03

minus02

minus01

00

01

fw = minus05

(b)

0 1 2 3 4 5 6

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

000

005

010

015

020

025

030

120579

fw = minus05

(c)

0 3 6 9 12

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

00

02

04

06

08

10

120601

fw = minus05

(d)

Figure 4 Effect of 119891119908on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Given that convective heating increases with Biot numberBi rarr infin simulates the isothermal surface which is clearlyseen from Figure 3(c) where 120579(0) = 1 as Bi rarr infin Infact a high Biot number indicates higher internal thermalresistance of the plate than boundary layer thermal resistanceIn this fluid temperature is maximum at the plate surfaceand decreases exponentially to zero value far out from theplate satisfying boundary conditions As a consequence anincrement in the Biot number leads to increase of fluidtemperature efficiency Figure 3(d) illustrates the variation ofdimensionless concentration for different values of Bi It isclear that the concentration of fluid decreases with increaseof Bi

The effect of 119891119908

on velocity profile is depicted inFigure 4(a) Here 119891

119908gt 0 represents suction and 119891

119908lt 0

denotes injectionThe lower velocity is noticed in case of suc-tion when compared to case of injection From Figure 4(b)we note that microrotation is showing reverse rotation neartwo boundaries with both suction and injection parame-ter The dimensionless temperature for different values ofsuctioninjection parameters is drawn in Figure 4(c) It isreadable that the temperature of the fluid is more in case ofinjection whereas it is less in case of suction in comparisonwith the impermeable surface case (119891

119908= 0) Figure 4(d)

demonstrates dimensionless concentration for different val-ues of suctioninjection parameters It is determined that the

12 Advances in High Energy Physics

00

01

02

03

04

05

06f998400

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(a)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10

120578

minus10

minus08

minus06

minus04

minus02

00

g

(b)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

120578

0 1 2 3 4 5

120579

000

003

006

009

012

015

(c)

00

02

04

06

08

10

120601

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(d)

Figure 5 Effect of material parameter 119899 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

concentration of fluid ismore with injection whereas it is lesswith suctionwhen compared to the impermeable surface case(119891119908= 0) As a finale the thermal and solutal boundary layer

thicknesses increase in case of injection compared to case ofsuction as displayed in Figures 4(c) and 4(d)

In Figures 5(a)ndash5(d) the effect of material parameter119899 on dimensionless velocity microrotation temperatureand concentration is presented for fixed values of otherparameters since the material parameter 119899 signifies themicrorotation effects (ie for 119899 = 0 particles are not freeto revolve near the surface whereas as 119899 increases from 0 to1 the microrotation term gets augmented and induces flowenhancement) As 119899 increases it is found from Figure 5(a)that the minimum velocity increases in amplitude and the

location of the minimum velocity moves farther away fromthe wall From Figure 5(b) we observe that themicrorotationis decreasing with increasing value of material parameter119899 within the boundary layer It is clear from Figures 5(c)-5(d) that the thermal and solutal boundary layer thicknessesdecrease with increase of material parameter 119899

The variations of minus1205791015840

(0) and minus1206011015840

(0) versus couplingnumber 119873 are shown in Figures 6ndash8 It can be noticed fromthese figures that the heat and mass transfer coefficientsare less in case of micropolar fluids when compared to theviscous fluids This is because as 119873 increases the thermaland solutal boundary layer thicknesses become larger thusgiving rise to a small value of local heat and mass transferrates The effect of the Biot number Bi with fixed 119891

119908= 05

Advances in High Energy Physics 13

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

fw = 05 n = 05

01

00

02

03

04

05

06

07Nu x

Grminus

14

x

(a)

fw = 05 n = 05

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

024

026

028

030

032

ShxGrminus

14

x

(b)

Figure 6 Effect of Bi on (a) heat transfer rate and (b) mass transfer rate

005

006

007

008

009

010

Nu x

Grminus

14

x

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

(a)

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

01

02

03

04

05

06

ShxGrminus

14

x

(b)

Figure 7 Effect of 119891119908on (a) heat transfer rate and (b) mass transfer rate

and 119899 = 05 on local heat transfer coefficient is exhibitedin Figure 6 It is found from Figure 6 that local heat transferrate increases nonlinearly with the increase in Biot numberBiThe influence of the Biot number Bi on local mass transfercoefficient is shown in Figure 6 Figure 6 reveals that the localmass transfer coefficient is enhanced by the growth in theBiot number Bi In Figure 7 the effect of the suctioninjectionparameter 119891

119908with fixed Bi = 01 and 119899 = 05 on local heat

and mass transfer coefficients is displayed It is found fromFigure 7 that the local heat and mass transfer coefficients areless in the case of injection 119891

119908lt 0 in comparison with the

case of suction 119891119908

gt 0 Figure 8 is prepared to analyze theeffect of the material parameter 119899 with fixed Bi = 01 and

119891119908= 05 on local heat andmass transfer coefficients Figure 8

reveals that the local heat and mass transfer coefficients areenhanced by the increase in material parameter 119899 This isbecause when 119899 increases from 0 to 1 the microrotation termgets augmented and induces flow enhancement

The variations of 119862119891Gr14119909

and 119872119908Gr12119909

which areproportional to the coefficients of skin friction and wallcouple stress are shown in Table 3 with different values ofthe coupling number 119873 for fixed 119899 = 05 119891

119908= 05 and

Bi = 01 It indicates that the skin friction factor is higherfor micropolar fluid than the viscous fluids (119873 = 0) Sincemicropolar fluids offer a heavy resistance (resulting fromvortex viscosity) to fluid movement and cause larger skin

14 Advances in High Energy Physics

0084

0085

0086

0087Nu x

Grminus

14

x

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

fw = 05 Bi = 01

(a)

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

024

026

028

030

ShxGrminus

14

x

fw = 05 Bi = 01

(b)

Figure 8 Effect of material parameter 119899 on (a) heat transfer rate and (b) mass transfer rate

Table 3 Effects of skin friction and wall couple stress for varyingvalues of Biot numbers Bi micropolar parameter 119873 materialparameter 119899 and suctioninjection parameter 119891

119908

119873 Bi 119891119908

119899 119862119891Gr14119909

119872119908Gr12119909

01 01 05 05 2424227 06327903 01 05 05 2502716 063996605 01 05 05 2647987 063086307 01 05 05 2943561 060387909 01 05 05 3874631 055089105 01 05 05 2647987 063086305 10 05 05 3211755 080587905 50 05 05 3533141 090807805 200 05 05 3629791 093913705 01 minus05 05 2426265 033621905 01 00 05 2530014 046542905 01 10 05 2737553 082089905 01 20 05 2721244 119359605 01 05 00 2916655 minus028933905 01 05 05 2647987 063086305 01 05 10 2232556 2066191

friction factor compared to viscous fluid the results as wellsuggest larger values of coupling number 119873 and lower wallcouple stresses Since the skin friction coefficient 11989110158401015840(0) andwall couple stress coefficient as well as high temperature andmass transport rates are more depressed in the micropolarfluid comparing to the viscous fluid which may be beneficialin flow temperature and concentration control of polymerprocessing thus the presence of microscopic effects arisingfrom the local structure and of the fluid elements reduces thehigh temperature and mass transfer coefficients The effectof Bi on 119862

119891Gr14119909

and 119872119908Gr12119909

for 119891119908

= 05 119899 = 05 and

119873 = 05 is illustrated in Table 3 It can be noticed that theskin friction and wall couple stress coefficients are increasingwith increase of Bi for fixed values of other parametersThis notice is consistent with physical profiles presented inFigure 3 The effects of suctioninjection parameter on theskin friction andwall couple stress coefficients are also shownin Table 3 It is noted that the skin friction and wall couplestress coefficients are less with injection case whereas theyare more with suction case when compared to the case ofimpermeable surface Finally the detailed behavior of thematerial parameter 119899 is given in Table 3 The skin frictiondecreases and wall couple stress increases with increase ofmaterial parameter 119899

8 Conclusions

In this composition the similarity solution of the free convec-tion flow on a permeable vertical plate of a micropolar fluidunder the convective boundary condition is obtained usingLie group transformations Using the similarity variables thegoverning equations are transformed into a set of nondi-mensional parabolic equations These equations are solvednumerically using spectral quasi-linearisation method Thenumerical computation is carried out for various values ofnondimensional physical parameters The main findings aresummarized as follows

(i) The numerical results indicate that velocity distribu-tion is less near the plate but it is more far away fromthe plate the wall couple stress coefficient and rate ofheat andmass transfers are lower but the temperatureand concentration distributions and the skin frictioncoefficient are higher for the micropolar fluids incomparison with those of viscous fluids Also thereverse rotation ofmicrorotation near two boundariesis found with the increasing value of119873

Advances in High Energy Physics 15

(ii) An increase in Biot number Bi decreases concen-tration distribution whereas it causes an increasein temperature distribution skin friction and wallcouple stress coefficients and heat and mass transferrates within the boundary layer Further enhancingin Biot number Bi enhances velocity distributionnear the plate but shows the reverse behavior faraway from the plate We observe reverse rotationof microrotation near two boundaries within theboundary layer in the presence of Biot number

(iii) Less velocity temperature and concentration distri-butions are observed more skin friction and wallcouple stress coefficients and heat and mass transferrates in the case of suction compared to the case ofinjection Further microrotation decreases near thewall and depicts the opposite trend far away from thewall

(iv) It is found that microrotation temperature and con-centration distributions and skin friction coefficientsare more in the case of a micropolar fluid with strongconcentration (ie 119899 = 0) when compared to the caseof a micropolar fluid with weak concentration (ie119899 = 12) Further velocity is less in the case of 119899 = 0when compared to the case of 119899 = 12 near the walland shows the opposite trend far away from the wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their valuablesuggestions and comments

References

[1] J H Merkin ldquoNatural-convection boundary-layer flow on avertical surface with Newtonian heatingrdquo International Journalof Heat and Fluid Flow vol 15 no 5 pp 392ndash398 1994

[2] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009

[3] O DMakinde ldquoSimilarity solution for natural convection froma moving vertical plate with internal heat generation and aconvective boundary conditionrdquo Thermal Science vol 15 ppS137ndashS143 2011

[4] S M Ibrahim and N Bhashar Reddy ldquoSimilarity solution ofheat and mass transfer for natural convection over a movingvertical plate with internal heat generation and a convectiveboundary condition in the presence of thermal radiationviscous dissipation and chemical reactionrdquo ISRN Thermody-namics vol 2013 Article ID 790604 10 pages 2013

[5] C RamReddy P V S N Murthy A J Chamkha and A MRashad ldquoSoret effect on mixed convection flow in a nanofluidunder convective boundary conditionrdquo International Journal ofHeat and Mass Transfer vol 64 pp 384ndash392 2013

[6] H T Alkasasbeh M Z Salleh R M Tahar and R NazarldquoNumerical solutions of free convection boundary layer flow ona solid sphere with convective boundary conditionsrdquo Journal ofPhysics Conference Series vol 495 Article ID 012025 2014

[7] A Pantokratoras ldquoA note on natural convection along a con-vectively heated vertical platerdquo International Journal of ThermalSciences vol 76 pp 221ndash224 2014

[8] G Lukaszewicz Micropolar FluidsmdashTheory and ApplicationsBirkhaauser Basel Switzerland 1999

[9] V A Eremeyev L P Lebedev and H Altenbach Foundationsof Micropolar Mechanics Springer New York NY USA 2013

[10] G Ahmadi ldquoSelf-similar solution of imcompressible micropo-lar boundary layer flow over a semi-infinite platerdquo InternationalJournal of Engineering Science vol 14 no 7 pp 639ndash646 1976

[11] D A Rees and I Pop ldquoFree convection boundary-layer flowof a micropolar fluid from a vertical flat platerdquo IMA Journal ofApplied Mathematics vol 61 no 2 pp 179ndash197 1998

[12] D Srinivasacharya and C RamReddy ldquoEffect of double stratifi-cation onmixed convection in a micropolar fluidrdquoMatematikavol 28 no 2 pp 133ndash149 2012

[13] D Srinivasacharya and C Ramreddy ldquoMixed convection heatand mass transfer in a doubly stratified micropolar fluidrdquoComputationalThermal Sciences vol 5 no 4 pp 273ndash287 2013

[14] D Srinivasacharya and C RamReddy ldquoNatural convection heatand mass transfer in a micropolar fluid with thermal and massstratificationrdquo International Journal for Computational Methodsin Engineering Science andMechanics vol 14 no 5 pp 401ndash4132013

[15] N A Yacob and A Ishak ldquoStagnation point flow towardsa stretchingshrinking sheet in a micropolar fluid with aconvective surface boundary conditionrdquoThe Canadian Journalof Chemical Engineering vol 90 no 3 pp 621ndash626 2012

[16] K Zaimi and A Ishak ldquoStagnation-point flow and heat transferover a nonlinearly stretchingshrinking sheet in a micropolarfluidrdquo Abstract and Applied Analysis vol 2014 Article ID261630 6 pages 2014

[17] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[18] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[19] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer 2009

[20] T Tapanidis G Tsagas and H P Mazumdar ldquoApplication ofscaling group of transformations to viscoelastic second-gradefluid flowrdquo Nonlinear Functional Analysis and Applications vol8 no 3 pp 345ndash350 2003

[21] I A A Hassanien andM A Hamad ldquoGroup theoretic methodfor unsteady free convection flow of a micropolar fluid along avertical plate in a thermally stratified mediumrdquo Applied Mathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 32 no 6 pp 1099ndash1114 2008

[22] R Kandasamy K Gunasekaran and S B H Hasan ldquoScalinggroup transformation on fluid flow with variable stream con-ditionsrdquo International Journal of Non-Linear Mechanics vol 46no 7 pp 976ndash985 2011

[23] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat TransfermdashAsian Research vol41 no 3 pp 241ndash259 2012

16 Advances in High Energy Physics

[24] A A Mutlag M J Uddin M A A Hamad and A I M IsmailldquoHeat transfer analysis for falkner-skan boundary layer flowpast a stationary wedge with slips boundary conditions con-sidering temperature-dependent thermal conductivityrdquo SainsMalaysiana vol 42 no 6 pp 855ndash862 2013

[25] S C Cowin ldquoPolar fluidsrdquo Physics of Fluids vol 11 no 9 pp1919ndash1927 1968

[26] M A Seddeek M Y Akl and A M Al-Hanaya ldquoThermalradiation effects on mixed convection and mass transfer flowon vertical porous plate with heat generation and chemicalreaction by using scaling grouprdquo Journal of Natural Sciences andMathematics vol 4 pp 41ndash60 2010

[27] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[28] M J Uddin W A Khan and A I M Ismail ldquoMHD freeconvective boundary layer flow of a nanofluid past a flat verticalplate with Newtonian heating boundary conditionrdquo PLoS ONEvol 7 no 11 Article ID e49499 2012

[29] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012

[30] M J Uddin W A Khan and A I M Ismail ldquoScalinggroup transformation for MHD boundary layer slip flow of ananofluid over a convectively heated stretching sheet with heatgenerationrdquo Mathematical Problems in Engineering vol 2012Article ID 934964 20 pages 2012

[31] MM RashidiM Ferdows A B Parsa and S Abelman ldquoMHDnatural convection with convective surface boundary conditionover a at platerdquo Abstract and Applied Analysis vol 2014 ArticleID 923487 10 pages 2014

[32] R E Bellman and R E Kalaba Quasilinearisation and Non-Linear Boundary-Value Problems Elsevier New York NY USA1965

[33] S SMotsa P GDlamini andMKhumalo ldquoSpectral relaxationmethod and spectral quasilinearization method for solvingunsteady boundary layer flow problemsrdquo Advances in Mathe-matical Physics vol 2014 Article ID 341964 12 pages 2014

[34] S S Motsa P Sibanda J M Ngnotchouye and G T MarewoldquoA spectral relaxation approach for unsteady boundary-layerflow and heat transfer of a nanofluid over a permeable stretch-ingshrinking sheetrdquo Advances in Mathematical Physics vol2014 Article ID 564942 10 pages 2014

[35] C Canuto M Y Hussaini A Quarteroni and T ZangSpectral Methods Fundamentals in Single Domains ScientificComputation Springer Berlin Germany 2006

[36] J HMerkin ldquoFree convection boundary layer on an isothermalhorizontal cylinderrdquo in Proceedings of the American Societyof Mechanical Engineers and American Institute of ChemicalEngineersHeat Transfer Conference p 911 ASME St LouisMoUSA August 1976

[37] R Nazar N Amin and I Pop ldquoFree convection boundary layeron an isothermal horizontal circular cylinder in a micropolarfluidrdquo in Proceedings of the 12th International Heat TransferConference vol 2 pp 525ndash530 Elsevier Paris France August2002

[38] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

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AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Advances in High Energy Physics

119873

sum

119896=0D119873119896

120579119903+1 (120591119896) minusBi 120579

119903+1 (120591119873) = minusBi

120579119903+1 (1205910) = 0

120601119903+1 (120591119873) = 1120601119903+1 (1205910) = 0

(36)

In (35) 119860 is a (4119873+ 4) times (4119873+ 4) square matrix and119883 and 119877are (4119873 + 1) times 1 column vectors defined by

119860 =

[

[

[

[

[

[

11986011 11986012 11986013 11986014

11986021 11986022 11986023 11986024

11986031 11986032 11986033 11986034

11986041 11986042 11986043 11986044

]

]

]

]

]

]

119883 =

[

[

[

[

[

[

F119903+1

G119903+1

Θ119903+1

Φ119903+1

]

]

]

]

]

]

119877 =

[

[

[

[

[

[

R1

R2

R3

R4

]

]

]

]

]

]

(37)

whereF = [119891

119903+1 (1205910) 119891119903+1 (1205911) 119891119903+1 (120591119873)]119879

G = [119892119903+1 (1205910) 119892119903+1 (1205911) 119892119903+1 (120591119873)]

119879

Θ = [120579119903+1 (1205910) 120579119903+1 (1205911) 120579119903+1 (120591119873)]

119879

Φ = [120601119903+1 (1205910) 120601119903+1 (1205911) 120601119903+1 (120591119873)]

119879

11986011 = (

11 minus 119873

)D3+ diag [1198861119903]D

2+ diag [1198862119903]D

+ diag [1198863119903]

11986012 = (

119873

1 minus 119873

)D

11986013 = I

11986014 = BI

11986021 = minus(

119873

1 minus 119873

)D2+ diag [1198871119903]D+ diag [1198872119903]

11986022 = (

2 minus 119873

2 minus 2119873)D2

+ diag [1198873119903]D+ diag [1198874119903]

11986023 = 0

11986024 = 0

11986031 = diag [1198881119903]

11986032 = 0

11986033 =1Pr

D2+ diag [1198882119903]D

Table 1 Comparison of minus1205791015840(0) for free convection along a verticalflat plate in Newtonian fluid when 119873 = 0 119899 = 0 B = 0 Pr = 1Bi rarr infin and 119891

119908= 0

Merkin [36] Nazar et al [37] Molla et al [38] Present04214 04214 04214 04214313

11986034 = 0

11986041 = diag [1198891119903]

11986042 = 0

11986043 = 0

11986044 =1Sc

D2+ diag [1198892119903]D

R1 = 1198771119903

R2 = 1198772119903

R3 = 1198773119903

R4 = 1198774119903

(38)

and here I is an identity matrix the size of the matrix 0 is(119873+1)times1 and diag[ ] is a diagonalmatrix of size (119873+1)times(119873+

1) Subscript r denotes the iteration number After modifyingmatrix system (35) to incorporate boundary condition (36)the solution is obtained as

119883 = 119860minus1119877 (39)

7 Results and Discussions

It is noticed that the present problem reduces to free convec-tion heat transfer along an impermeable vertical plate in amicropolar fluid without the convective boundary conditionwhen119891

119908= 0 Bi rarr infin andB = 0 Also in the limit as119873 rarr

0 governing equations (2)ndash(5) reduce to the correspondingequations for a free convection heat and mass transfer ina viscous fluid In order to validate the code generated theresults of the present problem have been compared with theresults obtained by Merkin [36] Nazar et al [37] and Mollaet al [38] as a special case by taking 119873 = 0 119899 = 0 B = 0Pr = 1 Bi rarr infin and 119891

119908= 0 and it was found that

they are in good agreement as presented in Table 1 Also thecomparison of heat transfer coefficient has been made withthe results obtained by Nazar et al [37] as shown in Table 2when 119899 = 05 B = 0 Pr = 1 Bi rarr infin and 119891

119908= 0

To study the effects of coupling number119873 suctioninjectionparameter 119891

119908 Biot number Bi and material parameter 119899

computations were carried out in the cases of B = 10Pr = 071 and Sc = 022

The effects of the coupling number119873 on the dimension-less velocity microrotation temperature and concentrationare illustrated in Figures 2(a)ndash2(d) with fixed values ofother parameters The coupling number 119873 characterizes thecoupling of linear and rotational motion arising from the

Advances in High Energy Physics 9

00

01

02

03

04

05

06f998400

0 4 8 12 16

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(a)

g

minus06

minus05

minus04

minus03

minus02

minus01

00

01

0 2 4 6 8 10 12

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(b)

000

004

008

012

016

120579

0 1 2 3 4 5 6

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(c)

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

120601

00

02

04

06

08

10

120578

0 2 4 6 8 10 12

Bi = 01 fw = 05

(d)

Figure 2 Effect of119873 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Table 2 Comparison of minus1205791015840

(0) for free convection flow in amicropolar fluid obtained by Nazar et al [37] when 119899 = 05B = 0Pr = 1 Bi rarr infin and 119891

119908= 0

119873 Nazar et al [37] Present000 04214 04214033 03991 03990050 03834 03834060 03709 03709066 03608 03608071 03522 03522075 03447 03447

fluid particles In the case of 119873 = 0 (ie as 120581 tends tozero) the micropolarity is absent and fluid becomes nonpolarfluid With a large value of 119873 effect of microstructurebecomes significant whereas with a diminished value of 119873the individuality of the substructure is much less articulatedAs 119873 increases it is found from Figure 2(a) that the max-imum velocity decreases in amplitude and the location ofthe maximum velocity moves farther away from the wallSince 119873 rarr 0 corresponds to viscous fluid the velocity incase of a micropolar fluid has been less compared to thatof viscous fluid case It can be observed from Figure 2(b)that as 119873 increases initially microrotation profiles tend tobecome flatter and then approach their free stream valuesfar away from the wall This happens due to the vanishing

10 Advances in High Energy Physics

00

01

02

03

04

05

06

f998400

120578

0 2 4 6 8 10 12

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(a)

minus06

minus04

minus03

minus02

minus05

minus01

00

01

g

0 2 4 6 8 10

120578

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(b)

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

120578

0 1 2 3 4 5

120579

00

02

04

06

08

10

(c)

120601

00

02

04

06

08

10

fw = 05 n = 05N = 05

0 2 4 6 8 10 12

120578

Bi = 01

Bi = 10

Bi = 50Bi = 200

(d)

Figure 3 Effect of Bi on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

of antisymmetric part of the stress on the boundary thatcorresponds to a weak concentration of microelements Thisis because an increment in the value of 119873 implies a highervortex viscosity of fluid which promotes the microrotationof micropolar fluids It is seen from Figures 2(c) and 2(d)that thermal and concentration boundary layers of thefluid increase with increase in coupling number 119873 Hencetemperature and concentration in case of micropolar fluidsare more than those of the viscous fluid case

The Biot number Bi is the ratio of internal thermalresistance of a solid to boundary layer thermal resistanceWhen Bi = 0 the plate is totally insulated internal thermalresistance of the plate is extremely high and no convectiveheat transfer to the cold fluid on the upper part of the

plate takes place Figure 3(a) depicts fluid velocity profilesfor different values of the Biot number with 119873 = 05 119891

119908=

05 and 119899 = 05 Generally fluid velocity is zero at platesurface and increases gradually away fromplate to free streamvalue satisfying boundary conditions It is interesting to notethat an increase in the intensity of convective surface heattransfer Bi produces significant enhancement in fluid velocitywithin the momentum boundary layer In Figure 3(b) webring out the behavior of microrotation with different valuesof Biot number Bi for fixed values of other parameters As theparameter value Bi increases microrotation showing reverserotation near the two boundaries Hence the condition ofvanishing of antisymmetric part of the stress on the boundaryresults in a drastic change of the microrotation profiles

Advances in High Energy Physics 11

07

06

05

04

03

02

01

00

f998400

fw = 00

fw = 10

fw = 20

0 3 6 9 12 15

120578

N = 05 Bi = 01 n = 05

fw = minus05

(a)

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

120578

0 2 4 6 8 10 12

g

minus04

minus03

minus02

minus01

00

01

fw = minus05

(b)

0 1 2 3 4 5 6

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

000

005

010

015

020

025

030

120579

fw = minus05

(c)

0 3 6 9 12

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

00

02

04

06

08

10

120601

fw = minus05

(d)

Figure 4 Effect of 119891119908on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Given that convective heating increases with Biot numberBi rarr infin simulates the isothermal surface which is clearlyseen from Figure 3(c) where 120579(0) = 1 as Bi rarr infin Infact a high Biot number indicates higher internal thermalresistance of the plate than boundary layer thermal resistanceIn this fluid temperature is maximum at the plate surfaceand decreases exponentially to zero value far out from theplate satisfying boundary conditions As a consequence anincrement in the Biot number leads to increase of fluidtemperature efficiency Figure 3(d) illustrates the variation ofdimensionless concentration for different values of Bi It isclear that the concentration of fluid decreases with increaseof Bi

The effect of 119891119908

on velocity profile is depicted inFigure 4(a) Here 119891

119908gt 0 represents suction and 119891

119908lt 0

denotes injectionThe lower velocity is noticed in case of suc-tion when compared to case of injection From Figure 4(b)we note that microrotation is showing reverse rotation neartwo boundaries with both suction and injection parame-ter The dimensionless temperature for different values ofsuctioninjection parameters is drawn in Figure 4(c) It isreadable that the temperature of the fluid is more in case ofinjection whereas it is less in case of suction in comparisonwith the impermeable surface case (119891

119908= 0) Figure 4(d)

demonstrates dimensionless concentration for different val-ues of suctioninjection parameters It is determined that the

12 Advances in High Energy Physics

00

01

02

03

04

05

06f998400

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(a)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10

120578

minus10

minus08

minus06

minus04

minus02

00

g

(b)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

120578

0 1 2 3 4 5

120579

000

003

006

009

012

015

(c)

00

02

04

06

08

10

120601

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(d)

Figure 5 Effect of material parameter 119899 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

concentration of fluid ismore with injection whereas it is lesswith suctionwhen compared to the impermeable surface case(119891119908= 0) As a finale the thermal and solutal boundary layer

thicknesses increase in case of injection compared to case ofsuction as displayed in Figures 4(c) and 4(d)

In Figures 5(a)ndash5(d) the effect of material parameter119899 on dimensionless velocity microrotation temperatureand concentration is presented for fixed values of otherparameters since the material parameter 119899 signifies themicrorotation effects (ie for 119899 = 0 particles are not freeto revolve near the surface whereas as 119899 increases from 0 to1 the microrotation term gets augmented and induces flowenhancement) As 119899 increases it is found from Figure 5(a)that the minimum velocity increases in amplitude and the

location of the minimum velocity moves farther away fromthe wall From Figure 5(b) we observe that themicrorotationis decreasing with increasing value of material parameter119899 within the boundary layer It is clear from Figures 5(c)-5(d) that the thermal and solutal boundary layer thicknessesdecrease with increase of material parameter 119899

The variations of minus1205791015840

(0) and minus1206011015840

(0) versus couplingnumber 119873 are shown in Figures 6ndash8 It can be noticed fromthese figures that the heat and mass transfer coefficientsare less in case of micropolar fluids when compared to theviscous fluids This is because as 119873 increases the thermaland solutal boundary layer thicknesses become larger thusgiving rise to a small value of local heat and mass transferrates The effect of the Biot number Bi with fixed 119891

119908= 05

Advances in High Energy Physics 13

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

fw = 05 n = 05

01

00

02

03

04

05

06

07Nu x

Grminus

14

x

(a)

fw = 05 n = 05

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

024

026

028

030

032

ShxGrminus

14

x

(b)

Figure 6 Effect of Bi on (a) heat transfer rate and (b) mass transfer rate

005

006

007

008

009

010

Nu x

Grminus

14

x

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

(a)

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

01

02

03

04

05

06

ShxGrminus

14

x

(b)

Figure 7 Effect of 119891119908on (a) heat transfer rate and (b) mass transfer rate

and 119899 = 05 on local heat transfer coefficient is exhibitedin Figure 6 It is found from Figure 6 that local heat transferrate increases nonlinearly with the increase in Biot numberBiThe influence of the Biot number Bi on local mass transfercoefficient is shown in Figure 6 Figure 6 reveals that the localmass transfer coefficient is enhanced by the growth in theBiot number Bi In Figure 7 the effect of the suctioninjectionparameter 119891

119908with fixed Bi = 01 and 119899 = 05 on local heat

and mass transfer coefficients is displayed It is found fromFigure 7 that the local heat and mass transfer coefficients areless in the case of injection 119891

119908lt 0 in comparison with the

case of suction 119891119908

gt 0 Figure 8 is prepared to analyze theeffect of the material parameter 119899 with fixed Bi = 01 and

119891119908= 05 on local heat andmass transfer coefficients Figure 8

reveals that the local heat and mass transfer coefficients areenhanced by the increase in material parameter 119899 This isbecause when 119899 increases from 0 to 1 the microrotation termgets augmented and induces flow enhancement

The variations of 119862119891Gr14119909

and 119872119908Gr12119909

which areproportional to the coefficients of skin friction and wallcouple stress are shown in Table 3 with different values ofthe coupling number 119873 for fixed 119899 = 05 119891

119908= 05 and

Bi = 01 It indicates that the skin friction factor is higherfor micropolar fluid than the viscous fluids (119873 = 0) Sincemicropolar fluids offer a heavy resistance (resulting fromvortex viscosity) to fluid movement and cause larger skin

14 Advances in High Energy Physics

0084

0085

0086

0087Nu x

Grminus

14

x

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

fw = 05 Bi = 01

(a)

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

024

026

028

030

ShxGrminus

14

x

fw = 05 Bi = 01

(b)

Figure 8 Effect of material parameter 119899 on (a) heat transfer rate and (b) mass transfer rate

Table 3 Effects of skin friction and wall couple stress for varyingvalues of Biot numbers Bi micropolar parameter 119873 materialparameter 119899 and suctioninjection parameter 119891

119908

119873 Bi 119891119908

119899 119862119891Gr14119909

119872119908Gr12119909

01 01 05 05 2424227 06327903 01 05 05 2502716 063996605 01 05 05 2647987 063086307 01 05 05 2943561 060387909 01 05 05 3874631 055089105 01 05 05 2647987 063086305 10 05 05 3211755 080587905 50 05 05 3533141 090807805 200 05 05 3629791 093913705 01 minus05 05 2426265 033621905 01 00 05 2530014 046542905 01 10 05 2737553 082089905 01 20 05 2721244 119359605 01 05 00 2916655 minus028933905 01 05 05 2647987 063086305 01 05 10 2232556 2066191

friction factor compared to viscous fluid the results as wellsuggest larger values of coupling number 119873 and lower wallcouple stresses Since the skin friction coefficient 11989110158401015840(0) andwall couple stress coefficient as well as high temperature andmass transport rates are more depressed in the micropolarfluid comparing to the viscous fluid which may be beneficialin flow temperature and concentration control of polymerprocessing thus the presence of microscopic effects arisingfrom the local structure and of the fluid elements reduces thehigh temperature and mass transfer coefficients The effectof Bi on 119862

119891Gr14119909

and 119872119908Gr12119909

for 119891119908

= 05 119899 = 05 and

119873 = 05 is illustrated in Table 3 It can be noticed that theskin friction and wall couple stress coefficients are increasingwith increase of Bi for fixed values of other parametersThis notice is consistent with physical profiles presented inFigure 3 The effects of suctioninjection parameter on theskin friction andwall couple stress coefficients are also shownin Table 3 It is noted that the skin friction and wall couplestress coefficients are less with injection case whereas theyare more with suction case when compared to the case ofimpermeable surface Finally the detailed behavior of thematerial parameter 119899 is given in Table 3 The skin frictiondecreases and wall couple stress increases with increase ofmaterial parameter 119899

8 Conclusions

In this composition the similarity solution of the free convec-tion flow on a permeable vertical plate of a micropolar fluidunder the convective boundary condition is obtained usingLie group transformations Using the similarity variables thegoverning equations are transformed into a set of nondi-mensional parabolic equations These equations are solvednumerically using spectral quasi-linearisation method Thenumerical computation is carried out for various values ofnondimensional physical parameters The main findings aresummarized as follows

(i) The numerical results indicate that velocity distribu-tion is less near the plate but it is more far away fromthe plate the wall couple stress coefficient and rate ofheat andmass transfers are lower but the temperatureand concentration distributions and the skin frictioncoefficient are higher for the micropolar fluids incomparison with those of viscous fluids Also thereverse rotation ofmicrorotation near two boundariesis found with the increasing value of119873

Advances in High Energy Physics 15

(ii) An increase in Biot number Bi decreases concen-tration distribution whereas it causes an increasein temperature distribution skin friction and wallcouple stress coefficients and heat and mass transferrates within the boundary layer Further enhancingin Biot number Bi enhances velocity distributionnear the plate but shows the reverse behavior faraway from the plate We observe reverse rotationof microrotation near two boundaries within theboundary layer in the presence of Biot number

(iii) Less velocity temperature and concentration distri-butions are observed more skin friction and wallcouple stress coefficients and heat and mass transferrates in the case of suction compared to the case ofinjection Further microrotation decreases near thewall and depicts the opposite trend far away from thewall

(iv) It is found that microrotation temperature and con-centration distributions and skin friction coefficientsare more in the case of a micropolar fluid with strongconcentration (ie 119899 = 0) when compared to the caseof a micropolar fluid with weak concentration (ie119899 = 12) Further velocity is less in the case of 119899 = 0when compared to the case of 119899 = 12 near the walland shows the opposite trend far away from the wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their valuablesuggestions and comments

References

[1] J H Merkin ldquoNatural-convection boundary-layer flow on avertical surface with Newtonian heatingrdquo International Journalof Heat and Fluid Flow vol 15 no 5 pp 392ndash398 1994

[2] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009

[3] O DMakinde ldquoSimilarity solution for natural convection froma moving vertical plate with internal heat generation and aconvective boundary conditionrdquo Thermal Science vol 15 ppS137ndashS143 2011

[4] S M Ibrahim and N Bhashar Reddy ldquoSimilarity solution ofheat and mass transfer for natural convection over a movingvertical plate with internal heat generation and a convectiveboundary condition in the presence of thermal radiationviscous dissipation and chemical reactionrdquo ISRN Thermody-namics vol 2013 Article ID 790604 10 pages 2013

[5] C RamReddy P V S N Murthy A J Chamkha and A MRashad ldquoSoret effect on mixed convection flow in a nanofluidunder convective boundary conditionrdquo International Journal ofHeat and Mass Transfer vol 64 pp 384ndash392 2013

[6] H T Alkasasbeh M Z Salleh R M Tahar and R NazarldquoNumerical solutions of free convection boundary layer flow ona solid sphere with convective boundary conditionsrdquo Journal ofPhysics Conference Series vol 495 Article ID 012025 2014

[7] A Pantokratoras ldquoA note on natural convection along a con-vectively heated vertical platerdquo International Journal of ThermalSciences vol 76 pp 221ndash224 2014

[8] G Lukaszewicz Micropolar FluidsmdashTheory and ApplicationsBirkhaauser Basel Switzerland 1999

[9] V A Eremeyev L P Lebedev and H Altenbach Foundationsof Micropolar Mechanics Springer New York NY USA 2013

[10] G Ahmadi ldquoSelf-similar solution of imcompressible micropo-lar boundary layer flow over a semi-infinite platerdquo InternationalJournal of Engineering Science vol 14 no 7 pp 639ndash646 1976

[11] D A Rees and I Pop ldquoFree convection boundary-layer flowof a micropolar fluid from a vertical flat platerdquo IMA Journal ofApplied Mathematics vol 61 no 2 pp 179ndash197 1998

[12] D Srinivasacharya and C RamReddy ldquoEffect of double stratifi-cation onmixed convection in a micropolar fluidrdquoMatematikavol 28 no 2 pp 133ndash149 2012

[13] D Srinivasacharya and C Ramreddy ldquoMixed convection heatand mass transfer in a doubly stratified micropolar fluidrdquoComputationalThermal Sciences vol 5 no 4 pp 273ndash287 2013

[14] D Srinivasacharya and C RamReddy ldquoNatural convection heatand mass transfer in a micropolar fluid with thermal and massstratificationrdquo International Journal for Computational Methodsin Engineering Science andMechanics vol 14 no 5 pp 401ndash4132013

[15] N A Yacob and A Ishak ldquoStagnation point flow towardsa stretchingshrinking sheet in a micropolar fluid with aconvective surface boundary conditionrdquoThe Canadian Journalof Chemical Engineering vol 90 no 3 pp 621ndash626 2012

[16] K Zaimi and A Ishak ldquoStagnation-point flow and heat transferover a nonlinearly stretchingshrinking sheet in a micropolarfluidrdquo Abstract and Applied Analysis vol 2014 Article ID261630 6 pages 2014

[17] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[18] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[19] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer 2009

[20] T Tapanidis G Tsagas and H P Mazumdar ldquoApplication ofscaling group of transformations to viscoelastic second-gradefluid flowrdquo Nonlinear Functional Analysis and Applications vol8 no 3 pp 345ndash350 2003

[21] I A A Hassanien andM A Hamad ldquoGroup theoretic methodfor unsteady free convection flow of a micropolar fluid along avertical plate in a thermally stratified mediumrdquo Applied Mathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 32 no 6 pp 1099ndash1114 2008

[22] R Kandasamy K Gunasekaran and S B H Hasan ldquoScalinggroup transformation on fluid flow with variable stream con-ditionsrdquo International Journal of Non-Linear Mechanics vol 46no 7 pp 976ndash985 2011

[23] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat TransfermdashAsian Research vol41 no 3 pp 241ndash259 2012

16 Advances in High Energy Physics

[24] A A Mutlag M J Uddin M A A Hamad and A I M IsmailldquoHeat transfer analysis for falkner-skan boundary layer flowpast a stationary wedge with slips boundary conditions con-sidering temperature-dependent thermal conductivityrdquo SainsMalaysiana vol 42 no 6 pp 855ndash862 2013

[25] S C Cowin ldquoPolar fluidsrdquo Physics of Fluids vol 11 no 9 pp1919ndash1927 1968

[26] M A Seddeek M Y Akl and A M Al-Hanaya ldquoThermalradiation effects on mixed convection and mass transfer flowon vertical porous plate with heat generation and chemicalreaction by using scaling grouprdquo Journal of Natural Sciences andMathematics vol 4 pp 41ndash60 2010

[27] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[28] M J Uddin W A Khan and A I M Ismail ldquoMHD freeconvective boundary layer flow of a nanofluid past a flat verticalplate with Newtonian heating boundary conditionrdquo PLoS ONEvol 7 no 11 Article ID e49499 2012

[29] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012

[30] M J Uddin W A Khan and A I M Ismail ldquoScalinggroup transformation for MHD boundary layer slip flow of ananofluid over a convectively heated stretching sheet with heatgenerationrdquo Mathematical Problems in Engineering vol 2012Article ID 934964 20 pages 2012

[31] MM RashidiM Ferdows A B Parsa and S Abelman ldquoMHDnatural convection with convective surface boundary conditionover a at platerdquo Abstract and Applied Analysis vol 2014 ArticleID 923487 10 pages 2014

[32] R E Bellman and R E Kalaba Quasilinearisation and Non-Linear Boundary-Value Problems Elsevier New York NY USA1965

[33] S SMotsa P GDlamini andMKhumalo ldquoSpectral relaxationmethod and spectral quasilinearization method for solvingunsteady boundary layer flow problemsrdquo Advances in Mathe-matical Physics vol 2014 Article ID 341964 12 pages 2014

[34] S S Motsa P Sibanda J M Ngnotchouye and G T MarewoldquoA spectral relaxation approach for unsteady boundary-layerflow and heat transfer of a nanofluid over a permeable stretch-ingshrinking sheetrdquo Advances in Mathematical Physics vol2014 Article ID 564942 10 pages 2014

[35] C Canuto M Y Hussaini A Quarteroni and T ZangSpectral Methods Fundamentals in Single Domains ScientificComputation Springer Berlin Germany 2006

[36] J HMerkin ldquoFree convection boundary layer on an isothermalhorizontal cylinderrdquo in Proceedings of the American Societyof Mechanical Engineers and American Institute of ChemicalEngineersHeat Transfer Conference p 911 ASME St LouisMoUSA August 1976

[37] R Nazar N Amin and I Pop ldquoFree convection boundary layeron an isothermal horizontal circular cylinder in a micropolarfluidrdquo in Proceedings of the 12th International Heat TransferConference vol 2 pp 525ndash530 Elsevier Paris France August2002

[38] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

Submit your manuscripts athttpwwwhindawicom

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ThermodynamicsJournal of

Advances in High Energy Physics 9

00

01

02

03

04

05

06f998400

0 4 8 12 16

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(a)

g

minus06

minus05

minus04

minus03

minus02

minus01

00

01

0 2 4 6 8 10 12

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(b)

000

004

008

012

016

120579

0 1 2 3 4 5 6

120578

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

Bi = 01 fw = 05

(c)

n = 00N = 00

n = 05N = 01

n = 05N = 03

n = 05N = 06

n = 05N = 09

120601

00

02

04

06

08

10

120578

0 2 4 6 8 10 12

Bi = 01 fw = 05

(d)

Figure 2 Effect of119873 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Table 2 Comparison of minus1205791015840

(0) for free convection flow in amicropolar fluid obtained by Nazar et al [37] when 119899 = 05B = 0Pr = 1 Bi rarr infin and 119891

119908= 0

119873 Nazar et al [37] Present000 04214 04214033 03991 03990050 03834 03834060 03709 03709066 03608 03608071 03522 03522075 03447 03447

fluid particles In the case of 119873 = 0 (ie as 120581 tends tozero) the micropolarity is absent and fluid becomes nonpolarfluid With a large value of 119873 effect of microstructurebecomes significant whereas with a diminished value of 119873the individuality of the substructure is much less articulatedAs 119873 increases it is found from Figure 2(a) that the max-imum velocity decreases in amplitude and the location ofthe maximum velocity moves farther away from the wallSince 119873 rarr 0 corresponds to viscous fluid the velocity incase of a micropolar fluid has been less compared to thatof viscous fluid case It can be observed from Figure 2(b)that as 119873 increases initially microrotation profiles tend tobecome flatter and then approach their free stream valuesfar away from the wall This happens due to the vanishing

10 Advances in High Energy Physics

00

01

02

03

04

05

06

f998400

120578

0 2 4 6 8 10 12

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(a)

minus06

minus04

minus03

minus02

minus05

minus01

00

01

g

0 2 4 6 8 10

120578

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(b)

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

120578

0 1 2 3 4 5

120579

00

02

04

06

08

10

(c)

120601

00

02

04

06

08

10

fw = 05 n = 05N = 05

0 2 4 6 8 10 12

120578

Bi = 01

Bi = 10

Bi = 50Bi = 200

(d)

Figure 3 Effect of Bi on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

of antisymmetric part of the stress on the boundary thatcorresponds to a weak concentration of microelements Thisis because an increment in the value of 119873 implies a highervortex viscosity of fluid which promotes the microrotationof micropolar fluids It is seen from Figures 2(c) and 2(d)that thermal and concentration boundary layers of thefluid increase with increase in coupling number 119873 Hencetemperature and concentration in case of micropolar fluidsare more than those of the viscous fluid case

The Biot number Bi is the ratio of internal thermalresistance of a solid to boundary layer thermal resistanceWhen Bi = 0 the plate is totally insulated internal thermalresistance of the plate is extremely high and no convectiveheat transfer to the cold fluid on the upper part of the

plate takes place Figure 3(a) depicts fluid velocity profilesfor different values of the Biot number with 119873 = 05 119891

119908=

05 and 119899 = 05 Generally fluid velocity is zero at platesurface and increases gradually away fromplate to free streamvalue satisfying boundary conditions It is interesting to notethat an increase in the intensity of convective surface heattransfer Bi produces significant enhancement in fluid velocitywithin the momentum boundary layer In Figure 3(b) webring out the behavior of microrotation with different valuesof Biot number Bi for fixed values of other parameters As theparameter value Bi increases microrotation showing reverserotation near the two boundaries Hence the condition ofvanishing of antisymmetric part of the stress on the boundaryresults in a drastic change of the microrotation profiles

Advances in High Energy Physics 11

07

06

05

04

03

02

01

00

f998400

fw = 00

fw = 10

fw = 20

0 3 6 9 12 15

120578

N = 05 Bi = 01 n = 05

fw = minus05

(a)

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

120578

0 2 4 6 8 10 12

g

minus04

minus03

minus02

minus01

00

01

fw = minus05

(b)

0 1 2 3 4 5 6

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

000

005

010

015

020

025

030

120579

fw = minus05

(c)

0 3 6 9 12

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

00

02

04

06

08

10

120601

fw = minus05

(d)

Figure 4 Effect of 119891119908on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Given that convective heating increases with Biot numberBi rarr infin simulates the isothermal surface which is clearlyseen from Figure 3(c) where 120579(0) = 1 as Bi rarr infin Infact a high Biot number indicates higher internal thermalresistance of the plate than boundary layer thermal resistanceIn this fluid temperature is maximum at the plate surfaceand decreases exponentially to zero value far out from theplate satisfying boundary conditions As a consequence anincrement in the Biot number leads to increase of fluidtemperature efficiency Figure 3(d) illustrates the variation ofdimensionless concentration for different values of Bi It isclear that the concentration of fluid decreases with increaseof Bi

The effect of 119891119908

on velocity profile is depicted inFigure 4(a) Here 119891

119908gt 0 represents suction and 119891

119908lt 0

denotes injectionThe lower velocity is noticed in case of suc-tion when compared to case of injection From Figure 4(b)we note that microrotation is showing reverse rotation neartwo boundaries with both suction and injection parame-ter The dimensionless temperature for different values ofsuctioninjection parameters is drawn in Figure 4(c) It isreadable that the temperature of the fluid is more in case ofinjection whereas it is less in case of suction in comparisonwith the impermeable surface case (119891

119908= 0) Figure 4(d)

demonstrates dimensionless concentration for different val-ues of suctioninjection parameters It is determined that the

12 Advances in High Energy Physics

00

01

02

03

04

05

06f998400

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(a)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10

120578

minus10

minus08

minus06

minus04

minus02

00

g

(b)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

120578

0 1 2 3 4 5

120579

000

003

006

009

012

015

(c)

00

02

04

06

08

10

120601

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(d)

Figure 5 Effect of material parameter 119899 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

concentration of fluid ismore with injection whereas it is lesswith suctionwhen compared to the impermeable surface case(119891119908= 0) As a finale the thermal and solutal boundary layer

thicknesses increase in case of injection compared to case ofsuction as displayed in Figures 4(c) and 4(d)

In Figures 5(a)ndash5(d) the effect of material parameter119899 on dimensionless velocity microrotation temperatureand concentration is presented for fixed values of otherparameters since the material parameter 119899 signifies themicrorotation effects (ie for 119899 = 0 particles are not freeto revolve near the surface whereas as 119899 increases from 0 to1 the microrotation term gets augmented and induces flowenhancement) As 119899 increases it is found from Figure 5(a)that the minimum velocity increases in amplitude and the

location of the minimum velocity moves farther away fromthe wall From Figure 5(b) we observe that themicrorotationis decreasing with increasing value of material parameter119899 within the boundary layer It is clear from Figures 5(c)-5(d) that the thermal and solutal boundary layer thicknessesdecrease with increase of material parameter 119899

The variations of minus1205791015840

(0) and minus1206011015840

(0) versus couplingnumber 119873 are shown in Figures 6ndash8 It can be noticed fromthese figures that the heat and mass transfer coefficientsare less in case of micropolar fluids when compared to theviscous fluids This is because as 119873 increases the thermaland solutal boundary layer thicknesses become larger thusgiving rise to a small value of local heat and mass transferrates The effect of the Biot number Bi with fixed 119891

119908= 05

Advances in High Energy Physics 13

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

fw = 05 n = 05

01

00

02

03

04

05

06

07Nu x

Grminus

14

x

(a)

fw = 05 n = 05

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

024

026

028

030

032

ShxGrminus

14

x

(b)

Figure 6 Effect of Bi on (a) heat transfer rate and (b) mass transfer rate

005

006

007

008

009

010

Nu x

Grminus

14

x

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

(a)

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

01

02

03

04

05

06

ShxGrminus

14

x

(b)

Figure 7 Effect of 119891119908on (a) heat transfer rate and (b) mass transfer rate

and 119899 = 05 on local heat transfer coefficient is exhibitedin Figure 6 It is found from Figure 6 that local heat transferrate increases nonlinearly with the increase in Biot numberBiThe influence of the Biot number Bi on local mass transfercoefficient is shown in Figure 6 Figure 6 reveals that the localmass transfer coefficient is enhanced by the growth in theBiot number Bi In Figure 7 the effect of the suctioninjectionparameter 119891

119908with fixed Bi = 01 and 119899 = 05 on local heat

and mass transfer coefficients is displayed It is found fromFigure 7 that the local heat and mass transfer coefficients areless in the case of injection 119891

119908lt 0 in comparison with the

case of suction 119891119908

gt 0 Figure 8 is prepared to analyze theeffect of the material parameter 119899 with fixed Bi = 01 and

119891119908= 05 on local heat andmass transfer coefficients Figure 8

reveals that the local heat and mass transfer coefficients areenhanced by the increase in material parameter 119899 This isbecause when 119899 increases from 0 to 1 the microrotation termgets augmented and induces flow enhancement

The variations of 119862119891Gr14119909

and 119872119908Gr12119909

which areproportional to the coefficients of skin friction and wallcouple stress are shown in Table 3 with different values ofthe coupling number 119873 for fixed 119899 = 05 119891

119908= 05 and

Bi = 01 It indicates that the skin friction factor is higherfor micropolar fluid than the viscous fluids (119873 = 0) Sincemicropolar fluids offer a heavy resistance (resulting fromvortex viscosity) to fluid movement and cause larger skin

14 Advances in High Energy Physics

0084

0085

0086

0087Nu x

Grminus

14

x

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

fw = 05 Bi = 01

(a)

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

024

026

028

030

ShxGrminus

14

x

fw = 05 Bi = 01

(b)

Figure 8 Effect of material parameter 119899 on (a) heat transfer rate and (b) mass transfer rate

Table 3 Effects of skin friction and wall couple stress for varyingvalues of Biot numbers Bi micropolar parameter 119873 materialparameter 119899 and suctioninjection parameter 119891

119908

119873 Bi 119891119908

119899 119862119891Gr14119909

119872119908Gr12119909

01 01 05 05 2424227 06327903 01 05 05 2502716 063996605 01 05 05 2647987 063086307 01 05 05 2943561 060387909 01 05 05 3874631 055089105 01 05 05 2647987 063086305 10 05 05 3211755 080587905 50 05 05 3533141 090807805 200 05 05 3629791 093913705 01 minus05 05 2426265 033621905 01 00 05 2530014 046542905 01 10 05 2737553 082089905 01 20 05 2721244 119359605 01 05 00 2916655 minus028933905 01 05 05 2647987 063086305 01 05 10 2232556 2066191

friction factor compared to viscous fluid the results as wellsuggest larger values of coupling number 119873 and lower wallcouple stresses Since the skin friction coefficient 11989110158401015840(0) andwall couple stress coefficient as well as high temperature andmass transport rates are more depressed in the micropolarfluid comparing to the viscous fluid which may be beneficialin flow temperature and concentration control of polymerprocessing thus the presence of microscopic effects arisingfrom the local structure and of the fluid elements reduces thehigh temperature and mass transfer coefficients The effectof Bi on 119862

119891Gr14119909

and 119872119908Gr12119909

for 119891119908

= 05 119899 = 05 and

119873 = 05 is illustrated in Table 3 It can be noticed that theskin friction and wall couple stress coefficients are increasingwith increase of Bi for fixed values of other parametersThis notice is consistent with physical profiles presented inFigure 3 The effects of suctioninjection parameter on theskin friction andwall couple stress coefficients are also shownin Table 3 It is noted that the skin friction and wall couplestress coefficients are less with injection case whereas theyare more with suction case when compared to the case ofimpermeable surface Finally the detailed behavior of thematerial parameter 119899 is given in Table 3 The skin frictiondecreases and wall couple stress increases with increase ofmaterial parameter 119899

8 Conclusions

In this composition the similarity solution of the free convec-tion flow on a permeable vertical plate of a micropolar fluidunder the convective boundary condition is obtained usingLie group transformations Using the similarity variables thegoverning equations are transformed into a set of nondi-mensional parabolic equations These equations are solvednumerically using spectral quasi-linearisation method Thenumerical computation is carried out for various values ofnondimensional physical parameters The main findings aresummarized as follows

(i) The numerical results indicate that velocity distribu-tion is less near the plate but it is more far away fromthe plate the wall couple stress coefficient and rate ofheat andmass transfers are lower but the temperatureand concentration distributions and the skin frictioncoefficient are higher for the micropolar fluids incomparison with those of viscous fluids Also thereverse rotation ofmicrorotation near two boundariesis found with the increasing value of119873

Advances in High Energy Physics 15

(ii) An increase in Biot number Bi decreases concen-tration distribution whereas it causes an increasein temperature distribution skin friction and wallcouple stress coefficients and heat and mass transferrates within the boundary layer Further enhancingin Biot number Bi enhances velocity distributionnear the plate but shows the reverse behavior faraway from the plate We observe reverse rotationof microrotation near two boundaries within theboundary layer in the presence of Biot number

(iii) Less velocity temperature and concentration distri-butions are observed more skin friction and wallcouple stress coefficients and heat and mass transferrates in the case of suction compared to the case ofinjection Further microrotation decreases near thewall and depicts the opposite trend far away from thewall

(iv) It is found that microrotation temperature and con-centration distributions and skin friction coefficientsare more in the case of a micropolar fluid with strongconcentration (ie 119899 = 0) when compared to the caseof a micropolar fluid with weak concentration (ie119899 = 12) Further velocity is less in the case of 119899 = 0when compared to the case of 119899 = 12 near the walland shows the opposite trend far away from the wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their valuablesuggestions and comments

References

[1] J H Merkin ldquoNatural-convection boundary-layer flow on avertical surface with Newtonian heatingrdquo International Journalof Heat and Fluid Flow vol 15 no 5 pp 392ndash398 1994

[2] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009

[3] O DMakinde ldquoSimilarity solution for natural convection froma moving vertical plate with internal heat generation and aconvective boundary conditionrdquo Thermal Science vol 15 ppS137ndashS143 2011

[4] S M Ibrahim and N Bhashar Reddy ldquoSimilarity solution ofheat and mass transfer for natural convection over a movingvertical plate with internal heat generation and a convectiveboundary condition in the presence of thermal radiationviscous dissipation and chemical reactionrdquo ISRN Thermody-namics vol 2013 Article ID 790604 10 pages 2013

[5] C RamReddy P V S N Murthy A J Chamkha and A MRashad ldquoSoret effect on mixed convection flow in a nanofluidunder convective boundary conditionrdquo International Journal ofHeat and Mass Transfer vol 64 pp 384ndash392 2013

[6] H T Alkasasbeh M Z Salleh R M Tahar and R NazarldquoNumerical solutions of free convection boundary layer flow ona solid sphere with convective boundary conditionsrdquo Journal ofPhysics Conference Series vol 495 Article ID 012025 2014

[7] A Pantokratoras ldquoA note on natural convection along a con-vectively heated vertical platerdquo International Journal of ThermalSciences vol 76 pp 221ndash224 2014

[8] G Lukaszewicz Micropolar FluidsmdashTheory and ApplicationsBirkhaauser Basel Switzerland 1999

[9] V A Eremeyev L P Lebedev and H Altenbach Foundationsof Micropolar Mechanics Springer New York NY USA 2013

[10] G Ahmadi ldquoSelf-similar solution of imcompressible micropo-lar boundary layer flow over a semi-infinite platerdquo InternationalJournal of Engineering Science vol 14 no 7 pp 639ndash646 1976

[11] D A Rees and I Pop ldquoFree convection boundary-layer flowof a micropolar fluid from a vertical flat platerdquo IMA Journal ofApplied Mathematics vol 61 no 2 pp 179ndash197 1998

[12] D Srinivasacharya and C RamReddy ldquoEffect of double stratifi-cation onmixed convection in a micropolar fluidrdquoMatematikavol 28 no 2 pp 133ndash149 2012

[13] D Srinivasacharya and C Ramreddy ldquoMixed convection heatand mass transfer in a doubly stratified micropolar fluidrdquoComputationalThermal Sciences vol 5 no 4 pp 273ndash287 2013

[14] D Srinivasacharya and C RamReddy ldquoNatural convection heatand mass transfer in a micropolar fluid with thermal and massstratificationrdquo International Journal for Computational Methodsin Engineering Science andMechanics vol 14 no 5 pp 401ndash4132013

[15] N A Yacob and A Ishak ldquoStagnation point flow towardsa stretchingshrinking sheet in a micropolar fluid with aconvective surface boundary conditionrdquoThe Canadian Journalof Chemical Engineering vol 90 no 3 pp 621ndash626 2012

[16] K Zaimi and A Ishak ldquoStagnation-point flow and heat transferover a nonlinearly stretchingshrinking sheet in a micropolarfluidrdquo Abstract and Applied Analysis vol 2014 Article ID261630 6 pages 2014

[17] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[18] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[19] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer 2009

[20] T Tapanidis G Tsagas and H P Mazumdar ldquoApplication ofscaling group of transformations to viscoelastic second-gradefluid flowrdquo Nonlinear Functional Analysis and Applications vol8 no 3 pp 345ndash350 2003

[21] I A A Hassanien andM A Hamad ldquoGroup theoretic methodfor unsteady free convection flow of a micropolar fluid along avertical plate in a thermally stratified mediumrdquo Applied Mathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 32 no 6 pp 1099ndash1114 2008

[22] R Kandasamy K Gunasekaran and S B H Hasan ldquoScalinggroup transformation on fluid flow with variable stream con-ditionsrdquo International Journal of Non-Linear Mechanics vol 46no 7 pp 976ndash985 2011

[23] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat TransfermdashAsian Research vol41 no 3 pp 241ndash259 2012

16 Advances in High Energy Physics

[24] A A Mutlag M J Uddin M A A Hamad and A I M IsmailldquoHeat transfer analysis for falkner-skan boundary layer flowpast a stationary wedge with slips boundary conditions con-sidering temperature-dependent thermal conductivityrdquo SainsMalaysiana vol 42 no 6 pp 855ndash862 2013

[25] S C Cowin ldquoPolar fluidsrdquo Physics of Fluids vol 11 no 9 pp1919ndash1927 1968

[26] M A Seddeek M Y Akl and A M Al-Hanaya ldquoThermalradiation effects on mixed convection and mass transfer flowon vertical porous plate with heat generation and chemicalreaction by using scaling grouprdquo Journal of Natural Sciences andMathematics vol 4 pp 41ndash60 2010

[27] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[28] M J Uddin W A Khan and A I M Ismail ldquoMHD freeconvective boundary layer flow of a nanofluid past a flat verticalplate with Newtonian heating boundary conditionrdquo PLoS ONEvol 7 no 11 Article ID e49499 2012

[29] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012

[30] M J Uddin W A Khan and A I M Ismail ldquoScalinggroup transformation for MHD boundary layer slip flow of ananofluid over a convectively heated stretching sheet with heatgenerationrdquo Mathematical Problems in Engineering vol 2012Article ID 934964 20 pages 2012

[31] MM RashidiM Ferdows A B Parsa and S Abelman ldquoMHDnatural convection with convective surface boundary conditionover a at platerdquo Abstract and Applied Analysis vol 2014 ArticleID 923487 10 pages 2014

[32] R E Bellman and R E Kalaba Quasilinearisation and Non-Linear Boundary-Value Problems Elsevier New York NY USA1965

[33] S SMotsa P GDlamini andMKhumalo ldquoSpectral relaxationmethod and spectral quasilinearization method for solvingunsteady boundary layer flow problemsrdquo Advances in Mathe-matical Physics vol 2014 Article ID 341964 12 pages 2014

[34] S S Motsa P Sibanda J M Ngnotchouye and G T MarewoldquoA spectral relaxation approach for unsteady boundary-layerflow and heat transfer of a nanofluid over a permeable stretch-ingshrinking sheetrdquo Advances in Mathematical Physics vol2014 Article ID 564942 10 pages 2014

[35] C Canuto M Y Hussaini A Quarteroni and T ZangSpectral Methods Fundamentals in Single Domains ScientificComputation Springer Berlin Germany 2006

[36] J HMerkin ldquoFree convection boundary layer on an isothermalhorizontal cylinderrdquo in Proceedings of the American Societyof Mechanical Engineers and American Institute of ChemicalEngineersHeat Transfer Conference p 911 ASME St LouisMoUSA August 1976

[37] R Nazar N Amin and I Pop ldquoFree convection boundary layeron an isothermal horizontal circular cylinder in a micropolarfluidrdquo in Proceedings of the 12th International Heat TransferConference vol 2 pp 525ndash530 Elsevier Paris France August2002

[38] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

Submit your manuscripts athttpwwwhindawicom

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FluidsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances in Condensed Matter Physics

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Superconductivity

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AstrophysicsJournal of

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Physics Research International

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Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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PhotonicsJournal of

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Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

10 Advances in High Energy Physics

00

01

02

03

04

05

06

f998400

120578

0 2 4 6 8 10 12

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(a)

minus06

minus04

minus03

minus02

minus05

minus01

00

01

g

0 2 4 6 8 10

120578

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

(b)

fw = 05 n = 05N = 05

Bi = 01

Bi = 10

Bi = 50Bi = 200

120578

0 1 2 3 4 5

120579

00

02

04

06

08

10

(c)

120601

00

02

04

06

08

10

fw = 05 n = 05N = 05

0 2 4 6 8 10 12

120578

Bi = 01

Bi = 10

Bi = 50Bi = 200

(d)

Figure 3 Effect of Bi on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

of antisymmetric part of the stress on the boundary thatcorresponds to a weak concentration of microelements Thisis because an increment in the value of 119873 implies a highervortex viscosity of fluid which promotes the microrotationof micropolar fluids It is seen from Figures 2(c) and 2(d)that thermal and concentration boundary layers of thefluid increase with increase in coupling number 119873 Hencetemperature and concentration in case of micropolar fluidsare more than those of the viscous fluid case

The Biot number Bi is the ratio of internal thermalresistance of a solid to boundary layer thermal resistanceWhen Bi = 0 the plate is totally insulated internal thermalresistance of the plate is extremely high and no convectiveheat transfer to the cold fluid on the upper part of the

plate takes place Figure 3(a) depicts fluid velocity profilesfor different values of the Biot number with 119873 = 05 119891

119908=

05 and 119899 = 05 Generally fluid velocity is zero at platesurface and increases gradually away fromplate to free streamvalue satisfying boundary conditions It is interesting to notethat an increase in the intensity of convective surface heattransfer Bi produces significant enhancement in fluid velocitywithin the momentum boundary layer In Figure 3(b) webring out the behavior of microrotation with different valuesof Biot number Bi for fixed values of other parameters As theparameter value Bi increases microrotation showing reverserotation near the two boundaries Hence the condition ofvanishing of antisymmetric part of the stress on the boundaryresults in a drastic change of the microrotation profiles

Advances in High Energy Physics 11

07

06

05

04

03

02

01

00

f998400

fw = 00

fw = 10

fw = 20

0 3 6 9 12 15

120578

N = 05 Bi = 01 n = 05

fw = minus05

(a)

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

120578

0 2 4 6 8 10 12

g

minus04

minus03

minus02

minus01

00

01

fw = minus05

(b)

0 1 2 3 4 5 6

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

000

005

010

015

020

025

030

120579

fw = minus05

(c)

0 3 6 9 12

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

00

02

04

06

08

10

120601

fw = minus05

(d)

Figure 4 Effect of 119891119908on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Given that convective heating increases with Biot numberBi rarr infin simulates the isothermal surface which is clearlyseen from Figure 3(c) where 120579(0) = 1 as Bi rarr infin Infact a high Biot number indicates higher internal thermalresistance of the plate than boundary layer thermal resistanceIn this fluid temperature is maximum at the plate surfaceand decreases exponentially to zero value far out from theplate satisfying boundary conditions As a consequence anincrement in the Biot number leads to increase of fluidtemperature efficiency Figure 3(d) illustrates the variation ofdimensionless concentration for different values of Bi It isclear that the concentration of fluid decreases with increaseof Bi

The effect of 119891119908

on velocity profile is depicted inFigure 4(a) Here 119891

119908gt 0 represents suction and 119891

119908lt 0

denotes injectionThe lower velocity is noticed in case of suc-tion when compared to case of injection From Figure 4(b)we note that microrotation is showing reverse rotation neartwo boundaries with both suction and injection parame-ter The dimensionless temperature for different values ofsuctioninjection parameters is drawn in Figure 4(c) It isreadable that the temperature of the fluid is more in case ofinjection whereas it is less in case of suction in comparisonwith the impermeable surface case (119891

119908= 0) Figure 4(d)

demonstrates dimensionless concentration for different val-ues of suctioninjection parameters It is determined that the

12 Advances in High Energy Physics

00

01

02

03

04

05

06f998400

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(a)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10

120578

minus10

minus08

minus06

minus04

minus02

00

g

(b)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

120578

0 1 2 3 4 5

120579

000

003

006

009

012

015

(c)

00

02

04

06

08

10

120601

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(d)

Figure 5 Effect of material parameter 119899 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

concentration of fluid ismore with injection whereas it is lesswith suctionwhen compared to the impermeable surface case(119891119908= 0) As a finale the thermal and solutal boundary layer

thicknesses increase in case of injection compared to case ofsuction as displayed in Figures 4(c) and 4(d)

In Figures 5(a)ndash5(d) the effect of material parameter119899 on dimensionless velocity microrotation temperatureand concentration is presented for fixed values of otherparameters since the material parameter 119899 signifies themicrorotation effects (ie for 119899 = 0 particles are not freeto revolve near the surface whereas as 119899 increases from 0 to1 the microrotation term gets augmented and induces flowenhancement) As 119899 increases it is found from Figure 5(a)that the minimum velocity increases in amplitude and the

location of the minimum velocity moves farther away fromthe wall From Figure 5(b) we observe that themicrorotationis decreasing with increasing value of material parameter119899 within the boundary layer It is clear from Figures 5(c)-5(d) that the thermal and solutal boundary layer thicknessesdecrease with increase of material parameter 119899

The variations of minus1205791015840

(0) and minus1206011015840

(0) versus couplingnumber 119873 are shown in Figures 6ndash8 It can be noticed fromthese figures that the heat and mass transfer coefficientsare less in case of micropolar fluids when compared to theviscous fluids This is because as 119873 increases the thermaland solutal boundary layer thicknesses become larger thusgiving rise to a small value of local heat and mass transferrates The effect of the Biot number Bi with fixed 119891

119908= 05

Advances in High Energy Physics 13

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

fw = 05 n = 05

01

00

02

03

04

05

06

07Nu x

Grminus

14

x

(a)

fw = 05 n = 05

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

024

026

028

030

032

ShxGrminus

14

x

(b)

Figure 6 Effect of Bi on (a) heat transfer rate and (b) mass transfer rate

005

006

007

008

009

010

Nu x

Grminus

14

x

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

(a)

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

01

02

03

04

05

06

ShxGrminus

14

x

(b)

Figure 7 Effect of 119891119908on (a) heat transfer rate and (b) mass transfer rate

and 119899 = 05 on local heat transfer coefficient is exhibitedin Figure 6 It is found from Figure 6 that local heat transferrate increases nonlinearly with the increase in Biot numberBiThe influence of the Biot number Bi on local mass transfercoefficient is shown in Figure 6 Figure 6 reveals that the localmass transfer coefficient is enhanced by the growth in theBiot number Bi In Figure 7 the effect of the suctioninjectionparameter 119891

119908with fixed Bi = 01 and 119899 = 05 on local heat

and mass transfer coefficients is displayed It is found fromFigure 7 that the local heat and mass transfer coefficients areless in the case of injection 119891

119908lt 0 in comparison with the

case of suction 119891119908

gt 0 Figure 8 is prepared to analyze theeffect of the material parameter 119899 with fixed Bi = 01 and

119891119908= 05 on local heat andmass transfer coefficients Figure 8

reveals that the local heat and mass transfer coefficients areenhanced by the increase in material parameter 119899 This isbecause when 119899 increases from 0 to 1 the microrotation termgets augmented and induces flow enhancement

The variations of 119862119891Gr14119909

and 119872119908Gr12119909

which areproportional to the coefficients of skin friction and wallcouple stress are shown in Table 3 with different values ofthe coupling number 119873 for fixed 119899 = 05 119891

119908= 05 and

Bi = 01 It indicates that the skin friction factor is higherfor micropolar fluid than the viscous fluids (119873 = 0) Sincemicropolar fluids offer a heavy resistance (resulting fromvortex viscosity) to fluid movement and cause larger skin

14 Advances in High Energy Physics

0084

0085

0086

0087Nu x

Grminus

14

x

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

fw = 05 Bi = 01

(a)

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

024

026

028

030

ShxGrminus

14

x

fw = 05 Bi = 01

(b)

Figure 8 Effect of material parameter 119899 on (a) heat transfer rate and (b) mass transfer rate

Table 3 Effects of skin friction and wall couple stress for varyingvalues of Biot numbers Bi micropolar parameter 119873 materialparameter 119899 and suctioninjection parameter 119891

119908

119873 Bi 119891119908

119899 119862119891Gr14119909

119872119908Gr12119909

01 01 05 05 2424227 06327903 01 05 05 2502716 063996605 01 05 05 2647987 063086307 01 05 05 2943561 060387909 01 05 05 3874631 055089105 01 05 05 2647987 063086305 10 05 05 3211755 080587905 50 05 05 3533141 090807805 200 05 05 3629791 093913705 01 minus05 05 2426265 033621905 01 00 05 2530014 046542905 01 10 05 2737553 082089905 01 20 05 2721244 119359605 01 05 00 2916655 minus028933905 01 05 05 2647987 063086305 01 05 10 2232556 2066191

friction factor compared to viscous fluid the results as wellsuggest larger values of coupling number 119873 and lower wallcouple stresses Since the skin friction coefficient 11989110158401015840(0) andwall couple stress coefficient as well as high temperature andmass transport rates are more depressed in the micropolarfluid comparing to the viscous fluid which may be beneficialin flow temperature and concentration control of polymerprocessing thus the presence of microscopic effects arisingfrom the local structure and of the fluid elements reduces thehigh temperature and mass transfer coefficients The effectof Bi on 119862

119891Gr14119909

and 119872119908Gr12119909

for 119891119908

= 05 119899 = 05 and

119873 = 05 is illustrated in Table 3 It can be noticed that theskin friction and wall couple stress coefficients are increasingwith increase of Bi for fixed values of other parametersThis notice is consistent with physical profiles presented inFigure 3 The effects of suctioninjection parameter on theskin friction andwall couple stress coefficients are also shownin Table 3 It is noted that the skin friction and wall couplestress coefficients are less with injection case whereas theyare more with suction case when compared to the case ofimpermeable surface Finally the detailed behavior of thematerial parameter 119899 is given in Table 3 The skin frictiondecreases and wall couple stress increases with increase ofmaterial parameter 119899

8 Conclusions

In this composition the similarity solution of the free convec-tion flow on a permeable vertical plate of a micropolar fluidunder the convective boundary condition is obtained usingLie group transformations Using the similarity variables thegoverning equations are transformed into a set of nondi-mensional parabolic equations These equations are solvednumerically using spectral quasi-linearisation method Thenumerical computation is carried out for various values ofnondimensional physical parameters The main findings aresummarized as follows

(i) The numerical results indicate that velocity distribu-tion is less near the plate but it is more far away fromthe plate the wall couple stress coefficient and rate ofheat andmass transfers are lower but the temperatureand concentration distributions and the skin frictioncoefficient are higher for the micropolar fluids incomparison with those of viscous fluids Also thereverse rotation ofmicrorotation near two boundariesis found with the increasing value of119873

Advances in High Energy Physics 15

(ii) An increase in Biot number Bi decreases concen-tration distribution whereas it causes an increasein temperature distribution skin friction and wallcouple stress coefficients and heat and mass transferrates within the boundary layer Further enhancingin Biot number Bi enhances velocity distributionnear the plate but shows the reverse behavior faraway from the plate We observe reverse rotationof microrotation near two boundaries within theboundary layer in the presence of Biot number

(iii) Less velocity temperature and concentration distri-butions are observed more skin friction and wallcouple stress coefficients and heat and mass transferrates in the case of suction compared to the case ofinjection Further microrotation decreases near thewall and depicts the opposite trend far away from thewall

(iv) It is found that microrotation temperature and con-centration distributions and skin friction coefficientsare more in the case of a micropolar fluid with strongconcentration (ie 119899 = 0) when compared to the caseof a micropolar fluid with weak concentration (ie119899 = 12) Further velocity is less in the case of 119899 = 0when compared to the case of 119899 = 12 near the walland shows the opposite trend far away from the wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their valuablesuggestions and comments

References

[1] J H Merkin ldquoNatural-convection boundary-layer flow on avertical surface with Newtonian heatingrdquo International Journalof Heat and Fluid Flow vol 15 no 5 pp 392ndash398 1994

[2] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009

[3] O DMakinde ldquoSimilarity solution for natural convection froma moving vertical plate with internal heat generation and aconvective boundary conditionrdquo Thermal Science vol 15 ppS137ndashS143 2011

[4] S M Ibrahim and N Bhashar Reddy ldquoSimilarity solution ofheat and mass transfer for natural convection over a movingvertical plate with internal heat generation and a convectiveboundary condition in the presence of thermal radiationviscous dissipation and chemical reactionrdquo ISRN Thermody-namics vol 2013 Article ID 790604 10 pages 2013

[5] C RamReddy P V S N Murthy A J Chamkha and A MRashad ldquoSoret effect on mixed convection flow in a nanofluidunder convective boundary conditionrdquo International Journal ofHeat and Mass Transfer vol 64 pp 384ndash392 2013

[6] H T Alkasasbeh M Z Salleh R M Tahar and R NazarldquoNumerical solutions of free convection boundary layer flow ona solid sphere with convective boundary conditionsrdquo Journal ofPhysics Conference Series vol 495 Article ID 012025 2014

[7] A Pantokratoras ldquoA note on natural convection along a con-vectively heated vertical platerdquo International Journal of ThermalSciences vol 76 pp 221ndash224 2014

[8] G Lukaszewicz Micropolar FluidsmdashTheory and ApplicationsBirkhaauser Basel Switzerland 1999

[9] V A Eremeyev L P Lebedev and H Altenbach Foundationsof Micropolar Mechanics Springer New York NY USA 2013

[10] G Ahmadi ldquoSelf-similar solution of imcompressible micropo-lar boundary layer flow over a semi-infinite platerdquo InternationalJournal of Engineering Science vol 14 no 7 pp 639ndash646 1976

[11] D A Rees and I Pop ldquoFree convection boundary-layer flowof a micropolar fluid from a vertical flat platerdquo IMA Journal ofApplied Mathematics vol 61 no 2 pp 179ndash197 1998

[12] D Srinivasacharya and C RamReddy ldquoEffect of double stratifi-cation onmixed convection in a micropolar fluidrdquoMatematikavol 28 no 2 pp 133ndash149 2012

[13] D Srinivasacharya and C Ramreddy ldquoMixed convection heatand mass transfer in a doubly stratified micropolar fluidrdquoComputationalThermal Sciences vol 5 no 4 pp 273ndash287 2013

[14] D Srinivasacharya and C RamReddy ldquoNatural convection heatand mass transfer in a micropolar fluid with thermal and massstratificationrdquo International Journal for Computational Methodsin Engineering Science andMechanics vol 14 no 5 pp 401ndash4132013

[15] N A Yacob and A Ishak ldquoStagnation point flow towardsa stretchingshrinking sheet in a micropolar fluid with aconvective surface boundary conditionrdquoThe Canadian Journalof Chemical Engineering vol 90 no 3 pp 621ndash626 2012

[16] K Zaimi and A Ishak ldquoStagnation-point flow and heat transferover a nonlinearly stretchingshrinking sheet in a micropolarfluidrdquo Abstract and Applied Analysis vol 2014 Article ID261630 6 pages 2014

[17] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[18] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[19] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer 2009

[20] T Tapanidis G Tsagas and H P Mazumdar ldquoApplication ofscaling group of transformations to viscoelastic second-gradefluid flowrdquo Nonlinear Functional Analysis and Applications vol8 no 3 pp 345ndash350 2003

[21] I A A Hassanien andM A Hamad ldquoGroup theoretic methodfor unsteady free convection flow of a micropolar fluid along avertical plate in a thermally stratified mediumrdquo Applied Mathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 32 no 6 pp 1099ndash1114 2008

[22] R Kandasamy K Gunasekaran and S B H Hasan ldquoScalinggroup transformation on fluid flow with variable stream con-ditionsrdquo International Journal of Non-Linear Mechanics vol 46no 7 pp 976ndash985 2011

[23] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat TransfermdashAsian Research vol41 no 3 pp 241ndash259 2012

16 Advances in High Energy Physics

[24] A A Mutlag M J Uddin M A A Hamad and A I M IsmailldquoHeat transfer analysis for falkner-skan boundary layer flowpast a stationary wedge with slips boundary conditions con-sidering temperature-dependent thermal conductivityrdquo SainsMalaysiana vol 42 no 6 pp 855ndash862 2013

[25] S C Cowin ldquoPolar fluidsrdquo Physics of Fluids vol 11 no 9 pp1919ndash1927 1968

[26] M A Seddeek M Y Akl and A M Al-Hanaya ldquoThermalradiation effects on mixed convection and mass transfer flowon vertical porous plate with heat generation and chemicalreaction by using scaling grouprdquo Journal of Natural Sciences andMathematics vol 4 pp 41ndash60 2010

[27] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[28] M J Uddin W A Khan and A I M Ismail ldquoMHD freeconvective boundary layer flow of a nanofluid past a flat verticalplate with Newtonian heating boundary conditionrdquo PLoS ONEvol 7 no 11 Article ID e49499 2012

[29] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012

[30] M J Uddin W A Khan and A I M Ismail ldquoScalinggroup transformation for MHD boundary layer slip flow of ananofluid over a convectively heated stretching sheet with heatgenerationrdquo Mathematical Problems in Engineering vol 2012Article ID 934964 20 pages 2012

[31] MM RashidiM Ferdows A B Parsa and S Abelman ldquoMHDnatural convection with convective surface boundary conditionover a at platerdquo Abstract and Applied Analysis vol 2014 ArticleID 923487 10 pages 2014

[32] R E Bellman and R E Kalaba Quasilinearisation and Non-Linear Boundary-Value Problems Elsevier New York NY USA1965

[33] S SMotsa P GDlamini andMKhumalo ldquoSpectral relaxationmethod and spectral quasilinearization method for solvingunsteady boundary layer flow problemsrdquo Advances in Mathe-matical Physics vol 2014 Article ID 341964 12 pages 2014

[34] S S Motsa P Sibanda J M Ngnotchouye and G T MarewoldquoA spectral relaxation approach for unsteady boundary-layerflow and heat transfer of a nanofluid over a permeable stretch-ingshrinking sheetrdquo Advances in Mathematical Physics vol2014 Article ID 564942 10 pages 2014

[35] C Canuto M Y Hussaini A Quarteroni and T ZangSpectral Methods Fundamentals in Single Domains ScientificComputation Springer Berlin Germany 2006

[36] J HMerkin ldquoFree convection boundary layer on an isothermalhorizontal cylinderrdquo in Proceedings of the American Societyof Mechanical Engineers and American Institute of ChemicalEngineersHeat Transfer Conference p 911 ASME St LouisMoUSA August 1976

[37] R Nazar N Amin and I Pop ldquoFree convection boundary layeron an isothermal horizontal circular cylinder in a micropolarfluidrdquo in Proceedings of the 12th International Heat TransferConference vol 2 pp 525ndash530 Elsevier Paris France August2002

[38] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 11

07

06

05

04

03

02

01

00

f998400

fw = 00

fw = 10

fw = 20

0 3 6 9 12 15

120578

N = 05 Bi = 01 n = 05

fw = minus05

(a)

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

120578

0 2 4 6 8 10 12

g

minus04

minus03

minus02

minus01

00

01

fw = minus05

(b)

0 1 2 3 4 5 6

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

000

005

010

015

020

025

030

120579

fw = minus05

(c)

0 3 6 9 12

120578

fw = 00

fw = 10

fw = 20

N = 05 Bi = 01 n = 05

00

02

04

06

08

10

120601

fw = minus05

(d)

Figure 4 Effect of 119891119908on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

Given that convective heating increases with Biot numberBi rarr infin simulates the isothermal surface which is clearlyseen from Figure 3(c) where 120579(0) = 1 as Bi rarr infin Infact a high Biot number indicates higher internal thermalresistance of the plate than boundary layer thermal resistanceIn this fluid temperature is maximum at the plate surfaceand decreases exponentially to zero value far out from theplate satisfying boundary conditions As a consequence anincrement in the Biot number leads to increase of fluidtemperature efficiency Figure 3(d) illustrates the variation ofdimensionless concentration for different values of Bi It isclear that the concentration of fluid decreases with increaseof Bi

The effect of 119891119908

on velocity profile is depicted inFigure 4(a) Here 119891

119908gt 0 represents suction and 119891

119908lt 0

denotes injectionThe lower velocity is noticed in case of suc-tion when compared to case of injection From Figure 4(b)we note that microrotation is showing reverse rotation neartwo boundaries with both suction and injection parame-ter The dimensionless temperature for different values ofsuctioninjection parameters is drawn in Figure 4(c) It isreadable that the temperature of the fluid is more in case ofinjection whereas it is less in case of suction in comparisonwith the impermeable surface case (119891

119908= 0) Figure 4(d)

demonstrates dimensionless concentration for different val-ues of suctioninjection parameters It is determined that the

12 Advances in High Energy Physics

00

01

02

03

04

05

06f998400

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(a)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10

120578

minus10

minus08

minus06

minus04

minus02

00

g

(b)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

120578

0 1 2 3 4 5

120579

000

003

006

009

012

015

(c)

00

02

04

06

08

10

120601

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(d)

Figure 5 Effect of material parameter 119899 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

concentration of fluid ismore with injection whereas it is lesswith suctionwhen compared to the impermeable surface case(119891119908= 0) As a finale the thermal and solutal boundary layer

thicknesses increase in case of injection compared to case ofsuction as displayed in Figures 4(c) and 4(d)

In Figures 5(a)ndash5(d) the effect of material parameter119899 on dimensionless velocity microrotation temperatureand concentration is presented for fixed values of otherparameters since the material parameter 119899 signifies themicrorotation effects (ie for 119899 = 0 particles are not freeto revolve near the surface whereas as 119899 increases from 0 to1 the microrotation term gets augmented and induces flowenhancement) As 119899 increases it is found from Figure 5(a)that the minimum velocity increases in amplitude and the

location of the minimum velocity moves farther away fromthe wall From Figure 5(b) we observe that themicrorotationis decreasing with increasing value of material parameter119899 within the boundary layer It is clear from Figures 5(c)-5(d) that the thermal and solutal boundary layer thicknessesdecrease with increase of material parameter 119899

The variations of minus1205791015840

(0) and minus1206011015840

(0) versus couplingnumber 119873 are shown in Figures 6ndash8 It can be noticed fromthese figures that the heat and mass transfer coefficientsare less in case of micropolar fluids when compared to theviscous fluids This is because as 119873 increases the thermaland solutal boundary layer thicknesses become larger thusgiving rise to a small value of local heat and mass transferrates The effect of the Biot number Bi with fixed 119891

119908= 05

Advances in High Energy Physics 13

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

fw = 05 n = 05

01

00

02

03

04

05

06

07Nu x

Grminus

14

x

(a)

fw = 05 n = 05

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

024

026

028

030

032

ShxGrminus

14

x

(b)

Figure 6 Effect of Bi on (a) heat transfer rate and (b) mass transfer rate

005

006

007

008

009

010

Nu x

Grminus

14

x

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

(a)

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

01

02

03

04

05

06

ShxGrminus

14

x

(b)

Figure 7 Effect of 119891119908on (a) heat transfer rate and (b) mass transfer rate

and 119899 = 05 on local heat transfer coefficient is exhibitedin Figure 6 It is found from Figure 6 that local heat transferrate increases nonlinearly with the increase in Biot numberBiThe influence of the Biot number Bi on local mass transfercoefficient is shown in Figure 6 Figure 6 reveals that the localmass transfer coefficient is enhanced by the growth in theBiot number Bi In Figure 7 the effect of the suctioninjectionparameter 119891

119908with fixed Bi = 01 and 119899 = 05 on local heat

and mass transfer coefficients is displayed It is found fromFigure 7 that the local heat and mass transfer coefficients areless in the case of injection 119891

119908lt 0 in comparison with the

case of suction 119891119908

gt 0 Figure 8 is prepared to analyze theeffect of the material parameter 119899 with fixed Bi = 01 and

119891119908= 05 on local heat andmass transfer coefficients Figure 8

reveals that the local heat and mass transfer coefficients areenhanced by the increase in material parameter 119899 This isbecause when 119899 increases from 0 to 1 the microrotation termgets augmented and induces flow enhancement

The variations of 119862119891Gr14119909

and 119872119908Gr12119909

which areproportional to the coefficients of skin friction and wallcouple stress are shown in Table 3 with different values ofthe coupling number 119873 for fixed 119899 = 05 119891

119908= 05 and

Bi = 01 It indicates that the skin friction factor is higherfor micropolar fluid than the viscous fluids (119873 = 0) Sincemicropolar fluids offer a heavy resistance (resulting fromvortex viscosity) to fluid movement and cause larger skin

14 Advances in High Energy Physics

0084

0085

0086

0087Nu x

Grminus

14

x

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

fw = 05 Bi = 01

(a)

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

024

026

028

030

ShxGrminus

14

x

fw = 05 Bi = 01

(b)

Figure 8 Effect of material parameter 119899 on (a) heat transfer rate and (b) mass transfer rate

Table 3 Effects of skin friction and wall couple stress for varyingvalues of Biot numbers Bi micropolar parameter 119873 materialparameter 119899 and suctioninjection parameter 119891

119908

119873 Bi 119891119908

119899 119862119891Gr14119909

119872119908Gr12119909

01 01 05 05 2424227 06327903 01 05 05 2502716 063996605 01 05 05 2647987 063086307 01 05 05 2943561 060387909 01 05 05 3874631 055089105 01 05 05 2647987 063086305 10 05 05 3211755 080587905 50 05 05 3533141 090807805 200 05 05 3629791 093913705 01 minus05 05 2426265 033621905 01 00 05 2530014 046542905 01 10 05 2737553 082089905 01 20 05 2721244 119359605 01 05 00 2916655 minus028933905 01 05 05 2647987 063086305 01 05 10 2232556 2066191

friction factor compared to viscous fluid the results as wellsuggest larger values of coupling number 119873 and lower wallcouple stresses Since the skin friction coefficient 11989110158401015840(0) andwall couple stress coefficient as well as high temperature andmass transport rates are more depressed in the micropolarfluid comparing to the viscous fluid which may be beneficialin flow temperature and concentration control of polymerprocessing thus the presence of microscopic effects arisingfrom the local structure and of the fluid elements reduces thehigh temperature and mass transfer coefficients The effectof Bi on 119862

119891Gr14119909

and 119872119908Gr12119909

for 119891119908

= 05 119899 = 05 and

119873 = 05 is illustrated in Table 3 It can be noticed that theskin friction and wall couple stress coefficients are increasingwith increase of Bi for fixed values of other parametersThis notice is consistent with physical profiles presented inFigure 3 The effects of suctioninjection parameter on theskin friction andwall couple stress coefficients are also shownin Table 3 It is noted that the skin friction and wall couplestress coefficients are less with injection case whereas theyare more with suction case when compared to the case ofimpermeable surface Finally the detailed behavior of thematerial parameter 119899 is given in Table 3 The skin frictiondecreases and wall couple stress increases with increase ofmaterial parameter 119899

8 Conclusions

In this composition the similarity solution of the free convec-tion flow on a permeable vertical plate of a micropolar fluidunder the convective boundary condition is obtained usingLie group transformations Using the similarity variables thegoverning equations are transformed into a set of nondi-mensional parabolic equations These equations are solvednumerically using spectral quasi-linearisation method Thenumerical computation is carried out for various values ofnondimensional physical parameters The main findings aresummarized as follows

(i) The numerical results indicate that velocity distribu-tion is less near the plate but it is more far away fromthe plate the wall couple stress coefficient and rate ofheat andmass transfers are lower but the temperatureand concentration distributions and the skin frictioncoefficient are higher for the micropolar fluids incomparison with those of viscous fluids Also thereverse rotation ofmicrorotation near two boundariesis found with the increasing value of119873

Advances in High Energy Physics 15

(ii) An increase in Biot number Bi decreases concen-tration distribution whereas it causes an increasein temperature distribution skin friction and wallcouple stress coefficients and heat and mass transferrates within the boundary layer Further enhancingin Biot number Bi enhances velocity distributionnear the plate but shows the reverse behavior faraway from the plate We observe reverse rotationof microrotation near two boundaries within theboundary layer in the presence of Biot number

(iii) Less velocity temperature and concentration distri-butions are observed more skin friction and wallcouple stress coefficients and heat and mass transferrates in the case of suction compared to the case ofinjection Further microrotation decreases near thewall and depicts the opposite trend far away from thewall

(iv) It is found that microrotation temperature and con-centration distributions and skin friction coefficientsare more in the case of a micropolar fluid with strongconcentration (ie 119899 = 0) when compared to the caseof a micropolar fluid with weak concentration (ie119899 = 12) Further velocity is less in the case of 119899 = 0when compared to the case of 119899 = 12 near the walland shows the opposite trend far away from the wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their valuablesuggestions and comments

References

[1] J H Merkin ldquoNatural-convection boundary-layer flow on avertical surface with Newtonian heatingrdquo International Journalof Heat and Fluid Flow vol 15 no 5 pp 392ndash398 1994

[2] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009

[3] O DMakinde ldquoSimilarity solution for natural convection froma moving vertical plate with internal heat generation and aconvective boundary conditionrdquo Thermal Science vol 15 ppS137ndashS143 2011

[4] S M Ibrahim and N Bhashar Reddy ldquoSimilarity solution ofheat and mass transfer for natural convection over a movingvertical plate with internal heat generation and a convectiveboundary condition in the presence of thermal radiationviscous dissipation and chemical reactionrdquo ISRN Thermody-namics vol 2013 Article ID 790604 10 pages 2013

[5] C RamReddy P V S N Murthy A J Chamkha and A MRashad ldquoSoret effect on mixed convection flow in a nanofluidunder convective boundary conditionrdquo International Journal ofHeat and Mass Transfer vol 64 pp 384ndash392 2013

[6] H T Alkasasbeh M Z Salleh R M Tahar and R NazarldquoNumerical solutions of free convection boundary layer flow ona solid sphere with convective boundary conditionsrdquo Journal ofPhysics Conference Series vol 495 Article ID 012025 2014

[7] A Pantokratoras ldquoA note on natural convection along a con-vectively heated vertical platerdquo International Journal of ThermalSciences vol 76 pp 221ndash224 2014

[8] G Lukaszewicz Micropolar FluidsmdashTheory and ApplicationsBirkhaauser Basel Switzerland 1999

[9] V A Eremeyev L P Lebedev and H Altenbach Foundationsof Micropolar Mechanics Springer New York NY USA 2013

[10] G Ahmadi ldquoSelf-similar solution of imcompressible micropo-lar boundary layer flow over a semi-infinite platerdquo InternationalJournal of Engineering Science vol 14 no 7 pp 639ndash646 1976

[11] D A Rees and I Pop ldquoFree convection boundary-layer flowof a micropolar fluid from a vertical flat platerdquo IMA Journal ofApplied Mathematics vol 61 no 2 pp 179ndash197 1998

[12] D Srinivasacharya and C RamReddy ldquoEffect of double stratifi-cation onmixed convection in a micropolar fluidrdquoMatematikavol 28 no 2 pp 133ndash149 2012

[13] D Srinivasacharya and C Ramreddy ldquoMixed convection heatand mass transfer in a doubly stratified micropolar fluidrdquoComputationalThermal Sciences vol 5 no 4 pp 273ndash287 2013

[14] D Srinivasacharya and C RamReddy ldquoNatural convection heatand mass transfer in a micropolar fluid with thermal and massstratificationrdquo International Journal for Computational Methodsin Engineering Science andMechanics vol 14 no 5 pp 401ndash4132013

[15] N A Yacob and A Ishak ldquoStagnation point flow towardsa stretchingshrinking sheet in a micropolar fluid with aconvective surface boundary conditionrdquoThe Canadian Journalof Chemical Engineering vol 90 no 3 pp 621ndash626 2012

[16] K Zaimi and A Ishak ldquoStagnation-point flow and heat transferover a nonlinearly stretchingshrinking sheet in a micropolarfluidrdquo Abstract and Applied Analysis vol 2014 Article ID261630 6 pages 2014

[17] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[18] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[19] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer 2009

[20] T Tapanidis G Tsagas and H P Mazumdar ldquoApplication ofscaling group of transformations to viscoelastic second-gradefluid flowrdquo Nonlinear Functional Analysis and Applications vol8 no 3 pp 345ndash350 2003

[21] I A A Hassanien andM A Hamad ldquoGroup theoretic methodfor unsteady free convection flow of a micropolar fluid along avertical plate in a thermally stratified mediumrdquo Applied Mathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 32 no 6 pp 1099ndash1114 2008

[22] R Kandasamy K Gunasekaran and S B H Hasan ldquoScalinggroup transformation on fluid flow with variable stream con-ditionsrdquo International Journal of Non-Linear Mechanics vol 46no 7 pp 976ndash985 2011

[23] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat TransfermdashAsian Research vol41 no 3 pp 241ndash259 2012

16 Advances in High Energy Physics

[24] A A Mutlag M J Uddin M A A Hamad and A I M IsmailldquoHeat transfer analysis for falkner-skan boundary layer flowpast a stationary wedge with slips boundary conditions con-sidering temperature-dependent thermal conductivityrdquo SainsMalaysiana vol 42 no 6 pp 855ndash862 2013

[25] S C Cowin ldquoPolar fluidsrdquo Physics of Fluids vol 11 no 9 pp1919ndash1927 1968

[26] M A Seddeek M Y Akl and A M Al-Hanaya ldquoThermalradiation effects on mixed convection and mass transfer flowon vertical porous plate with heat generation and chemicalreaction by using scaling grouprdquo Journal of Natural Sciences andMathematics vol 4 pp 41ndash60 2010

[27] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[28] M J Uddin W A Khan and A I M Ismail ldquoMHD freeconvective boundary layer flow of a nanofluid past a flat verticalplate with Newtonian heating boundary conditionrdquo PLoS ONEvol 7 no 11 Article ID e49499 2012

[29] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012

[30] M J Uddin W A Khan and A I M Ismail ldquoScalinggroup transformation for MHD boundary layer slip flow of ananofluid over a convectively heated stretching sheet with heatgenerationrdquo Mathematical Problems in Engineering vol 2012Article ID 934964 20 pages 2012

[31] MM RashidiM Ferdows A B Parsa and S Abelman ldquoMHDnatural convection with convective surface boundary conditionover a at platerdquo Abstract and Applied Analysis vol 2014 ArticleID 923487 10 pages 2014

[32] R E Bellman and R E Kalaba Quasilinearisation and Non-Linear Boundary-Value Problems Elsevier New York NY USA1965

[33] S SMotsa P GDlamini andMKhumalo ldquoSpectral relaxationmethod and spectral quasilinearization method for solvingunsteady boundary layer flow problemsrdquo Advances in Mathe-matical Physics vol 2014 Article ID 341964 12 pages 2014

[34] S S Motsa P Sibanda J M Ngnotchouye and G T MarewoldquoA spectral relaxation approach for unsteady boundary-layerflow and heat transfer of a nanofluid over a permeable stretch-ingshrinking sheetrdquo Advances in Mathematical Physics vol2014 Article ID 564942 10 pages 2014

[35] C Canuto M Y Hussaini A Quarteroni and T ZangSpectral Methods Fundamentals in Single Domains ScientificComputation Springer Berlin Germany 2006

[36] J HMerkin ldquoFree convection boundary layer on an isothermalhorizontal cylinderrdquo in Proceedings of the American Societyof Mechanical Engineers and American Institute of ChemicalEngineersHeat Transfer Conference p 911 ASME St LouisMoUSA August 1976

[37] R Nazar N Amin and I Pop ldquoFree convection boundary layeron an isothermal horizontal circular cylinder in a micropolarfluidrdquo in Proceedings of the 12th International Heat TransferConference vol 2 pp 525ndash530 Elsevier Paris France August2002

[38] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

12 Advances in High Energy Physics

00

01

02

03

04

05

06f998400

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(a)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10

120578

minus10

minus08

minus06

minus04

minus02

00

g

(b)

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

120578

0 1 2 3 4 5

120579

000

003

006

009

012

015

(c)

00

02

04

06

08

10

120601

n = 00

n = 05

n = 10

Bi = 01 fw = 05N = 05

0 2 4 6 8 10 12

120578

(d)

Figure 5 Effect of material parameter 119899 on (a) velocity (b) microrotation (c) temperature and (d) concentration profiles

concentration of fluid ismore with injection whereas it is lesswith suctionwhen compared to the impermeable surface case(119891119908= 0) As a finale the thermal and solutal boundary layer

thicknesses increase in case of injection compared to case ofsuction as displayed in Figures 4(c) and 4(d)

In Figures 5(a)ndash5(d) the effect of material parameter119899 on dimensionless velocity microrotation temperatureand concentration is presented for fixed values of otherparameters since the material parameter 119899 signifies themicrorotation effects (ie for 119899 = 0 particles are not freeto revolve near the surface whereas as 119899 increases from 0 to1 the microrotation term gets augmented and induces flowenhancement) As 119899 increases it is found from Figure 5(a)that the minimum velocity increases in amplitude and the

location of the minimum velocity moves farther away fromthe wall From Figure 5(b) we observe that themicrorotationis decreasing with increasing value of material parameter119899 within the boundary layer It is clear from Figures 5(c)-5(d) that the thermal and solutal boundary layer thicknessesdecrease with increase of material parameter 119899

The variations of minus1205791015840

(0) and minus1206011015840

(0) versus couplingnumber 119873 are shown in Figures 6ndash8 It can be noticed fromthese figures that the heat and mass transfer coefficientsare less in case of micropolar fluids when compared to theviscous fluids This is because as 119873 increases the thermaland solutal boundary layer thicknesses become larger thusgiving rise to a small value of local heat and mass transferrates The effect of the Biot number Bi with fixed 119891

119908= 05

Advances in High Energy Physics 13

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

fw = 05 n = 05

01

00

02

03

04

05

06

07Nu x

Grminus

14

x

(a)

fw = 05 n = 05

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

024

026

028

030

032

ShxGrminus

14

x

(b)

Figure 6 Effect of Bi on (a) heat transfer rate and (b) mass transfer rate

005

006

007

008

009

010

Nu x

Grminus

14

x

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

(a)

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

01

02

03

04

05

06

ShxGrminus

14

x

(b)

Figure 7 Effect of 119891119908on (a) heat transfer rate and (b) mass transfer rate

and 119899 = 05 on local heat transfer coefficient is exhibitedin Figure 6 It is found from Figure 6 that local heat transferrate increases nonlinearly with the increase in Biot numberBiThe influence of the Biot number Bi on local mass transfercoefficient is shown in Figure 6 Figure 6 reveals that the localmass transfer coefficient is enhanced by the growth in theBiot number Bi In Figure 7 the effect of the suctioninjectionparameter 119891

119908with fixed Bi = 01 and 119899 = 05 on local heat

and mass transfer coefficients is displayed It is found fromFigure 7 that the local heat and mass transfer coefficients areless in the case of injection 119891

119908lt 0 in comparison with the

case of suction 119891119908

gt 0 Figure 8 is prepared to analyze theeffect of the material parameter 119899 with fixed Bi = 01 and

119891119908= 05 on local heat andmass transfer coefficients Figure 8

reveals that the local heat and mass transfer coefficients areenhanced by the increase in material parameter 119899 This isbecause when 119899 increases from 0 to 1 the microrotation termgets augmented and induces flow enhancement

The variations of 119862119891Gr14119909

and 119872119908Gr12119909

which areproportional to the coefficients of skin friction and wallcouple stress are shown in Table 3 with different values ofthe coupling number 119873 for fixed 119899 = 05 119891

119908= 05 and

Bi = 01 It indicates that the skin friction factor is higherfor micropolar fluid than the viscous fluids (119873 = 0) Sincemicropolar fluids offer a heavy resistance (resulting fromvortex viscosity) to fluid movement and cause larger skin

14 Advances in High Energy Physics

0084

0085

0086

0087Nu x

Grminus

14

x

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

fw = 05 Bi = 01

(a)

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

024

026

028

030

ShxGrminus

14

x

fw = 05 Bi = 01

(b)

Figure 8 Effect of material parameter 119899 on (a) heat transfer rate and (b) mass transfer rate

Table 3 Effects of skin friction and wall couple stress for varyingvalues of Biot numbers Bi micropolar parameter 119873 materialparameter 119899 and suctioninjection parameter 119891

119908

119873 Bi 119891119908

119899 119862119891Gr14119909

119872119908Gr12119909

01 01 05 05 2424227 06327903 01 05 05 2502716 063996605 01 05 05 2647987 063086307 01 05 05 2943561 060387909 01 05 05 3874631 055089105 01 05 05 2647987 063086305 10 05 05 3211755 080587905 50 05 05 3533141 090807805 200 05 05 3629791 093913705 01 minus05 05 2426265 033621905 01 00 05 2530014 046542905 01 10 05 2737553 082089905 01 20 05 2721244 119359605 01 05 00 2916655 minus028933905 01 05 05 2647987 063086305 01 05 10 2232556 2066191

friction factor compared to viscous fluid the results as wellsuggest larger values of coupling number 119873 and lower wallcouple stresses Since the skin friction coefficient 11989110158401015840(0) andwall couple stress coefficient as well as high temperature andmass transport rates are more depressed in the micropolarfluid comparing to the viscous fluid which may be beneficialin flow temperature and concentration control of polymerprocessing thus the presence of microscopic effects arisingfrom the local structure and of the fluid elements reduces thehigh temperature and mass transfer coefficients The effectof Bi on 119862

119891Gr14119909

and 119872119908Gr12119909

for 119891119908

= 05 119899 = 05 and

119873 = 05 is illustrated in Table 3 It can be noticed that theskin friction and wall couple stress coefficients are increasingwith increase of Bi for fixed values of other parametersThis notice is consistent with physical profiles presented inFigure 3 The effects of suctioninjection parameter on theskin friction andwall couple stress coefficients are also shownin Table 3 It is noted that the skin friction and wall couplestress coefficients are less with injection case whereas theyare more with suction case when compared to the case ofimpermeable surface Finally the detailed behavior of thematerial parameter 119899 is given in Table 3 The skin frictiondecreases and wall couple stress increases with increase ofmaterial parameter 119899

8 Conclusions

In this composition the similarity solution of the free convec-tion flow on a permeable vertical plate of a micropolar fluidunder the convective boundary condition is obtained usingLie group transformations Using the similarity variables thegoverning equations are transformed into a set of nondi-mensional parabolic equations These equations are solvednumerically using spectral quasi-linearisation method Thenumerical computation is carried out for various values ofnondimensional physical parameters The main findings aresummarized as follows

(i) The numerical results indicate that velocity distribu-tion is less near the plate but it is more far away fromthe plate the wall couple stress coefficient and rate ofheat andmass transfers are lower but the temperatureand concentration distributions and the skin frictioncoefficient are higher for the micropolar fluids incomparison with those of viscous fluids Also thereverse rotation ofmicrorotation near two boundariesis found with the increasing value of119873

Advances in High Energy Physics 15

(ii) An increase in Biot number Bi decreases concen-tration distribution whereas it causes an increasein temperature distribution skin friction and wallcouple stress coefficients and heat and mass transferrates within the boundary layer Further enhancingin Biot number Bi enhances velocity distributionnear the plate but shows the reverse behavior faraway from the plate We observe reverse rotationof microrotation near two boundaries within theboundary layer in the presence of Biot number

(iii) Less velocity temperature and concentration distri-butions are observed more skin friction and wallcouple stress coefficients and heat and mass transferrates in the case of suction compared to the case ofinjection Further microrotation decreases near thewall and depicts the opposite trend far away from thewall

(iv) It is found that microrotation temperature and con-centration distributions and skin friction coefficientsare more in the case of a micropolar fluid with strongconcentration (ie 119899 = 0) when compared to the caseof a micropolar fluid with weak concentration (ie119899 = 12) Further velocity is less in the case of 119899 = 0when compared to the case of 119899 = 12 near the walland shows the opposite trend far away from the wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their valuablesuggestions and comments

References

[1] J H Merkin ldquoNatural-convection boundary-layer flow on avertical surface with Newtonian heatingrdquo International Journalof Heat and Fluid Flow vol 15 no 5 pp 392ndash398 1994

[2] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009

[3] O DMakinde ldquoSimilarity solution for natural convection froma moving vertical plate with internal heat generation and aconvective boundary conditionrdquo Thermal Science vol 15 ppS137ndashS143 2011

[4] S M Ibrahim and N Bhashar Reddy ldquoSimilarity solution ofheat and mass transfer for natural convection over a movingvertical plate with internal heat generation and a convectiveboundary condition in the presence of thermal radiationviscous dissipation and chemical reactionrdquo ISRN Thermody-namics vol 2013 Article ID 790604 10 pages 2013

[5] C RamReddy P V S N Murthy A J Chamkha and A MRashad ldquoSoret effect on mixed convection flow in a nanofluidunder convective boundary conditionrdquo International Journal ofHeat and Mass Transfer vol 64 pp 384ndash392 2013

[6] H T Alkasasbeh M Z Salleh R M Tahar and R NazarldquoNumerical solutions of free convection boundary layer flow ona solid sphere with convective boundary conditionsrdquo Journal ofPhysics Conference Series vol 495 Article ID 012025 2014

[7] A Pantokratoras ldquoA note on natural convection along a con-vectively heated vertical platerdquo International Journal of ThermalSciences vol 76 pp 221ndash224 2014

[8] G Lukaszewicz Micropolar FluidsmdashTheory and ApplicationsBirkhaauser Basel Switzerland 1999

[9] V A Eremeyev L P Lebedev and H Altenbach Foundationsof Micropolar Mechanics Springer New York NY USA 2013

[10] G Ahmadi ldquoSelf-similar solution of imcompressible micropo-lar boundary layer flow over a semi-infinite platerdquo InternationalJournal of Engineering Science vol 14 no 7 pp 639ndash646 1976

[11] D A Rees and I Pop ldquoFree convection boundary-layer flowof a micropolar fluid from a vertical flat platerdquo IMA Journal ofApplied Mathematics vol 61 no 2 pp 179ndash197 1998

[12] D Srinivasacharya and C RamReddy ldquoEffect of double stratifi-cation onmixed convection in a micropolar fluidrdquoMatematikavol 28 no 2 pp 133ndash149 2012

[13] D Srinivasacharya and C Ramreddy ldquoMixed convection heatand mass transfer in a doubly stratified micropolar fluidrdquoComputationalThermal Sciences vol 5 no 4 pp 273ndash287 2013

[14] D Srinivasacharya and C RamReddy ldquoNatural convection heatand mass transfer in a micropolar fluid with thermal and massstratificationrdquo International Journal for Computational Methodsin Engineering Science andMechanics vol 14 no 5 pp 401ndash4132013

[15] N A Yacob and A Ishak ldquoStagnation point flow towardsa stretchingshrinking sheet in a micropolar fluid with aconvective surface boundary conditionrdquoThe Canadian Journalof Chemical Engineering vol 90 no 3 pp 621ndash626 2012

[16] K Zaimi and A Ishak ldquoStagnation-point flow and heat transferover a nonlinearly stretchingshrinking sheet in a micropolarfluidrdquo Abstract and Applied Analysis vol 2014 Article ID261630 6 pages 2014

[17] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[18] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[19] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer 2009

[20] T Tapanidis G Tsagas and H P Mazumdar ldquoApplication ofscaling group of transformations to viscoelastic second-gradefluid flowrdquo Nonlinear Functional Analysis and Applications vol8 no 3 pp 345ndash350 2003

[21] I A A Hassanien andM A Hamad ldquoGroup theoretic methodfor unsteady free convection flow of a micropolar fluid along avertical plate in a thermally stratified mediumrdquo Applied Mathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 32 no 6 pp 1099ndash1114 2008

[22] R Kandasamy K Gunasekaran and S B H Hasan ldquoScalinggroup transformation on fluid flow with variable stream con-ditionsrdquo International Journal of Non-Linear Mechanics vol 46no 7 pp 976ndash985 2011

[23] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat TransfermdashAsian Research vol41 no 3 pp 241ndash259 2012

16 Advances in High Energy Physics

[24] A A Mutlag M J Uddin M A A Hamad and A I M IsmailldquoHeat transfer analysis for falkner-skan boundary layer flowpast a stationary wedge with slips boundary conditions con-sidering temperature-dependent thermal conductivityrdquo SainsMalaysiana vol 42 no 6 pp 855ndash862 2013

[25] S C Cowin ldquoPolar fluidsrdquo Physics of Fluids vol 11 no 9 pp1919ndash1927 1968

[26] M A Seddeek M Y Akl and A M Al-Hanaya ldquoThermalradiation effects on mixed convection and mass transfer flowon vertical porous plate with heat generation and chemicalreaction by using scaling grouprdquo Journal of Natural Sciences andMathematics vol 4 pp 41ndash60 2010

[27] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[28] M J Uddin W A Khan and A I M Ismail ldquoMHD freeconvective boundary layer flow of a nanofluid past a flat verticalplate with Newtonian heating boundary conditionrdquo PLoS ONEvol 7 no 11 Article ID e49499 2012

[29] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012

[30] M J Uddin W A Khan and A I M Ismail ldquoScalinggroup transformation for MHD boundary layer slip flow of ananofluid over a convectively heated stretching sheet with heatgenerationrdquo Mathematical Problems in Engineering vol 2012Article ID 934964 20 pages 2012

[31] MM RashidiM Ferdows A B Parsa and S Abelman ldquoMHDnatural convection with convective surface boundary conditionover a at platerdquo Abstract and Applied Analysis vol 2014 ArticleID 923487 10 pages 2014

[32] R E Bellman and R E Kalaba Quasilinearisation and Non-Linear Boundary-Value Problems Elsevier New York NY USA1965

[33] S SMotsa P GDlamini andMKhumalo ldquoSpectral relaxationmethod and spectral quasilinearization method for solvingunsteady boundary layer flow problemsrdquo Advances in Mathe-matical Physics vol 2014 Article ID 341964 12 pages 2014

[34] S S Motsa P Sibanda J M Ngnotchouye and G T MarewoldquoA spectral relaxation approach for unsteady boundary-layerflow and heat transfer of a nanofluid over a permeable stretch-ingshrinking sheetrdquo Advances in Mathematical Physics vol2014 Article ID 564942 10 pages 2014

[35] C Canuto M Y Hussaini A Quarteroni and T ZangSpectral Methods Fundamentals in Single Domains ScientificComputation Springer Berlin Germany 2006

[36] J HMerkin ldquoFree convection boundary layer on an isothermalhorizontal cylinderrdquo in Proceedings of the American Societyof Mechanical Engineers and American Institute of ChemicalEngineersHeat Transfer Conference p 911 ASME St LouisMoUSA August 1976

[37] R Nazar N Amin and I Pop ldquoFree convection boundary layeron an isothermal horizontal circular cylinder in a micropolarfluidrdquo in Proceedings of the 12th International Heat TransferConference vol 2 pp 525ndash530 Elsevier Paris France August2002

[38] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 13

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

fw = 05 n = 05

01

00

02

03

04

05

06

07Nu x

Grminus

14

x

(a)

fw = 05 n = 05

01 02 03 04 05 06 07 08 09

N

Bi = 01Bi = 10

Bi = 50Bi = 200

024

026

028

030

032

ShxGrminus

14

x

(b)

Figure 6 Effect of Bi on (a) heat transfer rate and (b) mass transfer rate

005

006

007

008

009

010

Nu x

Grminus

14

x

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

(a)

Bi = 01 n = 05

fw = minus05

fw = 00

fw = 10

fw = 20

01 02 03 04 05 06 07 08 09

N

01

02

03

04

05

06

ShxGrminus

14

x

(b)

Figure 7 Effect of 119891119908on (a) heat transfer rate and (b) mass transfer rate

and 119899 = 05 on local heat transfer coefficient is exhibitedin Figure 6 It is found from Figure 6 that local heat transferrate increases nonlinearly with the increase in Biot numberBiThe influence of the Biot number Bi on local mass transfercoefficient is shown in Figure 6 Figure 6 reveals that the localmass transfer coefficient is enhanced by the growth in theBiot number Bi In Figure 7 the effect of the suctioninjectionparameter 119891

119908with fixed Bi = 01 and 119899 = 05 on local heat

and mass transfer coefficients is displayed It is found fromFigure 7 that the local heat and mass transfer coefficients areless in the case of injection 119891

119908lt 0 in comparison with the

case of suction 119891119908

gt 0 Figure 8 is prepared to analyze theeffect of the material parameter 119899 with fixed Bi = 01 and

119891119908= 05 on local heat andmass transfer coefficients Figure 8

reveals that the local heat and mass transfer coefficients areenhanced by the increase in material parameter 119899 This isbecause when 119899 increases from 0 to 1 the microrotation termgets augmented and induces flow enhancement

The variations of 119862119891Gr14119909

and 119872119908Gr12119909

which areproportional to the coefficients of skin friction and wallcouple stress are shown in Table 3 with different values ofthe coupling number 119873 for fixed 119899 = 05 119891

119908= 05 and

Bi = 01 It indicates that the skin friction factor is higherfor micropolar fluid than the viscous fluids (119873 = 0) Sincemicropolar fluids offer a heavy resistance (resulting fromvortex viscosity) to fluid movement and cause larger skin

14 Advances in High Energy Physics

0084

0085

0086

0087Nu x

Grminus

14

x

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

fw = 05 Bi = 01

(a)

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

024

026

028

030

ShxGrminus

14

x

fw = 05 Bi = 01

(b)

Figure 8 Effect of material parameter 119899 on (a) heat transfer rate and (b) mass transfer rate

Table 3 Effects of skin friction and wall couple stress for varyingvalues of Biot numbers Bi micropolar parameter 119873 materialparameter 119899 and suctioninjection parameter 119891

119908

119873 Bi 119891119908

119899 119862119891Gr14119909

119872119908Gr12119909

01 01 05 05 2424227 06327903 01 05 05 2502716 063996605 01 05 05 2647987 063086307 01 05 05 2943561 060387909 01 05 05 3874631 055089105 01 05 05 2647987 063086305 10 05 05 3211755 080587905 50 05 05 3533141 090807805 200 05 05 3629791 093913705 01 minus05 05 2426265 033621905 01 00 05 2530014 046542905 01 10 05 2737553 082089905 01 20 05 2721244 119359605 01 05 00 2916655 minus028933905 01 05 05 2647987 063086305 01 05 10 2232556 2066191

friction factor compared to viscous fluid the results as wellsuggest larger values of coupling number 119873 and lower wallcouple stresses Since the skin friction coefficient 11989110158401015840(0) andwall couple stress coefficient as well as high temperature andmass transport rates are more depressed in the micropolarfluid comparing to the viscous fluid which may be beneficialin flow temperature and concentration control of polymerprocessing thus the presence of microscopic effects arisingfrom the local structure and of the fluid elements reduces thehigh temperature and mass transfer coefficients The effectof Bi on 119862

119891Gr14119909

and 119872119908Gr12119909

for 119891119908

= 05 119899 = 05 and

119873 = 05 is illustrated in Table 3 It can be noticed that theskin friction and wall couple stress coefficients are increasingwith increase of Bi for fixed values of other parametersThis notice is consistent with physical profiles presented inFigure 3 The effects of suctioninjection parameter on theskin friction andwall couple stress coefficients are also shownin Table 3 It is noted that the skin friction and wall couplestress coefficients are less with injection case whereas theyare more with suction case when compared to the case ofimpermeable surface Finally the detailed behavior of thematerial parameter 119899 is given in Table 3 The skin frictiondecreases and wall couple stress increases with increase ofmaterial parameter 119899

8 Conclusions

In this composition the similarity solution of the free convec-tion flow on a permeable vertical plate of a micropolar fluidunder the convective boundary condition is obtained usingLie group transformations Using the similarity variables thegoverning equations are transformed into a set of nondi-mensional parabolic equations These equations are solvednumerically using spectral quasi-linearisation method Thenumerical computation is carried out for various values ofnondimensional physical parameters The main findings aresummarized as follows

(i) The numerical results indicate that velocity distribu-tion is less near the plate but it is more far away fromthe plate the wall couple stress coefficient and rate ofheat andmass transfers are lower but the temperatureand concentration distributions and the skin frictioncoefficient are higher for the micropolar fluids incomparison with those of viscous fluids Also thereverse rotation ofmicrorotation near two boundariesis found with the increasing value of119873

Advances in High Energy Physics 15

(ii) An increase in Biot number Bi decreases concen-tration distribution whereas it causes an increasein temperature distribution skin friction and wallcouple stress coefficients and heat and mass transferrates within the boundary layer Further enhancingin Biot number Bi enhances velocity distributionnear the plate but shows the reverse behavior faraway from the plate We observe reverse rotationof microrotation near two boundaries within theboundary layer in the presence of Biot number

(iii) Less velocity temperature and concentration distri-butions are observed more skin friction and wallcouple stress coefficients and heat and mass transferrates in the case of suction compared to the case ofinjection Further microrotation decreases near thewall and depicts the opposite trend far away from thewall

(iv) It is found that microrotation temperature and con-centration distributions and skin friction coefficientsare more in the case of a micropolar fluid with strongconcentration (ie 119899 = 0) when compared to the caseof a micropolar fluid with weak concentration (ie119899 = 12) Further velocity is less in the case of 119899 = 0when compared to the case of 119899 = 12 near the walland shows the opposite trend far away from the wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their valuablesuggestions and comments

References

[1] J H Merkin ldquoNatural-convection boundary-layer flow on avertical surface with Newtonian heatingrdquo International Journalof Heat and Fluid Flow vol 15 no 5 pp 392ndash398 1994

[2] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009

[3] O DMakinde ldquoSimilarity solution for natural convection froma moving vertical plate with internal heat generation and aconvective boundary conditionrdquo Thermal Science vol 15 ppS137ndashS143 2011

[4] S M Ibrahim and N Bhashar Reddy ldquoSimilarity solution ofheat and mass transfer for natural convection over a movingvertical plate with internal heat generation and a convectiveboundary condition in the presence of thermal radiationviscous dissipation and chemical reactionrdquo ISRN Thermody-namics vol 2013 Article ID 790604 10 pages 2013

[5] C RamReddy P V S N Murthy A J Chamkha and A MRashad ldquoSoret effect on mixed convection flow in a nanofluidunder convective boundary conditionrdquo International Journal ofHeat and Mass Transfer vol 64 pp 384ndash392 2013

[6] H T Alkasasbeh M Z Salleh R M Tahar and R NazarldquoNumerical solutions of free convection boundary layer flow ona solid sphere with convective boundary conditionsrdquo Journal ofPhysics Conference Series vol 495 Article ID 012025 2014

[7] A Pantokratoras ldquoA note on natural convection along a con-vectively heated vertical platerdquo International Journal of ThermalSciences vol 76 pp 221ndash224 2014

[8] G Lukaszewicz Micropolar FluidsmdashTheory and ApplicationsBirkhaauser Basel Switzerland 1999

[9] V A Eremeyev L P Lebedev and H Altenbach Foundationsof Micropolar Mechanics Springer New York NY USA 2013

[10] G Ahmadi ldquoSelf-similar solution of imcompressible micropo-lar boundary layer flow over a semi-infinite platerdquo InternationalJournal of Engineering Science vol 14 no 7 pp 639ndash646 1976

[11] D A Rees and I Pop ldquoFree convection boundary-layer flowof a micropolar fluid from a vertical flat platerdquo IMA Journal ofApplied Mathematics vol 61 no 2 pp 179ndash197 1998

[12] D Srinivasacharya and C RamReddy ldquoEffect of double stratifi-cation onmixed convection in a micropolar fluidrdquoMatematikavol 28 no 2 pp 133ndash149 2012

[13] D Srinivasacharya and C Ramreddy ldquoMixed convection heatand mass transfer in a doubly stratified micropolar fluidrdquoComputationalThermal Sciences vol 5 no 4 pp 273ndash287 2013

[14] D Srinivasacharya and C RamReddy ldquoNatural convection heatand mass transfer in a micropolar fluid with thermal and massstratificationrdquo International Journal for Computational Methodsin Engineering Science andMechanics vol 14 no 5 pp 401ndash4132013

[15] N A Yacob and A Ishak ldquoStagnation point flow towardsa stretchingshrinking sheet in a micropolar fluid with aconvective surface boundary conditionrdquoThe Canadian Journalof Chemical Engineering vol 90 no 3 pp 621ndash626 2012

[16] K Zaimi and A Ishak ldquoStagnation-point flow and heat transferover a nonlinearly stretchingshrinking sheet in a micropolarfluidrdquo Abstract and Applied Analysis vol 2014 Article ID261630 6 pages 2014

[17] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[18] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[19] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer 2009

[20] T Tapanidis G Tsagas and H P Mazumdar ldquoApplication ofscaling group of transformations to viscoelastic second-gradefluid flowrdquo Nonlinear Functional Analysis and Applications vol8 no 3 pp 345ndash350 2003

[21] I A A Hassanien andM A Hamad ldquoGroup theoretic methodfor unsteady free convection flow of a micropolar fluid along avertical plate in a thermally stratified mediumrdquo Applied Mathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 32 no 6 pp 1099ndash1114 2008

[22] R Kandasamy K Gunasekaran and S B H Hasan ldquoScalinggroup transformation on fluid flow with variable stream con-ditionsrdquo International Journal of Non-Linear Mechanics vol 46no 7 pp 976ndash985 2011

[23] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat TransfermdashAsian Research vol41 no 3 pp 241ndash259 2012

16 Advances in High Energy Physics

[24] A A Mutlag M J Uddin M A A Hamad and A I M IsmailldquoHeat transfer analysis for falkner-skan boundary layer flowpast a stationary wedge with slips boundary conditions con-sidering temperature-dependent thermal conductivityrdquo SainsMalaysiana vol 42 no 6 pp 855ndash862 2013

[25] S C Cowin ldquoPolar fluidsrdquo Physics of Fluids vol 11 no 9 pp1919ndash1927 1968

[26] M A Seddeek M Y Akl and A M Al-Hanaya ldquoThermalradiation effects on mixed convection and mass transfer flowon vertical porous plate with heat generation and chemicalreaction by using scaling grouprdquo Journal of Natural Sciences andMathematics vol 4 pp 41ndash60 2010

[27] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[28] M J Uddin W A Khan and A I M Ismail ldquoMHD freeconvective boundary layer flow of a nanofluid past a flat verticalplate with Newtonian heating boundary conditionrdquo PLoS ONEvol 7 no 11 Article ID e49499 2012

[29] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012

[30] M J Uddin W A Khan and A I M Ismail ldquoScalinggroup transformation for MHD boundary layer slip flow of ananofluid over a convectively heated stretching sheet with heatgenerationrdquo Mathematical Problems in Engineering vol 2012Article ID 934964 20 pages 2012

[31] MM RashidiM Ferdows A B Parsa and S Abelman ldquoMHDnatural convection with convective surface boundary conditionover a at platerdquo Abstract and Applied Analysis vol 2014 ArticleID 923487 10 pages 2014

[32] R E Bellman and R E Kalaba Quasilinearisation and Non-Linear Boundary-Value Problems Elsevier New York NY USA1965

[33] S SMotsa P GDlamini andMKhumalo ldquoSpectral relaxationmethod and spectral quasilinearization method for solvingunsteady boundary layer flow problemsrdquo Advances in Mathe-matical Physics vol 2014 Article ID 341964 12 pages 2014

[34] S S Motsa P Sibanda J M Ngnotchouye and G T MarewoldquoA spectral relaxation approach for unsteady boundary-layerflow and heat transfer of a nanofluid over a permeable stretch-ingshrinking sheetrdquo Advances in Mathematical Physics vol2014 Article ID 564942 10 pages 2014

[35] C Canuto M Y Hussaini A Quarteroni and T ZangSpectral Methods Fundamentals in Single Domains ScientificComputation Springer Berlin Germany 2006

[36] J HMerkin ldquoFree convection boundary layer on an isothermalhorizontal cylinderrdquo in Proceedings of the American Societyof Mechanical Engineers and American Institute of ChemicalEngineersHeat Transfer Conference p 911 ASME St LouisMoUSA August 1976

[37] R Nazar N Amin and I Pop ldquoFree convection boundary layeron an isothermal horizontal circular cylinder in a micropolarfluidrdquo in Proceedings of the 12th International Heat TransferConference vol 2 pp 525ndash530 Elsevier Paris France August2002

[38] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

14 Advances in High Energy Physics

0084

0085

0086

0087Nu x

Grminus

14

x

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

fw = 05 Bi = 01

(a)

n = 00

n = 05

n = 10

01 02 03 04 05 06 07 08 09

N

024

026

028

030

ShxGrminus

14

x

fw = 05 Bi = 01

(b)

Figure 8 Effect of material parameter 119899 on (a) heat transfer rate and (b) mass transfer rate

Table 3 Effects of skin friction and wall couple stress for varyingvalues of Biot numbers Bi micropolar parameter 119873 materialparameter 119899 and suctioninjection parameter 119891

119908

119873 Bi 119891119908

119899 119862119891Gr14119909

119872119908Gr12119909

01 01 05 05 2424227 06327903 01 05 05 2502716 063996605 01 05 05 2647987 063086307 01 05 05 2943561 060387909 01 05 05 3874631 055089105 01 05 05 2647987 063086305 10 05 05 3211755 080587905 50 05 05 3533141 090807805 200 05 05 3629791 093913705 01 minus05 05 2426265 033621905 01 00 05 2530014 046542905 01 10 05 2737553 082089905 01 20 05 2721244 119359605 01 05 00 2916655 minus028933905 01 05 05 2647987 063086305 01 05 10 2232556 2066191

friction factor compared to viscous fluid the results as wellsuggest larger values of coupling number 119873 and lower wallcouple stresses Since the skin friction coefficient 11989110158401015840(0) andwall couple stress coefficient as well as high temperature andmass transport rates are more depressed in the micropolarfluid comparing to the viscous fluid which may be beneficialin flow temperature and concentration control of polymerprocessing thus the presence of microscopic effects arisingfrom the local structure and of the fluid elements reduces thehigh temperature and mass transfer coefficients The effectof Bi on 119862

119891Gr14119909

and 119872119908Gr12119909

for 119891119908

= 05 119899 = 05 and

119873 = 05 is illustrated in Table 3 It can be noticed that theskin friction and wall couple stress coefficients are increasingwith increase of Bi for fixed values of other parametersThis notice is consistent with physical profiles presented inFigure 3 The effects of suctioninjection parameter on theskin friction andwall couple stress coefficients are also shownin Table 3 It is noted that the skin friction and wall couplestress coefficients are less with injection case whereas theyare more with suction case when compared to the case ofimpermeable surface Finally the detailed behavior of thematerial parameter 119899 is given in Table 3 The skin frictiondecreases and wall couple stress increases with increase ofmaterial parameter 119899

8 Conclusions

In this composition the similarity solution of the free convec-tion flow on a permeable vertical plate of a micropolar fluidunder the convective boundary condition is obtained usingLie group transformations Using the similarity variables thegoverning equations are transformed into a set of nondi-mensional parabolic equations These equations are solvednumerically using spectral quasi-linearisation method Thenumerical computation is carried out for various values ofnondimensional physical parameters The main findings aresummarized as follows

(i) The numerical results indicate that velocity distribu-tion is less near the plate but it is more far away fromthe plate the wall couple stress coefficient and rate ofheat andmass transfers are lower but the temperatureand concentration distributions and the skin frictioncoefficient are higher for the micropolar fluids incomparison with those of viscous fluids Also thereverse rotation ofmicrorotation near two boundariesis found with the increasing value of119873

Advances in High Energy Physics 15

(ii) An increase in Biot number Bi decreases concen-tration distribution whereas it causes an increasein temperature distribution skin friction and wallcouple stress coefficients and heat and mass transferrates within the boundary layer Further enhancingin Biot number Bi enhances velocity distributionnear the plate but shows the reverse behavior faraway from the plate We observe reverse rotationof microrotation near two boundaries within theboundary layer in the presence of Biot number

(iii) Less velocity temperature and concentration distri-butions are observed more skin friction and wallcouple stress coefficients and heat and mass transferrates in the case of suction compared to the case ofinjection Further microrotation decreases near thewall and depicts the opposite trend far away from thewall

(iv) It is found that microrotation temperature and con-centration distributions and skin friction coefficientsare more in the case of a micropolar fluid with strongconcentration (ie 119899 = 0) when compared to the caseof a micropolar fluid with weak concentration (ie119899 = 12) Further velocity is less in the case of 119899 = 0when compared to the case of 119899 = 12 near the walland shows the opposite trend far away from the wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their valuablesuggestions and comments

References

[1] J H Merkin ldquoNatural-convection boundary-layer flow on avertical surface with Newtonian heatingrdquo International Journalof Heat and Fluid Flow vol 15 no 5 pp 392ndash398 1994

[2] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009

[3] O DMakinde ldquoSimilarity solution for natural convection froma moving vertical plate with internal heat generation and aconvective boundary conditionrdquo Thermal Science vol 15 ppS137ndashS143 2011

[4] S M Ibrahim and N Bhashar Reddy ldquoSimilarity solution ofheat and mass transfer for natural convection over a movingvertical plate with internal heat generation and a convectiveboundary condition in the presence of thermal radiationviscous dissipation and chemical reactionrdquo ISRN Thermody-namics vol 2013 Article ID 790604 10 pages 2013

[5] C RamReddy P V S N Murthy A J Chamkha and A MRashad ldquoSoret effect on mixed convection flow in a nanofluidunder convective boundary conditionrdquo International Journal ofHeat and Mass Transfer vol 64 pp 384ndash392 2013

[6] H T Alkasasbeh M Z Salleh R M Tahar and R NazarldquoNumerical solutions of free convection boundary layer flow ona solid sphere with convective boundary conditionsrdquo Journal ofPhysics Conference Series vol 495 Article ID 012025 2014

[7] A Pantokratoras ldquoA note on natural convection along a con-vectively heated vertical platerdquo International Journal of ThermalSciences vol 76 pp 221ndash224 2014

[8] G Lukaszewicz Micropolar FluidsmdashTheory and ApplicationsBirkhaauser Basel Switzerland 1999

[9] V A Eremeyev L P Lebedev and H Altenbach Foundationsof Micropolar Mechanics Springer New York NY USA 2013

[10] G Ahmadi ldquoSelf-similar solution of imcompressible micropo-lar boundary layer flow over a semi-infinite platerdquo InternationalJournal of Engineering Science vol 14 no 7 pp 639ndash646 1976

[11] D A Rees and I Pop ldquoFree convection boundary-layer flowof a micropolar fluid from a vertical flat platerdquo IMA Journal ofApplied Mathematics vol 61 no 2 pp 179ndash197 1998

[12] D Srinivasacharya and C RamReddy ldquoEffect of double stratifi-cation onmixed convection in a micropolar fluidrdquoMatematikavol 28 no 2 pp 133ndash149 2012

[13] D Srinivasacharya and C Ramreddy ldquoMixed convection heatand mass transfer in a doubly stratified micropolar fluidrdquoComputationalThermal Sciences vol 5 no 4 pp 273ndash287 2013

[14] D Srinivasacharya and C RamReddy ldquoNatural convection heatand mass transfer in a micropolar fluid with thermal and massstratificationrdquo International Journal for Computational Methodsin Engineering Science andMechanics vol 14 no 5 pp 401ndash4132013

[15] N A Yacob and A Ishak ldquoStagnation point flow towardsa stretchingshrinking sheet in a micropolar fluid with aconvective surface boundary conditionrdquoThe Canadian Journalof Chemical Engineering vol 90 no 3 pp 621ndash626 2012

[16] K Zaimi and A Ishak ldquoStagnation-point flow and heat transferover a nonlinearly stretchingshrinking sheet in a micropolarfluidrdquo Abstract and Applied Analysis vol 2014 Article ID261630 6 pages 2014

[17] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[18] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[19] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer 2009

[20] T Tapanidis G Tsagas and H P Mazumdar ldquoApplication ofscaling group of transformations to viscoelastic second-gradefluid flowrdquo Nonlinear Functional Analysis and Applications vol8 no 3 pp 345ndash350 2003

[21] I A A Hassanien andM A Hamad ldquoGroup theoretic methodfor unsteady free convection flow of a micropolar fluid along avertical plate in a thermally stratified mediumrdquo Applied Mathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 32 no 6 pp 1099ndash1114 2008

[22] R Kandasamy K Gunasekaran and S B H Hasan ldquoScalinggroup transformation on fluid flow with variable stream con-ditionsrdquo International Journal of Non-Linear Mechanics vol 46no 7 pp 976ndash985 2011

[23] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat TransfermdashAsian Research vol41 no 3 pp 241ndash259 2012

16 Advances in High Energy Physics

[24] A A Mutlag M J Uddin M A A Hamad and A I M IsmailldquoHeat transfer analysis for falkner-skan boundary layer flowpast a stationary wedge with slips boundary conditions con-sidering temperature-dependent thermal conductivityrdquo SainsMalaysiana vol 42 no 6 pp 855ndash862 2013

[25] S C Cowin ldquoPolar fluidsrdquo Physics of Fluids vol 11 no 9 pp1919ndash1927 1968

[26] M A Seddeek M Y Akl and A M Al-Hanaya ldquoThermalradiation effects on mixed convection and mass transfer flowon vertical porous plate with heat generation and chemicalreaction by using scaling grouprdquo Journal of Natural Sciences andMathematics vol 4 pp 41ndash60 2010

[27] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[28] M J Uddin W A Khan and A I M Ismail ldquoMHD freeconvective boundary layer flow of a nanofluid past a flat verticalplate with Newtonian heating boundary conditionrdquo PLoS ONEvol 7 no 11 Article ID e49499 2012

[29] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012

[30] M J Uddin W A Khan and A I M Ismail ldquoScalinggroup transformation for MHD boundary layer slip flow of ananofluid over a convectively heated stretching sheet with heatgenerationrdquo Mathematical Problems in Engineering vol 2012Article ID 934964 20 pages 2012

[31] MM RashidiM Ferdows A B Parsa and S Abelman ldquoMHDnatural convection with convective surface boundary conditionover a at platerdquo Abstract and Applied Analysis vol 2014 ArticleID 923487 10 pages 2014

[32] R E Bellman and R E Kalaba Quasilinearisation and Non-Linear Boundary-Value Problems Elsevier New York NY USA1965

[33] S SMotsa P GDlamini andMKhumalo ldquoSpectral relaxationmethod and spectral quasilinearization method for solvingunsteady boundary layer flow problemsrdquo Advances in Mathe-matical Physics vol 2014 Article ID 341964 12 pages 2014

[34] S S Motsa P Sibanda J M Ngnotchouye and G T MarewoldquoA spectral relaxation approach for unsteady boundary-layerflow and heat transfer of a nanofluid over a permeable stretch-ingshrinking sheetrdquo Advances in Mathematical Physics vol2014 Article ID 564942 10 pages 2014

[35] C Canuto M Y Hussaini A Quarteroni and T ZangSpectral Methods Fundamentals in Single Domains ScientificComputation Springer Berlin Germany 2006

[36] J HMerkin ldquoFree convection boundary layer on an isothermalhorizontal cylinderrdquo in Proceedings of the American Societyof Mechanical Engineers and American Institute of ChemicalEngineersHeat Transfer Conference p 911 ASME St LouisMoUSA August 1976

[37] R Nazar N Amin and I Pop ldquoFree convection boundary layeron an isothermal horizontal circular cylinder in a micropolarfluidrdquo in Proceedings of the 12th International Heat TransferConference vol 2 pp 525ndash530 Elsevier Paris France August2002

[38] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 15

(ii) An increase in Biot number Bi decreases concen-tration distribution whereas it causes an increasein temperature distribution skin friction and wallcouple stress coefficients and heat and mass transferrates within the boundary layer Further enhancingin Biot number Bi enhances velocity distributionnear the plate but shows the reverse behavior faraway from the plate We observe reverse rotationof microrotation near two boundaries within theboundary layer in the presence of Biot number

(iii) Less velocity temperature and concentration distri-butions are observed more skin friction and wallcouple stress coefficients and heat and mass transferrates in the case of suction compared to the case ofinjection Further microrotation decreases near thewall and depicts the opposite trend far away from thewall

(iv) It is found that microrotation temperature and con-centration distributions and skin friction coefficientsare more in the case of a micropolar fluid with strongconcentration (ie 119899 = 0) when compared to the caseof a micropolar fluid with weak concentration (ie119899 = 12) Further velocity is less in the case of 119899 = 0when compared to the case of 119899 = 12 near the walland shows the opposite trend far away from the wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their valuablesuggestions and comments

References

[1] J H Merkin ldquoNatural-convection boundary-layer flow on avertical surface with Newtonian heatingrdquo International Journalof Heat and Fluid Flow vol 15 no 5 pp 392ndash398 1994

[2] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009

[3] O DMakinde ldquoSimilarity solution for natural convection froma moving vertical plate with internal heat generation and aconvective boundary conditionrdquo Thermal Science vol 15 ppS137ndashS143 2011

[4] S M Ibrahim and N Bhashar Reddy ldquoSimilarity solution ofheat and mass transfer for natural convection over a movingvertical plate with internal heat generation and a convectiveboundary condition in the presence of thermal radiationviscous dissipation and chemical reactionrdquo ISRN Thermody-namics vol 2013 Article ID 790604 10 pages 2013

[5] C RamReddy P V S N Murthy A J Chamkha and A MRashad ldquoSoret effect on mixed convection flow in a nanofluidunder convective boundary conditionrdquo International Journal ofHeat and Mass Transfer vol 64 pp 384ndash392 2013

[6] H T Alkasasbeh M Z Salleh R M Tahar and R NazarldquoNumerical solutions of free convection boundary layer flow ona solid sphere with convective boundary conditionsrdquo Journal ofPhysics Conference Series vol 495 Article ID 012025 2014

[7] A Pantokratoras ldquoA note on natural convection along a con-vectively heated vertical platerdquo International Journal of ThermalSciences vol 76 pp 221ndash224 2014

[8] G Lukaszewicz Micropolar FluidsmdashTheory and ApplicationsBirkhaauser Basel Switzerland 1999

[9] V A Eremeyev L P Lebedev and H Altenbach Foundationsof Micropolar Mechanics Springer New York NY USA 2013

[10] G Ahmadi ldquoSelf-similar solution of imcompressible micropo-lar boundary layer flow over a semi-infinite platerdquo InternationalJournal of Engineering Science vol 14 no 7 pp 639ndash646 1976

[11] D A Rees and I Pop ldquoFree convection boundary-layer flowof a micropolar fluid from a vertical flat platerdquo IMA Journal ofApplied Mathematics vol 61 no 2 pp 179ndash197 1998

[12] D Srinivasacharya and C RamReddy ldquoEffect of double stratifi-cation onmixed convection in a micropolar fluidrdquoMatematikavol 28 no 2 pp 133ndash149 2012

[13] D Srinivasacharya and C Ramreddy ldquoMixed convection heatand mass transfer in a doubly stratified micropolar fluidrdquoComputationalThermal Sciences vol 5 no 4 pp 273ndash287 2013

[14] D Srinivasacharya and C RamReddy ldquoNatural convection heatand mass transfer in a micropolar fluid with thermal and massstratificationrdquo International Journal for Computational Methodsin Engineering Science andMechanics vol 14 no 5 pp 401ndash4132013

[15] N A Yacob and A Ishak ldquoStagnation point flow towardsa stretchingshrinking sheet in a micropolar fluid with aconvective surface boundary conditionrdquoThe Canadian Journalof Chemical Engineering vol 90 no 3 pp 621ndash626 2012

[16] K Zaimi and A Ishak ldquoStagnation-point flow and heat transferover a nonlinearly stretchingshrinking sheet in a micropolarfluidrdquo Abstract and Applied Analysis vol 2014 Article ID261630 6 pages 2014

[17] L V Ovsiannikov Group Analysis of Differential EquationsAcademic Press New York NY USA 1982

[18] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[19] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer 2009

[20] T Tapanidis G Tsagas and H P Mazumdar ldquoApplication ofscaling group of transformations to viscoelastic second-gradefluid flowrdquo Nonlinear Functional Analysis and Applications vol8 no 3 pp 345ndash350 2003

[21] I A A Hassanien andM A Hamad ldquoGroup theoretic methodfor unsteady free convection flow of a micropolar fluid along avertical plate in a thermally stratified mediumrdquo Applied Mathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 32 no 6 pp 1099ndash1114 2008

[22] R Kandasamy K Gunasekaran and S B H Hasan ldquoScalinggroup transformation on fluid flow with variable stream con-ditionsrdquo International Journal of Non-Linear Mechanics vol 46no 7 pp 976ndash985 2011

[23] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat TransfermdashAsian Research vol41 no 3 pp 241ndash259 2012

16 Advances in High Energy Physics

[24] A A Mutlag M J Uddin M A A Hamad and A I M IsmailldquoHeat transfer analysis for falkner-skan boundary layer flowpast a stationary wedge with slips boundary conditions con-sidering temperature-dependent thermal conductivityrdquo SainsMalaysiana vol 42 no 6 pp 855ndash862 2013

[25] S C Cowin ldquoPolar fluidsrdquo Physics of Fluids vol 11 no 9 pp1919ndash1927 1968

[26] M A Seddeek M Y Akl and A M Al-Hanaya ldquoThermalradiation effects on mixed convection and mass transfer flowon vertical porous plate with heat generation and chemicalreaction by using scaling grouprdquo Journal of Natural Sciences andMathematics vol 4 pp 41ndash60 2010

[27] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[28] M J Uddin W A Khan and A I M Ismail ldquoMHD freeconvective boundary layer flow of a nanofluid past a flat verticalplate with Newtonian heating boundary conditionrdquo PLoS ONEvol 7 no 11 Article ID e49499 2012

[29] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012

[30] M J Uddin W A Khan and A I M Ismail ldquoScalinggroup transformation for MHD boundary layer slip flow of ananofluid over a convectively heated stretching sheet with heatgenerationrdquo Mathematical Problems in Engineering vol 2012Article ID 934964 20 pages 2012

[31] MM RashidiM Ferdows A B Parsa and S Abelman ldquoMHDnatural convection with convective surface boundary conditionover a at platerdquo Abstract and Applied Analysis vol 2014 ArticleID 923487 10 pages 2014

[32] R E Bellman and R E Kalaba Quasilinearisation and Non-Linear Boundary-Value Problems Elsevier New York NY USA1965

[33] S SMotsa P GDlamini andMKhumalo ldquoSpectral relaxationmethod and spectral quasilinearization method for solvingunsteady boundary layer flow problemsrdquo Advances in Mathe-matical Physics vol 2014 Article ID 341964 12 pages 2014

[34] S S Motsa P Sibanda J M Ngnotchouye and G T MarewoldquoA spectral relaxation approach for unsteady boundary-layerflow and heat transfer of a nanofluid over a permeable stretch-ingshrinking sheetrdquo Advances in Mathematical Physics vol2014 Article ID 564942 10 pages 2014

[35] C Canuto M Y Hussaini A Quarteroni and T ZangSpectral Methods Fundamentals in Single Domains ScientificComputation Springer Berlin Germany 2006

[36] J HMerkin ldquoFree convection boundary layer on an isothermalhorizontal cylinderrdquo in Proceedings of the American Societyof Mechanical Engineers and American Institute of ChemicalEngineersHeat Transfer Conference p 911 ASME St LouisMoUSA August 1976

[37] R Nazar N Amin and I Pop ldquoFree convection boundary layeron an isothermal horizontal circular cylinder in a micropolarfluidrdquo in Proceedings of the 12th International Heat TransferConference vol 2 pp 525ndash530 Elsevier Paris France August2002

[38] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

16 Advances in High Energy Physics

[24] A A Mutlag M J Uddin M A A Hamad and A I M IsmailldquoHeat transfer analysis for falkner-skan boundary layer flowpast a stationary wedge with slips boundary conditions con-sidering temperature-dependent thermal conductivityrdquo SainsMalaysiana vol 42 no 6 pp 855ndash862 2013

[25] S C Cowin ldquoPolar fluidsrdquo Physics of Fluids vol 11 no 9 pp1919ndash1927 1968

[26] M A Seddeek M Y Akl and A M Al-Hanaya ldquoThermalradiation effects on mixed convection and mass transfer flowon vertical porous plate with heat generation and chemicalreaction by using scaling grouprdquo Journal of Natural Sciences andMathematics vol 4 pp 41ndash60 2010

[27] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[28] M J Uddin W A Khan and A I M Ismail ldquoMHD freeconvective boundary layer flow of a nanofluid past a flat verticalplate with Newtonian heating boundary conditionrdquo PLoS ONEvol 7 no 11 Article ID e49499 2012

[29] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012

[30] M J Uddin W A Khan and A I M Ismail ldquoScalinggroup transformation for MHD boundary layer slip flow of ananofluid over a convectively heated stretching sheet with heatgenerationrdquo Mathematical Problems in Engineering vol 2012Article ID 934964 20 pages 2012

[31] MM RashidiM Ferdows A B Parsa and S Abelman ldquoMHDnatural convection with convective surface boundary conditionover a at platerdquo Abstract and Applied Analysis vol 2014 ArticleID 923487 10 pages 2014

[32] R E Bellman and R E Kalaba Quasilinearisation and Non-Linear Boundary-Value Problems Elsevier New York NY USA1965

[33] S SMotsa P GDlamini andMKhumalo ldquoSpectral relaxationmethod and spectral quasilinearization method for solvingunsteady boundary layer flow problemsrdquo Advances in Mathe-matical Physics vol 2014 Article ID 341964 12 pages 2014

[34] S S Motsa P Sibanda J M Ngnotchouye and G T MarewoldquoA spectral relaxation approach for unsteady boundary-layerflow and heat transfer of a nanofluid over a permeable stretch-ingshrinking sheetrdquo Advances in Mathematical Physics vol2014 Article ID 564942 10 pages 2014

[35] C Canuto M Y Hussaini A Quarteroni and T ZangSpectral Methods Fundamentals in Single Domains ScientificComputation Springer Berlin Germany 2006

[36] J HMerkin ldquoFree convection boundary layer on an isothermalhorizontal cylinderrdquo in Proceedings of the American Societyof Mechanical Engineers and American Institute of ChemicalEngineersHeat Transfer Conference p 911 ASME St LouisMoUSA August 1976

[37] R Nazar N Amin and I Pop ldquoFree convection boundary layeron an isothermal horizontal circular cylinder in a micropolarfluidrdquo in Proceedings of the 12th International Heat TransferConference vol 2 pp 525ndash530 Elsevier Paris France August2002

[38] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of