Research ArticleAnalytical Modelling of Three-Dimensional Squeezing NanofluidFlow in a Rotating Channel on a Lower Stretching Porous Wall

Navid Freidoonimehr1 Behnam Rostami1

Mohammad Mehdi Rashidi2 and Ebrahim Momoniat3

1 Young Researchers amp Elite Club Islamic Azad University Hamedan Branch Hamedan 65181 15743 Iran2Mechanical Engineering Department Engineering Faculty of Bu-Ali Sina University Hamedan 65178 38695 Iran3 Centre for Differential Equations Continuum Mechanics and Applications School of Computational and Applied MathematicsUniversity of the Witwatersrand Private Bag 3 Johannesburg 2050 South Africa

Correspondence should be addressed to Mohammad Mehdi Rashidi mm rashidiyahoocom

Received 3 July 2014 Accepted 30 July 2014 Published 17 August 2014

Academic Editor Sandile Motsa

Copyright copy 2014 Navid Freidoonimehr et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A coupled system of nonlinear ordinary differential equations that models the three-dimensional flow of a nanofluid in a rotatingchannel on a lower permeable stretching porous wall is derived The mathematical equations are derived from the Navier-Stokes equations where the governing equations are normalized by suitable similarity transformations The fluid in the rotatingchannel is water that contains different nanoparticles silver copper copper oxide titanium oxide and aluminum oxide Thedifferential transform method (DTM) is employed to solve the coupled system of nonlinear ordinary differential equations Theeffects of the following physical parameters on the flow are investigated characteristic parameter of the flow rotation parameterthe magnetic parameter nanoparticle volume fraction the suction parameter and different types of nanoparticles Results areillustrated graphically and discussed in detail

1 Introduction

Due to the vast number of applications in chemical as well asmechanical engineering processes such as the manufactureof thin plastic sheets insulating materials paper fabricationand other various processes this paper pays particularattention to the investigation of rotating flow over stretchingsurfaces [1]

Sakiadis [2 3] initiated the study of boundary-layer flowover a continuous solid surface that moves with constantspeed Considering the effect of suctioninjection Ericksonet al [4] studied heat and mass transfer over a movingsurface with constant surface velocity and temperature Tsouet al [5] studied heat transfer effects of a moving solidsurface with constant velocity and temperature Crane [6]investigated the two-dimensional flow of a viscous fluid overa stretching wall Andersson [7] investigated MHD effects

on boundary-layer flow of a viscoelastic fluid flow past astretching sheet Prasad et al [8] employed a fourth orderRunge-Kutta integration scheme to investigate the effect ofvariable fluid viscosity magnetic parameter Prandtl numbervariable thermal conductivity heat sourcesink parameterand thermal radiation parameter on MHD fluid flow over astretching sheetThe generalized three-dimensional flow andheat transfer over a stretching sheet and in a channel boundedby the lower stretching plate and upper permeable wall werestudied by Mehmood and Ali [9 10] Three-dimensionalflow in a channel with a stretching wall was investigated byBorkakoti and Bharali [11] Munawar et al [12] analyzed theslip effect on the flow in a channel bounded by two stretchingdisks analytically

Flow squeezed by two parallel plates has been investigatedby various researchers Chamkha et al [13] determined ananalytical solution for the problem of fully developed free

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 692728 14 pageshttpdxdoiorg1011552014692728

2 Mathematical Problems in Engineering

convective flow of a micropolar fluid between two verticalparallel plates Bhargava et al [14] investigated the fullydeveloped flow and heat transfer of an electrically conductingmicropolar fluid (a strong cross magnetic field) between twoparallel and porous plates in which the temperature has beenconsidered to be dependent on the heat source including theeffect of frictional heating The quasilinearization method isused to solve the governing system of ordinary differentialequations Ariel [15] presented solutions for two problemsof laminar forced convection of a second-grade (viscoelastic)fluid through two parallel porous walls considering rectan-gular and cylindrical geometries Applying similarity trans-formations to the governing nonlinear partial differentialequations Hayat and Abbas [16] derived nonlinear ordi-nary differential equations and studied the two-dimensionalboundary-layer flow of an upper-convected Maxwell fluid ina channel with chemical reaction Domairry and Aziz [17]obtained an analytic solution for unsteady MHD squeezingflowwith suction and injection effects by the use of homotopyperturbation method

To the best of authorsrsquo knowledge Choi and Eastman[18] were probably the first researchers who employed amixture of nanoparticles and base fluid and called thismixture a ldquonanofluidrdquo A wide range of review papers havebeen published on nanofluids in recent years Xuan and Li[19] considered the Reynolds number and volume fraction ofnanoparticle influences in turbulent flows for nanofluids intubes experimentally Bachok et al [20] employed a numer-ical Keller-Box technique for steady nanofluid flow over aporous rotating disk Khan and Pop [21] studied laminarflow for a nanofluid across a stretching flat surface usingan implicit finite difference method Abolbashari et al [22]employed HAM to study the entropy analysis in an unsteadyMHDnanofluid regime adjacent to an accelerating stretchingpermeable surface Beg et al [23] presented a comparativenumerical solution for both single- and two-phase modelsfor bionanofluid transport phenomena Rashidi et al [24]compared the two phase and single phase of heat transferand flow field of copper-water nanofluid in a wavy channelnumerically Abu-Nada et al [25] illustrated the impactsof variable properties in natural convection nanofluid flowRashidi et al [26] showed how the second law of thermo-dynamics can be applied to MHD incompressible nanofluidflow over a porous rotating disk The stagnation flow for ananofluid over a stretching sheet was studied by Mustafa etal [27] analytically The interested reader is referred to thefollowing papers for further reading on the application ofnanoparticles in fluid flow [28ndash30]

In this paper we derive a coupled system of non-linear ordinary differential equations to model the three-dimensional flow of a nanofluid in a rotating channel ona lower permeable stretching wall The resulting system ofequations is solved using the differential transform method(DTM) [31 32] The DTM method has been applied suc-cessfully to solve nonlinear differential equations withoutrequiring linearization or discretization [33] The presentDTM code is benchmarked with numerical results basedon a shooting technique and previously published results

y = h(t)

B0

Nanofluid

y = 0

z w

V0

Vh(t)

x u

U0

Ω

y

Figure 1 Schematic diagram of the flow configuration and thecoordinate system for the considered flow

The DTM method shows excellent correlation with resultsobtained using these other methods

The paper is divided up as follows in Section 2 theproblem is formulated and a coupled system of nonlinearordinary differential equations is derived The DTM methodis applied to solve the resulting system of nonlinear ordinarydifferential equations in Section 3 Results are discussed inSection 4 Concluding remarks are made in Section 5

2 Mathematical Formulation of the Problem

Consider unsteady 3D rotating nanofluid flow of an incom-pressible electrically conducting viscous fluid between twoinfinite horizontal plane walls The lower plane is placed at119910 = 0 and is stretched with a time-dependent velocity119880

0(119905) =

119886119909(1 minus 120574119905) in x-direction The upper plane is also placedat a variable distance ℎ(119905) = radic]

119891(1 minus 120574119905)119886 and the fluid is

squeezed with a time-dependent velocity 119881ℎ= 119889ℎ119889119905 in the

negative y-direction The fluid and the channel are rotatingaround y-axis with an angular velocity Ω = 120596119869(1 minus 120574119905)

and also the lower plate intakes the flow with the velocityminus1198810(1 minus 120574119905) A magnetic field with density 119861

0radic(1 minus 120574119905) is

applied along the y-axis about the system which is rotatingThese velocities and magnetic fields are introduced to obtainsimilarity solutions by reducing governing equations intoordinary differential equations (ODEs) The physical modelof the considered problemalongwith the coordinate system isillustrated in Figure 1 The governing equations of continuityand momentum of nanofluid flow in a rotating frame ofreference are given as [34 35]

nabla sdot V = 0

120588nf [120597V120597119905

+ (V sdot nabla)V + 2Ω times V] = nabla sdot T + J times B(1)

Mathematical Problems in Engineering 3

Table 1 Thermophysical properties of the base fluid and different nanoparticles [39]

Physical properties Fluid phase (water) Cu CuO Ag Al2O3 TiO2

120588 (kgm3) 9971 8933 6320 10500 3970 4250

where T is the Cauchy stress tensor J the magnetic flux andB the current density The above governing equations can bealso described by the following set of Navier-Stokes equations[36 37]

120597119906

120597119909+120597V120597119910

= 0

120597119906

120597119905+ 119906

120597119906

120597119909+ V

120597119906

120597119910+ 2

120596

1 minus 120574119905119908

= minus1

120588nf

120597119901

120597119909+ ]nf [

1205972119906

1205971199092+1205972119906

1205971199102] minus

1205901198612

0

120588nf (1 minus 120574119905)119906

120597V120597119905

+ 119906120597V120597119909

+ V120597V120597119910

= minus1

120588nf

120597119901

120597119910+ ]nf [

1205972V1205971199092

+1205972V1205971199102

]

120597119908

120597119905+ 119906

120597119908

120597119909+ V

120597119908

120597119910minus 2

120596

1 minus 120574119905119906

= ]nf [1205972119908

1205971199092+1205972119908

1205971199102] minus

1205901198612

0

120588nf (1 minus 120574119905)119908

(2)

where 120588nf is the nanofluid density ]nf (= 120583nf120588nf) is thenanofluid kinematic viscosity where 120583nf has been proposedby Brinkman [38] 120590 is the electrical conductivity 119861

0is the

magnetic field and 120574 is the characteristic parameter withdimension of (time)minus1 and 120574119905 lt 1 The above nanofluidconstants are defined as follows

120583nf =120583119891

(1 minus 120593)25 120588nf = (1 minus 120593) 120588

119891+ 120593120588119904 (3)

where 120583119891

is the viscosity of the fluid fraction 120593 is thenanoparticle volume fraction and 120588

119891and 120588119904are the densities

of the fluid and of the solid fractions respectively Thethermophysical properties of the base fluid (water) and dif-ferent nanoparticles are given in Table 1 [39]The appropriateboundary conditions are introduced as follows

119906 (119909 119910 119905) = 1198800=

119886119909

1 minus 120574119905

V (119909 119910 119905) = minus1198810

1 minus 120574119905

119908 (119909 119910 119905) = 0

at 119910 = 0

119906 (119909 119910 119905) = 0

V (119909 119910 119905) = 119881ℎ=119889ℎ

119889119905=minus120574

2radic

]119891

119886 (1 minus 120574119905)

119908 (119909 119910 119905) = 0

at 119910 = ℎ (119905)

(4)

where 119886 is the stretching rate of the lower plateThe followingappropriate similarity transformations are employed to con-vert the above governing equations (2)ndash(4) into a system of

ordinary differential equations in terms of a stream function120595

120595 = radic119886]119891

1 minus 120574119905119909119891 (120578) 120578 =

119910

ℎ (119905) (5)

119906 =120597120595

120597119910= 11988001198911015840(120578) V = minus

120597120595

120597119909= minusradic

119886]119891

1 minus 120574119905119891 (120578)

119908 = 1198800119892 (120578)

(6)

Substituting the similarity transformations (5) and (6) into(2)ndash(4) we obtain the following system of nonlinear ordinarydifferential equations

119891101584010158401015840+]119891

]nf(11989111989110158401015840minus (1198911015840)2

minus 120573(1198911015840+120578

211989110158401015840)

minus2Ω119892 minus1

120588nf12058811989111987221198911015840) =

(1 minus 120574119905)2]119891

120588nf1198862119909]nf

120597119901

120597119909

minus11989110158401015840+]119891

]nf(minus119891119891

1015840+120573

2(119891 + 120578119891

1015840)) =

1 minus 120574119905

120588nf119886]nf

120597119901

120597120578

(7)

11989210158401015840+]119891

]nf(1198911198921015840minus 1198911015840119892 minus 120573(119892 +

120578

21198921015840) + 2Ω119891

1015840)

minus1

120583nf1205831198911198722119892 = 0

(8)

where 120573 = 120574119886 is the characteristic parameter of the flowΩ = 120596119886 is the rotation parameter 1198722 = 120590119861

2

0120588119891119886 is the

magnetic parameter and prime denotes differentiation withrespect to 120578 In order to squeeze the flow we take 120573 gt 0 forwhich the upper plate moves downward with velocity119881

ℎlt 0

For 120573 lt 0 the upper plate moves apart with respect to theplane 119910 = 0 and 120573 = 0 corresponds to the steady statecase of the considered problem or stationary upper plate Inorder to reduce the number of independent variables andto retain the similarity solution (7) are simplified by crossdifferentiation and thus we obtain the following system ofdifferential equations

119891119894Vminus (1 minus 120593 + 120593 (120588

119904120588119891)) (1 minus 120593)

25

times (119891101584011989110158401015840minus 119891119891101584010158401015840+ 2Ω119892

1015840+120573

2(311989110158401015840+ 120578119891101584010158401015840))

minus (1 minus 120593)25119872211989110158401015840= 0

11989210158401015840+ (1 minus 120593 + 120593 (120588

119904120588119891)) (1 minus 120593)

25

times (1198911198921015840minus 1198911015840119892 minus 120573(119892 +

120578

21198921015840) + 2Ω119891

1015840)

minus (1 minus 120593)251198722119892 = 0

(9)

4 Mathematical Problems in Engineering

The transformed boundary conditions take the form

119891 (0) = 1199080 119891

1015840(0) = 1 119892 (0) = 0

119891 (1) =120573

2 119891

1015840(1) = 0 119892 (1) = 0

(10)

where 1199080= 1198810119886ℎ is the suction parameter

In this problem the physical quantity of interest is theskin friction coefficients 119862

119891along the stretching wall at the

lower and upper walls which are defined as [40]

119862119891lower =

120583nf(120597119906120597119910)119910=0

120588nf1198802

0

119862119891upper =

120583nf(120597119906120597119910)119910=ℎ(119905)

120588nf1198802

0

(11)

Substituting (6) into (11) we obtain

119862119891lower = 119862

119891lowerRe119909 =11989110158401015840(0)

(1 minus 120593 + 120593 (120588119904120588119891)) (1 minus 120593)

25

119862119891upper = 119862

119891upperRe119909 =11989110158401015840(1)

(1 minus 120593 + 120593 (120588119904120588119891)) (1 minus 120593)

25

(12)

where Re119909= 1205881198911198800ℎ120583119891is the local Reynolds number

In this section we have derived the coupled system ofnonlinear ordinary differential equations (9) to model thethree-dimensional flow of a nanofluid in a rotating channelon a lower permeable stretchingwallWe next apply theDTMto solve the coupled system

3 Analytical Approximation by Means of DTM

Taking the differential transform of (9) one can obtain (formore details see [41 42])

(119896 + 1) (119896 + 2) (119896 + 3) (119896 + 4) 119891 (119896 + 4)

minus (1 minus 120593)251198722(119896 + 1) (119896 + 2) 119891 (119896 + 2)

minus (1 minus 120593 + 120593(120588119904

120588119891

)) (1 minus 120593)25

times (

119896

sum

119903=0

((119896 minus 119903 + 1) (119896 minus 119903 + 2) (119903 + 1) 119891 (119903 + 1)

times 119891 (119896 minus 119903 + 2) minus (119896 minus 119903 + 1) (119896 minus 119903 + 2)

times (119896 minus 119903 + 3) 119891 (119903) 119891 (119896 minus 119903 + 3))

+120573

2(3 (119896 + 1) (119896 + 2) 119891 (119896 + 2)

+

119896

sum

119903=0

(

(119896 minus 119903 + 1) (119896 minus 119903 + 2)

(119896 minus 119903 + 3) 120575 (119903) 119891 (119896 minus 119903 + 3)

))

+2Ω (119896 + 1) 119892 (119896 + 1)) = 0

(119896 + 1) (119896 + 2) 119892 (119896 + 2) minus (1 minus 120593)251198722119892 (119896)

+ (1 minus 120593 + 120593(120588119904

120588119891

)) (1 minus 120593)25

times ((

119896

sum

119903=0

((119896 minus 119903 + 1) 119891 (119903) 119892 (119896 minus 119903 + 1)

minus (119896 minus 119903 + 1) 119892 (119903) 119891 (119896 minus 119903 + 1)))

+ 2Ω (119896 + 1) 119891 (119896 + 1)

minus 120573(119892 (119896) + (1

2)

times

119896

sum

119903=0

((119896 minus 119903 + 1) 120575 (119903) 119892 (119896 minus 119903 + 1)))) = 0

(13)

where 119891(119896) and 119892(119896) are differential transforms of 119891(120578) and119892(120578) given by

119891 (120578) =

infin

sum

119896=0

119891 (119896) 120578119896 (14)

119892 (120578) =

infin

sum

119896=0

119892 (119896) 120578119896 (15)

119891 (0) = 1199080 119891 (1) = 1 119891 (2) = 120572

119891 (3) = 120575 119892 (0) = 0 119892 (1) = 120576

(16)

where (16) is the transformed boundary conditions and 120572 120575and 120576 are constants By substituting (16) into (13) usingrecursion and then substituting them in (14)-(15) we obtainthe values of 119891(120578) and 119892(120578) given by

Mathematical Problems in Engineering 5

119891 (120578) = 1199080+ 120578 + 120572120578

2+ 1205751205783+

1

24

times(21198722(1 minus 120593)

25120572 + (1 minus 120593)

25(1 minus 120593 +

120588119904

120588119891

120593)

(2120572 + 3120572120573 minus 61199080120575 + 2Ω120576)

)

times 1205784+ sdot sdot sdot

119892 (120578) = 120578120576 +1

2(1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)

times (minus2Ω minus 1199080120576) 1205782+1

6

times(

1198722(1 minus 120593)

25120576 + (1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)

(minus4Ω120572 +3120573120576

2minus (1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)1199080(minus2Ω minus 119908

0120576))

)

times 1205783+

1

12

times((

(

1

21198722(1 minus 120593)

5(1 minus 120593 + (

120588119904

120588119891

)120593) (minus2Ω minus 1199080120576)

+ (1 minus 120593)25(1 minus 120593 + (

120588119904

120588119891

)120593)

times((

(

minus 6Ω120575 minus1

2(1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593) (minus2Ω minus 1199080120576)

+ 120573(1 minus 120593)25(1 minus 120593 + (

120588119904

120588119891

)120593) (minus2Ω minus 1199080120576) + 120572120576 minus

1

21199080

times((

(

1198722(1 minus 120593)

25120576 + (1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)

(

minus4Ω120572 +3120573120576

2

minus (1 minus 120593)25(1 minus 120593 + (

120588119904

120588119891

)120593)1199080(minus2Ω minus 119908

0120576)

)

))

)

))

)

))

)

times1205784+ sdot sdot sdot

(17)

The constants 120572 120575 and 120576 can be determined by applyingthe remaining boundary conditions to (17) The numberof required terms is determined by the convergence of the

numerical values to onersquos desired accuracy We obtain theapproximants using the computational software MATHE-MATICA The effect of different orders of the DTM solution

6 Mathematical Problems in Engineering

0 02 04 06 08 1

0

05

1

DTM n = 5DTM n = 10DTM n = 20

DTM n = 30DTM n = 40Numerical

g(120578

)

120578

minus05

minus1

minus15

minus2

minus25

Figure 2 The obtained results of transverse velocity of 119892(120578) fordifferent orders of DTM solution in comparison with the numericalsolution when 120573 = minus1Ω = 3119872 = 2 120593 = 01 and 119908

0= 05

on the convergence radius of the transverse velocity of 119892(120578) isdisplayed in Figure 2

In order to highlight the validity of the presented DTMsolution we compare some of our results with the obtainedHAM results of [1] A very good validation of the presentanalytical results has been achieved with the previouslypublished study as shown in Table 2

4 Results and Discussion

In the previous section the nonlinear ordinary differentialequations (9) are solved subject to the boundary condi-tions (10) via the DTM method for values of the five keyparameters characteristic parameter of the flow (120573) rotationparameter (Ω) the magnetic parameter (119872) nanoparticlevolume fraction (120593) the suction parameter (119908

0) and different

nanoparticles on the different velocity components It shouldbe stated that the copper nanoparticle is used in all of the casesin this section except for those figures which focus on theinfluences of the types of applied nanoparticles on the velocitycomponent profiles In addition we assume that the valuesof the volume fraction parameter 120593 vary from 0 (regularNewtonian fluid) to 02 Representative values are employedto simulate the physically realistic flows Tables 2 and 3 displaythe comparison between the DTM and numerical solutionresults based on a shooting technique for the shear stressesat lower and upper walls for different values of 119908

0and 120573

The influence of nanoparticle volume fraction (120593) onthe normal axial and transverse velocity components isshown in Figures 3 4 and 5The normal velocity component

0 02 04 06 08 1048

051

054

057

06

063

066

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020f

(120578)

120578

Figure 3 Effect of 120593 on the velocity component of 119891(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

reduces for higher values of the nanoparticle volume fractionThe axial velocity profile decreases with the increase innanoparticle volume fraction in the lower half channel whilein the upper half of the channel 1198911015840(120578) enhances with theincrease in120593 In addition transverse velocity profile increasesnear the lower surface for the large value of nanoparticlevolume fraction but the adverse trend happens on the otherside (in the center and the upper half of the channel)

The effects of the characteristic parameter of flow (120573)

on the velocity component 119891(120578) are plotted in Figure 6 Asone can see with the increase in 120573 the velocity profile 119891(120578)augments in the vicinity of the upper plane It can be easilyunderstood that it accumulates the squeezing effects on theflow In Figure 7 the effect of characteristic parameter of theflow on the velocity component parallel to x-axis 1198911015840(120578) isdepicted and it is clearly obvious that the velocity increaseswith the increase in the value of 120573 It should be noted that thehigher values of 120573 amortize the reverse flow but the negativevalues of 120573 support the reverse flow because of the squeezingand extricating effects of the upper wall The same trendcan be observed for transverse velocity component Figure 8shows that the transverse velocity component 119892(120578) increasesas 120573 increases

Figure 9 shows the effect of rotation parameter (Ω) onthe normal velocity component (119891(120578)) The normal velocitycomponent reduces for large values of the rotation parameterThe effect of the rotation parameter on the axial velocitycomponent (1198911015840(120578)) is presented in Figure 10 The resultsshow that in the lower half channel the velocity componentparallel to x-direction decreases with the increase in rotationparameter while in the upper half of the channel 1198911015840(120578)

Mathematical Problems in Engineering 7

Table 2 Comparison results of shear stresses at lower and upper walls for different values of 1199080when 120573 = Ω = 2119872 = 05 and 120593 = 0

1199080

11989110158401015840(0) 119891

10158401015840(1)

DTM Result HAM result of [1] DTM result HAM result of [1]00 +200373789 +20037379 minus458270559 minus4582705603 +006127071 +00612707 minus262278886 minus2622788806 minus198233944 minus19823394 minus073129902 minus0731299009 minus413494482 minus41349448 +109241241 +1092412412 minus640476369 minus64047636 +284916544 +28491655

Table 3 Comparison results of shear stresses at lower and upper walls or different values of 1199080and 120573 when Ω = 1119872 = 05 and 120593 = 01

1199080

120573119862119891lower 119862

119891upper

DTM result Numerical result DTM result Numerical result

00

minus30 minus852098772 minus852098769 +753714763 +753714759minus20 minus676815369 minus676815367 +555472891 +555472890minus10 minus497050617 minus497050616 +354801649 +35480164800 minus301634611 minus301634612 +139835301 +139835302+10 minus089270759 minus089270759 minus090954349 minus090954349+20 +139749899 +139749899 minus337418335 minus337418335+30 +384773079 +384773079 minus599011133 minus599011133

05

minus30 minus117761059 minus117761055 +947810573 +947810571minus20 minus951967384 minus951967382 +718294804 +718294803minus10 minus759674602 minus759674601 +533619342 +53361934200 minus556301306 minus556301307 +338177815 +338177815+10 minus337407009 minus337407009 +126581626 +126581626+20 minus102490197 minus102490197 minus101698614 minus101698614+30 +148110397 +148110397 minus346202219 minus346202219

10

minus30 minus177188652 minus177188648 +148771382 +148771378minus20 minus127797205 minus127797203 +872969245 +872969242minus10 minus106266507 minus106266506 +692986184 +69298618200 minus846845956 minus846845957 +516113154 +516113153+10 minus618186918 minus618186918 +324194096 +324194096+20 minus374763053 minus374763053 +114862333 +114862333+30 minus116400368 minus116400368 minus111763550 minus111763550

increases with the increase in Ω This reverse trend occursapproximately in the center of the channel (120578 cong 04) Theeffect of rotation parameter (Ω) on the transverse velocity119892(120578) is shown in Figure 11 It is clear that although the largerotation causes an increase in transverse velocity near lowersurface the adverse trend occurs on the other side In themain nanoflow regime two diverse trends can be noticedIn the lower half channel the transverse velocity increases inthe center and the upper half of the channel the transversevelocity decreases

The effect of magnetic parameter (119872) on the normalaxial and transverse velocity components can be observedin Figures 12ndash14 From Figure 12 it is seen that the velocitydecreases as magnetic parameter increases because Lorentzforce tends to decrease the velocity profile 119891(120578) In Figure 13two different trends can be seen In the lower half of the

channel the velocity component parallel to the x-directiondecreases with the increase in the magnetic parameter whilein the upper half of the channel 1198911015840(120578) increases with theincrease in 119872 This reverse trend occurs approximately inthe center of the channel The transverse velocity showsincreasing behavior with the increase in magnetic parameterin all region of the channel (Figure 14)

The consequence of suction parameter (1199080) on all the

velocity profiles is illustrated in Figures 15 16 and 17Figure 15 depicts that as the suction parameter increasesthe normal velocity component (119891(120578)) increases and thevariation in the velocity profile confines in the vicinity of theupper plate It is known that the large suction values cause1198911015840(120578) to decrease which results in the occurrence of reverse

flowThe reverse flow is more prominent near the upper platerather than the lower plate The reverse flow near the lower

8 Mathematical Problems in Engineering

Table 4 Effect of nanoparticle volume fraction and nanoparticles material on the shear stresses at lower and upper walls when Ω = 1199080= 1

and 120573 = 119872 = 05

120593Ag Cu CuO TiO2 Al2O3

119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper

000 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272005 minus772463 +4511947 minus808637 +4787926 minus879156 +5326106 minus946511 +5840246 minus95657 +5917038010 minus685916 +385395 minus734313 +4222859 minus837683 +5011777 minus949215 +5863511 minus967086 +5999997015 minus645632 +3548727 minus700493 +3966591 minus824921 +4916332 minus971233 +6034163 minus995938 +6222928020 minus631569 +344302 minus691686 +3900868 minus83391 +4986844 minus101216 +6349316 minus10435 +6588886

0 02 04 06 08 1

minus03

0

03

06

09

12

120578

f998400 (120578

)

minus06

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020

Figure 4 Effect of 120593 on the velocity component of 1198911015840(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

plate is produced by the adverse pressure gradient because thelarge amount of fluid particles escapes from the lower wallFigure 17 shows the same behavior for the transverse velocitycomponent 119892(120578)

Figures 18 19 and 20 demonstrate the effect of differ-ent nanoparticle materials on the normal velocity compo-nent (119891(120578)) axial velocity component (1198911015840(120578)) and also thetransverse velocity profile (119892(120578)) As the results show themaximum amount of normal velocity component belongsto Al2O3nanoparticle Moreover the effects of nanoparticle

volume fraction and the types of nanoparticles on the shearstresses at lower and upper walls are depicted in Table 4

5 Conclusions

In this paper we have used DTM to solve a coupled sys-tem of nonlinear ordinary differential equations for three-dimensional flow of a nanofluid in a rotating channel on

0 02 04 06 08 1

0

002

120578

g(120578

)minus002

minus004

minus006

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020

Figure 5 Effect of 120593 on the velocity component of 119892(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

a lower permeable stretching wall We have consideredwater the base fluid and four different types of nanoparti-cles copper copper oxide aluminum oxide and titaniumdioxide have been examined in this simulation The upperwall is moving along the direction normal to the surfacewith time dependent velocity The transformed dimen-sionless equations have been formulated and solved withrobust boundary conditions Important physical parame-ters have been investigated graphically These parametersinclude the flow parameter the rotation parameter themagnetic parameter the nanoparticle volume fraction thesuction parameter and also different types of nanoparti-cles Results show that the vertical motion of the upperplate interrupts the velocity in the channel remarkablyThe downward motion of the upper plate augments theforward flow whereas the upward motion reverses theflow

Mathematical Problems in Engineering 9

0 02 04 06 08 1

0

1

2

minus1

minus2

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

f(120578

)

120578

Figure 6 Effect of120573 on the velocity component of119891(120578)whenΩ = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1

0

2

minus2

minus4

minus6

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

120578

f998400 (120578

)

Figure 7 Effect of 120573 on the velocity component of 1198911015840(120578) when Ω =

1 120593 = 01 and119872 = 1199080= 05

0 02 04 06 08 1

0

2

minus2

minus4

minus6

minus8

f(120578

)

120578

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

Figure 8 Effect of 120573 on the velocity component of 119892(120578)whenΩ = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1048

051

054

057

06

063

066

g(120578

)

120578

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 9 Effect ofΩ on the transverse velocity of 119891(120578) when 120573 = 1120593 = 01 and119872 = 119908

0= 05

10 Mathematical Problems in Engineering

0 02 04 06 08 1

0

03

06

09

12

minus03

120578

f998400 (120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 10 Effect ofΩ on the transverse velocity of1198911015840(120578)when120573 = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1

0

005

minus005

minus01

minus015

minus02

minus025

120578

g(120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 11 Effect ofΩ on the transverse velocity of 119892(120578)when 120573 = 1120593 = 01 and119872 = 119908

0= 05

048

051

054

057

06

063

066

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1

f(120578

)

120578

Figure 12 Effect of119872 on the velocity component of 119891(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

0

03

06

09

12

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

minus03

120578

f998400 (120578

)

0 02 04 06 08 1

Figure 13 Effect of 119872 on the velocity component of 1198911015840(120578) when120573 = Ω = 1 120593 = 01 and 119908

0= 05

Mathematical Problems in Engineering 11

0

002

minus002

minus004

minus006

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1120578

g(120578

)

Figure 14 Effect of119872 on the velocity component of 119892(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

025

05

075

1

125

00 02 04 06 08 1

120578

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

f(120578

)

Figure 15 Effect of1199080on the velocity component of 119891(120578)when 120573 =

Ω = 1119872 = 05 and 120593 = 01

0

05

1

15

minus05

minus10 02 04 06 08 1

120578

f998400 (120578

)w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

Figure 16 Effect of 1199080on the velocity component of 1198911015840(120578) when

120573 = Ω = 1119872 = 05 and 120593 = 01

0

01

02

minus02

minus03

minus01

0 02 04 06 08 1

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

120578

g(120578

)

Figure 17 Effect of1199080on the velocity component of 119892(120578) when 120573 =

Ω = 1119872 = 05 and 120593 = 01

12 Mathematical Problems in Engineering

048

051

054

057

06

063

066

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1

f(120578

)

120578

Figure 18 Effect of nanoparticle types on the velocity componentof 119891(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0

03

06

09

12

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1120578

f998400 (120578

)

minus03

minus06

Figure 19 Effect of nanoparticle types on the velocity componentof 1198911015840(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0 02 04 06 08 1

0

002

minus006

minus004

minus002

AgCuCuO

TiO2Al2O3

120578

g(120578

)

Figure 20 Effect of nanoparticle types on the velocity componentof 119892(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

Nomenclature

119886 Stretching rate of the lower plate119861 External uniform magnetic field1198610 Constant magnetic flux density

119862119891 Skin friction coefficient

119891 119892 Self-similar velocitiesℎ(119905) Upper plane distance119869 Magnetic flux119901 Pressure119905 Time119879 Cauchy stress tensor119906 Velocity component in the x directionV Velocity component in the y direction119908 Velocity component in the z direction

Dimensionless Parameters

120573 Characteristic parameter of the flow (120574119886)

Ω Rotation parameter (120596119886)1198722 Magnetic parameter (1205901198612

0120588119891119886)

1199080 Suction parameter (119881

0119886ℎ)

Re119909 Local Reynolds number (120588

1198911198800ℎ120583119891)

Greek Symbols

120572 120575 and 120576 DTM constant parameters120590 Electrical conductivity120588 Density

Mathematical Problems in Engineering 13

120583 Dynamic viscosity120578 A scaled boundary-layer coordinate] Kinematic viscosity120574 Characteristic constant parameter120595 Stream function120596 Constant angular velocity120593 Nanoparticle volume fraction

Subscripts

119891 Fluid phase119899119891 Nanofluid119904 Solid phase

Conflict of Interests

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

Acknowledgment

EbrahimMomoniat acknowledges support from theNationalResearch foundation of South Africa under Grant number76868

References

[1] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquoComputers andMathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I Boundary layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 26ndash28 1961

[3] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer on a continuous flat surfacerdquoAIChEJournal vol 7 pp 221ndash225 1961

[4] L E Erickson L T Fan and V G Fox ldquoHeat and mass transferon a moving continuous flat plate with suction or injectionrdquoIndustrial and Engineering Chemistry Fundamentals vol 5 no1 pp 19ndash25 1966

[5] F K Tsou E M Sparrow and R J Goldstein ldquoFlow and heattransfer in the boundary layer on a continuousmoving surfacerdquoInternational Journal ofHeat andMass Transfer vol 10 no 2 pp219ndash235 1967

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[7] H I Andersson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquo Acta Mechanica vol 95 no 1ndash4 pp 227ndash230 1992

[8] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[9] A Mehmood and A Ali ldquoAnalytic solution of generalizedthree-dimensional flow and heat transfer over a stretching plane

wallrdquo International Communications in Heat andMass Transfervol 33 no 10 pp 1243ndash1252 2006

[10] A Mehmood and A Ali ldquoAnalytic homotopy solution ofgeneralized three-dimensional channel flow due to uniformstretching of the platerdquoActaMechanica Sinica vol 23 no 5 pp503ndash510 2007

[11] A K Borkakoti and A Bharali ldquoHydromagnetic flow and heattransfer between two horizontal plates the lower plate being astretching sheetrdquo Quarterly of Applied Mathematics vol 40 no4 pp 461ndash467 1983

[12] S Munawar A Mehmood and A Ali ldquoEffects of slip on flowbetween two stretchable disks using optimal homotopy analysismethodrdquo Canadian Journal of Applied Science vol 1 pp 50ndash682011

[13] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicropolar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 no 8 pp1119ndash1127 2002

[14] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[15] P D Ariel ldquoOn exact solutions of flow problems of a secondgrade fluid through two parallel porous wallsrdquo InternationalJournal of Engineering Science vol 40 no 8 pp 913ndash941 2002

[16] T Hayat and Z Abbas ldquoChannel flow of a Maxwell fluid withchemical reactionrdquo Zeitschrift fur angewandte Mathematik undPhysik vol 59 no 1 pp 124ndash144 2008

[17] G Domairry and A Aziz ldquoApproximate analysis of MHDdqueeze flow between two parallel disks with suction orinjection by homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2009 Article ID 603916 19 pages2009

[18] S U S Choi and J A Eastman ldquoEnhancing thermal conductiv-ity of fluids with nanoparticlesrdquo Materials Science vol 231 pp99ndash105 1995

[19] Y Xuan and Q Li ldquoInvestigation on convective heat transferand flow features of nanofluidsrdquo Journal of Heat Transfer vol125 no 1 pp 151ndash155 2003

[20] N Bachok A Ishak and I Pop ldquoFlow and heat transfer overa rotating porous disk in a nanofluidrdquo Physica B CondensedMatter vol 406 no 9 pp 1767ndash1772 2011

[21] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[22] M H Abolbashari N Freidoonimehr F Nazari and M MRashidi ldquoEntropy analysis for an unsteady MHD flow past astretching permeable surface in nano-fluidrdquoPowder Technologyvol 267 pp 256ndash267 2014

[23] O A Beg M M Rashidi M Akbari and A HosseinildquoComparative numerical study of single-phase and two-phasemodels for bio-nanofluid transport phenomenardquo Journal ofMechanics in Medicine and Biology vol 14 Article ID 145001131 pages 2014

[24] M M Rashidi A Hosseini I Pop S Kumar and N Freidoon-imehr ldquoComparative numerical study of single and two-phasemodels of nanofluid heat transfer in wavy channelrdquo AppliedMathematics and Mechanics vol 35 pp 831ndash848 2014

[25] E Abu-Nada Z Masoud H F Oztop and A Campo ldquoEffectof nanofluid variable properties on natural convection inenclosuresrdquo International Journal of Thermal Sciences vol 49no 3 pp 479ndash491 2010

14 Mathematical Problems in Engineering

[26] M M Rashidi S Abelman and N Freidoonimehr ldquoEntropygeneration in steady MHD flow due to a rotating porous diskin a nanofluidrdquo International Journal of Heat andMass Transfervol 62 no 1 pp 515ndash525 2013

[27] M Mustafa T Hayat I Pop S Asghar and S ObaidatldquoStagnation-point flow of a nanofluid towards a stretchingsheetrdquo International Journal of Heat and Mass Transfer vol 54no 25-26 pp 5588ndash5594 2011

[28] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica vol 49 pp 469ndash482 2014

[29] M Sheikholeslami and D D Ganji ldquoThree dimensional heatand mass transfer in a rotating system using nanofluidrdquo PowderTechnology vol 253 pp 789ndash796 2014

[30] M Sheikholeslami M Gorji-Bandpy and D D Ganji ldquoLatticeBoltzmann method for MHD natural convection heat transferusing nanofluidrdquo Powder Technology vol 254 pp 82ndash93 2014

[31] F Ayaz ldquoApplications of differential transform method to diff-erential-algebraic equationsrdquoAppliedMathematics and Compu-tation vol 152 no 3 pp 649ndash657 2004

[32] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004

[33] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[34] M Kurosaka ldquoThe oscillatory boundary layer growth over thetop and bottom plates of a rota ting channelrdquo Journal of FluidsEngineering Transactions of the ASME vol 95 no 1 pp 68ndash741973

[35] H S Takhar A J Chamkha and G Nath ldquoMHD flow overa moving plate in a rotating fluid with magnetic field hallcurrents and free stream velocityrdquo International Journal ofEngineering Science vol 40 no 13 pp 1511ndash1527 2002

[36] K Vajravelu and B V R Kumar ldquoAnalytical and numericalsolutions of a coupled non-linear system arising in a three-dimensional rotating flowrdquo International Journal of Non-LinearMechanics vol 39 no 1 pp 13ndash24 2004

[37] G Domairry and A Aziz ldquoApproximate analysis of MHDsqueeze flow between two parallel disks with suction or in-jection by homotopy perturbation methodrdquoMathematical Pro-blems in Engineering vol 2009 Article ID 603916 19 pages2009

[38] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4p 571 1952

[39] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled withnanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[40] M Sheikholeslami and D D Ganji ldquoHeat transfer of Cu-waternanofluid flow between parallel platesrdquo Powder Technology vol235 pp 873ndash879 2013

[41] M M Rashidi ldquoThe modified differential transform methodfor solving MHD boundary-layer equationsrdquo Computer PhysicsCommunications vol 180 no 11 pp 2210ndash2217 2009

[42] M M Rashidi and N Freidoonimehr ldquoSeries solutions forthe flow in the vicinity of the equator of an MHD boundary-layer over a porous rotating sphere with heat transferrdquoThermalScience vol 18 pp S527ndashS537 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

convective flow of a micropolar fluid between two verticalparallel plates Bhargava et al [14] investigated the fullydeveloped flow and heat transfer of an electrically conductingmicropolar fluid (a strong cross magnetic field) between twoparallel and porous plates in which the temperature has beenconsidered to be dependent on the heat source including theeffect of frictional heating The quasilinearization method isused to solve the governing system of ordinary differentialequations Ariel [15] presented solutions for two problemsof laminar forced convection of a second-grade (viscoelastic)fluid through two parallel porous walls considering rectan-gular and cylindrical geometries Applying similarity trans-formations to the governing nonlinear partial differentialequations Hayat and Abbas [16] derived nonlinear ordi-nary differential equations and studied the two-dimensionalboundary-layer flow of an upper-convected Maxwell fluid ina channel with chemical reaction Domairry and Aziz [17]obtained an analytic solution for unsteady MHD squeezingflowwith suction and injection effects by the use of homotopyperturbation method

To the best of authorsrsquo knowledge Choi and Eastman[18] were probably the first researchers who employed amixture of nanoparticles and base fluid and called thismixture a ldquonanofluidrdquo A wide range of review papers havebeen published on nanofluids in recent years Xuan and Li[19] considered the Reynolds number and volume fraction ofnanoparticle influences in turbulent flows for nanofluids intubes experimentally Bachok et al [20] employed a numer-ical Keller-Box technique for steady nanofluid flow over aporous rotating disk Khan and Pop [21] studied laminarflow for a nanofluid across a stretching flat surface usingan implicit finite difference method Abolbashari et al [22]employed HAM to study the entropy analysis in an unsteadyMHDnanofluid regime adjacent to an accelerating stretchingpermeable surface Beg et al [23] presented a comparativenumerical solution for both single- and two-phase modelsfor bionanofluid transport phenomena Rashidi et al [24]compared the two phase and single phase of heat transferand flow field of copper-water nanofluid in a wavy channelnumerically Abu-Nada et al [25] illustrated the impactsof variable properties in natural convection nanofluid flowRashidi et al [26] showed how the second law of thermo-dynamics can be applied to MHD incompressible nanofluidflow over a porous rotating disk The stagnation flow for ananofluid over a stretching sheet was studied by Mustafa etal [27] analytically The interested reader is referred to thefollowing papers for further reading on the application ofnanoparticles in fluid flow [28ndash30]

In this paper we derive a coupled system of non-linear ordinary differential equations to model the three-dimensional flow of a nanofluid in a rotating channel ona lower permeable stretching wall The resulting system ofequations is solved using the differential transform method(DTM) [31 32] The DTM method has been applied suc-cessfully to solve nonlinear differential equations withoutrequiring linearization or discretization [33] The presentDTM code is benchmarked with numerical results basedon a shooting technique and previously published results

y = h(t)

B0

Nanofluid

y = 0

z w

V0

Vh(t)

x u

U0

Ω

y

Figure 1 Schematic diagram of the flow configuration and thecoordinate system for the considered flow

The DTM method shows excellent correlation with resultsobtained using these other methods

The paper is divided up as follows in Section 2 theproblem is formulated and a coupled system of nonlinearordinary differential equations is derived The DTM methodis applied to solve the resulting system of nonlinear ordinarydifferential equations in Section 3 Results are discussed inSection 4 Concluding remarks are made in Section 5

2 Mathematical Formulation of the Problem

Consider unsteady 3D rotating nanofluid flow of an incom-pressible electrically conducting viscous fluid between twoinfinite horizontal plane walls The lower plane is placed at119910 = 0 and is stretched with a time-dependent velocity119880

0(119905) =

119886119909(1 minus 120574119905) in x-direction The upper plane is also placedat a variable distance ℎ(119905) = radic]

119891(1 minus 120574119905)119886 and the fluid is

squeezed with a time-dependent velocity 119881ℎ= 119889ℎ119889119905 in the

negative y-direction The fluid and the channel are rotatingaround y-axis with an angular velocity Ω = 120596119869(1 minus 120574119905)

and also the lower plate intakes the flow with the velocityminus1198810(1 minus 120574119905) A magnetic field with density 119861

0radic(1 minus 120574119905) is

applied along the y-axis about the system which is rotatingThese velocities and magnetic fields are introduced to obtainsimilarity solutions by reducing governing equations intoordinary differential equations (ODEs) The physical modelof the considered problemalongwith the coordinate system isillustrated in Figure 1 The governing equations of continuityand momentum of nanofluid flow in a rotating frame ofreference are given as [34 35]

nabla sdot V = 0

120588nf [120597V120597119905

+ (V sdot nabla)V + 2Ω times V] = nabla sdot T + J times B(1)

Mathematical Problems in Engineering 3

Table 1 Thermophysical properties of the base fluid and different nanoparticles [39]

Physical properties Fluid phase (water) Cu CuO Ag Al2O3 TiO2

120588 (kgm3) 9971 8933 6320 10500 3970 4250

where T is the Cauchy stress tensor J the magnetic flux andB the current density The above governing equations can bealso described by the following set of Navier-Stokes equations[36 37]

120597119906

120597119909+120597V120597119910

= 0

120597119906

120597119905+ 119906

120597119906

120597119909+ V

120597119906

120597119910+ 2

120596

1 minus 120574119905119908

= minus1

120588nf

120597119901

120597119909+ ]nf [

1205972119906

1205971199092+1205972119906

1205971199102] minus

1205901198612

0

120588nf (1 minus 120574119905)119906

120597V120597119905

+ 119906120597V120597119909

+ V120597V120597119910

= minus1

120588nf

120597119901

120597119910+ ]nf [

1205972V1205971199092

+1205972V1205971199102

]

120597119908

120597119905+ 119906

120597119908

120597119909+ V

120597119908

120597119910minus 2

120596

1 minus 120574119905119906

= ]nf [1205972119908

1205971199092+1205972119908

1205971199102] minus

1205901198612

0

120588nf (1 minus 120574119905)119908

(2)

where 120588nf is the nanofluid density ]nf (= 120583nf120588nf) is thenanofluid kinematic viscosity where 120583nf has been proposedby Brinkman [38] 120590 is the electrical conductivity 119861

0is the

magnetic field and 120574 is the characteristic parameter withdimension of (time)minus1 and 120574119905 lt 1 The above nanofluidconstants are defined as follows

120583nf =120583119891

(1 minus 120593)25 120588nf = (1 minus 120593) 120588

119891+ 120593120588119904 (3)

where 120583119891

is the viscosity of the fluid fraction 120593 is thenanoparticle volume fraction and 120588

119891and 120588119904are the densities

of the fluid and of the solid fractions respectively Thethermophysical properties of the base fluid (water) and dif-ferent nanoparticles are given in Table 1 [39]The appropriateboundary conditions are introduced as follows

119906 (119909 119910 119905) = 1198800=

119886119909

1 minus 120574119905

V (119909 119910 119905) = minus1198810

1 minus 120574119905

119908 (119909 119910 119905) = 0

at 119910 = 0

119906 (119909 119910 119905) = 0

V (119909 119910 119905) = 119881ℎ=119889ℎ

119889119905=minus120574

2radic

]119891

119886 (1 minus 120574119905)

119908 (119909 119910 119905) = 0

at 119910 = ℎ (119905)

(4)

where 119886 is the stretching rate of the lower plateThe followingappropriate similarity transformations are employed to con-vert the above governing equations (2)ndash(4) into a system of

ordinary differential equations in terms of a stream function120595

120595 = radic119886]119891

1 minus 120574119905119909119891 (120578) 120578 =

119910

ℎ (119905) (5)

119906 =120597120595

120597119910= 11988001198911015840(120578) V = minus

120597120595

120597119909= minusradic

119886]119891

1 minus 120574119905119891 (120578)

119908 = 1198800119892 (120578)

(6)

Substituting the similarity transformations (5) and (6) into(2)ndash(4) we obtain the following system of nonlinear ordinarydifferential equations

119891101584010158401015840+]119891

]nf(11989111989110158401015840minus (1198911015840)2

minus 120573(1198911015840+120578

211989110158401015840)

minus2Ω119892 minus1

120588nf12058811989111987221198911015840) =

(1 minus 120574119905)2]119891

120588nf1198862119909]nf

120597119901

120597119909

minus11989110158401015840+]119891

]nf(minus119891119891

1015840+120573

2(119891 + 120578119891

1015840)) =

1 minus 120574119905

120588nf119886]nf

120597119901

120597120578

(7)

11989210158401015840+]119891

]nf(1198911198921015840minus 1198911015840119892 minus 120573(119892 +

120578

21198921015840) + 2Ω119891

1015840)

minus1

120583nf1205831198911198722119892 = 0

(8)

where 120573 = 120574119886 is the characteristic parameter of the flowΩ = 120596119886 is the rotation parameter 1198722 = 120590119861

2

0120588119891119886 is the

magnetic parameter and prime denotes differentiation withrespect to 120578 In order to squeeze the flow we take 120573 gt 0 forwhich the upper plate moves downward with velocity119881

ℎlt 0

For 120573 lt 0 the upper plate moves apart with respect to theplane 119910 = 0 and 120573 = 0 corresponds to the steady statecase of the considered problem or stationary upper plate Inorder to reduce the number of independent variables andto retain the similarity solution (7) are simplified by crossdifferentiation and thus we obtain the following system ofdifferential equations

119891119894Vminus (1 minus 120593 + 120593 (120588

119904120588119891)) (1 minus 120593)

25

times (119891101584011989110158401015840minus 119891119891101584010158401015840+ 2Ω119892

1015840+120573

2(311989110158401015840+ 120578119891101584010158401015840))

minus (1 minus 120593)25119872211989110158401015840= 0

11989210158401015840+ (1 minus 120593 + 120593 (120588

119904120588119891)) (1 minus 120593)

25

times (1198911198921015840minus 1198911015840119892 minus 120573(119892 +

120578

21198921015840) + 2Ω119891

1015840)

minus (1 minus 120593)251198722119892 = 0

(9)

4 Mathematical Problems in Engineering

The transformed boundary conditions take the form

119891 (0) = 1199080 119891

1015840(0) = 1 119892 (0) = 0

119891 (1) =120573

2 119891

1015840(1) = 0 119892 (1) = 0

(10)

where 1199080= 1198810119886ℎ is the suction parameter

In this problem the physical quantity of interest is theskin friction coefficients 119862

119891along the stretching wall at the

lower and upper walls which are defined as [40]

119862119891lower =

120583nf(120597119906120597119910)119910=0

120588nf1198802

0

119862119891upper =

120583nf(120597119906120597119910)119910=ℎ(119905)

120588nf1198802

0

(11)

Substituting (6) into (11) we obtain

119862119891lower = 119862

119891lowerRe119909 =11989110158401015840(0)

(1 minus 120593 + 120593 (120588119904120588119891)) (1 minus 120593)

25

119862119891upper = 119862

119891upperRe119909 =11989110158401015840(1)

(1 minus 120593 + 120593 (120588119904120588119891)) (1 minus 120593)

25

(12)

where Re119909= 1205881198911198800ℎ120583119891is the local Reynolds number

In this section we have derived the coupled system ofnonlinear ordinary differential equations (9) to model thethree-dimensional flow of a nanofluid in a rotating channelon a lower permeable stretchingwallWe next apply theDTMto solve the coupled system

3 Analytical Approximation by Means of DTM

Taking the differential transform of (9) one can obtain (formore details see [41 42])

(119896 + 1) (119896 + 2) (119896 + 3) (119896 + 4) 119891 (119896 + 4)

minus (1 minus 120593)251198722(119896 + 1) (119896 + 2) 119891 (119896 + 2)

minus (1 minus 120593 + 120593(120588119904

120588119891

)) (1 minus 120593)25

times (

119896

sum

119903=0

((119896 minus 119903 + 1) (119896 minus 119903 + 2) (119903 + 1) 119891 (119903 + 1)

times 119891 (119896 minus 119903 + 2) minus (119896 minus 119903 + 1) (119896 minus 119903 + 2)

times (119896 minus 119903 + 3) 119891 (119903) 119891 (119896 minus 119903 + 3))

+120573

2(3 (119896 + 1) (119896 + 2) 119891 (119896 + 2)

+

119896

sum

119903=0

(

(119896 minus 119903 + 1) (119896 minus 119903 + 2)

(119896 minus 119903 + 3) 120575 (119903) 119891 (119896 minus 119903 + 3)

))

+2Ω (119896 + 1) 119892 (119896 + 1)) = 0

(119896 + 1) (119896 + 2) 119892 (119896 + 2) minus (1 minus 120593)251198722119892 (119896)

+ (1 minus 120593 + 120593(120588119904

120588119891

)) (1 minus 120593)25

times ((

119896

sum

119903=0

((119896 minus 119903 + 1) 119891 (119903) 119892 (119896 minus 119903 + 1)

minus (119896 minus 119903 + 1) 119892 (119903) 119891 (119896 minus 119903 + 1)))

+ 2Ω (119896 + 1) 119891 (119896 + 1)

minus 120573(119892 (119896) + (1

2)

times

119896

sum

119903=0

((119896 minus 119903 + 1) 120575 (119903) 119892 (119896 minus 119903 + 1)))) = 0

(13)

where 119891(119896) and 119892(119896) are differential transforms of 119891(120578) and119892(120578) given by

119891 (120578) =

infin

sum

119896=0

119891 (119896) 120578119896 (14)

119892 (120578) =

infin

sum

119896=0

119892 (119896) 120578119896 (15)

119891 (0) = 1199080 119891 (1) = 1 119891 (2) = 120572

119891 (3) = 120575 119892 (0) = 0 119892 (1) = 120576

(16)

where (16) is the transformed boundary conditions and 120572 120575and 120576 are constants By substituting (16) into (13) usingrecursion and then substituting them in (14)-(15) we obtainthe values of 119891(120578) and 119892(120578) given by

Mathematical Problems in Engineering 5

119891 (120578) = 1199080+ 120578 + 120572120578

2+ 1205751205783+

1

24

times(21198722(1 minus 120593)

25120572 + (1 minus 120593)

25(1 minus 120593 +

120588119904

120588119891

120593)

(2120572 + 3120572120573 minus 61199080120575 + 2Ω120576)

)

times 1205784+ sdot sdot sdot

119892 (120578) = 120578120576 +1

2(1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)

times (minus2Ω minus 1199080120576) 1205782+1

6

times(

1198722(1 minus 120593)

25120576 + (1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)

(minus4Ω120572 +3120573120576

2minus (1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)1199080(minus2Ω minus 119908

0120576))

)

times 1205783+

1

12

times((

(

1

21198722(1 minus 120593)

5(1 minus 120593 + (

120588119904

120588119891

)120593) (minus2Ω minus 1199080120576)

+ (1 minus 120593)25(1 minus 120593 + (

120588119904

120588119891

)120593)

times((

(

minus 6Ω120575 minus1

2(1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593) (minus2Ω minus 1199080120576)

+ 120573(1 minus 120593)25(1 minus 120593 + (

120588119904

120588119891

)120593) (minus2Ω minus 1199080120576) + 120572120576 minus

1

21199080

times((

(

1198722(1 minus 120593)

25120576 + (1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)

(

minus4Ω120572 +3120573120576

2

minus (1 minus 120593)25(1 minus 120593 + (

120588119904

120588119891

)120593)1199080(minus2Ω minus 119908

0120576)

)

))

)

))

)

))

)

times1205784+ sdot sdot sdot

(17)

The constants 120572 120575 and 120576 can be determined by applyingthe remaining boundary conditions to (17) The numberof required terms is determined by the convergence of the

numerical values to onersquos desired accuracy We obtain theapproximants using the computational software MATHE-MATICA The effect of different orders of the DTM solution

6 Mathematical Problems in Engineering

0 02 04 06 08 1

0

05

1

DTM n = 5DTM n = 10DTM n = 20

DTM n = 30DTM n = 40Numerical

g(120578

)

120578

minus05

minus1

minus15

minus2

minus25

Figure 2 The obtained results of transverse velocity of 119892(120578) fordifferent orders of DTM solution in comparison with the numericalsolution when 120573 = minus1Ω = 3119872 = 2 120593 = 01 and 119908

0= 05

on the convergence radius of the transverse velocity of 119892(120578) isdisplayed in Figure 2

In order to highlight the validity of the presented DTMsolution we compare some of our results with the obtainedHAM results of [1] A very good validation of the presentanalytical results has been achieved with the previouslypublished study as shown in Table 2

4 Results and Discussion

In the previous section the nonlinear ordinary differentialequations (9) are solved subject to the boundary condi-tions (10) via the DTM method for values of the five keyparameters characteristic parameter of the flow (120573) rotationparameter (Ω) the magnetic parameter (119872) nanoparticlevolume fraction (120593) the suction parameter (119908

0) and different

nanoparticles on the different velocity components It shouldbe stated that the copper nanoparticle is used in all of the casesin this section except for those figures which focus on theinfluences of the types of applied nanoparticles on the velocitycomponent profiles In addition we assume that the valuesof the volume fraction parameter 120593 vary from 0 (regularNewtonian fluid) to 02 Representative values are employedto simulate the physically realistic flows Tables 2 and 3 displaythe comparison between the DTM and numerical solutionresults based on a shooting technique for the shear stressesat lower and upper walls for different values of 119908

0and 120573

The influence of nanoparticle volume fraction (120593) onthe normal axial and transverse velocity components isshown in Figures 3 4 and 5The normal velocity component

0 02 04 06 08 1048

051

054

057

06

063

066

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020f

(120578)

120578

Figure 3 Effect of 120593 on the velocity component of 119891(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

reduces for higher values of the nanoparticle volume fractionThe axial velocity profile decreases with the increase innanoparticle volume fraction in the lower half channel whilein the upper half of the channel 1198911015840(120578) enhances with theincrease in120593 In addition transverse velocity profile increasesnear the lower surface for the large value of nanoparticlevolume fraction but the adverse trend happens on the otherside (in the center and the upper half of the channel)

The effects of the characteristic parameter of flow (120573)

on the velocity component 119891(120578) are plotted in Figure 6 Asone can see with the increase in 120573 the velocity profile 119891(120578)augments in the vicinity of the upper plane It can be easilyunderstood that it accumulates the squeezing effects on theflow In Figure 7 the effect of characteristic parameter of theflow on the velocity component parallel to x-axis 1198911015840(120578) isdepicted and it is clearly obvious that the velocity increaseswith the increase in the value of 120573 It should be noted that thehigher values of 120573 amortize the reverse flow but the negativevalues of 120573 support the reverse flow because of the squeezingand extricating effects of the upper wall The same trendcan be observed for transverse velocity component Figure 8shows that the transverse velocity component 119892(120578) increasesas 120573 increases

Figure 9 shows the effect of rotation parameter (Ω) onthe normal velocity component (119891(120578)) The normal velocitycomponent reduces for large values of the rotation parameterThe effect of the rotation parameter on the axial velocitycomponent (1198911015840(120578)) is presented in Figure 10 The resultsshow that in the lower half channel the velocity componentparallel to x-direction decreases with the increase in rotationparameter while in the upper half of the channel 1198911015840(120578)

Mathematical Problems in Engineering 7

Table 2 Comparison results of shear stresses at lower and upper walls for different values of 1199080when 120573 = Ω = 2119872 = 05 and 120593 = 0

1199080

11989110158401015840(0) 119891

10158401015840(1)

DTM Result HAM result of [1] DTM result HAM result of [1]00 +200373789 +20037379 minus458270559 minus4582705603 +006127071 +00612707 minus262278886 minus2622788806 minus198233944 minus19823394 minus073129902 minus0731299009 minus413494482 minus41349448 +109241241 +1092412412 minus640476369 minus64047636 +284916544 +28491655

Table 3 Comparison results of shear stresses at lower and upper walls or different values of 1199080and 120573 when Ω = 1119872 = 05 and 120593 = 01

1199080

120573119862119891lower 119862

119891upper

DTM result Numerical result DTM result Numerical result

00

minus30 minus852098772 minus852098769 +753714763 +753714759minus20 minus676815369 minus676815367 +555472891 +555472890minus10 minus497050617 minus497050616 +354801649 +35480164800 minus301634611 minus301634612 +139835301 +139835302+10 minus089270759 minus089270759 minus090954349 minus090954349+20 +139749899 +139749899 minus337418335 minus337418335+30 +384773079 +384773079 minus599011133 minus599011133

05

minus30 minus117761059 minus117761055 +947810573 +947810571minus20 minus951967384 minus951967382 +718294804 +718294803minus10 minus759674602 minus759674601 +533619342 +53361934200 minus556301306 minus556301307 +338177815 +338177815+10 minus337407009 minus337407009 +126581626 +126581626+20 minus102490197 minus102490197 minus101698614 minus101698614+30 +148110397 +148110397 minus346202219 minus346202219

10

minus30 minus177188652 minus177188648 +148771382 +148771378minus20 minus127797205 minus127797203 +872969245 +872969242minus10 minus106266507 minus106266506 +692986184 +69298618200 minus846845956 minus846845957 +516113154 +516113153+10 minus618186918 minus618186918 +324194096 +324194096+20 minus374763053 minus374763053 +114862333 +114862333+30 minus116400368 minus116400368 minus111763550 minus111763550

increases with the increase in Ω This reverse trend occursapproximately in the center of the channel (120578 cong 04) Theeffect of rotation parameter (Ω) on the transverse velocity119892(120578) is shown in Figure 11 It is clear that although the largerotation causes an increase in transverse velocity near lowersurface the adverse trend occurs on the other side In themain nanoflow regime two diverse trends can be noticedIn the lower half channel the transverse velocity increases inthe center and the upper half of the channel the transversevelocity decreases

The effect of magnetic parameter (119872) on the normalaxial and transverse velocity components can be observedin Figures 12ndash14 From Figure 12 it is seen that the velocitydecreases as magnetic parameter increases because Lorentzforce tends to decrease the velocity profile 119891(120578) In Figure 13two different trends can be seen In the lower half of the

channel the velocity component parallel to the x-directiondecreases with the increase in the magnetic parameter whilein the upper half of the channel 1198911015840(120578) increases with theincrease in 119872 This reverse trend occurs approximately inthe center of the channel The transverse velocity showsincreasing behavior with the increase in magnetic parameterin all region of the channel (Figure 14)

The consequence of suction parameter (1199080) on all the

velocity profiles is illustrated in Figures 15 16 and 17Figure 15 depicts that as the suction parameter increasesthe normal velocity component (119891(120578)) increases and thevariation in the velocity profile confines in the vicinity of theupper plate It is known that the large suction values cause1198911015840(120578) to decrease which results in the occurrence of reverse

flowThe reverse flow is more prominent near the upper platerather than the lower plate The reverse flow near the lower

8 Mathematical Problems in Engineering

Table 4 Effect of nanoparticle volume fraction and nanoparticles material on the shear stresses at lower and upper walls when Ω = 1199080= 1

and 120573 = 119872 = 05

120593Ag Cu CuO TiO2 Al2O3

119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper

000 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272005 minus772463 +4511947 minus808637 +4787926 minus879156 +5326106 minus946511 +5840246 minus95657 +5917038010 minus685916 +385395 minus734313 +4222859 minus837683 +5011777 minus949215 +5863511 minus967086 +5999997015 minus645632 +3548727 minus700493 +3966591 minus824921 +4916332 minus971233 +6034163 minus995938 +6222928020 minus631569 +344302 minus691686 +3900868 minus83391 +4986844 minus101216 +6349316 minus10435 +6588886

0 02 04 06 08 1

minus03

0

03

06

09

12

120578

f998400 (120578

)

minus06

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020

Figure 4 Effect of 120593 on the velocity component of 1198911015840(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

plate is produced by the adverse pressure gradient because thelarge amount of fluid particles escapes from the lower wallFigure 17 shows the same behavior for the transverse velocitycomponent 119892(120578)

Figures 18 19 and 20 demonstrate the effect of differ-ent nanoparticle materials on the normal velocity compo-nent (119891(120578)) axial velocity component (1198911015840(120578)) and also thetransverse velocity profile (119892(120578)) As the results show themaximum amount of normal velocity component belongsto Al2O3nanoparticle Moreover the effects of nanoparticle

volume fraction and the types of nanoparticles on the shearstresses at lower and upper walls are depicted in Table 4

5 Conclusions

In this paper we have used DTM to solve a coupled sys-tem of nonlinear ordinary differential equations for three-dimensional flow of a nanofluid in a rotating channel on

0 02 04 06 08 1

0

002

120578

g(120578

)minus002

minus004

minus006

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020

Figure 5 Effect of 120593 on the velocity component of 119892(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

a lower permeable stretching wall We have consideredwater the base fluid and four different types of nanoparti-cles copper copper oxide aluminum oxide and titaniumdioxide have been examined in this simulation The upperwall is moving along the direction normal to the surfacewith time dependent velocity The transformed dimen-sionless equations have been formulated and solved withrobust boundary conditions Important physical parame-ters have been investigated graphically These parametersinclude the flow parameter the rotation parameter themagnetic parameter the nanoparticle volume fraction thesuction parameter and also different types of nanoparti-cles Results show that the vertical motion of the upperplate interrupts the velocity in the channel remarkablyThe downward motion of the upper plate augments theforward flow whereas the upward motion reverses theflow

Mathematical Problems in Engineering 9

0 02 04 06 08 1

0

1

2

minus1

minus2

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

f(120578

)

120578

Figure 6 Effect of120573 on the velocity component of119891(120578)whenΩ = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1

0

2

minus2

minus4

minus6

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

120578

f998400 (120578

)

Figure 7 Effect of 120573 on the velocity component of 1198911015840(120578) when Ω =

1 120593 = 01 and119872 = 1199080= 05

0 02 04 06 08 1

0

2

minus2

minus4

minus6

minus8

f(120578

)

120578

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

Figure 8 Effect of 120573 on the velocity component of 119892(120578)whenΩ = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1048

051

054

057

06

063

066

g(120578

)

120578

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 9 Effect ofΩ on the transverse velocity of 119891(120578) when 120573 = 1120593 = 01 and119872 = 119908

0= 05

10 Mathematical Problems in Engineering

0 02 04 06 08 1

0

03

06

09

12

minus03

120578

f998400 (120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 10 Effect ofΩ on the transverse velocity of1198911015840(120578)when120573 = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1

0

005

minus005

minus01

minus015

minus02

minus025

120578

g(120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 11 Effect ofΩ on the transverse velocity of 119892(120578)when 120573 = 1120593 = 01 and119872 = 119908

0= 05

048

051

054

057

06

063

066

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1

f(120578

)

120578

Figure 12 Effect of119872 on the velocity component of 119891(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

0

03

06

09

12

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

minus03

120578

f998400 (120578

)

0 02 04 06 08 1

Figure 13 Effect of 119872 on the velocity component of 1198911015840(120578) when120573 = Ω = 1 120593 = 01 and 119908

0= 05

Mathematical Problems in Engineering 11

0

002

minus002

minus004

minus006

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1120578

g(120578

)

Figure 14 Effect of119872 on the velocity component of 119892(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

025

05

075

1

125

00 02 04 06 08 1

120578

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

f(120578

)

Figure 15 Effect of1199080on the velocity component of 119891(120578)when 120573 =

Ω = 1119872 = 05 and 120593 = 01

0

05

1

15

minus05

minus10 02 04 06 08 1

120578

f998400 (120578

)w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

Figure 16 Effect of 1199080on the velocity component of 1198911015840(120578) when

120573 = Ω = 1119872 = 05 and 120593 = 01

0

01

02

minus02

minus03

minus01

0 02 04 06 08 1

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

120578

g(120578

)

Figure 17 Effect of1199080on the velocity component of 119892(120578) when 120573 =

Ω = 1119872 = 05 and 120593 = 01

12 Mathematical Problems in Engineering

048

051

054

057

06

063

066

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1

f(120578

)

120578

Figure 18 Effect of nanoparticle types on the velocity componentof 119891(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0

03

06

09

12

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1120578

f998400 (120578

)

minus03

minus06

Figure 19 Effect of nanoparticle types on the velocity componentof 1198911015840(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0 02 04 06 08 1

0

002

minus006

minus004

minus002

AgCuCuO

TiO2Al2O3

120578

g(120578

)

Figure 20 Effect of nanoparticle types on the velocity componentof 119892(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

Nomenclature

119886 Stretching rate of the lower plate119861 External uniform magnetic field1198610 Constant magnetic flux density

119862119891 Skin friction coefficient

119891 119892 Self-similar velocitiesℎ(119905) Upper plane distance119869 Magnetic flux119901 Pressure119905 Time119879 Cauchy stress tensor119906 Velocity component in the x directionV Velocity component in the y direction119908 Velocity component in the z direction

Dimensionless Parameters

120573 Characteristic parameter of the flow (120574119886)

Ω Rotation parameter (120596119886)1198722 Magnetic parameter (1205901198612

0120588119891119886)

1199080 Suction parameter (119881

0119886ℎ)

Re119909 Local Reynolds number (120588

1198911198800ℎ120583119891)

Greek Symbols

120572 120575 and 120576 DTM constant parameters120590 Electrical conductivity120588 Density

Mathematical Problems in Engineering 13

120583 Dynamic viscosity120578 A scaled boundary-layer coordinate] Kinematic viscosity120574 Characteristic constant parameter120595 Stream function120596 Constant angular velocity120593 Nanoparticle volume fraction

Subscripts

119891 Fluid phase119899119891 Nanofluid119904 Solid phase

Conflict of Interests

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

Acknowledgment

EbrahimMomoniat acknowledges support from theNationalResearch foundation of South Africa under Grant number76868

References

[1] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquoComputers andMathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I Boundary layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 26ndash28 1961

[3] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer on a continuous flat surfacerdquoAIChEJournal vol 7 pp 221ndash225 1961

[4] L E Erickson L T Fan and V G Fox ldquoHeat and mass transferon a moving continuous flat plate with suction or injectionrdquoIndustrial and Engineering Chemistry Fundamentals vol 5 no1 pp 19ndash25 1966

[5] F K Tsou E M Sparrow and R J Goldstein ldquoFlow and heattransfer in the boundary layer on a continuousmoving surfacerdquoInternational Journal ofHeat andMass Transfer vol 10 no 2 pp219ndash235 1967

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[7] H I Andersson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquo Acta Mechanica vol 95 no 1ndash4 pp 227ndash230 1992

[8] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[9] A Mehmood and A Ali ldquoAnalytic solution of generalizedthree-dimensional flow and heat transfer over a stretching plane

wallrdquo International Communications in Heat andMass Transfervol 33 no 10 pp 1243ndash1252 2006

[10] A Mehmood and A Ali ldquoAnalytic homotopy solution ofgeneralized three-dimensional channel flow due to uniformstretching of the platerdquoActaMechanica Sinica vol 23 no 5 pp503ndash510 2007

[11] A K Borkakoti and A Bharali ldquoHydromagnetic flow and heattransfer between two horizontal plates the lower plate being astretching sheetrdquo Quarterly of Applied Mathematics vol 40 no4 pp 461ndash467 1983

[12] S Munawar A Mehmood and A Ali ldquoEffects of slip on flowbetween two stretchable disks using optimal homotopy analysismethodrdquo Canadian Journal of Applied Science vol 1 pp 50ndash682011

[13] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicropolar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 no 8 pp1119ndash1127 2002

[14] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[15] P D Ariel ldquoOn exact solutions of flow problems of a secondgrade fluid through two parallel porous wallsrdquo InternationalJournal of Engineering Science vol 40 no 8 pp 913ndash941 2002

[16] T Hayat and Z Abbas ldquoChannel flow of a Maxwell fluid withchemical reactionrdquo Zeitschrift fur angewandte Mathematik undPhysik vol 59 no 1 pp 124ndash144 2008

[17] G Domairry and A Aziz ldquoApproximate analysis of MHDdqueeze flow between two parallel disks with suction orinjection by homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2009 Article ID 603916 19 pages2009

[18] S U S Choi and J A Eastman ldquoEnhancing thermal conductiv-ity of fluids with nanoparticlesrdquo Materials Science vol 231 pp99ndash105 1995

[19] Y Xuan and Q Li ldquoInvestigation on convective heat transferand flow features of nanofluidsrdquo Journal of Heat Transfer vol125 no 1 pp 151ndash155 2003

[20] N Bachok A Ishak and I Pop ldquoFlow and heat transfer overa rotating porous disk in a nanofluidrdquo Physica B CondensedMatter vol 406 no 9 pp 1767ndash1772 2011

[21] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[22] M H Abolbashari N Freidoonimehr F Nazari and M MRashidi ldquoEntropy analysis for an unsteady MHD flow past astretching permeable surface in nano-fluidrdquoPowder Technologyvol 267 pp 256ndash267 2014

[23] O A Beg M M Rashidi M Akbari and A HosseinildquoComparative numerical study of single-phase and two-phasemodels for bio-nanofluid transport phenomenardquo Journal ofMechanics in Medicine and Biology vol 14 Article ID 145001131 pages 2014

[24] M M Rashidi A Hosseini I Pop S Kumar and N Freidoon-imehr ldquoComparative numerical study of single and two-phasemodels of nanofluid heat transfer in wavy channelrdquo AppliedMathematics and Mechanics vol 35 pp 831ndash848 2014

[25] E Abu-Nada Z Masoud H F Oztop and A Campo ldquoEffectof nanofluid variable properties on natural convection inenclosuresrdquo International Journal of Thermal Sciences vol 49no 3 pp 479ndash491 2010

14 Mathematical Problems in Engineering

[26] M M Rashidi S Abelman and N Freidoonimehr ldquoEntropygeneration in steady MHD flow due to a rotating porous diskin a nanofluidrdquo International Journal of Heat andMass Transfervol 62 no 1 pp 515ndash525 2013

[27] M Mustafa T Hayat I Pop S Asghar and S ObaidatldquoStagnation-point flow of a nanofluid towards a stretchingsheetrdquo International Journal of Heat and Mass Transfer vol 54no 25-26 pp 5588ndash5594 2011

[28] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica vol 49 pp 469ndash482 2014

[29] M Sheikholeslami and D D Ganji ldquoThree dimensional heatand mass transfer in a rotating system using nanofluidrdquo PowderTechnology vol 253 pp 789ndash796 2014

[30] M Sheikholeslami M Gorji-Bandpy and D D Ganji ldquoLatticeBoltzmann method for MHD natural convection heat transferusing nanofluidrdquo Powder Technology vol 254 pp 82ndash93 2014

[31] F Ayaz ldquoApplications of differential transform method to diff-erential-algebraic equationsrdquoAppliedMathematics and Compu-tation vol 152 no 3 pp 649ndash657 2004

[32] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004

[33] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[34] M Kurosaka ldquoThe oscillatory boundary layer growth over thetop and bottom plates of a rota ting channelrdquo Journal of FluidsEngineering Transactions of the ASME vol 95 no 1 pp 68ndash741973

[35] H S Takhar A J Chamkha and G Nath ldquoMHD flow overa moving plate in a rotating fluid with magnetic field hallcurrents and free stream velocityrdquo International Journal ofEngineering Science vol 40 no 13 pp 1511ndash1527 2002

[36] K Vajravelu and B V R Kumar ldquoAnalytical and numericalsolutions of a coupled non-linear system arising in a three-dimensional rotating flowrdquo International Journal of Non-LinearMechanics vol 39 no 1 pp 13ndash24 2004

[37] G Domairry and A Aziz ldquoApproximate analysis of MHDsqueeze flow between two parallel disks with suction or in-jection by homotopy perturbation methodrdquoMathematical Pro-blems in Engineering vol 2009 Article ID 603916 19 pages2009

[38] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4p 571 1952

[39] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled withnanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[40] M Sheikholeslami and D D Ganji ldquoHeat transfer of Cu-waternanofluid flow between parallel platesrdquo Powder Technology vol235 pp 873ndash879 2013

[41] M M Rashidi ldquoThe modified differential transform methodfor solving MHD boundary-layer equationsrdquo Computer PhysicsCommunications vol 180 no 11 pp 2210ndash2217 2009

[42] M M Rashidi and N Freidoonimehr ldquoSeries solutions forthe flow in the vicinity of the equator of an MHD boundary-layer over a porous rotating sphere with heat transferrdquoThermalScience vol 18 pp S527ndashS537 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Table 1 Thermophysical properties of the base fluid and different nanoparticles [39]

Physical properties Fluid phase (water) Cu CuO Ag Al2O3 TiO2

120588 (kgm3) 9971 8933 6320 10500 3970 4250

where T is the Cauchy stress tensor J the magnetic flux andB the current density The above governing equations can bealso described by the following set of Navier-Stokes equations[36 37]

120597119906

120597119909+120597V120597119910

= 0

120597119906

120597119905+ 119906

120597119906

120597119909+ V

120597119906

120597119910+ 2

120596

1 minus 120574119905119908

= minus1

120588nf

120597119901

120597119909+ ]nf [

1205972119906

1205971199092+1205972119906

1205971199102] minus

1205901198612

0

120588nf (1 minus 120574119905)119906

120597V120597119905

+ 119906120597V120597119909

+ V120597V120597119910

= minus1

120588nf

120597119901

120597119910+ ]nf [

1205972V1205971199092

+1205972V1205971199102

]

120597119908

120597119905+ 119906

120597119908

120597119909+ V

120597119908

120597119910minus 2

120596

1 minus 120574119905119906

= ]nf [1205972119908

1205971199092+1205972119908

1205971199102] minus

1205901198612

0

120588nf (1 minus 120574119905)119908

(2)

where 120588nf is the nanofluid density ]nf (= 120583nf120588nf) is thenanofluid kinematic viscosity where 120583nf has been proposedby Brinkman [38] 120590 is the electrical conductivity 119861

0is the

magnetic field and 120574 is the characteristic parameter withdimension of (time)minus1 and 120574119905 lt 1 The above nanofluidconstants are defined as follows

120583nf =120583119891

(1 minus 120593)25 120588nf = (1 minus 120593) 120588

119891+ 120593120588119904 (3)

where 120583119891

is the viscosity of the fluid fraction 120593 is thenanoparticle volume fraction and 120588

119891and 120588119904are the densities

of the fluid and of the solid fractions respectively Thethermophysical properties of the base fluid (water) and dif-ferent nanoparticles are given in Table 1 [39]The appropriateboundary conditions are introduced as follows

119906 (119909 119910 119905) = 1198800=

119886119909

1 minus 120574119905

V (119909 119910 119905) = minus1198810

1 minus 120574119905

119908 (119909 119910 119905) = 0

at 119910 = 0

119906 (119909 119910 119905) = 0

V (119909 119910 119905) = 119881ℎ=119889ℎ

119889119905=minus120574

2radic

]119891

119886 (1 minus 120574119905)

119908 (119909 119910 119905) = 0

at 119910 = ℎ (119905)

(4)

where 119886 is the stretching rate of the lower plateThe followingappropriate similarity transformations are employed to con-vert the above governing equations (2)ndash(4) into a system of

ordinary differential equations in terms of a stream function120595

120595 = radic119886]119891

1 minus 120574119905119909119891 (120578) 120578 =

119910

ℎ (119905) (5)

119906 =120597120595

120597119910= 11988001198911015840(120578) V = minus

120597120595

120597119909= minusradic

119886]119891

1 minus 120574119905119891 (120578)

119908 = 1198800119892 (120578)

(6)

Substituting the similarity transformations (5) and (6) into(2)ndash(4) we obtain the following system of nonlinear ordinarydifferential equations

119891101584010158401015840+]119891

]nf(11989111989110158401015840minus (1198911015840)2

minus 120573(1198911015840+120578

211989110158401015840)

minus2Ω119892 minus1

120588nf12058811989111987221198911015840) =

(1 minus 120574119905)2]119891

120588nf1198862119909]nf

120597119901

120597119909

minus11989110158401015840+]119891

]nf(minus119891119891

1015840+120573

2(119891 + 120578119891

1015840)) =

1 minus 120574119905

120588nf119886]nf

120597119901

120597120578

(7)

11989210158401015840+]119891

]nf(1198911198921015840minus 1198911015840119892 minus 120573(119892 +

120578

21198921015840) + 2Ω119891

1015840)

minus1

120583nf1205831198911198722119892 = 0

(8)

where 120573 = 120574119886 is the characteristic parameter of the flowΩ = 120596119886 is the rotation parameter 1198722 = 120590119861

2

0120588119891119886 is the

magnetic parameter and prime denotes differentiation withrespect to 120578 In order to squeeze the flow we take 120573 gt 0 forwhich the upper plate moves downward with velocity119881

ℎlt 0

For 120573 lt 0 the upper plate moves apart with respect to theplane 119910 = 0 and 120573 = 0 corresponds to the steady statecase of the considered problem or stationary upper plate Inorder to reduce the number of independent variables andto retain the similarity solution (7) are simplified by crossdifferentiation and thus we obtain the following system ofdifferential equations

119891119894Vminus (1 minus 120593 + 120593 (120588

119904120588119891)) (1 minus 120593)

25

times (119891101584011989110158401015840minus 119891119891101584010158401015840+ 2Ω119892

1015840+120573

2(311989110158401015840+ 120578119891101584010158401015840))

minus (1 minus 120593)25119872211989110158401015840= 0

11989210158401015840+ (1 minus 120593 + 120593 (120588

119904120588119891)) (1 minus 120593)

25

times (1198911198921015840minus 1198911015840119892 minus 120573(119892 +

120578

21198921015840) + 2Ω119891

1015840)

minus (1 minus 120593)251198722119892 = 0

(9)

4 Mathematical Problems in Engineering

The transformed boundary conditions take the form

119891 (0) = 1199080 119891

1015840(0) = 1 119892 (0) = 0

119891 (1) =120573

2 119891

1015840(1) = 0 119892 (1) = 0

(10)

where 1199080= 1198810119886ℎ is the suction parameter

In this problem the physical quantity of interest is theskin friction coefficients 119862

119891along the stretching wall at the

lower and upper walls which are defined as [40]

119862119891lower =

120583nf(120597119906120597119910)119910=0

120588nf1198802

0

119862119891upper =

120583nf(120597119906120597119910)119910=ℎ(119905)

120588nf1198802

0

(11)

Substituting (6) into (11) we obtain

119862119891lower = 119862

119891lowerRe119909 =11989110158401015840(0)

(1 minus 120593 + 120593 (120588119904120588119891)) (1 minus 120593)

25

119862119891upper = 119862

119891upperRe119909 =11989110158401015840(1)

(1 minus 120593 + 120593 (120588119904120588119891)) (1 minus 120593)

25

(12)

where Re119909= 1205881198911198800ℎ120583119891is the local Reynolds number

In this section we have derived the coupled system ofnonlinear ordinary differential equations (9) to model thethree-dimensional flow of a nanofluid in a rotating channelon a lower permeable stretchingwallWe next apply theDTMto solve the coupled system

3 Analytical Approximation by Means of DTM

Taking the differential transform of (9) one can obtain (formore details see [41 42])

(119896 + 1) (119896 + 2) (119896 + 3) (119896 + 4) 119891 (119896 + 4)

minus (1 minus 120593)251198722(119896 + 1) (119896 + 2) 119891 (119896 + 2)

minus (1 minus 120593 + 120593(120588119904

120588119891

)) (1 minus 120593)25

times (

119896

sum

119903=0

((119896 minus 119903 + 1) (119896 minus 119903 + 2) (119903 + 1) 119891 (119903 + 1)

times 119891 (119896 minus 119903 + 2) minus (119896 minus 119903 + 1) (119896 minus 119903 + 2)

times (119896 minus 119903 + 3) 119891 (119903) 119891 (119896 minus 119903 + 3))

+120573

2(3 (119896 + 1) (119896 + 2) 119891 (119896 + 2)

+

119896

sum

119903=0

(

(119896 minus 119903 + 1) (119896 minus 119903 + 2)

(119896 minus 119903 + 3) 120575 (119903) 119891 (119896 minus 119903 + 3)

))

+2Ω (119896 + 1) 119892 (119896 + 1)) = 0

(119896 + 1) (119896 + 2) 119892 (119896 + 2) minus (1 minus 120593)251198722119892 (119896)

+ (1 minus 120593 + 120593(120588119904

120588119891

)) (1 minus 120593)25

times ((

119896

sum

119903=0

((119896 minus 119903 + 1) 119891 (119903) 119892 (119896 minus 119903 + 1)

minus (119896 minus 119903 + 1) 119892 (119903) 119891 (119896 minus 119903 + 1)))

+ 2Ω (119896 + 1) 119891 (119896 + 1)

minus 120573(119892 (119896) + (1

2)

times

119896

sum

119903=0

((119896 minus 119903 + 1) 120575 (119903) 119892 (119896 minus 119903 + 1)))) = 0

(13)

where 119891(119896) and 119892(119896) are differential transforms of 119891(120578) and119892(120578) given by

119891 (120578) =

infin

sum

119896=0

119891 (119896) 120578119896 (14)

119892 (120578) =

infin

sum

119896=0

119892 (119896) 120578119896 (15)

119891 (0) = 1199080 119891 (1) = 1 119891 (2) = 120572

119891 (3) = 120575 119892 (0) = 0 119892 (1) = 120576

(16)

where (16) is the transformed boundary conditions and 120572 120575and 120576 are constants By substituting (16) into (13) usingrecursion and then substituting them in (14)-(15) we obtainthe values of 119891(120578) and 119892(120578) given by

Mathematical Problems in Engineering 5

119891 (120578) = 1199080+ 120578 + 120572120578

2+ 1205751205783+

1

24

times(21198722(1 minus 120593)

25120572 + (1 minus 120593)

25(1 minus 120593 +

120588119904

120588119891

120593)

(2120572 + 3120572120573 minus 61199080120575 + 2Ω120576)

)

times 1205784+ sdot sdot sdot

119892 (120578) = 120578120576 +1

2(1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)

times (minus2Ω minus 1199080120576) 1205782+1

6

times(

1198722(1 minus 120593)

25120576 + (1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)

(minus4Ω120572 +3120573120576

2minus (1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)1199080(minus2Ω minus 119908

0120576))

)

times 1205783+

1

12

times((

(

1

21198722(1 minus 120593)

5(1 minus 120593 + (

120588119904

120588119891

)120593) (minus2Ω minus 1199080120576)

+ (1 minus 120593)25(1 minus 120593 + (

120588119904

120588119891

)120593)

times((

(

minus 6Ω120575 minus1

2(1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593) (minus2Ω minus 1199080120576)

+ 120573(1 minus 120593)25(1 minus 120593 + (

120588119904

120588119891

)120593) (minus2Ω minus 1199080120576) + 120572120576 minus

1

21199080

times((

(

1198722(1 minus 120593)

25120576 + (1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)

(

minus4Ω120572 +3120573120576

2

minus (1 minus 120593)25(1 minus 120593 + (

120588119904

120588119891

)120593)1199080(minus2Ω minus 119908

0120576)

)

))

)

))

)

))

)

times1205784+ sdot sdot sdot

(17)

The constants 120572 120575 and 120576 can be determined by applyingthe remaining boundary conditions to (17) The numberof required terms is determined by the convergence of the

numerical values to onersquos desired accuracy We obtain theapproximants using the computational software MATHE-MATICA The effect of different orders of the DTM solution

6 Mathematical Problems in Engineering

0 02 04 06 08 1

0

05

1

DTM n = 5DTM n = 10DTM n = 20

DTM n = 30DTM n = 40Numerical

g(120578

)

120578

minus05

minus1

minus15

minus2

minus25

Figure 2 The obtained results of transverse velocity of 119892(120578) fordifferent orders of DTM solution in comparison with the numericalsolution when 120573 = minus1Ω = 3119872 = 2 120593 = 01 and 119908

0= 05

on the convergence radius of the transverse velocity of 119892(120578) isdisplayed in Figure 2

In order to highlight the validity of the presented DTMsolution we compare some of our results with the obtainedHAM results of [1] A very good validation of the presentanalytical results has been achieved with the previouslypublished study as shown in Table 2

4 Results and Discussion

In the previous section the nonlinear ordinary differentialequations (9) are solved subject to the boundary condi-tions (10) via the DTM method for values of the five keyparameters characteristic parameter of the flow (120573) rotationparameter (Ω) the magnetic parameter (119872) nanoparticlevolume fraction (120593) the suction parameter (119908

0) and different

nanoparticles on the different velocity components It shouldbe stated that the copper nanoparticle is used in all of the casesin this section except for those figures which focus on theinfluences of the types of applied nanoparticles on the velocitycomponent profiles In addition we assume that the valuesof the volume fraction parameter 120593 vary from 0 (regularNewtonian fluid) to 02 Representative values are employedto simulate the physically realistic flows Tables 2 and 3 displaythe comparison between the DTM and numerical solutionresults based on a shooting technique for the shear stressesat lower and upper walls for different values of 119908

0and 120573

The influence of nanoparticle volume fraction (120593) onthe normal axial and transverse velocity components isshown in Figures 3 4 and 5The normal velocity component

0 02 04 06 08 1048

051

054

057

06

063

066

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020f

(120578)

120578

Figure 3 Effect of 120593 on the velocity component of 119891(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

reduces for higher values of the nanoparticle volume fractionThe axial velocity profile decreases with the increase innanoparticle volume fraction in the lower half channel whilein the upper half of the channel 1198911015840(120578) enhances with theincrease in120593 In addition transverse velocity profile increasesnear the lower surface for the large value of nanoparticlevolume fraction but the adverse trend happens on the otherside (in the center and the upper half of the channel)

The effects of the characteristic parameter of flow (120573)

on the velocity component 119891(120578) are plotted in Figure 6 Asone can see with the increase in 120573 the velocity profile 119891(120578)augments in the vicinity of the upper plane It can be easilyunderstood that it accumulates the squeezing effects on theflow In Figure 7 the effect of characteristic parameter of theflow on the velocity component parallel to x-axis 1198911015840(120578) isdepicted and it is clearly obvious that the velocity increaseswith the increase in the value of 120573 It should be noted that thehigher values of 120573 amortize the reverse flow but the negativevalues of 120573 support the reverse flow because of the squeezingand extricating effects of the upper wall The same trendcan be observed for transverse velocity component Figure 8shows that the transverse velocity component 119892(120578) increasesas 120573 increases

Figure 9 shows the effect of rotation parameter (Ω) onthe normal velocity component (119891(120578)) The normal velocitycomponent reduces for large values of the rotation parameterThe effect of the rotation parameter on the axial velocitycomponent (1198911015840(120578)) is presented in Figure 10 The resultsshow that in the lower half channel the velocity componentparallel to x-direction decreases with the increase in rotationparameter while in the upper half of the channel 1198911015840(120578)

Mathematical Problems in Engineering 7

Table 2 Comparison results of shear stresses at lower and upper walls for different values of 1199080when 120573 = Ω = 2119872 = 05 and 120593 = 0

1199080

11989110158401015840(0) 119891

10158401015840(1)

DTM Result HAM result of [1] DTM result HAM result of [1]00 +200373789 +20037379 minus458270559 minus4582705603 +006127071 +00612707 minus262278886 minus2622788806 minus198233944 minus19823394 minus073129902 minus0731299009 minus413494482 minus41349448 +109241241 +1092412412 minus640476369 minus64047636 +284916544 +28491655

Table 3 Comparison results of shear stresses at lower and upper walls or different values of 1199080and 120573 when Ω = 1119872 = 05 and 120593 = 01

1199080

120573119862119891lower 119862

119891upper

DTM result Numerical result DTM result Numerical result

00

minus30 minus852098772 minus852098769 +753714763 +753714759minus20 minus676815369 minus676815367 +555472891 +555472890minus10 minus497050617 minus497050616 +354801649 +35480164800 minus301634611 minus301634612 +139835301 +139835302+10 minus089270759 minus089270759 minus090954349 minus090954349+20 +139749899 +139749899 minus337418335 minus337418335+30 +384773079 +384773079 minus599011133 minus599011133

05

minus30 minus117761059 minus117761055 +947810573 +947810571minus20 minus951967384 minus951967382 +718294804 +718294803minus10 minus759674602 minus759674601 +533619342 +53361934200 minus556301306 minus556301307 +338177815 +338177815+10 minus337407009 minus337407009 +126581626 +126581626+20 minus102490197 minus102490197 minus101698614 minus101698614+30 +148110397 +148110397 minus346202219 minus346202219

10

minus30 minus177188652 minus177188648 +148771382 +148771378minus20 minus127797205 minus127797203 +872969245 +872969242minus10 minus106266507 minus106266506 +692986184 +69298618200 minus846845956 minus846845957 +516113154 +516113153+10 minus618186918 minus618186918 +324194096 +324194096+20 minus374763053 minus374763053 +114862333 +114862333+30 minus116400368 minus116400368 minus111763550 minus111763550

increases with the increase in Ω This reverse trend occursapproximately in the center of the channel (120578 cong 04) Theeffect of rotation parameter (Ω) on the transverse velocity119892(120578) is shown in Figure 11 It is clear that although the largerotation causes an increase in transverse velocity near lowersurface the adverse trend occurs on the other side In themain nanoflow regime two diverse trends can be noticedIn the lower half channel the transverse velocity increases inthe center and the upper half of the channel the transversevelocity decreases

The effect of magnetic parameter (119872) on the normalaxial and transverse velocity components can be observedin Figures 12ndash14 From Figure 12 it is seen that the velocitydecreases as magnetic parameter increases because Lorentzforce tends to decrease the velocity profile 119891(120578) In Figure 13two different trends can be seen In the lower half of the

channel the velocity component parallel to the x-directiondecreases with the increase in the magnetic parameter whilein the upper half of the channel 1198911015840(120578) increases with theincrease in 119872 This reverse trend occurs approximately inthe center of the channel The transverse velocity showsincreasing behavior with the increase in magnetic parameterin all region of the channel (Figure 14)

The consequence of suction parameter (1199080) on all the

velocity profiles is illustrated in Figures 15 16 and 17Figure 15 depicts that as the suction parameter increasesthe normal velocity component (119891(120578)) increases and thevariation in the velocity profile confines in the vicinity of theupper plate It is known that the large suction values cause1198911015840(120578) to decrease which results in the occurrence of reverse

flowThe reverse flow is more prominent near the upper platerather than the lower plate The reverse flow near the lower

8 Mathematical Problems in Engineering

Table 4 Effect of nanoparticle volume fraction and nanoparticles material on the shear stresses at lower and upper walls when Ω = 1199080= 1

and 120573 = 119872 = 05

120593Ag Cu CuO TiO2 Al2O3

119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper

000 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272005 minus772463 +4511947 minus808637 +4787926 minus879156 +5326106 minus946511 +5840246 minus95657 +5917038010 minus685916 +385395 minus734313 +4222859 minus837683 +5011777 minus949215 +5863511 minus967086 +5999997015 minus645632 +3548727 minus700493 +3966591 minus824921 +4916332 minus971233 +6034163 minus995938 +6222928020 minus631569 +344302 minus691686 +3900868 minus83391 +4986844 minus101216 +6349316 minus10435 +6588886

0 02 04 06 08 1

minus03

0

03

06

09

12

120578

f998400 (120578

)

minus06

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020

Figure 4 Effect of 120593 on the velocity component of 1198911015840(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

plate is produced by the adverse pressure gradient because thelarge amount of fluid particles escapes from the lower wallFigure 17 shows the same behavior for the transverse velocitycomponent 119892(120578)

Figures 18 19 and 20 demonstrate the effect of differ-ent nanoparticle materials on the normal velocity compo-nent (119891(120578)) axial velocity component (1198911015840(120578)) and also thetransverse velocity profile (119892(120578)) As the results show themaximum amount of normal velocity component belongsto Al2O3nanoparticle Moreover the effects of nanoparticle

volume fraction and the types of nanoparticles on the shearstresses at lower and upper walls are depicted in Table 4

5 Conclusions

In this paper we have used DTM to solve a coupled sys-tem of nonlinear ordinary differential equations for three-dimensional flow of a nanofluid in a rotating channel on

0 02 04 06 08 1

0

002

120578

g(120578

)minus002

minus004

minus006

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020

Figure 5 Effect of 120593 on the velocity component of 119892(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

a lower permeable stretching wall We have consideredwater the base fluid and four different types of nanoparti-cles copper copper oxide aluminum oxide and titaniumdioxide have been examined in this simulation The upperwall is moving along the direction normal to the surfacewith time dependent velocity The transformed dimen-sionless equations have been formulated and solved withrobust boundary conditions Important physical parame-ters have been investigated graphically These parametersinclude the flow parameter the rotation parameter themagnetic parameter the nanoparticle volume fraction thesuction parameter and also different types of nanoparti-cles Results show that the vertical motion of the upperplate interrupts the velocity in the channel remarkablyThe downward motion of the upper plate augments theforward flow whereas the upward motion reverses theflow

Mathematical Problems in Engineering 9

0 02 04 06 08 1

0

1

2

minus1

minus2

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

f(120578

)

120578

Figure 6 Effect of120573 on the velocity component of119891(120578)whenΩ = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1

0

2

minus2

minus4

minus6

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

120578

f998400 (120578

)

Figure 7 Effect of 120573 on the velocity component of 1198911015840(120578) when Ω =

1 120593 = 01 and119872 = 1199080= 05

0 02 04 06 08 1

0

2

minus2

minus4

minus6

minus8

f(120578

)

120578

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

Figure 8 Effect of 120573 on the velocity component of 119892(120578)whenΩ = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1048

051

054

057

06

063

066

g(120578

)

120578

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 9 Effect ofΩ on the transverse velocity of 119891(120578) when 120573 = 1120593 = 01 and119872 = 119908

0= 05

10 Mathematical Problems in Engineering

0 02 04 06 08 1

0

03

06

09

12

minus03

120578

f998400 (120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 10 Effect ofΩ on the transverse velocity of1198911015840(120578)when120573 = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1

0

005

minus005

minus01

minus015

minus02

minus025

120578

g(120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 11 Effect ofΩ on the transverse velocity of 119892(120578)when 120573 = 1120593 = 01 and119872 = 119908

0= 05

048

051

054

057

06

063

066

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1

f(120578

)

120578

Figure 12 Effect of119872 on the velocity component of 119891(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

0

03

06

09

12

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

minus03

120578

f998400 (120578

)

0 02 04 06 08 1

Figure 13 Effect of 119872 on the velocity component of 1198911015840(120578) when120573 = Ω = 1 120593 = 01 and 119908

0= 05

Mathematical Problems in Engineering 11

0

002

minus002

minus004

minus006

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1120578

g(120578

)

Figure 14 Effect of119872 on the velocity component of 119892(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

025

05

075

1

125

00 02 04 06 08 1

120578

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

f(120578

)

Figure 15 Effect of1199080on the velocity component of 119891(120578)when 120573 =

Ω = 1119872 = 05 and 120593 = 01

0

05

1

15

minus05

minus10 02 04 06 08 1

120578

f998400 (120578

)w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

Figure 16 Effect of 1199080on the velocity component of 1198911015840(120578) when

120573 = Ω = 1119872 = 05 and 120593 = 01

0

01

02

minus02

minus03

minus01

0 02 04 06 08 1

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

120578

g(120578

)

Figure 17 Effect of1199080on the velocity component of 119892(120578) when 120573 =

Ω = 1119872 = 05 and 120593 = 01

12 Mathematical Problems in Engineering

048

051

054

057

06

063

066

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1

f(120578

)

120578

Figure 18 Effect of nanoparticle types on the velocity componentof 119891(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0

03

06

09

12

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1120578

f998400 (120578

)

minus03

minus06

Figure 19 Effect of nanoparticle types on the velocity componentof 1198911015840(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0 02 04 06 08 1

0

002

minus006

minus004

minus002

AgCuCuO

TiO2Al2O3

120578

g(120578

)

Figure 20 Effect of nanoparticle types on the velocity componentof 119892(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

Nomenclature

119886 Stretching rate of the lower plate119861 External uniform magnetic field1198610 Constant magnetic flux density

119862119891 Skin friction coefficient

119891 119892 Self-similar velocitiesℎ(119905) Upper plane distance119869 Magnetic flux119901 Pressure119905 Time119879 Cauchy stress tensor119906 Velocity component in the x directionV Velocity component in the y direction119908 Velocity component in the z direction

Dimensionless Parameters

120573 Characteristic parameter of the flow (120574119886)

Ω Rotation parameter (120596119886)1198722 Magnetic parameter (1205901198612

0120588119891119886)

1199080 Suction parameter (119881

0119886ℎ)

Re119909 Local Reynolds number (120588

1198911198800ℎ120583119891)

Greek Symbols

120572 120575 and 120576 DTM constant parameters120590 Electrical conductivity120588 Density

Mathematical Problems in Engineering 13

120583 Dynamic viscosity120578 A scaled boundary-layer coordinate] Kinematic viscosity120574 Characteristic constant parameter120595 Stream function120596 Constant angular velocity120593 Nanoparticle volume fraction

Subscripts

119891 Fluid phase119899119891 Nanofluid119904 Solid phase

Conflict of Interests

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

Acknowledgment

EbrahimMomoniat acknowledges support from theNationalResearch foundation of South Africa under Grant number76868

References

[1] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquoComputers andMathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I Boundary layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 26ndash28 1961

[3] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer on a continuous flat surfacerdquoAIChEJournal vol 7 pp 221ndash225 1961

[4] L E Erickson L T Fan and V G Fox ldquoHeat and mass transferon a moving continuous flat plate with suction or injectionrdquoIndustrial and Engineering Chemistry Fundamentals vol 5 no1 pp 19ndash25 1966

[5] F K Tsou E M Sparrow and R J Goldstein ldquoFlow and heattransfer in the boundary layer on a continuousmoving surfacerdquoInternational Journal ofHeat andMass Transfer vol 10 no 2 pp219ndash235 1967

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[7] H I Andersson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquo Acta Mechanica vol 95 no 1ndash4 pp 227ndash230 1992

[8] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[9] A Mehmood and A Ali ldquoAnalytic solution of generalizedthree-dimensional flow and heat transfer over a stretching plane

wallrdquo International Communications in Heat andMass Transfervol 33 no 10 pp 1243ndash1252 2006

[10] A Mehmood and A Ali ldquoAnalytic homotopy solution ofgeneralized three-dimensional channel flow due to uniformstretching of the platerdquoActaMechanica Sinica vol 23 no 5 pp503ndash510 2007

[11] A K Borkakoti and A Bharali ldquoHydromagnetic flow and heattransfer between two horizontal plates the lower plate being astretching sheetrdquo Quarterly of Applied Mathematics vol 40 no4 pp 461ndash467 1983

[12] S Munawar A Mehmood and A Ali ldquoEffects of slip on flowbetween two stretchable disks using optimal homotopy analysismethodrdquo Canadian Journal of Applied Science vol 1 pp 50ndash682011

[13] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicropolar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 no 8 pp1119ndash1127 2002

[14] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[15] P D Ariel ldquoOn exact solutions of flow problems of a secondgrade fluid through two parallel porous wallsrdquo InternationalJournal of Engineering Science vol 40 no 8 pp 913ndash941 2002

[16] T Hayat and Z Abbas ldquoChannel flow of a Maxwell fluid withchemical reactionrdquo Zeitschrift fur angewandte Mathematik undPhysik vol 59 no 1 pp 124ndash144 2008

[17] G Domairry and A Aziz ldquoApproximate analysis of MHDdqueeze flow between two parallel disks with suction orinjection by homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2009 Article ID 603916 19 pages2009

[18] S U S Choi and J A Eastman ldquoEnhancing thermal conductiv-ity of fluids with nanoparticlesrdquo Materials Science vol 231 pp99ndash105 1995

[19] Y Xuan and Q Li ldquoInvestigation on convective heat transferand flow features of nanofluidsrdquo Journal of Heat Transfer vol125 no 1 pp 151ndash155 2003

[20] N Bachok A Ishak and I Pop ldquoFlow and heat transfer overa rotating porous disk in a nanofluidrdquo Physica B CondensedMatter vol 406 no 9 pp 1767ndash1772 2011

[21] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[22] M H Abolbashari N Freidoonimehr F Nazari and M MRashidi ldquoEntropy analysis for an unsteady MHD flow past astretching permeable surface in nano-fluidrdquoPowder Technologyvol 267 pp 256ndash267 2014

[23] O A Beg M M Rashidi M Akbari and A HosseinildquoComparative numerical study of single-phase and two-phasemodels for bio-nanofluid transport phenomenardquo Journal ofMechanics in Medicine and Biology vol 14 Article ID 145001131 pages 2014

[24] M M Rashidi A Hosseini I Pop S Kumar and N Freidoon-imehr ldquoComparative numerical study of single and two-phasemodels of nanofluid heat transfer in wavy channelrdquo AppliedMathematics and Mechanics vol 35 pp 831ndash848 2014

[25] E Abu-Nada Z Masoud H F Oztop and A Campo ldquoEffectof nanofluid variable properties on natural convection inenclosuresrdquo International Journal of Thermal Sciences vol 49no 3 pp 479ndash491 2010

14 Mathematical Problems in Engineering

[26] M M Rashidi S Abelman and N Freidoonimehr ldquoEntropygeneration in steady MHD flow due to a rotating porous diskin a nanofluidrdquo International Journal of Heat andMass Transfervol 62 no 1 pp 515ndash525 2013

[27] M Mustafa T Hayat I Pop S Asghar and S ObaidatldquoStagnation-point flow of a nanofluid towards a stretchingsheetrdquo International Journal of Heat and Mass Transfer vol 54no 25-26 pp 5588ndash5594 2011

[28] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica vol 49 pp 469ndash482 2014

[29] M Sheikholeslami and D D Ganji ldquoThree dimensional heatand mass transfer in a rotating system using nanofluidrdquo PowderTechnology vol 253 pp 789ndash796 2014

[30] M Sheikholeslami M Gorji-Bandpy and D D Ganji ldquoLatticeBoltzmann method for MHD natural convection heat transferusing nanofluidrdquo Powder Technology vol 254 pp 82ndash93 2014

[31] F Ayaz ldquoApplications of differential transform method to diff-erential-algebraic equationsrdquoAppliedMathematics and Compu-tation vol 152 no 3 pp 649ndash657 2004

[32] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004

[33] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[34] M Kurosaka ldquoThe oscillatory boundary layer growth over thetop and bottom plates of a rota ting channelrdquo Journal of FluidsEngineering Transactions of the ASME vol 95 no 1 pp 68ndash741973

[35] H S Takhar A J Chamkha and G Nath ldquoMHD flow overa moving plate in a rotating fluid with magnetic field hallcurrents and free stream velocityrdquo International Journal ofEngineering Science vol 40 no 13 pp 1511ndash1527 2002

[36] K Vajravelu and B V R Kumar ldquoAnalytical and numericalsolutions of a coupled non-linear system arising in a three-dimensional rotating flowrdquo International Journal of Non-LinearMechanics vol 39 no 1 pp 13ndash24 2004

[37] G Domairry and A Aziz ldquoApproximate analysis of MHDsqueeze flow between two parallel disks with suction or in-jection by homotopy perturbation methodrdquoMathematical Pro-blems in Engineering vol 2009 Article ID 603916 19 pages2009

[38] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4p 571 1952

[39] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled withnanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[40] M Sheikholeslami and D D Ganji ldquoHeat transfer of Cu-waternanofluid flow between parallel platesrdquo Powder Technology vol235 pp 873ndash879 2013

[41] M M Rashidi ldquoThe modified differential transform methodfor solving MHD boundary-layer equationsrdquo Computer PhysicsCommunications vol 180 no 11 pp 2210ndash2217 2009

[42] M M Rashidi and N Freidoonimehr ldquoSeries solutions forthe flow in the vicinity of the equator of an MHD boundary-layer over a porous rotating sphere with heat transferrdquoThermalScience vol 18 pp S527ndashS537 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

The transformed boundary conditions take the form

119891 (0) = 1199080 119891

1015840(0) = 1 119892 (0) = 0

119891 (1) =120573

2 119891

1015840(1) = 0 119892 (1) = 0

(10)

where 1199080= 1198810119886ℎ is the suction parameter

In this problem the physical quantity of interest is theskin friction coefficients 119862

119891along the stretching wall at the

lower and upper walls which are defined as [40]

119862119891lower =

120583nf(120597119906120597119910)119910=0

120588nf1198802

0

119862119891upper =

120583nf(120597119906120597119910)119910=ℎ(119905)

120588nf1198802

0

(11)

Substituting (6) into (11) we obtain

119862119891lower = 119862

119891lowerRe119909 =11989110158401015840(0)

(1 minus 120593 + 120593 (120588119904120588119891)) (1 minus 120593)

25

119862119891upper = 119862

119891upperRe119909 =11989110158401015840(1)

(1 minus 120593 + 120593 (120588119904120588119891)) (1 minus 120593)

25

(12)

where Re119909= 1205881198911198800ℎ120583119891is the local Reynolds number

In this section we have derived the coupled system ofnonlinear ordinary differential equations (9) to model thethree-dimensional flow of a nanofluid in a rotating channelon a lower permeable stretchingwallWe next apply theDTMto solve the coupled system

3 Analytical Approximation by Means of DTM

Taking the differential transform of (9) one can obtain (formore details see [41 42])

(119896 + 1) (119896 + 2) (119896 + 3) (119896 + 4) 119891 (119896 + 4)

minus (1 minus 120593)251198722(119896 + 1) (119896 + 2) 119891 (119896 + 2)

minus (1 minus 120593 + 120593(120588119904

120588119891

)) (1 minus 120593)25

times (

119896

sum

119903=0

((119896 minus 119903 + 1) (119896 minus 119903 + 2) (119903 + 1) 119891 (119903 + 1)

times 119891 (119896 minus 119903 + 2) minus (119896 minus 119903 + 1) (119896 minus 119903 + 2)

times (119896 minus 119903 + 3) 119891 (119903) 119891 (119896 minus 119903 + 3))

+120573

2(3 (119896 + 1) (119896 + 2) 119891 (119896 + 2)

+

119896

sum

119903=0

(

(119896 minus 119903 + 1) (119896 minus 119903 + 2)

(119896 minus 119903 + 3) 120575 (119903) 119891 (119896 minus 119903 + 3)

))

+2Ω (119896 + 1) 119892 (119896 + 1)) = 0

(119896 + 1) (119896 + 2) 119892 (119896 + 2) minus (1 minus 120593)251198722119892 (119896)

+ (1 minus 120593 + 120593(120588119904

120588119891

)) (1 minus 120593)25

times ((

119896

sum

119903=0

((119896 minus 119903 + 1) 119891 (119903) 119892 (119896 minus 119903 + 1)

minus (119896 minus 119903 + 1) 119892 (119903) 119891 (119896 minus 119903 + 1)))

+ 2Ω (119896 + 1) 119891 (119896 + 1)

minus 120573(119892 (119896) + (1

2)

times

119896

sum

119903=0

((119896 minus 119903 + 1) 120575 (119903) 119892 (119896 minus 119903 + 1)))) = 0

(13)

where 119891(119896) and 119892(119896) are differential transforms of 119891(120578) and119892(120578) given by

119891 (120578) =

infin

sum

119896=0

119891 (119896) 120578119896 (14)

119892 (120578) =

infin

sum

119896=0

119892 (119896) 120578119896 (15)

119891 (0) = 1199080 119891 (1) = 1 119891 (2) = 120572

119891 (3) = 120575 119892 (0) = 0 119892 (1) = 120576

(16)

where (16) is the transformed boundary conditions and 120572 120575and 120576 are constants By substituting (16) into (13) usingrecursion and then substituting them in (14)-(15) we obtainthe values of 119891(120578) and 119892(120578) given by

Mathematical Problems in Engineering 5

119891 (120578) = 1199080+ 120578 + 120572120578

2+ 1205751205783+

1

24

times(21198722(1 minus 120593)

25120572 + (1 minus 120593)

25(1 minus 120593 +

120588119904

120588119891

120593)

(2120572 + 3120572120573 minus 61199080120575 + 2Ω120576)

)

times 1205784+ sdot sdot sdot

119892 (120578) = 120578120576 +1

2(1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)

times (minus2Ω minus 1199080120576) 1205782+1

6

times(

1198722(1 minus 120593)

25120576 + (1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)

(minus4Ω120572 +3120573120576

2minus (1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)1199080(minus2Ω minus 119908

0120576))

)

times 1205783+

1

12

times((

(

1

21198722(1 minus 120593)

5(1 minus 120593 + (

120588119904

120588119891

)120593) (minus2Ω minus 1199080120576)

+ (1 minus 120593)25(1 minus 120593 + (

120588119904

120588119891

)120593)

times((

(

minus 6Ω120575 minus1

2(1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593) (minus2Ω minus 1199080120576)

+ 120573(1 minus 120593)25(1 minus 120593 + (

120588119904

120588119891

)120593) (minus2Ω minus 1199080120576) + 120572120576 minus

1

21199080

times((

(

1198722(1 minus 120593)

25120576 + (1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)

(

minus4Ω120572 +3120573120576

2

minus (1 minus 120593)25(1 minus 120593 + (

120588119904

120588119891

)120593)1199080(minus2Ω minus 119908

0120576)

)

))

)

))

)

))

)

times1205784+ sdot sdot sdot

(17)

The constants 120572 120575 and 120576 can be determined by applyingthe remaining boundary conditions to (17) The numberof required terms is determined by the convergence of the

numerical values to onersquos desired accuracy We obtain theapproximants using the computational software MATHE-MATICA The effect of different orders of the DTM solution

6 Mathematical Problems in Engineering

0 02 04 06 08 1

0

05

1

DTM n = 5DTM n = 10DTM n = 20

DTM n = 30DTM n = 40Numerical

g(120578

)

120578

minus05

minus1

minus15

minus2

minus25

Figure 2 The obtained results of transverse velocity of 119892(120578) fordifferent orders of DTM solution in comparison with the numericalsolution when 120573 = minus1Ω = 3119872 = 2 120593 = 01 and 119908

0= 05

on the convergence radius of the transverse velocity of 119892(120578) isdisplayed in Figure 2

In order to highlight the validity of the presented DTMsolution we compare some of our results with the obtainedHAM results of [1] A very good validation of the presentanalytical results has been achieved with the previouslypublished study as shown in Table 2

4 Results and Discussion

In the previous section the nonlinear ordinary differentialequations (9) are solved subject to the boundary condi-tions (10) via the DTM method for values of the five keyparameters characteristic parameter of the flow (120573) rotationparameter (Ω) the magnetic parameter (119872) nanoparticlevolume fraction (120593) the suction parameter (119908

0) and different

nanoparticles on the different velocity components It shouldbe stated that the copper nanoparticle is used in all of the casesin this section except for those figures which focus on theinfluences of the types of applied nanoparticles on the velocitycomponent profiles In addition we assume that the valuesof the volume fraction parameter 120593 vary from 0 (regularNewtonian fluid) to 02 Representative values are employedto simulate the physically realistic flows Tables 2 and 3 displaythe comparison between the DTM and numerical solutionresults based on a shooting technique for the shear stressesat lower and upper walls for different values of 119908

0and 120573

The influence of nanoparticle volume fraction (120593) onthe normal axial and transverse velocity components isshown in Figures 3 4 and 5The normal velocity component

0 02 04 06 08 1048

051

054

057

06

063

066

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020f

(120578)

120578

Figure 3 Effect of 120593 on the velocity component of 119891(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

reduces for higher values of the nanoparticle volume fractionThe axial velocity profile decreases with the increase innanoparticle volume fraction in the lower half channel whilein the upper half of the channel 1198911015840(120578) enhances with theincrease in120593 In addition transverse velocity profile increasesnear the lower surface for the large value of nanoparticlevolume fraction but the adverse trend happens on the otherside (in the center and the upper half of the channel)

The effects of the characteristic parameter of flow (120573)

on the velocity component 119891(120578) are plotted in Figure 6 Asone can see with the increase in 120573 the velocity profile 119891(120578)augments in the vicinity of the upper plane It can be easilyunderstood that it accumulates the squeezing effects on theflow In Figure 7 the effect of characteristic parameter of theflow on the velocity component parallel to x-axis 1198911015840(120578) isdepicted and it is clearly obvious that the velocity increaseswith the increase in the value of 120573 It should be noted that thehigher values of 120573 amortize the reverse flow but the negativevalues of 120573 support the reverse flow because of the squeezingand extricating effects of the upper wall The same trendcan be observed for transverse velocity component Figure 8shows that the transverse velocity component 119892(120578) increasesas 120573 increases

Figure 9 shows the effect of rotation parameter (Ω) onthe normal velocity component (119891(120578)) The normal velocitycomponent reduces for large values of the rotation parameterThe effect of the rotation parameter on the axial velocitycomponent (1198911015840(120578)) is presented in Figure 10 The resultsshow that in the lower half channel the velocity componentparallel to x-direction decreases with the increase in rotationparameter while in the upper half of the channel 1198911015840(120578)

Mathematical Problems in Engineering 7

Table 2 Comparison results of shear stresses at lower and upper walls for different values of 1199080when 120573 = Ω = 2119872 = 05 and 120593 = 0

1199080

11989110158401015840(0) 119891

10158401015840(1)

DTM Result HAM result of [1] DTM result HAM result of [1]00 +200373789 +20037379 minus458270559 minus4582705603 +006127071 +00612707 minus262278886 minus2622788806 minus198233944 minus19823394 minus073129902 minus0731299009 minus413494482 minus41349448 +109241241 +1092412412 minus640476369 minus64047636 +284916544 +28491655

Table 3 Comparison results of shear stresses at lower and upper walls or different values of 1199080and 120573 when Ω = 1119872 = 05 and 120593 = 01

1199080

120573119862119891lower 119862

119891upper

DTM result Numerical result DTM result Numerical result

00

minus30 minus852098772 minus852098769 +753714763 +753714759minus20 minus676815369 minus676815367 +555472891 +555472890minus10 minus497050617 minus497050616 +354801649 +35480164800 minus301634611 minus301634612 +139835301 +139835302+10 minus089270759 minus089270759 minus090954349 minus090954349+20 +139749899 +139749899 minus337418335 minus337418335+30 +384773079 +384773079 minus599011133 minus599011133

05

minus30 minus117761059 minus117761055 +947810573 +947810571minus20 minus951967384 minus951967382 +718294804 +718294803minus10 minus759674602 minus759674601 +533619342 +53361934200 minus556301306 minus556301307 +338177815 +338177815+10 minus337407009 minus337407009 +126581626 +126581626+20 minus102490197 minus102490197 minus101698614 minus101698614+30 +148110397 +148110397 minus346202219 minus346202219

10

minus30 minus177188652 minus177188648 +148771382 +148771378minus20 minus127797205 minus127797203 +872969245 +872969242minus10 minus106266507 minus106266506 +692986184 +69298618200 minus846845956 minus846845957 +516113154 +516113153+10 minus618186918 minus618186918 +324194096 +324194096+20 minus374763053 minus374763053 +114862333 +114862333+30 minus116400368 minus116400368 minus111763550 minus111763550

increases with the increase in Ω This reverse trend occursapproximately in the center of the channel (120578 cong 04) Theeffect of rotation parameter (Ω) on the transverse velocity119892(120578) is shown in Figure 11 It is clear that although the largerotation causes an increase in transverse velocity near lowersurface the adverse trend occurs on the other side In themain nanoflow regime two diverse trends can be noticedIn the lower half channel the transverse velocity increases inthe center and the upper half of the channel the transversevelocity decreases

The effect of magnetic parameter (119872) on the normalaxial and transverse velocity components can be observedin Figures 12ndash14 From Figure 12 it is seen that the velocitydecreases as magnetic parameter increases because Lorentzforce tends to decrease the velocity profile 119891(120578) In Figure 13two different trends can be seen In the lower half of the

channel the velocity component parallel to the x-directiondecreases with the increase in the magnetic parameter whilein the upper half of the channel 1198911015840(120578) increases with theincrease in 119872 This reverse trend occurs approximately inthe center of the channel The transverse velocity showsincreasing behavior with the increase in magnetic parameterin all region of the channel (Figure 14)

The consequence of suction parameter (1199080) on all the

velocity profiles is illustrated in Figures 15 16 and 17Figure 15 depicts that as the suction parameter increasesthe normal velocity component (119891(120578)) increases and thevariation in the velocity profile confines in the vicinity of theupper plate It is known that the large suction values cause1198911015840(120578) to decrease which results in the occurrence of reverse

flowThe reverse flow is more prominent near the upper platerather than the lower plate The reverse flow near the lower

8 Mathematical Problems in Engineering

Table 4 Effect of nanoparticle volume fraction and nanoparticles material on the shear stresses at lower and upper walls when Ω = 1199080= 1

and 120573 = 119872 = 05

120593Ag Cu CuO TiO2 Al2O3

119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper

000 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272005 minus772463 +4511947 minus808637 +4787926 minus879156 +5326106 minus946511 +5840246 minus95657 +5917038010 minus685916 +385395 minus734313 +4222859 minus837683 +5011777 minus949215 +5863511 minus967086 +5999997015 minus645632 +3548727 minus700493 +3966591 minus824921 +4916332 minus971233 +6034163 minus995938 +6222928020 minus631569 +344302 minus691686 +3900868 minus83391 +4986844 minus101216 +6349316 minus10435 +6588886

0 02 04 06 08 1

minus03

0

03

06

09

12

120578

f998400 (120578

)

minus06

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020

Figure 4 Effect of 120593 on the velocity component of 1198911015840(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

plate is produced by the adverse pressure gradient because thelarge amount of fluid particles escapes from the lower wallFigure 17 shows the same behavior for the transverse velocitycomponent 119892(120578)

Figures 18 19 and 20 demonstrate the effect of differ-ent nanoparticle materials on the normal velocity compo-nent (119891(120578)) axial velocity component (1198911015840(120578)) and also thetransverse velocity profile (119892(120578)) As the results show themaximum amount of normal velocity component belongsto Al2O3nanoparticle Moreover the effects of nanoparticle

volume fraction and the types of nanoparticles on the shearstresses at lower and upper walls are depicted in Table 4

5 Conclusions

In this paper we have used DTM to solve a coupled sys-tem of nonlinear ordinary differential equations for three-dimensional flow of a nanofluid in a rotating channel on

0 02 04 06 08 1

0

002

120578

g(120578

)minus002

minus004

minus006

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020

Figure 5 Effect of 120593 on the velocity component of 119892(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

a lower permeable stretching wall We have consideredwater the base fluid and four different types of nanoparti-cles copper copper oxide aluminum oxide and titaniumdioxide have been examined in this simulation The upperwall is moving along the direction normal to the surfacewith time dependent velocity The transformed dimen-sionless equations have been formulated and solved withrobust boundary conditions Important physical parame-ters have been investigated graphically These parametersinclude the flow parameter the rotation parameter themagnetic parameter the nanoparticle volume fraction thesuction parameter and also different types of nanoparti-cles Results show that the vertical motion of the upperplate interrupts the velocity in the channel remarkablyThe downward motion of the upper plate augments theforward flow whereas the upward motion reverses theflow

Mathematical Problems in Engineering 9

0 02 04 06 08 1

0

1

2

minus1

minus2

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

f(120578

)

120578

Figure 6 Effect of120573 on the velocity component of119891(120578)whenΩ = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1

0

2

minus2

minus4

minus6

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

120578

f998400 (120578

)

Figure 7 Effect of 120573 on the velocity component of 1198911015840(120578) when Ω =

1 120593 = 01 and119872 = 1199080= 05

0 02 04 06 08 1

0

2

minus2

minus4

minus6

minus8

f(120578

)

120578

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

Figure 8 Effect of 120573 on the velocity component of 119892(120578)whenΩ = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1048

051

054

057

06

063

066

g(120578

)

120578

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 9 Effect ofΩ on the transverse velocity of 119891(120578) when 120573 = 1120593 = 01 and119872 = 119908

0= 05

10 Mathematical Problems in Engineering

0 02 04 06 08 1

0

03

06

09

12

minus03

120578

f998400 (120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 10 Effect ofΩ on the transverse velocity of1198911015840(120578)when120573 = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1

0

005

minus005

minus01

minus015

minus02

minus025

120578

g(120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 11 Effect ofΩ on the transverse velocity of 119892(120578)when 120573 = 1120593 = 01 and119872 = 119908

0= 05

048

051

054

057

06

063

066

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1

f(120578

)

120578

Figure 12 Effect of119872 on the velocity component of 119891(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

0

03

06

09

12

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

minus03

120578

f998400 (120578

)

0 02 04 06 08 1

Figure 13 Effect of 119872 on the velocity component of 1198911015840(120578) when120573 = Ω = 1 120593 = 01 and 119908

0= 05

Mathematical Problems in Engineering 11

0

002

minus002

minus004

minus006

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1120578

g(120578

)

Figure 14 Effect of119872 on the velocity component of 119892(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

025

05

075

1

125

00 02 04 06 08 1

120578

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

f(120578

)

Figure 15 Effect of1199080on the velocity component of 119891(120578)when 120573 =

Ω = 1119872 = 05 and 120593 = 01

0

05

1

15

minus05

minus10 02 04 06 08 1

120578

f998400 (120578

)w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

Figure 16 Effect of 1199080on the velocity component of 1198911015840(120578) when

120573 = Ω = 1119872 = 05 and 120593 = 01

0

01

02

minus02

minus03

minus01

0 02 04 06 08 1

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

120578

g(120578

)

Figure 17 Effect of1199080on the velocity component of 119892(120578) when 120573 =

Ω = 1119872 = 05 and 120593 = 01

12 Mathematical Problems in Engineering

048

051

054

057

06

063

066

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1

f(120578

)

120578

Figure 18 Effect of nanoparticle types on the velocity componentof 119891(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0

03

06

09

12

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1120578

f998400 (120578

)

minus03

minus06

Figure 19 Effect of nanoparticle types on the velocity componentof 1198911015840(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0 02 04 06 08 1

0

002

minus006

minus004

minus002

AgCuCuO

TiO2Al2O3

120578

g(120578

)

Figure 20 Effect of nanoparticle types on the velocity componentof 119892(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

Nomenclature

119886 Stretching rate of the lower plate119861 External uniform magnetic field1198610 Constant magnetic flux density

119862119891 Skin friction coefficient

119891 119892 Self-similar velocitiesℎ(119905) Upper plane distance119869 Magnetic flux119901 Pressure119905 Time119879 Cauchy stress tensor119906 Velocity component in the x directionV Velocity component in the y direction119908 Velocity component in the z direction

Dimensionless Parameters

120573 Characteristic parameter of the flow (120574119886)

Ω Rotation parameter (120596119886)1198722 Magnetic parameter (1205901198612

0120588119891119886)

1199080 Suction parameter (119881

0119886ℎ)

Re119909 Local Reynolds number (120588

1198911198800ℎ120583119891)

Greek Symbols

120572 120575 and 120576 DTM constant parameters120590 Electrical conductivity120588 Density

Mathematical Problems in Engineering 13

120583 Dynamic viscosity120578 A scaled boundary-layer coordinate] Kinematic viscosity120574 Characteristic constant parameter120595 Stream function120596 Constant angular velocity120593 Nanoparticle volume fraction

Subscripts

119891 Fluid phase119899119891 Nanofluid119904 Solid phase

Conflict of Interests

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

Acknowledgment

EbrahimMomoniat acknowledges support from theNationalResearch foundation of South Africa under Grant number76868

References

[1] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquoComputers andMathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I Boundary layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 26ndash28 1961

[3] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer on a continuous flat surfacerdquoAIChEJournal vol 7 pp 221ndash225 1961

[4] L E Erickson L T Fan and V G Fox ldquoHeat and mass transferon a moving continuous flat plate with suction or injectionrdquoIndustrial and Engineering Chemistry Fundamentals vol 5 no1 pp 19ndash25 1966

[5] F K Tsou E M Sparrow and R J Goldstein ldquoFlow and heattransfer in the boundary layer on a continuousmoving surfacerdquoInternational Journal ofHeat andMass Transfer vol 10 no 2 pp219ndash235 1967

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[7] H I Andersson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquo Acta Mechanica vol 95 no 1ndash4 pp 227ndash230 1992

[8] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[9] A Mehmood and A Ali ldquoAnalytic solution of generalizedthree-dimensional flow and heat transfer over a stretching plane

wallrdquo International Communications in Heat andMass Transfervol 33 no 10 pp 1243ndash1252 2006

[10] A Mehmood and A Ali ldquoAnalytic homotopy solution ofgeneralized three-dimensional channel flow due to uniformstretching of the platerdquoActaMechanica Sinica vol 23 no 5 pp503ndash510 2007

[11] A K Borkakoti and A Bharali ldquoHydromagnetic flow and heattransfer between two horizontal plates the lower plate being astretching sheetrdquo Quarterly of Applied Mathematics vol 40 no4 pp 461ndash467 1983

[12] S Munawar A Mehmood and A Ali ldquoEffects of slip on flowbetween two stretchable disks using optimal homotopy analysismethodrdquo Canadian Journal of Applied Science vol 1 pp 50ndash682011

[13] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicropolar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 no 8 pp1119ndash1127 2002

[14] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[15] P D Ariel ldquoOn exact solutions of flow problems of a secondgrade fluid through two parallel porous wallsrdquo InternationalJournal of Engineering Science vol 40 no 8 pp 913ndash941 2002

[16] T Hayat and Z Abbas ldquoChannel flow of a Maxwell fluid withchemical reactionrdquo Zeitschrift fur angewandte Mathematik undPhysik vol 59 no 1 pp 124ndash144 2008

[17] G Domairry and A Aziz ldquoApproximate analysis of MHDdqueeze flow between two parallel disks with suction orinjection by homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2009 Article ID 603916 19 pages2009

[18] S U S Choi and J A Eastman ldquoEnhancing thermal conductiv-ity of fluids with nanoparticlesrdquo Materials Science vol 231 pp99ndash105 1995

[19] Y Xuan and Q Li ldquoInvestigation on convective heat transferand flow features of nanofluidsrdquo Journal of Heat Transfer vol125 no 1 pp 151ndash155 2003

[20] N Bachok A Ishak and I Pop ldquoFlow and heat transfer overa rotating porous disk in a nanofluidrdquo Physica B CondensedMatter vol 406 no 9 pp 1767ndash1772 2011

[21] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[22] M H Abolbashari N Freidoonimehr F Nazari and M MRashidi ldquoEntropy analysis for an unsteady MHD flow past astretching permeable surface in nano-fluidrdquoPowder Technologyvol 267 pp 256ndash267 2014

[23] O A Beg M M Rashidi M Akbari and A HosseinildquoComparative numerical study of single-phase and two-phasemodels for bio-nanofluid transport phenomenardquo Journal ofMechanics in Medicine and Biology vol 14 Article ID 145001131 pages 2014

[24] M M Rashidi A Hosseini I Pop S Kumar and N Freidoon-imehr ldquoComparative numerical study of single and two-phasemodels of nanofluid heat transfer in wavy channelrdquo AppliedMathematics and Mechanics vol 35 pp 831ndash848 2014

[25] E Abu-Nada Z Masoud H F Oztop and A Campo ldquoEffectof nanofluid variable properties on natural convection inenclosuresrdquo International Journal of Thermal Sciences vol 49no 3 pp 479ndash491 2010

14 Mathematical Problems in Engineering

[26] M M Rashidi S Abelman and N Freidoonimehr ldquoEntropygeneration in steady MHD flow due to a rotating porous diskin a nanofluidrdquo International Journal of Heat andMass Transfervol 62 no 1 pp 515ndash525 2013

[27] M Mustafa T Hayat I Pop S Asghar and S ObaidatldquoStagnation-point flow of a nanofluid towards a stretchingsheetrdquo International Journal of Heat and Mass Transfer vol 54no 25-26 pp 5588ndash5594 2011

[28] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica vol 49 pp 469ndash482 2014

[29] M Sheikholeslami and D D Ganji ldquoThree dimensional heatand mass transfer in a rotating system using nanofluidrdquo PowderTechnology vol 253 pp 789ndash796 2014

[30] M Sheikholeslami M Gorji-Bandpy and D D Ganji ldquoLatticeBoltzmann method for MHD natural convection heat transferusing nanofluidrdquo Powder Technology vol 254 pp 82ndash93 2014

[31] F Ayaz ldquoApplications of differential transform method to diff-erential-algebraic equationsrdquoAppliedMathematics and Compu-tation vol 152 no 3 pp 649ndash657 2004

[32] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004

[33] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[34] M Kurosaka ldquoThe oscillatory boundary layer growth over thetop and bottom plates of a rota ting channelrdquo Journal of FluidsEngineering Transactions of the ASME vol 95 no 1 pp 68ndash741973

[35] H S Takhar A J Chamkha and G Nath ldquoMHD flow overa moving plate in a rotating fluid with magnetic field hallcurrents and free stream velocityrdquo International Journal ofEngineering Science vol 40 no 13 pp 1511ndash1527 2002

[36] K Vajravelu and B V R Kumar ldquoAnalytical and numericalsolutions of a coupled non-linear system arising in a three-dimensional rotating flowrdquo International Journal of Non-LinearMechanics vol 39 no 1 pp 13ndash24 2004

[37] G Domairry and A Aziz ldquoApproximate analysis of MHDsqueeze flow between two parallel disks with suction or in-jection by homotopy perturbation methodrdquoMathematical Pro-blems in Engineering vol 2009 Article ID 603916 19 pages2009

[38] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4p 571 1952

[39] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled withnanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[40] M Sheikholeslami and D D Ganji ldquoHeat transfer of Cu-waternanofluid flow between parallel platesrdquo Powder Technology vol235 pp 873ndash879 2013

[41] M M Rashidi ldquoThe modified differential transform methodfor solving MHD boundary-layer equationsrdquo Computer PhysicsCommunications vol 180 no 11 pp 2210ndash2217 2009

[42] M M Rashidi and N Freidoonimehr ldquoSeries solutions forthe flow in the vicinity of the equator of an MHD boundary-layer over a porous rotating sphere with heat transferrdquoThermalScience vol 18 pp S527ndashS537 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

119891 (120578) = 1199080+ 120578 + 120572120578

2+ 1205751205783+

1

24

times(21198722(1 minus 120593)

25120572 + (1 minus 120593)

25(1 minus 120593 +

120588119904

120588119891

120593)

(2120572 + 3120572120573 minus 61199080120575 + 2Ω120576)

)

times 1205784+ sdot sdot sdot

119892 (120578) = 120578120576 +1

2(1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)

times (minus2Ω minus 1199080120576) 1205782+1

6

times(

1198722(1 minus 120593)

25120576 + (1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)

(minus4Ω120572 +3120573120576

2minus (1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)1199080(minus2Ω minus 119908

0120576))

)

times 1205783+

1

12

times((

(

1

21198722(1 minus 120593)

5(1 minus 120593 + (

120588119904

120588119891

)120593) (minus2Ω minus 1199080120576)

+ (1 minus 120593)25(1 minus 120593 + (

120588119904

120588119891

)120593)

times((

(

minus 6Ω120575 minus1

2(1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593) (minus2Ω minus 1199080120576)

+ 120573(1 minus 120593)25(1 minus 120593 + (

120588119904

120588119891

)120593) (minus2Ω minus 1199080120576) + 120572120576 minus

1

21199080

times((

(

1198722(1 minus 120593)

25120576 + (1 minus 120593)

25(1 minus 120593 + (

120588119904

120588119891

)120593)

(

minus4Ω120572 +3120573120576

2

minus (1 minus 120593)25(1 minus 120593 + (

120588119904

120588119891

)120593)1199080(minus2Ω minus 119908

0120576)

)

))

)

))

)

))

)

times1205784+ sdot sdot sdot

(17)

The constants 120572 120575 and 120576 can be determined by applyingthe remaining boundary conditions to (17) The numberof required terms is determined by the convergence of the

numerical values to onersquos desired accuracy We obtain theapproximants using the computational software MATHE-MATICA The effect of different orders of the DTM solution

6 Mathematical Problems in Engineering

0 02 04 06 08 1

0

05

1

DTM n = 5DTM n = 10DTM n = 20

DTM n = 30DTM n = 40Numerical

g(120578

)

120578

minus05

minus1

minus15

minus2

minus25

Figure 2 The obtained results of transverse velocity of 119892(120578) fordifferent orders of DTM solution in comparison with the numericalsolution when 120573 = minus1Ω = 3119872 = 2 120593 = 01 and 119908

0= 05

on the convergence radius of the transverse velocity of 119892(120578) isdisplayed in Figure 2

In order to highlight the validity of the presented DTMsolution we compare some of our results with the obtainedHAM results of [1] A very good validation of the presentanalytical results has been achieved with the previouslypublished study as shown in Table 2

4 Results and Discussion

In the previous section the nonlinear ordinary differentialequations (9) are solved subject to the boundary condi-tions (10) via the DTM method for values of the five keyparameters characteristic parameter of the flow (120573) rotationparameter (Ω) the magnetic parameter (119872) nanoparticlevolume fraction (120593) the suction parameter (119908

0) and different

nanoparticles on the different velocity components It shouldbe stated that the copper nanoparticle is used in all of the casesin this section except for those figures which focus on theinfluences of the types of applied nanoparticles on the velocitycomponent profiles In addition we assume that the valuesof the volume fraction parameter 120593 vary from 0 (regularNewtonian fluid) to 02 Representative values are employedto simulate the physically realistic flows Tables 2 and 3 displaythe comparison between the DTM and numerical solutionresults based on a shooting technique for the shear stressesat lower and upper walls for different values of 119908

0and 120573

The influence of nanoparticle volume fraction (120593) onthe normal axial and transverse velocity components isshown in Figures 3 4 and 5The normal velocity component

0 02 04 06 08 1048

051

054

057

06

063

066

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020f

(120578)

120578

Figure 3 Effect of 120593 on the velocity component of 119891(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

reduces for higher values of the nanoparticle volume fractionThe axial velocity profile decreases with the increase innanoparticle volume fraction in the lower half channel whilein the upper half of the channel 1198911015840(120578) enhances with theincrease in120593 In addition transverse velocity profile increasesnear the lower surface for the large value of nanoparticlevolume fraction but the adverse trend happens on the otherside (in the center and the upper half of the channel)

The effects of the characteristic parameter of flow (120573)

on the velocity component 119891(120578) are plotted in Figure 6 Asone can see with the increase in 120573 the velocity profile 119891(120578)augments in the vicinity of the upper plane It can be easilyunderstood that it accumulates the squeezing effects on theflow In Figure 7 the effect of characteristic parameter of theflow on the velocity component parallel to x-axis 1198911015840(120578) isdepicted and it is clearly obvious that the velocity increaseswith the increase in the value of 120573 It should be noted that thehigher values of 120573 amortize the reverse flow but the negativevalues of 120573 support the reverse flow because of the squeezingand extricating effects of the upper wall The same trendcan be observed for transverse velocity component Figure 8shows that the transverse velocity component 119892(120578) increasesas 120573 increases

Figure 9 shows the effect of rotation parameter (Ω) onthe normal velocity component (119891(120578)) The normal velocitycomponent reduces for large values of the rotation parameterThe effect of the rotation parameter on the axial velocitycomponent (1198911015840(120578)) is presented in Figure 10 The resultsshow that in the lower half channel the velocity componentparallel to x-direction decreases with the increase in rotationparameter while in the upper half of the channel 1198911015840(120578)

Mathematical Problems in Engineering 7

Table 2 Comparison results of shear stresses at lower and upper walls for different values of 1199080when 120573 = Ω = 2119872 = 05 and 120593 = 0

1199080

11989110158401015840(0) 119891

10158401015840(1)

DTM Result HAM result of [1] DTM result HAM result of [1]00 +200373789 +20037379 minus458270559 minus4582705603 +006127071 +00612707 minus262278886 minus2622788806 minus198233944 minus19823394 minus073129902 minus0731299009 minus413494482 minus41349448 +109241241 +1092412412 minus640476369 minus64047636 +284916544 +28491655

Table 3 Comparison results of shear stresses at lower and upper walls or different values of 1199080and 120573 when Ω = 1119872 = 05 and 120593 = 01

1199080

120573119862119891lower 119862

119891upper

DTM result Numerical result DTM result Numerical result

00

minus30 minus852098772 minus852098769 +753714763 +753714759minus20 minus676815369 minus676815367 +555472891 +555472890minus10 minus497050617 minus497050616 +354801649 +35480164800 minus301634611 minus301634612 +139835301 +139835302+10 minus089270759 minus089270759 minus090954349 minus090954349+20 +139749899 +139749899 minus337418335 minus337418335+30 +384773079 +384773079 minus599011133 minus599011133

05

minus30 minus117761059 minus117761055 +947810573 +947810571minus20 minus951967384 minus951967382 +718294804 +718294803minus10 minus759674602 minus759674601 +533619342 +53361934200 minus556301306 minus556301307 +338177815 +338177815+10 minus337407009 minus337407009 +126581626 +126581626+20 minus102490197 minus102490197 minus101698614 minus101698614+30 +148110397 +148110397 minus346202219 minus346202219

10

minus30 minus177188652 minus177188648 +148771382 +148771378minus20 minus127797205 minus127797203 +872969245 +872969242minus10 minus106266507 minus106266506 +692986184 +69298618200 minus846845956 minus846845957 +516113154 +516113153+10 minus618186918 minus618186918 +324194096 +324194096+20 minus374763053 minus374763053 +114862333 +114862333+30 minus116400368 minus116400368 minus111763550 minus111763550

increases with the increase in Ω This reverse trend occursapproximately in the center of the channel (120578 cong 04) Theeffect of rotation parameter (Ω) on the transverse velocity119892(120578) is shown in Figure 11 It is clear that although the largerotation causes an increase in transverse velocity near lowersurface the adverse trend occurs on the other side In themain nanoflow regime two diverse trends can be noticedIn the lower half channel the transverse velocity increases inthe center and the upper half of the channel the transversevelocity decreases

The effect of magnetic parameter (119872) on the normalaxial and transverse velocity components can be observedin Figures 12ndash14 From Figure 12 it is seen that the velocitydecreases as magnetic parameter increases because Lorentzforce tends to decrease the velocity profile 119891(120578) In Figure 13two different trends can be seen In the lower half of the

channel the velocity component parallel to the x-directiondecreases with the increase in the magnetic parameter whilein the upper half of the channel 1198911015840(120578) increases with theincrease in 119872 This reverse trend occurs approximately inthe center of the channel The transverse velocity showsincreasing behavior with the increase in magnetic parameterin all region of the channel (Figure 14)

The consequence of suction parameter (1199080) on all the

velocity profiles is illustrated in Figures 15 16 and 17Figure 15 depicts that as the suction parameter increasesthe normal velocity component (119891(120578)) increases and thevariation in the velocity profile confines in the vicinity of theupper plate It is known that the large suction values cause1198911015840(120578) to decrease which results in the occurrence of reverse

flowThe reverse flow is more prominent near the upper platerather than the lower plate The reverse flow near the lower

8 Mathematical Problems in Engineering

Table 4 Effect of nanoparticle volume fraction and nanoparticles material on the shear stresses at lower and upper walls when Ω = 1199080= 1

and 120573 = 119872 = 05

120593Ag Cu CuO TiO2 Al2O3

119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper

000 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272005 minus772463 +4511947 minus808637 +4787926 minus879156 +5326106 minus946511 +5840246 minus95657 +5917038010 minus685916 +385395 minus734313 +4222859 minus837683 +5011777 minus949215 +5863511 minus967086 +5999997015 minus645632 +3548727 minus700493 +3966591 minus824921 +4916332 minus971233 +6034163 minus995938 +6222928020 minus631569 +344302 minus691686 +3900868 minus83391 +4986844 minus101216 +6349316 minus10435 +6588886

0 02 04 06 08 1

minus03

0

03

06

09

12

120578

f998400 (120578

)

minus06

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020

Figure 4 Effect of 120593 on the velocity component of 1198911015840(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

plate is produced by the adverse pressure gradient because thelarge amount of fluid particles escapes from the lower wallFigure 17 shows the same behavior for the transverse velocitycomponent 119892(120578)

Figures 18 19 and 20 demonstrate the effect of differ-ent nanoparticle materials on the normal velocity compo-nent (119891(120578)) axial velocity component (1198911015840(120578)) and also thetransverse velocity profile (119892(120578)) As the results show themaximum amount of normal velocity component belongsto Al2O3nanoparticle Moreover the effects of nanoparticle

volume fraction and the types of nanoparticles on the shearstresses at lower and upper walls are depicted in Table 4

5 Conclusions

In this paper we have used DTM to solve a coupled sys-tem of nonlinear ordinary differential equations for three-dimensional flow of a nanofluid in a rotating channel on

0 02 04 06 08 1

0

002

120578

g(120578

)minus002

minus004

minus006

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020

Figure 5 Effect of 120593 on the velocity component of 119892(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

a lower permeable stretching wall We have consideredwater the base fluid and four different types of nanoparti-cles copper copper oxide aluminum oxide and titaniumdioxide have been examined in this simulation The upperwall is moving along the direction normal to the surfacewith time dependent velocity The transformed dimen-sionless equations have been formulated and solved withrobust boundary conditions Important physical parame-ters have been investigated graphically These parametersinclude the flow parameter the rotation parameter themagnetic parameter the nanoparticle volume fraction thesuction parameter and also different types of nanoparti-cles Results show that the vertical motion of the upperplate interrupts the velocity in the channel remarkablyThe downward motion of the upper plate augments theforward flow whereas the upward motion reverses theflow

Mathematical Problems in Engineering 9

0 02 04 06 08 1

0

1

2

minus1

minus2

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

f(120578

)

120578

Figure 6 Effect of120573 on the velocity component of119891(120578)whenΩ = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1

0

2

minus2

minus4

minus6

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

120578

f998400 (120578

)

Figure 7 Effect of 120573 on the velocity component of 1198911015840(120578) when Ω =

1 120593 = 01 and119872 = 1199080= 05

0 02 04 06 08 1

0

2

minus2

minus4

minus6

minus8

f(120578

)

120578

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

Figure 8 Effect of 120573 on the velocity component of 119892(120578)whenΩ = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1048

051

054

057

06

063

066

g(120578

)

120578

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 9 Effect ofΩ on the transverse velocity of 119891(120578) when 120573 = 1120593 = 01 and119872 = 119908

0= 05

10 Mathematical Problems in Engineering

0 02 04 06 08 1

0

03

06

09

12

minus03

120578

f998400 (120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 10 Effect ofΩ on the transverse velocity of1198911015840(120578)when120573 = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1

0

005

minus005

minus01

minus015

minus02

minus025

120578

g(120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 11 Effect ofΩ on the transverse velocity of 119892(120578)when 120573 = 1120593 = 01 and119872 = 119908

0= 05

048

051

054

057

06

063

066

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1

f(120578

)

120578

Figure 12 Effect of119872 on the velocity component of 119891(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

0

03

06

09

12

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

minus03

120578

f998400 (120578

)

0 02 04 06 08 1

Figure 13 Effect of 119872 on the velocity component of 1198911015840(120578) when120573 = Ω = 1 120593 = 01 and 119908

0= 05

Mathematical Problems in Engineering 11

0

002

minus002

minus004

minus006

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1120578

g(120578

)

Figure 14 Effect of119872 on the velocity component of 119892(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

025

05

075

1

125

00 02 04 06 08 1

120578

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

f(120578

)

Figure 15 Effect of1199080on the velocity component of 119891(120578)when 120573 =

Ω = 1119872 = 05 and 120593 = 01

0

05

1

15

minus05

minus10 02 04 06 08 1

120578

f998400 (120578

)w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

Figure 16 Effect of 1199080on the velocity component of 1198911015840(120578) when

120573 = Ω = 1119872 = 05 and 120593 = 01

0

01

02

minus02

minus03

minus01

0 02 04 06 08 1

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

120578

g(120578

)

Figure 17 Effect of1199080on the velocity component of 119892(120578) when 120573 =

Ω = 1119872 = 05 and 120593 = 01

12 Mathematical Problems in Engineering

048

051

054

057

06

063

066

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1

f(120578

)

120578

Figure 18 Effect of nanoparticle types on the velocity componentof 119891(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0

03

06

09

12

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1120578

f998400 (120578

)

minus03

minus06

Figure 19 Effect of nanoparticle types on the velocity componentof 1198911015840(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0 02 04 06 08 1

0

002

minus006

minus004

minus002

AgCuCuO

TiO2Al2O3

120578

g(120578

)

Figure 20 Effect of nanoparticle types on the velocity componentof 119892(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

Nomenclature

119886 Stretching rate of the lower plate119861 External uniform magnetic field1198610 Constant magnetic flux density

119862119891 Skin friction coefficient

119891 119892 Self-similar velocitiesℎ(119905) Upper plane distance119869 Magnetic flux119901 Pressure119905 Time119879 Cauchy stress tensor119906 Velocity component in the x directionV Velocity component in the y direction119908 Velocity component in the z direction

Dimensionless Parameters

120573 Characteristic parameter of the flow (120574119886)

Ω Rotation parameter (120596119886)1198722 Magnetic parameter (1205901198612

0120588119891119886)

1199080 Suction parameter (119881

0119886ℎ)

Re119909 Local Reynolds number (120588

1198911198800ℎ120583119891)

Greek Symbols

120572 120575 and 120576 DTM constant parameters120590 Electrical conductivity120588 Density

Mathematical Problems in Engineering 13

120583 Dynamic viscosity120578 A scaled boundary-layer coordinate] Kinematic viscosity120574 Characteristic constant parameter120595 Stream function120596 Constant angular velocity120593 Nanoparticle volume fraction

Subscripts

119891 Fluid phase119899119891 Nanofluid119904 Solid phase

Conflict of Interests

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

Acknowledgment

EbrahimMomoniat acknowledges support from theNationalResearch foundation of South Africa under Grant number76868

References

[1] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquoComputers andMathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I Boundary layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 26ndash28 1961

[3] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer on a continuous flat surfacerdquoAIChEJournal vol 7 pp 221ndash225 1961

[4] L E Erickson L T Fan and V G Fox ldquoHeat and mass transferon a moving continuous flat plate with suction or injectionrdquoIndustrial and Engineering Chemistry Fundamentals vol 5 no1 pp 19ndash25 1966

[5] F K Tsou E M Sparrow and R J Goldstein ldquoFlow and heattransfer in the boundary layer on a continuousmoving surfacerdquoInternational Journal ofHeat andMass Transfer vol 10 no 2 pp219ndash235 1967

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[7] H I Andersson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquo Acta Mechanica vol 95 no 1ndash4 pp 227ndash230 1992

[8] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[9] A Mehmood and A Ali ldquoAnalytic solution of generalizedthree-dimensional flow and heat transfer over a stretching plane

wallrdquo International Communications in Heat andMass Transfervol 33 no 10 pp 1243ndash1252 2006

[10] A Mehmood and A Ali ldquoAnalytic homotopy solution ofgeneralized three-dimensional channel flow due to uniformstretching of the platerdquoActaMechanica Sinica vol 23 no 5 pp503ndash510 2007

[11] A K Borkakoti and A Bharali ldquoHydromagnetic flow and heattransfer between two horizontal plates the lower plate being astretching sheetrdquo Quarterly of Applied Mathematics vol 40 no4 pp 461ndash467 1983

[12] S Munawar A Mehmood and A Ali ldquoEffects of slip on flowbetween two stretchable disks using optimal homotopy analysismethodrdquo Canadian Journal of Applied Science vol 1 pp 50ndash682011

[13] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicropolar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 no 8 pp1119ndash1127 2002

[14] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[15] P D Ariel ldquoOn exact solutions of flow problems of a secondgrade fluid through two parallel porous wallsrdquo InternationalJournal of Engineering Science vol 40 no 8 pp 913ndash941 2002

[16] T Hayat and Z Abbas ldquoChannel flow of a Maxwell fluid withchemical reactionrdquo Zeitschrift fur angewandte Mathematik undPhysik vol 59 no 1 pp 124ndash144 2008

[17] G Domairry and A Aziz ldquoApproximate analysis of MHDdqueeze flow between two parallel disks with suction orinjection by homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2009 Article ID 603916 19 pages2009

[18] S U S Choi and J A Eastman ldquoEnhancing thermal conductiv-ity of fluids with nanoparticlesrdquo Materials Science vol 231 pp99ndash105 1995

[19] Y Xuan and Q Li ldquoInvestigation on convective heat transferand flow features of nanofluidsrdquo Journal of Heat Transfer vol125 no 1 pp 151ndash155 2003

[20] N Bachok A Ishak and I Pop ldquoFlow and heat transfer overa rotating porous disk in a nanofluidrdquo Physica B CondensedMatter vol 406 no 9 pp 1767ndash1772 2011

[21] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[22] M H Abolbashari N Freidoonimehr F Nazari and M MRashidi ldquoEntropy analysis for an unsteady MHD flow past astretching permeable surface in nano-fluidrdquoPowder Technologyvol 267 pp 256ndash267 2014

[23] O A Beg M M Rashidi M Akbari and A HosseinildquoComparative numerical study of single-phase and two-phasemodels for bio-nanofluid transport phenomenardquo Journal ofMechanics in Medicine and Biology vol 14 Article ID 145001131 pages 2014

[24] M M Rashidi A Hosseini I Pop S Kumar and N Freidoon-imehr ldquoComparative numerical study of single and two-phasemodels of nanofluid heat transfer in wavy channelrdquo AppliedMathematics and Mechanics vol 35 pp 831ndash848 2014

[25] E Abu-Nada Z Masoud H F Oztop and A Campo ldquoEffectof nanofluid variable properties on natural convection inenclosuresrdquo International Journal of Thermal Sciences vol 49no 3 pp 479ndash491 2010

14 Mathematical Problems in Engineering

[26] M M Rashidi S Abelman and N Freidoonimehr ldquoEntropygeneration in steady MHD flow due to a rotating porous diskin a nanofluidrdquo International Journal of Heat andMass Transfervol 62 no 1 pp 515ndash525 2013

[27] M Mustafa T Hayat I Pop S Asghar and S ObaidatldquoStagnation-point flow of a nanofluid towards a stretchingsheetrdquo International Journal of Heat and Mass Transfer vol 54no 25-26 pp 5588ndash5594 2011

[28] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica vol 49 pp 469ndash482 2014

[29] M Sheikholeslami and D D Ganji ldquoThree dimensional heatand mass transfer in a rotating system using nanofluidrdquo PowderTechnology vol 253 pp 789ndash796 2014

[30] M Sheikholeslami M Gorji-Bandpy and D D Ganji ldquoLatticeBoltzmann method for MHD natural convection heat transferusing nanofluidrdquo Powder Technology vol 254 pp 82ndash93 2014

[31] F Ayaz ldquoApplications of differential transform method to diff-erential-algebraic equationsrdquoAppliedMathematics and Compu-tation vol 152 no 3 pp 649ndash657 2004

[32] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004

[33] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[34] M Kurosaka ldquoThe oscillatory boundary layer growth over thetop and bottom plates of a rota ting channelrdquo Journal of FluidsEngineering Transactions of the ASME vol 95 no 1 pp 68ndash741973

[35] H S Takhar A J Chamkha and G Nath ldquoMHD flow overa moving plate in a rotating fluid with magnetic field hallcurrents and free stream velocityrdquo International Journal ofEngineering Science vol 40 no 13 pp 1511ndash1527 2002

[36] K Vajravelu and B V R Kumar ldquoAnalytical and numericalsolutions of a coupled non-linear system arising in a three-dimensional rotating flowrdquo International Journal of Non-LinearMechanics vol 39 no 1 pp 13ndash24 2004

[37] G Domairry and A Aziz ldquoApproximate analysis of MHDsqueeze flow between two parallel disks with suction or in-jection by homotopy perturbation methodrdquoMathematical Pro-blems in Engineering vol 2009 Article ID 603916 19 pages2009

[38] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4p 571 1952

[39] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled withnanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[40] M Sheikholeslami and D D Ganji ldquoHeat transfer of Cu-waternanofluid flow between parallel platesrdquo Powder Technology vol235 pp 873ndash879 2013

[41] M M Rashidi ldquoThe modified differential transform methodfor solving MHD boundary-layer equationsrdquo Computer PhysicsCommunications vol 180 no 11 pp 2210ndash2217 2009

[42] M M Rashidi and N Freidoonimehr ldquoSeries solutions forthe flow in the vicinity of the equator of an MHD boundary-layer over a porous rotating sphere with heat transferrdquoThermalScience vol 18 pp S527ndashS537 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0 02 04 06 08 1

0

05

1

DTM n = 5DTM n = 10DTM n = 20

DTM n = 30DTM n = 40Numerical

g(120578

)

120578

minus05

minus1

minus15

minus2

minus25

Figure 2 The obtained results of transverse velocity of 119892(120578) fordifferent orders of DTM solution in comparison with the numericalsolution when 120573 = minus1Ω = 3119872 = 2 120593 = 01 and 119908

0= 05

on the convergence radius of the transverse velocity of 119892(120578) isdisplayed in Figure 2

In order to highlight the validity of the presented DTMsolution we compare some of our results with the obtainedHAM results of [1] A very good validation of the presentanalytical results has been achieved with the previouslypublished study as shown in Table 2

4 Results and Discussion

In the previous section the nonlinear ordinary differentialequations (9) are solved subject to the boundary condi-tions (10) via the DTM method for values of the five keyparameters characteristic parameter of the flow (120573) rotationparameter (Ω) the magnetic parameter (119872) nanoparticlevolume fraction (120593) the suction parameter (119908

0) and different

nanoparticles on the different velocity components It shouldbe stated that the copper nanoparticle is used in all of the casesin this section except for those figures which focus on theinfluences of the types of applied nanoparticles on the velocitycomponent profiles In addition we assume that the valuesof the volume fraction parameter 120593 vary from 0 (regularNewtonian fluid) to 02 Representative values are employedto simulate the physically realistic flows Tables 2 and 3 displaythe comparison between the DTM and numerical solutionresults based on a shooting technique for the shear stressesat lower and upper walls for different values of 119908

0and 120573

The influence of nanoparticle volume fraction (120593) onthe normal axial and transverse velocity components isshown in Figures 3 4 and 5The normal velocity component

0 02 04 06 08 1048

051

054

057

06

063

066

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020f

(120578)

120578

Figure 3 Effect of 120593 on the velocity component of 119891(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

reduces for higher values of the nanoparticle volume fractionThe axial velocity profile decreases with the increase innanoparticle volume fraction in the lower half channel whilein the upper half of the channel 1198911015840(120578) enhances with theincrease in120593 In addition transverse velocity profile increasesnear the lower surface for the large value of nanoparticlevolume fraction but the adverse trend happens on the otherside (in the center and the upper half of the channel)

The effects of the characteristic parameter of flow (120573)

on the velocity component 119891(120578) are plotted in Figure 6 Asone can see with the increase in 120573 the velocity profile 119891(120578)augments in the vicinity of the upper plane It can be easilyunderstood that it accumulates the squeezing effects on theflow In Figure 7 the effect of characteristic parameter of theflow on the velocity component parallel to x-axis 1198911015840(120578) isdepicted and it is clearly obvious that the velocity increaseswith the increase in the value of 120573 It should be noted that thehigher values of 120573 amortize the reverse flow but the negativevalues of 120573 support the reverse flow because of the squeezingand extricating effects of the upper wall The same trendcan be observed for transverse velocity component Figure 8shows that the transverse velocity component 119892(120578) increasesas 120573 increases

Figure 9 shows the effect of rotation parameter (Ω) onthe normal velocity component (119891(120578)) The normal velocitycomponent reduces for large values of the rotation parameterThe effect of the rotation parameter on the axial velocitycomponent (1198911015840(120578)) is presented in Figure 10 The resultsshow that in the lower half channel the velocity componentparallel to x-direction decreases with the increase in rotationparameter while in the upper half of the channel 1198911015840(120578)

Mathematical Problems in Engineering 7

Table 2 Comparison results of shear stresses at lower and upper walls for different values of 1199080when 120573 = Ω = 2119872 = 05 and 120593 = 0

1199080

11989110158401015840(0) 119891

10158401015840(1)

DTM Result HAM result of [1] DTM result HAM result of [1]00 +200373789 +20037379 minus458270559 minus4582705603 +006127071 +00612707 minus262278886 minus2622788806 minus198233944 minus19823394 minus073129902 minus0731299009 minus413494482 minus41349448 +109241241 +1092412412 minus640476369 minus64047636 +284916544 +28491655

Table 3 Comparison results of shear stresses at lower and upper walls or different values of 1199080and 120573 when Ω = 1119872 = 05 and 120593 = 01

1199080

120573119862119891lower 119862

119891upper

DTM result Numerical result DTM result Numerical result

00

minus30 minus852098772 minus852098769 +753714763 +753714759minus20 minus676815369 minus676815367 +555472891 +555472890minus10 minus497050617 minus497050616 +354801649 +35480164800 minus301634611 minus301634612 +139835301 +139835302+10 minus089270759 minus089270759 minus090954349 minus090954349+20 +139749899 +139749899 minus337418335 minus337418335+30 +384773079 +384773079 minus599011133 minus599011133

05

minus30 minus117761059 minus117761055 +947810573 +947810571minus20 minus951967384 minus951967382 +718294804 +718294803minus10 minus759674602 minus759674601 +533619342 +53361934200 minus556301306 minus556301307 +338177815 +338177815+10 minus337407009 minus337407009 +126581626 +126581626+20 minus102490197 minus102490197 minus101698614 minus101698614+30 +148110397 +148110397 minus346202219 minus346202219

10

minus30 minus177188652 minus177188648 +148771382 +148771378minus20 minus127797205 minus127797203 +872969245 +872969242minus10 minus106266507 minus106266506 +692986184 +69298618200 minus846845956 minus846845957 +516113154 +516113153+10 minus618186918 minus618186918 +324194096 +324194096+20 minus374763053 minus374763053 +114862333 +114862333+30 minus116400368 minus116400368 minus111763550 minus111763550

increases with the increase in Ω This reverse trend occursapproximately in the center of the channel (120578 cong 04) Theeffect of rotation parameter (Ω) on the transverse velocity119892(120578) is shown in Figure 11 It is clear that although the largerotation causes an increase in transverse velocity near lowersurface the adverse trend occurs on the other side In themain nanoflow regime two diverse trends can be noticedIn the lower half channel the transverse velocity increases inthe center and the upper half of the channel the transversevelocity decreases

The effect of magnetic parameter (119872) on the normalaxial and transverse velocity components can be observedin Figures 12ndash14 From Figure 12 it is seen that the velocitydecreases as magnetic parameter increases because Lorentzforce tends to decrease the velocity profile 119891(120578) In Figure 13two different trends can be seen In the lower half of the

channel the velocity component parallel to the x-directiondecreases with the increase in the magnetic parameter whilein the upper half of the channel 1198911015840(120578) increases with theincrease in 119872 This reverse trend occurs approximately inthe center of the channel The transverse velocity showsincreasing behavior with the increase in magnetic parameterin all region of the channel (Figure 14)

The consequence of suction parameter (1199080) on all the

velocity profiles is illustrated in Figures 15 16 and 17Figure 15 depicts that as the suction parameter increasesthe normal velocity component (119891(120578)) increases and thevariation in the velocity profile confines in the vicinity of theupper plate It is known that the large suction values cause1198911015840(120578) to decrease which results in the occurrence of reverse

flowThe reverse flow is more prominent near the upper platerather than the lower plate The reverse flow near the lower

8 Mathematical Problems in Engineering

Table 4 Effect of nanoparticle volume fraction and nanoparticles material on the shear stresses at lower and upper walls when Ω = 1199080= 1

and 120573 = 119872 = 05

120593Ag Cu CuO TiO2 Al2O3

119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper

000 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272005 minus772463 +4511947 minus808637 +4787926 minus879156 +5326106 minus946511 +5840246 minus95657 +5917038010 minus685916 +385395 minus734313 +4222859 minus837683 +5011777 minus949215 +5863511 minus967086 +5999997015 minus645632 +3548727 minus700493 +3966591 minus824921 +4916332 minus971233 +6034163 minus995938 +6222928020 minus631569 +344302 minus691686 +3900868 minus83391 +4986844 minus101216 +6349316 minus10435 +6588886

0 02 04 06 08 1

minus03

0

03

06

09

12

120578

f998400 (120578

)

minus06

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020

Figure 4 Effect of 120593 on the velocity component of 1198911015840(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

plate is produced by the adverse pressure gradient because thelarge amount of fluid particles escapes from the lower wallFigure 17 shows the same behavior for the transverse velocitycomponent 119892(120578)

Figures 18 19 and 20 demonstrate the effect of differ-ent nanoparticle materials on the normal velocity compo-nent (119891(120578)) axial velocity component (1198911015840(120578)) and also thetransverse velocity profile (119892(120578)) As the results show themaximum amount of normal velocity component belongsto Al2O3nanoparticle Moreover the effects of nanoparticle

volume fraction and the types of nanoparticles on the shearstresses at lower and upper walls are depicted in Table 4

5 Conclusions

In this paper we have used DTM to solve a coupled sys-tem of nonlinear ordinary differential equations for three-dimensional flow of a nanofluid in a rotating channel on

0 02 04 06 08 1

0

002

120578

g(120578

)minus002

minus004

minus006

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020

Figure 5 Effect of 120593 on the velocity component of 119892(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

a lower permeable stretching wall We have consideredwater the base fluid and four different types of nanoparti-cles copper copper oxide aluminum oxide and titaniumdioxide have been examined in this simulation The upperwall is moving along the direction normal to the surfacewith time dependent velocity The transformed dimen-sionless equations have been formulated and solved withrobust boundary conditions Important physical parame-ters have been investigated graphically These parametersinclude the flow parameter the rotation parameter themagnetic parameter the nanoparticle volume fraction thesuction parameter and also different types of nanoparti-cles Results show that the vertical motion of the upperplate interrupts the velocity in the channel remarkablyThe downward motion of the upper plate augments theforward flow whereas the upward motion reverses theflow

Mathematical Problems in Engineering 9

0 02 04 06 08 1

0

1

2

minus1

minus2

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

f(120578

)

120578

Figure 6 Effect of120573 on the velocity component of119891(120578)whenΩ = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1

0

2

minus2

minus4

minus6

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

120578

f998400 (120578

)

Figure 7 Effect of 120573 on the velocity component of 1198911015840(120578) when Ω =

1 120593 = 01 and119872 = 1199080= 05

0 02 04 06 08 1

0

2

minus2

minus4

minus6

minus8

f(120578

)

120578

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

Figure 8 Effect of 120573 on the velocity component of 119892(120578)whenΩ = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1048

051

054

057

06

063

066

g(120578

)

120578

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 9 Effect ofΩ on the transverse velocity of 119891(120578) when 120573 = 1120593 = 01 and119872 = 119908

0= 05

10 Mathematical Problems in Engineering

0 02 04 06 08 1

0

03

06

09

12

minus03

120578

f998400 (120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 10 Effect ofΩ on the transverse velocity of1198911015840(120578)when120573 = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1

0

005

minus005

minus01

minus015

minus02

minus025

120578

g(120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 11 Effect ofΩ on the transverse velocity of 119892(120578)when 120573 = 1120593 = 01 and119872 = 119908

0= 05

048

051

054

057

06

063

066

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1

f(120578

)

120578

Figure 12 Effect of119872 on the velocity component of 119891(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

0

03

06

09

12

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

minus03

120578

f998400 (120578

)

0 02 04 06 08 1

Figure 13 Effect of 119872 on the velocity component of 1198911015840(120578) when120573 = Ω = 1 120593 = 01 and 119908

0= 05

Mathematical Problems in Engineering 11

0

002

minus002

minus004

minus006

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1120578

g(120578

)

Figure 14 Effect of119872 on the velocity component of 119892(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

025

05

075

1

125

00 02 04 06 08 1

120578

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

f(120578

)

Figure 15 Effect of1199080on the velocity component of 119891(120578)when 120573 =

Ω = 1119872 = 05 and 120593 = 01

0

05

1

15

minus05

minus10 02 04 06 08 1

120578

f998400 (120578

)w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

Figure 16 Effect of 1199080on the velocity component of 1198911015840(120578) when

120573 = Ω = 1119872 = 05 and 120593 = 01

0

01

02

minus02

minus03

minus01

0 02 04 06 08 1

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

120578

g(120578

)

Figure 17 Effect of1199080on the velocity component of 119892(120578) when 120573 =

Ω = 1119872 = 05 and 120593 = 01

12 Mathematical Problems in Engineering

048

051

054

057

06

063

066

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1

f(120578

)

120578

Figure 18 Effect of nanoparticle types on the velocity componentof 119891(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0

03

06

09

12

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1120578

f998400 (120578

)

minus03

minus06

Figure 19 Effect of nanoparticle types on the velocity componentof 1198911015840(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0 02 04 06 08 1

0

002

minus006

minus004

minus002

AgCuCuO

TiO2Al2O3

120578

g(120578

)

Figure 20 Effect of nanoparticle types on the velocity componentof 119892(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

Nomenclature

119886 Stretching rate of the lower plate119861 External uniform magnetic field1198610 Constant magnetic flux density

119862119891 Skin friction coefficient

119891 119892 Self-similar velocitiesℎ(119905) Upper plane distance119869 Magnetic flux119901 Pressure119905 Time119879 Cauchy stress tensor119906 Velocity component in the x directionV Velocity component in the y direction119908 Velocity component in the z direction

Dimensionless Parameters

120573 Characteristic parameter of the flow (120574119886)

Ω Rotation parameter (120596119886)1198722 Magnetic parameter (1205901198612

0120588119891119886)

1199080 Suction parameter (119881

0119886ℎ)

Re119909 Local Reynolds number (120588

1198911198800ℎ120583119891)

Greek Symbols

120572 120575 and 120576 DTM constant parameters120590 Electrical conductivity120588 Density

Mathematical Problems in Engineering 13

120583 Dynamic viscosity120578 A scaled boundary-layer coordinate] Kinematic viscosity120574 Characteristic constant parameter120595 Stream function120596 Constant angular velocity120593 Nanoparticle volume fraction

Subscripts

119891 Fluid phase119899119891 Nanofluid119904 Solid phase

Conflict of Interests

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

Acknowledgment

EbrahimMomoniat acknowledges support from theNationalResearch foundation of South Africa under Grant number76868

References

[1] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquoComputers andMathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I Boundary layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 26ndash28 1961

[3] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer on a continuous flat surfacerdquoAIChEJournal vol 7 pp 221ndash225 1961

[4] L E Erickson L T Fan and V G Fox ldquoHeat and mass transferon a moving continuous flat plate with suction or injectionrdquoIndustrial and Engineering Chemistry Fundamentals vol 5 no1 pp 19ndash25 1966

[5] F K Tsou E M Sparrow and R J Goldstein ldquoFlow and heattransfer in the boundary layer on a continuousmoving surfacerdquoInternational Journal ofHeat andMass Transfer vol 10 no 2 pp219ndash235 1967

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[7] H I Andersson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquo Acta Mechanica vol 95 no 1ndash4 pp 227ndash230 1992

[8] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[9] A Mehmood and A Ali ldquoAnalytic solution of generalizedthree-dimensional flow and heat transfer over a stretching plane

wallrdquo International Communications in Heat andMass Transfervol 33 no 10 pp 1243ndash1252 2006

[10] A Mehmood and A Ali ldquoAnalytic homotopy solution ofgeneralized three-dimensional channel flow due to uniformstretching of the platerdquoActaMechanica Sinica vol 23 no 5 pp503ndash510 2007

[11] A K Borkakoti and A Bharali ldquoHydromagnetic flow and heattransfer between two horizontal plates the lower plate being astretching sheetrdquo Quarterly of Applied Mathematics vol 40 no4 pp 461ndash467 1983

[12] S Munawar A Mehmood and A Ali ldquoEffects of slip on flowbetween two stretchable disks using optimal homotopy analysismethodrdquo Canadian Journal of Applied Science vol 1 pp 50ndash682011

[13] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicropolar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 no 8 pp1119ndash1127 2002

[14] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[15] P D Ariel ldquoOn exact solutions of flow problems of a secondgrade fluid through two parallel porous wallsrdquo InternationalJournal of Engineering Science vol 40 no 8 pp 913ndash941 2002

[16] T Hayat and Z Abbas ldquoChannel flow of a Maxwell fluid withchemical reactionrdquo Zeitschrift fur angewandte Mathematik undPhysik vol 59 no 1 pp 124ndash144 2008

[17] G Domairry and A Aziz ldquoApproximate analysis of MHDdqueeze flow between two parallel disks with suction orinjection by homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2009 Article ID 603916 19 pages2009

[18] S U S Choi and J A Eastman ldquoEnhancing thermal conductiv-ity of fluids with nanoparticlesrdquo Materials Science vol 231 pp99ndash105 1995

[19] Y Xuan and Q Li ldquoInvestigation on convective heat transferand flow features of nanofluidsrdquo Journal of Heat Transfer vol125 no 1 pp 151ndash155 2003

[20] N Bachok A Ishak and I Pop ldquoFlow and heat transfer overa rotating porous disk in a nanofluidrdquo Physica B CondensedMatter vol 406 no 9 pp 1767ndash1772 2011

[21] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[22] M H Abolbashari N Freidoonimehr F Nazari and M MRashidi ldquoEntropy analysis for an unsteady MHD flow past astretching permeable surface in nano-fluidrdquoPowder Technologyvol 267 pp 256ndash267 2014

[23] O A Beg M M Rashidi M Akbari and A HosseinildquoComparative numerical study of single-phase and two-phasemodels for bio-nanofluid transport phenomenardquo Journal ofMechanics in Medicine and Biology vol 14 Article ID 145001131 pages 2014

[24] M M Rashidi A Hosseini I Pop S Kumar and N Freidoon-imehr ldquoComparative numerical study of single and two-phasemodels of nanofluid heat transfer in wavy channelrdquo AppliedMathematics and Mechanics vol 35 pp 831ndash848 2014

[25] E Abu-Nada Z Masoud H F Oztop and A Campo ldquoEffectof nanofluid variable properties on natural convection inenclosuresrdquo International Journal of Thermal Sciences vol 49no 3 pp 479ndash491 2010

14 Mathematical Problems in Engineering

[26] M M Rashidi S Abelman and N Freidoonimehr ldquoEntropygeneration in steady MHD flow due to a rotating porous diskin a nanofluidrdquo International Journal of Heat andMass Transfervol 62 no 1 pp 515ndash525 2013

[27] M Mustafa T Hayat I Pop S Asghar and S ObaidatldquoStagnation-point flow of a nanofluid towards a stretchingsheetrdquo International Journal of Heat and Mass Transfer vol 54no 25-26 pp 5588ndash5594 2011

[28] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica vol 49 pp 469ndash482 2014

[29] M Sheikholeslami and D D Ganji ldquoThree dimensional heatand mass transfer in a rotating system using nanofluidrdquo PowderTechnology vol 253 pp 789ndash796 2014

[30] M Sheikholeslami M Gorji-Bandpy and D D Ganji ldquoLatticeBoltzmann method for MHD natural convection heat transferusing nanofluidrdquo Powder Technology vol 254 pp 82ndash93 2014

[31] F Ayaz ldquoApplications of differential transform method to diff-erential-algebraic equationsrdquoAppliedMathematics and Compu-tation vol 152 no 3 pp 649ndash657 2004

[32] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004

[33] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[34] M Kurosaka ldquoThe oscillatory boundary layer growth over thetop and bottom plates of a rota ting channelrdquo Journal of FluidsEngineering Transactions of the ASME vol 95 no 1 pp 68ndash741973

[35] H S Takhar A J Chamkha and G Nath ldquoMHD flow overa moving plate in a rotating fluid with magnetic field hallcurrents and free stream velocityrdquo International Journal ofEngineering Science vol 40 no 13 pp 1511ndash1527 2002

[36] K Vajravelu and B V R Kumar ldquoAnalytical and numericalsolutions of a coupled non-linear system arising in a three-dimensional rotating flowrdquo International Journal of Non-LinearMechanics vol 39 no 1 pp 13ndash24 2004

[37] G Domairry and A Aziz ldquoApproximate analysis of MHDsqueeze flow between two parallel disks with suction or in-jection by homotopy perturbation methodrdquoMathematical Pro-blems in Engineering vol 2009 Article ID 603916 19 pages2009

[38] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4p 571 1952

[39] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled withnanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[40] M Sheikholeslami and D D Ganji ldquoHeat transfer of Cu-waternanofluid flow between parallel platesrdquo Powder Technology vol235 pp 873ndash879 2013

[41] M M Rashidi ldquoThe modified differential transform methodfor solving MHD boundary-layer equationsrdquo Computer PhysicsCommunications vol 180 no 11 pp 2210ndash2217 2009

[42] M M Rashidi and N Freidoonimehr ldquoSeries solutions forthe flow in the vicinity of the equator of an MHD boundary-layer over a porous rotating sphere with heat transferrdquoThermalScience vol 18 pp S527ndashS537 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 2 Comparison results of shear stresses at lower and upper walls for different values of 1199080when 120573 = Ω = 2119872 = 05 and 120593 = 0

1199080

11989110158401015840(0) 119891

10158401015840(1)

DTM Result HAM result of [1] DTM result HAM result of [1]00 +200373789 +20037379 minus458270559 minus4582705603 +006127071 +00612707 minus262278886 minus2622788806 minus198233944 minus19823394 minus073129902 minus0731299009 minus413494482 minus41349448 +109241241 +1092412412 minus640476369 minus64047636 +284916544 +28491655

Table 3 Comparison results of shear stresses at lower and upper walls or different values of 1199080and 120573 when Ω = 1119872 = 05 and 120593 = 01

1199080

120573119862119891lower 119862

119891upper

DTM result Numerical result DTM result Numerical result

00

minus30 minus852098772 minus852098769 +753714763 +753714759minus20 minus676815369 minus676815367 +555472891 +555472890minus10 minus497050617 minus497050616 +354801649 +35480164800 minus301634611 minus301634612 +139835301 +139835302+10 minus089270759 minus089270759 minus090954349 minus090954349+20 +139749899 +139749899 minus337418335 minus337418335+30 +384773079 +384773079 minus599011133 minus599011133

05

minus30 minus117761059 minus117761055 +947810573 +947810571minus20 minus951967384 minus951967382 +718294804 +718294803minus10 minus759674602 minus759674601 +533619342 +53361934200 minus556301306 minus556301307 +338177815 +338177815+10 minus337407009 minus337407009 +126581626 +126581626+20 minus102490197 minus102490197 minus101698614 minus101698614+30 +148110397 +148110397 minus346202219 minus346202219

10

minus30 minus177188652 minus177188648 +148771382 +148771378minus20 minus127797205 minus127797203 +872969245 +872969242minus10 minus106266507 minus106266506 +692986184 +69298618200 minus846845956 minus846845957 +516113154 +516113153+10 minus618186918 minus618186918 +324194096 +324194096+20 minus374763053 minus374763053 +114862333 +114862333+30 minus116400368 minus116400368 minus111763550 minus111763550

increases with the increase in Ω This reverse trend occursapproximately in the center of the channel (120578 cong 04) Theeffect of rotation parameter (Ω) on the transverse velocity119892(120578) is shown in Figure 11 It is clear that although the largerotation causes an increase in transverse velocity near lowersurface the adverse trend occurs on the other side In themain nanoflow regime two diverse trends can be noticedIn the lower half channel the transverse velocity increases inthe center and the upper half of the channel the transversevelocity decreases

The effect of magnetic parameter (119872) on the normalaxial and transverse velocity components can be observedin Figures 12ndash14 From Figure 12 it is seen that the velocitydecreases as magnetic parameter increases because Lorentzforce tends to decrease the velocity profile 119891(120578) In Figure 13two different trends can be seen In the lower half of the

channel the velocity component parallel to the x-directiondecreases with the increase in the magnetic parameter whilein the upper half of the channel 1198911015840(120578) increases with theincrease in 119872 This reverse trend occurs approximately inthe center of the channel The transverse velocity showsincreasing behavior with the increase in magnetic parameterin all region of the channel (Figure 14)

The consequence of suction parameter (1199080) on all the

velocity profiles is illustrated in Figures 15 16 and 17Figure 15 depicts that as the suction parameter increasesthe normal velocity component (119891(120578)) increases and thevariation in the velocity profile confines in the vicinity of theupper plate It is known that the large suction values cause1198911015840(120578) to decrease which results in the occurrence of reverse

flowThe reverse flow is more prominent near the upper platerather than the lower plate The reverse flow near the lower

8 Mathematical Problems in Engineering

Table 4 Effect of nanoparticle volume fraction and nanoparticles material on the shear stresses at lower and upper walls when Ω = 1199080= 1

and 120573 = 119872 = 05

120593Ag Cu CuO TiO2 Al2O3

119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper

000 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272005 minus772463 +4511947 minus808637 +4787926 minus879156 +5326106 minus946511 +5840246 minus95657 +5917038010 minus685916 +385395 minus734313 +4222859 minus837683 +5011777 minus949215 +5863511 minus967086 +5999997015 minus645632 +3548727 minus700493 +3966591 minus824921 +4916332 minus971233 +6034163 minus995938 +6222928020 minus631569 +344302 minus691686 +3900868 minus83391 +4986844 minus101216 +6349316 minus10435 +6588886

0 02 04 06 08 1

minus03

0

03

06

09

12

120578

f998400 (120578

)

minus06

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020

Figure 4 Effect of 120593 on the velocity component of 1198911015840(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

plate is produced by the adverse pressure gradient because thelarge amount of fluid particles escapes from the lower wallFigure 17 shows the same behavior for the transverse velocitycomponent 119892(120578)

Figures 18 19 and 20 demonstrate the effect of differ-ent nanoparticle materials on the normal velocity compo-nent (119891(120578)) axial velocity component (1198911015840(120578)) and also thetransverse velocity profile (119892(120578)) As the results show themaximum amount of normal velocity component belongsto Al2O3nanoparticle Moreover the effects of nanoparticle

volume fraction and the types of nanoparticles on the shearstresses at lower and upper walls are depicted in Table 4

5 Conclusions

In this paper we have used DTM to solve a coupled sys-tem of nonlinear ordinary differential equations for three-dimensional flow of a nanofluid in a rotating channel on

0 02 04 06 08 1

0

002

120578

g(120578

)minus002

minus004

minus006

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020

Figure 5 Effect of 120593 on the velocity component of 119892(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

a lower permeable stretching wall We have consideredwater the base fluid and four different types of nanoparti-cles copper copper oxide aluminum oxide and titaniumdioxide have been examined in this simulation The upperwall is moving along the direction normal to the surfacewith time dependent velocity The transformed dimen-sionless equations have been formulated and solved withrobust boundary conditions Important physical parame-ters have been investigated graphically These parametersinclude the flow parameter the rotation parameter themagnetic parameter the nanoparticle volume fraction thesuction parameter and also different types of nanoparti-cles Results show that the vertical motion of the upperplate interrupts the velocity in the channel remarkablyThe downward motion of the upper plate augments theforward flow whereas the upward motion reverses theflow

Mathematical Problems in Engineering 9

0 02 04 06 08 1

0

1

2

minus1

minus2

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

f(120578

)

120578

Figure 6 Effect of120573 on the velocity component of119891(120578)whenΩ = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1

0

2

minus2

minus4

minus6

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

120578

f998400 (120578

)

Figure 7 Effect of 120573 on the velocity component of 1198911015840(120578) when Ω =

1 120593 = 01 and119872 = 1199080= 05

0 02 04 06 08 1

0

2

minus2

minus4

minus6

minus8

f(120578

)

120578

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

Figure 8 Effect of 120573 on the velocity component of 119892(120578)whenΩ = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1048

051

054

057

06

063

066

g(120578

)

120578

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 9 Effect ofΩ on the transverse velocity of 119891(120578) when 120573 = 1120593 = 01 and119872 = 119908

0= 05

10 Mathematical Problems in Engineering

0 02 04 06 08 1

0

03

06

09

12

minus03

120578

f998400 (120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 10 Effect ofΩ on the transverse velocity of1198911015840(120578)when120573 = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1

0

005

minus005

minus01

minus015

minus02

minus025

120578

g(120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 11 Effect ofΩ on the transverse velocity of 119892(120578)when 120573 = 1120593 = 01 and119872 = 119908

0= 05

048

051

054

057

06

063

066

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1

f(120578

)

120578

Figure 12 Effect of119872 on the velocity component of 119891(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

0

03

06

09

12

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

minus03

120578

f998400 (120578

)

0 02 04 06 08 1

Figure 13 Effect of 119872 on the velocity component of 1198911015840(120578) when120573 = Ω = 1 120593 = 01 and 119908

0= 05

Mathematical Problems in Engineering 11

0

002

minus002

minus004

minus006

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1120578

g(120578

)

Figure 14 Effect of119872 on the velocity component of 119892(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

025

05

075

1

125

00 02 04 06 08 1

120578

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

f(120578

)

Figure 15 Effect of1199080on the velocity component of 119891(120578)when 120573 =

Ω = 1119872 = 05 and 120593 = 01

0

05

1

15

minus05

minus10 02 04 06 08 1

120578

f998400 (120578

)w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

Figure 16 Effect of 1199080on the velocity component of 1198911015840(120578) when

120573 = Ω = 1119872 = 05 and 120593 = 01

0

01

02

minus02

minus03

minus01

0 02 04 06 08 1

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

120578

g(120578

)

Figure 17 Effect of1199080on the velocity component of 119892(120578) when 120573 =

Ω = 1119872 = 05 and 120593 = 01

12 Mathematical Problems in Engineering

048

051

054

057

06

063

066

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1

f(120578

)

120578

Figure 18 Effect of nanoparticle types on the velocity componentof 119891(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0

03

06

09

12

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1120578

f998400 (120578

)

minus03

minus06

Figure 19 Effect of nanoparticle types on the velocity componentof 1198911015840(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0 02 04 06 08 1

0

002

minus006

minus004

minus002

AgCuCuO

TiO2Al2O3

120578

g(120578

)

Figure 20 Effect of nanoparticle types on the velocity componentof 119892(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

Nomenclature

119886 Stretching rate of the lower plate119861 External uniform magnetic field1198610 Constant magnetic flux density

119862119891 Skin friction coefficient

119891 119892 Self-similar velocitiesℎ(119905) Upper plane distance119869 Magnetic flux119901 Pressure119905 Time119879 Cauchy stress tensor119906 Velocity component in the x directionV Velocity component in the y direction119908 Velocity component in the z direction

Dimensionless Parameters

120573 Characteristic parameter of the flow (120574119886)

Ω Rotation parameter (120596119886)1198722 Magnetic parameter (1205901198612

0120588119891119886)

1199080 Suction parameter (119881

0119886ℎ)

Re119909 Local Reynolds number (120588

1198911198800ℎ120583119891)

Greek Symbols

120572 120575 and 120576 DTM constant parameters120590 Electrical conductivity120588 Density

Mathematical Problems in Engineering 13

120583 Dynamic viscosity120578 A scaled boundary-layer coordinate] Kinematic viscosity120574 Characteristic constant parameter120595 Stream function120596 Constant angular velocity120593 Nanoparticle volume fraction

Subscripts

119891 Fluid phase119899119891 Nanofluid119904 Solid phase

Conflict of Interests

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

Acknowledgment

EbrahimMomoniat acknowledges support from theNationalResearch foundation of South Africa under Grant number76868

References

[1] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquoComputers andMathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I Boundary layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 26ndash28 1961

[3] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer on a continuous flat surfacerdquoAIChEJournal vol 7 pp 221ndash225 1961

[4] L E Erickson L T Fan and V G Fox ldquoHeat and mass transferon a moving continuous flat plate with suction or injectionrdquoIndustrial and Engineering Chemistry Fundamentals vol 5 no1 pp 19ndash25 1966

[5] F K Tsou E M Sparrow and R J Goldstein ldquoFlow and heattransfer in the boundary layer on a continuousmoving surfacerdquoInternational Journal ofHeat andMass Transfer vol 10 no 2 pp219ndash235 1967

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[7] H I Andersson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquo Acta Mechanica vol 95 no 1ndash4 pp 227ndash230 1992

[8] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[9] A Mehmood and A Ali ldquoAnalytic solution of generalizedthree-dimensional flow and heat transfer over a stretching plane

wallrdquo International Communications in Heat andMass Transfervol 33 no 10 pp 1243ndash1252 2006

[10] A Mehmood and A Ali ldquoAnalytic homotopy solution ofgeneralized three-dimensional channel flow due to uniformstretching of the platerdquoActaMechanica Sinica vol 23 no 5 pp503ndash510 2007

[11] A K Borkakoti and A Bharali ldquoHydromagnetic flow and heattransfer between two horizontal plates the lower plate being astretching sheetrdquo Quarterly of Applied Mathematics vol 40 no4 pp 461ndash467 1983

[12] S Munawar A Mehmood and A Ali ldquoEffects of slip on flowbetween two stretchable disks using optimal homotopy analysismethodrdquo Canadian Journal of Applied Science vol 1 pp 50ndash682011

[13] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicropolar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 no 8 pp1119ndash1127 2002

[14] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[15] P D Ariel ldquoOn exact solutions of flow problems of a secondgrade fluid through two parallel porous wallsrdquo InternationalJournal of Engineering Science vol 40 no 8 pp 913ndash941 2002

[16] T Hayat and Z Abbas ldquoChannel flow of a Maxwell fluid withchemical reactionrdquo Zeitschrift fur angewandte Mathematik undPhysik vol 59 no 1 pp 124ndash144 2008

[17] G Domairry and A Aziz ldquoApproximate analysis of MHDdqueeze flow between two parallel disks with suction orinjection by homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2009 Article ID 603916 19 pages2009

[18] S U S Choi and J A Eastman ldquoEnhancing thermal conductiv-ity of fluids with nanoparticlesrdquo Materials Science vol 231 pp99ndash105 1995

[19] Y Xuan and Q Li ldquoInvestigation on convective heat transferand flow features of nanofluidsrdquo Journal of Heat Transfer vol125 no 1 pp 151ndash155 2003

[20] N Bachok A Ishak and I Pop ldquoFlow and heat transfer overa rotating porous disk in a nanofluidrdquo Physica B CondensedMatter vol 406 no 9 pp 1767ndash1772 2011

[21] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[22] M H Abolbashari N Freidoonimehr F Nazari and M MRashidi ldquoEntropy analysis for an unsteady MHD flow past astretching permeable surface in nano-fluidrdquoPowder Technologyvol 267 pp 256ndash267 2014

[23] O A Beg M M Rashidi M Akbari and A HosseinildquoComparative numerical study of single-phase and two-phasemodels for bio-nanofluid transport phenomenardquo Journal ofMechanics in Medicine and Biology vol 14 Article ID 145001131 pages 2014

[24] M M Rashidi A Hosseini I Pop S Kumar and N Freidoon-imehr ldquoComparative numerical study of single and two-phasemodels of nanofluid heat transfer in wavy channelrdquo AppliedMathematics and Mechanics vol 35 pp 831ndash848 2014

[25] E Abu-Nada Z Masoud H F Oztop and A Campo ldquoEffectof nanofluid variable properties on natural convection inenclosuresrdquo International Journal of Thermal Sciences vol 49no 3 pp 479ndash491 2010

14 Mathematical Problems in Engineering

[26] M M Rashidi S Abelman and N Freidoonimehr ldquoEntropygeneration in steady MHD flow due to a rotating porous diskin a nanofluidrdquo International Journal of Heat andMass Transfervol 62 no 1 pp 515ndash525 2013

[27] M Mustafa T Hayat I Pop S Asghar and S ObaidatldquoStagnation-point flow of a nanofluid towards a stretchingsheetrdquo International Journal of Heat and Mass Transfer vol 54no 25-26 pp 5588ndash5594 2011

[28] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica vol 49 pp 469ndash482 2014

[29] M Sheikholeslami and D D Ganji ldquoThree dimensional heatand mass transfer in a rotating system using nanofluidrdquo PowderTechnology vol 253 pp 789ndash796 2014

[30] M Sheikholeslami M Gorji-Bandpy and D D Ganji ldquoLatticeBoltzmann method for MHD natural convection heat transferusing nanofluidrdquo Powder Technology vol 254 pp 82ndash93 2014

[31] F Ayaz ldquoApplications of differential transform method to diff-erential-algebraic equationsrdquoAppliedMathematics and Compu-tation vol 152 no 3 pp 649ndash657 2004

[32] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004

[33] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[34] M Kurosaka ldquoThe oscillatory boundary layer growth over thetop and bottom plates of a rota ting channelrdquo Journal of FluidsEngineering Transactions of the ASME vol 95 no 1 pp 68ndash741973

[35] H S Takhar A J Chamkha and G Nath ldquoMHD flow overa moving plate in a rotating fluid with magnetic field hallcurrents and free stream velocityrdquo International Journal ofEngineering Science vol 40 no 13 pp 1511ndash1527 2002

[36] K Vajravelu and B V R Kumar ldquoAnalytical and numericalsolutions of a coupled non-linear system arising in a three-dimensional rotating flowrdquo International Journal of Non-LinearMechanics vol 39 no 1 pp 13ndash24 2004

[37] G Domairry and A Aziz ldquoApproximate analysis of MHDsqueeze flow between two parallel disks with suction or in-jection by homotopy perturbation methodrdquoMathematical Pro-blems in Engineering vol 2009 Article ID 603916 19 pages2009

[38] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4p 571 1952

[39] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled withnanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[40] M Sheikholeslami and D D Ganji ldquoHeat transfer of Cu-waternanofluid flow between parallel platesrdquo Powder Technology vol235 pp 873ndash879 2013

[41] M M Rashidi ldquoThe modified differential transform methodfor solving MHD boundary-layer equationsrdquo Computer PhysicsCommunications vol 180 no 11 pp 2210ndash2217 2009

[42] M M Rashidi and N Freidoonimehr ldquoSeries solutions forthe flow in the vicinity of the equator of an MHD boundary-layer over a porous rotating sphere with heat transferrdquoThermalScience vol 18 pp S527ndashS537 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 4 Effect of nanoparticle volume fraction and nanoparticles material on the shear stresses at lower and upper walls when Ω = 1199080= 1

and 120573 = 119872 = 05

120593Ag Cu CuO TiO2 Al2O3

119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper 119862119891lower 119862

119891upper

000 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272 minus966151 +5987272005 minus772463 +4511947 minus808637 +4787926 minus879156 +5326106 minus946511 +5840246 minus95657 +5917038010 minus685916 +385395 minus734313 +4222859 minus837683 +5011777 minus949215 +5863511 minus967086 +5999997015 minus645632 +3548727 minus700493 +3966591 minus824921 +4916332 minus971233 +6034163 minus995938 +6222928020 minus631569 +344302 minus691686 +3900868 minus83391 +4986844 minus101216 +6349316 minus10435 +6588886

0 02 04 06 08 1

minus03

0

03

06

09

12

120578

f998400 (120578

)

minus06

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020

Figure 4 Effect of 120593 on the velocity component of 1198911015840(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

plate is produced by the adverse pressure gradient because thelarge amount of fluid particles escapes from the lower wallFigure 17 shows the same behavior for the transverse velocitycomponent 119892(120578)

Figures 18 19 and 20 demonstrate the effect of differ-ent nanoparticle materials on the normal velocity compo-nent (119891(120578)) axial velocity component (1198911015840(120578)) and also thetransverse velocity profile (119892(120578)) As the results show themaximum amount of normal velocity component belongsto Al2O3nanoparticle Moreover the effects of nanoparticle

volume fraction and the types of nanoparticles on the shearstresses at lower and upper walls are depicted in Table 4

5 Conclusions

In this paper we have used DTM to solve a coupled sys-tem of nonlinear ordinary differential equations for three-dimensional flow of a nanofluid in a rotating channel on

0 02 04 06 08 1

0

002

120578

g(120578

)minus002

minus004

minus006

120593 = 000

120593 = 005120593 = 010

120593 = 015

120593 = 020

Figure 5 Effect of 120593 on the velocity component of 119892(120578) when 120573 =

Ω = 1 and119872 = 1199080= 05

a lower permeable stretching wall We have consideredwater the base fluid and four different types of nanoparti-cles copper copper oxide aluminum oxide and titaniumdioxide have been examined in this simulation The upperwall is moving along the direction normal to the surfacewith time dependent velocity The transformed dimen-sionless equations have been formulated and solved withrobust boundary conditions Important physical parame-ters have been investigated graphically These parametersinclude the flow parameter the rotation parameter themagnetic parameter the nanoparticle volume fraction thesuction parameter and also different types of nanoparti-cles Results show that the vertical motion of the upperplate interrupts the velocity in the channel remarkablyThe downward motion of the upper plate augments theforward flow whereas the upward motion reverses theflow

Mathematical Problems in Engineering 9

0 02 04 06 08 1

0

1

2

minus1

minus2

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

f(120578

)

120578

Figure 6 Effect of120573 on the velocity component of119891(120578)whenΩ = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1

0

2

minus2

minus4

minus6

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

120578

f998400 (120578

)

Figure 7 Effect of 120573 on the velocity component of 1198911015840(120578) when Ω =

1 120593 = 01 and119872 = 1199080= 05

0 02 04 06 08 1

0

2

minus2

minus4

minus6

minus8

f(120578

)

120578

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

Figure 8 Effect of 120573 on the velocity component of 119892(120578)whenΩ = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1048

051

054

057

06

063

066

g(120578

)

120578

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 9 Effect ofΩ on the transverse velocity of 119891(120578) when 120573 = 1120593 = 01 and119872 = 119908

0= 05

10 Mathematical Problems in Engineering

0 02 04 06 08 1

0

03

06

09

12

minus03

120578

f998400 (120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 10 Effect ofΩ on the transverse velocity of1198911015840(120578)when120573 = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1

0

005

minus005

minus01

minus015

minus02

minus025

120578

g(120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 11 Effect ofΩ on the transverse velocity of 119892(120578)when 120573 = 1120593 = 01 and119872 = 119908

0= 05

048

051

054

057

06

063

066

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1

f(120578

)

120578

Figure 12 Effect of119872 on the velocity component of 119891(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

0

03

06

09

12

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

minus03

120578

f998400 (120578

)

0 02 04 06 08 1

Figure 13 Effect of 119872 on the velocity component of 1198911015840(120578) when120573 = Ω = 1 120593 = 01 and 119908

0= 05

Mathematical Problems in Engineering 11

0

002

minus002

minus004

minus006

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1120578

g(120578

)

Figure 14 Effect of119872 on the velocity component of 119892(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

025

05

075

1

125

00 02 04 06 08 1

120578

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

f(120578

)

Figure 15 Effect of1199080on the velocity component of 119891(120578)when 120573 =

Ω = 1119872 = 05 and 120593 = 01

0

05

1

15

minus05

minus10 02 04 06 08 1

120578

f998400 (120578

)w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

Figure 16 Effect of 1199080on the velocity component of 1198911015840(120578) when

120573 = Ω = 1119872 = 05 and 120593 = 01

0

01

02

minus02

minus03

minus01

0 02 04 06 08 1

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

120578

g(120578

)

Figure 17 Effect of1199080on the velocity component of 119892(120578) when 120573 =

Ω = 1119872 = 05 and 120593 = 01

12 Mathematical Problems in Engineering

048

051

054

057

06

063

066

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1

f(120578

)

120578

Figure 18 Effect of nanoparticle types on the velocity componentof 119891(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0

03

06

09

12

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1120578

f998400 (120578

)

minus03

minus06

Figure 19 Effect of nanoparticle types on the velocity componentof 1198911015840(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0 02 04 06 08 1

0

002

minus006

minus004

minus002

AgCuCuO

TiO2Al2O3

120578

g(120578

)

Figure 20 Effect of nanoparticle types on the velocity componentof 119892(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

Nomenclature

119886 Stretching rate of the lower plate119861 External uniform magnetic field1198610 Constant magnetic flux density

119862119891 Skin friction coefficient

119891 119892 Self-similar velocitiesℎ(119905) Upper plane distance119869 Magnetic flux119901 Pressure119905 Time119879 Cauchy stress tensor119906 Velocity component in the x directionV Velocity component in the y direction119908 Velocity component in the z direction

Dimensionless Parameters

120573 Characteristic parameter of the flow (120574119886)

Ω Rotation parameter (120596119886)1198722 Magnetic parameter (1205901198612

0120588119891119886)

1199080 Suction parameter (119881

0119886ℎ)

Re119909 Local Reynolds number (120588

1198911198800ℎ120583119891)

Greek Symbols

120572 120575 and 120576 DTM constant parameters120590 Electrical conductivity120588 Density

Mathematical Problems in Engineering 13

120583 Dynamic viscosity120578 A scaled boundary-layer coordinate] Kinematic viscosity120574 Characteristic constant parameter120595 Stream function120596 Constant angular velocity120593 Nanoparticle volume fraction

Subscripts

119891 Fluid phase119899119891 Nanofluid119904 Solid phase

Conflict of Interests

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

Acknowledgment

EbrahimMomoniat acknowledges support from theNationalResearch foundation of South Africa under Grant number76868

References

[1] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquoComputers andMathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I Boundary layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 26ndash28 1961

[3] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer on a continuous flat surfacerdquoAIChEJournal vol 7 pp 221ndash225 1961

[4] L E Erickson L T Fan and V G Fox ldquoHeat and mass transferon a moving continuous flat plate with suction or injectionrdquoIndustrial and Engineering Chemistry Fundamentals vol 5 no1 pp 19ndash25 1966

[5] F K Tsou E M Sparrow and R J Goldstein ldquoFlow and heattransfer in the boundary layer on a continuousmoving surfacerdquoInternational Journal ofHeat andMass Transfer vol 10 no 2 pp219ndash235 1967

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[7] H I Andersson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquo Acta Mechanica vol 95 no 1ndash4 pp 227ndash230 1992

[8] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[9] A Mehmood and A Ali ldquoAnalytic solution of generalizedthree-dimensional flow and heat transfer over a stretching plane

wallrdquo International Communications in Heat andMass Transfervol 33 no 10 pp 1243ndash1252 2006

[10] A Mehmood and A Ali ldquoAnalytic homotopy solution ofgeneralized three-dimensional channel flow due to uniformstretching of the platerdquoActaMechanica Sinica vol 23 no 5 pp503ndash510 2007

[11] A K Borkakoti and A Bharali ldquoHydromagnetic flow and heattransfer between two horizontal plates the lower plate being astretching sheetrdquo Quarterly of Applied Mathematics vol 40 no4 pp 461ndash467 1983

[12] S Munawar A Mehmood and A Ali ldquoEffects of slip on flowbetween two stretchable disks using optimal homotopy analysismethodrdquo Canadian Journal of Applied Science vol 1 pp 50ndash682011

[13] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicropolar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 no 8 pp1119ndash1127 2002

[14] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[15] P D Ariel ldquoOn exact solutions of flow problems of a secondgrade fluid through two parallel porous wallsrdquo InternationalJournal of Engineering Science vol 40 no 8 pp 913ndash941 2002

[16] T Hayat and Z Abbas ldquoChannel flow of a Maxwell fluid withchemical reactionrdquo Zeitschrift fur angewandte Mathematik undPhysik vol 59 no 1 pp 124ndash144 2008

[17] G Domairry and A Aziz ldquoApproximate analysis of MHDdqueeze flow between two parallel disks with suction orinjection by homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2009 Article ID 603916 19 pages2009

[18] S U S Choi and J A Eastman ldquoEnhancing thermal conductiv-ity of fluids with nanoparticlesrdquo Materials Science vol 231 pp99ndash105 1995

[19] Y Xuan and Q Li ldquoInvestigation on convective heat transferand flow features of nanofluidsrdquo Journal of Heat Transfer vol125 no 1 pp 151ndash155 2003

[20] N Bachok A Ishak and I Pop ldquoFlow and heat transfer overa rotating porous disk in a nanofluidrdquo Physica B CondensedMatter vol 406 no 9 pp 1767ndash1772 2011

[21] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[22] M H Abolbashari N Freidoonimehr F Nazari and M MRashidi ldquoEntropy analysis for an unsteady MHD flow past astretching permeable surface in nano-fluidrdquoPowder Technologyvol 267 pp 256ndash267 2014

[23] O A Beg M M Rashidi M Akbari and A HosseinildquoComparative numerical study of single-phase and two-phasemodels for bio-nanofluid transport phenomenardquo Journal ofMechanics in Medicine and Biology vol 14 Article ID 145001131 pages 2014

[24] M M Rashidi A Hosseini I Pop S Kumar and N Freidoon-imehr ldquoComparative numerical study of single and two-phasemodels of nanofluid heat transfer in wavy channelrdquo AppliedMathematics and Mechanics vol 35 pp 831ndash848 2014

[25] E Abu-Nada Z Masoud H F Oztop and A Campo ldquoEffectof nanofluid variable properties on natural convection inenclosuresrdquo International Journal of Thermal Sciences vol 49no 3 pp 479ndash491 2010

14 Mathematical Problems in Engineering

[26] M M Rashidi S Abelman and N Freidoonimehr ldquoEntropygeneration in steady MHD flow due to a rotating porous diskin a nanofluidrdquo International Journal of Heat andMass Transfervol 62 no 1 pp 515ndash525 2013

[27] M Mustafa T Hayat I Pop S Asghar and S ObaidatldquoStagnation-point flow of a nanofluid towards a stretchingsheetrdquo International Journal of Heat and Mass Transfer vol 54no 25-26 pp 5588ndash5594 2011

[28] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica vol 49 pp 469ndash482 2014

[29] M Sheikholeslami and D D Ganji ldquoThree dimensional heatand mass transfer in a rotating system using nanofluidrdquo PowderTechnology vol 253 pp 789ndash796 2014

[30] M Sheikholeslami M Gorji-Bandpy and D D Ganji ldquoLatticeBoltzmann method for MHD natural convection heat transferusing nanofluidrdquo Powder Technology vol 254 pp 82ndash93 2014

[31] F Ayaz ldquoApplications of differential transform method to diff-erential-algebraic equationsrdquoAppliedMathematics and Compu-tation vol 152 no 3 pp 649ndash657 2004

[32] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004

[33] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[34] M Kurosaka ldquoThe oscillatory boundary layer growth over thetop and bottom plates of a rota ting channelrdquo Journal of FluidsEngineering Transactions of the ASME vol 95 no 1 pp 68ndash741973

[35] H S Takhar A J Chamkha and G Nath ldquoMHD flow overa moving plate in a rotating fluid with magnetic field hallcurrents and free stream velocityrdquo International Journal ofEngineering Science vol 40 no 13 pp 1511ndash1527 2002

[36] K Vajravelu and B V R Kumar ldquoAnalytical and numericalsolutions of a coupled non-linear system arising in a three-dimensional rotating flowrdquo International Journal of Non-LinearMechanics vol 39 no 1 pp 13ndash24 2004

[37] G Domairry and A Aziz ldquoApproximate analysis of MHDsqueeze flow between two parallel disks with suction or in-jection by homotopy perturbation methodrdquoMathematical Pro-blems in Engineering vol 2009 Article ID 603916 19 pages2009

[38] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4p 571 1952

[39] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled withnanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[40] M Sheikholeslami and D D Ganji ldquoHeat transfer of Cu-waternanofluid flow between parallel platesrdquo Powder Technology vol235 pp 873ndash879 2013

[41] M M Rashidi ldquoThe modified differential transform methodfor solving MHD boundary-layer equationsrdquo Computer PhysicsCommunications vol 180 no 11 pp 2210ndash2217 2009

[42] M M Rashidi and N Freidoonimehr ldquoSeries solutions forthe flow in the vicinity of the equator of an MHD boundary-layer over a porous rotating sphere with heat transferrdquoThermalScience vol 18 pp S527ndashS537 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 02 04 06 08 1

0

1

2

minus1

minus2

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

f(120578

)

120578

Figure 6 Effect of120573 on the velocity component of119891(120578)whenΩ = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1

0

2

minus2

minus4

minus6

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

120578

f998400 (120578

)

Figure 7 Effect of 120573 on the velocity component of 1198911015840(120578) when Ω =

1 120593 = 01 and119872 = 1199080= 05

0 02 04 06 08 1

0

2

minus2

minus4

minus6

minus8

f(120578

)

120578

120573 = minus3

120573 = minus2

120573 = minus1

120573 = 00

120573 = +1

120573 = +2

120573 = +3

Figure 8 Effect of 120573 on the velocity component of 119892(120578)whenΩ = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1048

051

054

057

06

063

066

g(120578

)

120578

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 9 Effect ofΩ on the transverse velocity of 119891(120578) when 120573 = 1120593 = 01 and119872 = 119908

0= 05

10 Mathematical Problems in Engineering

0 02 04 06 08 1

0

03

06

09

12

minus03

120578

f998400 (120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 10 Effect ofΩ on the transverse velocity of1198911015840(120578)when120573 = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1

0

005

minus005

minus01

minus015

minus02

minus025

120578

g(120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 11 Effect ofΩ on the transverse velocity of 119892(120578)when 120573 = 1120593 = 01 and119872 = 119908

0= 05

048

051

054

057

06

063

066

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1

f(120578

)

120578

Figure 12 Effect of119872 on the velocity component of 119891(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

0

03

06

09

12

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

minus03

120578

f998400 (120578

)

0 02 04 06 08 1

Figure 13 Effect of 119872 on the velocity component of 1198911015840(120578) when120573 = Ω = 1 120593 = 01 and 119908

0= 05

Mathematical Problems in Engineering 11

0

002

minus002

minus004

minus006

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1120578

g(120578

)

Figure 14 Effect of119872 on the velocity component of 119892(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

025

05

075

1

125

00 02 04 06 08 1

120578

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

f(120578

)

Figure 15 Effect of1199080on the velocity component of 119891(120578)when 120573 =

Ω = 1119872 = 05 and 120593 = 01

0

05

1

15

minus05

minus10 02 04 06 08 1

120578

f998400 (120578

)w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

Figure 16 Effect of 1199080on the velocity component of 1198911015840(120578) when

120573 = Ω = 1119872 = 05 and 120593 = 01

0

01

02

minus02

minus03

minus01

0 02 04 06 08 1

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

120578

g(120578

)

Figure 17 Effect of1199080on the velocity component of 119892(120578) when 120573 =

Ω = 1119872 = 05 and 120593 = 01

12 Mathematical Problems in Engineering

048

051

054

057

06

063

066

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1

f(120578

)

120578

Figure 18 Effect of nanoparticle types on the velocity componentof 119891(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0

03

06

09

12

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1120578

f998400 (120578

)

minus03

minus06

Figure 19 Effect of nanoparticle types on the velocity componentof 1198911015840(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0 02 04 06 08 1

0

002

minus006

minus004

minus002

AgCuCuO

TiO2Al2O3

120578

g(120578

)

Figure 20 Effect of nanoparticle types on the velocity componentof 119892(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

Nomenclature

119886 Stretching rate of the lower plate119861 External uniform magnetic field1198610 Constant magnetic flux density

119862119891 Skin friction coefficient

119891 119892 Self-similar velocitiesℎ(119905) Upper plane distance119869 Magnetic flux119901 Pressure119905 Time119879 Cauchy stress tensor119906 Velocity component in the x directionV Velocity component in the y direction119908 Velocity component in the z direction

Dimensionless Parameters

120573 Characteristic parameter of the flow (120574119886)

Ω Rotation parameter (120596119886)1198722 Magnetic parameter (1205901198612

0120588119891119886)

1199080 Suction parameter (119881

0119886ℎ)

Re119909 Local Reynolds number (120588

1198911198800ℎ120583119891)

Greek Symbols

120572 120575 and 120576 DTM constant parameters120590 Electrical conductivity120588 Density

Mathematical Problems in Engineering 13

120583 Dynamic viscosity120578 A scaled boundary-layer coordinate] Kinematic viscosity120574 Characteristic constant parameter120595 Stream function120596 Constant angular velocity120593 Nanoparticle volume fraction

Subscripts

119891 Fluid phase119899119891 Nanofluid119904 Solid phase

Conflict of Interests

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

Acknowledgment

EbrahimMomoniat acknowledges support from theNationalResearch foundation of South Africa under Grant number76868

References

[1] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquoComputers andMathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I Boundary layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 26ndash28 1961

[3] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer on a continuous flat surfacerdquoAIChEJournal vol 7 pp 221ndash225 1961

[4] L E Erickson L T Fan and V G Fox ldquoHeat and mass transferon a moving continuous flat plate with suction or injectionrdquoIndustrial and Engineering Chemistry Fundamentals vol 5 no1 pp 19ndash25 1966

[5] F K Tsou E M Sparrow and R J Goldstein ldquoFlow and heattransfer in the boundary layer on a continuousmoving surfacerdquoInternational Journal ofHeat andMass Transfer vol 10 no 2 pp219ndash235 1967

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[7] H I Andersson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquo Acta Mechanica vol 95 no 1ndash4 pp 227ndash230 1992

[8] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[9] A Mehmood and A Ali ldquoAnalytic solution of generalizedthree-dimensional flow and heat transfer over a stretching plane

wallrdquo International Communications in Heat andMass Transfervol 33 no 10 pp 1243ndash1252 2006

[10] A Mehmood and A Ali ldquoAnalytic homotopy solution ofgeneralized three-dimensional channel flow due to uniformstretching of the platerdquoActaMechanica Sinica vol 23 no 5 pp503ndash510 2007

[11] A K Borkakoti and A Bharali ldquoHydromagnetic flow and heattransfer between two horizontal plates the lower plate being astretching sheetrdquo Quarterly of Applied Mathematics vol 40 no4 pp 461ndash467 1983

[12] S Munawar A Mehmood and A Ali ldquoEffects of slip on flowbetween two stretchable disks using optimal homotopy analysismethodrdquo Canadian Journal of Applied Science vol 1 pp 50ndash682011

[13] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicropolar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 no 8 pp1119ndash1127 2002

[14] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[15] P D Ariel ldquoOn exact solutions of flow problems of a secondgrade fluid through two parallel porous wallsrdquo InternationalJournal of Engineering Science vol 40 no 8 pp 913ndash941 2002

[16] T Hayat and Z Abbas ldquoChannel flow of a Maxwell fluid withchemical reactionrdquo Zeitschrift fur angewandte Mathematik undPhysik vol 59 no 1 pp 124ndash144 2008

[17] G Domairry and A Aziz ldquoApproximate analysis of MHDdqueeze flow between two parallel disks with suction orinjection by homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2009 Article ID 603916 19 pages2009

[18] S U S Choi and J A Eastman ldquoEnhancing thermal conductiv-ity of fluids with nanoparticlesrdquo Materials Science vol 231 pp99ndash105 1995

[19] Y Xuan and Q Li ldquoInvestigation on convective heat transferand flow features of nanofluidsrdquo Journal of Heat Transfer vol125 no 1 pp 151ndash155 2003

[20] N Bachok A Ishak and I Pop ldquoFlow and heat transfer overa rotating porous disk in a nanofluidrdquo Physica B CondensedMatter vol 406 no 9 pp 1767ndash1772 2011

[21] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[22] M H Abolbashari N Freidoonimehr F Nazari and M MRashidi ldquoEntropy analysis for an unsteady MHD flow past astretching permeable surface in nano-fluidrdquoPowder Technologyvol 267 pp 256ndash267 2014

[23] O A Beg M M Rashidi M Akbari and A HosseinildquoComparative numerical study of single-phase and two-phasemodels for bio-nanofluid transport phenomenardquo Journal ofMechanics in Medicine and Biology vol 14 Article ID 145001131 pages 2014

[24] M M Rashidi A Hosseini I Pop S Kumar and N Freidoon-imehr ldquoComparative numerical study of single and two-phasemodels of nanofluid heat transfer in wavy channelrdquo AppliedMathematics and Mechanics vol 35 pp 831ndash848 2014

[25] E Abu-Nada Z Masoud H F Oztop and A Campo ldquoEffectof nanofluid variable properties on natural convection inenclosuresrdquo International Journal of Thermal Sciences vol 49no 3 pp 479ndash491 2010

14 Mathematical Problems in Engineering

[26] M M Rashidi S Abelman and N Freidoonimehr ldquoEntropygeneration in steady MHD flow due to a rotating porous diskin a nanofluidrdquo International Journal of Heat andMass Transfervol 62 no 1 pp 515ndash525 2013

[27] M Mustafa T Hayat I Pop S Asghar and S ObaidatldquoStagnation-point flow of a nanofluid towards a stretchingsheetrdquo International Journal of Heat and Mass Transfer vol 54no 25-26 pp 5588ndash5594 2011

[28] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica vol 49 pp 469ndash482 2014

[29] M Sheikholeslami and D D Ganji ldquoThree dimensional heatand mass transfer in a rotating system using nanofluidrdquo PowderTechnology vol 253 pp 789ndash796 2014

[30] M Sheikholeslami M Gorji-Bandpy and D D Ganji ldquoLatticeBoltzmann method for MHD natural convection heat transferusing nanofluidrdquo Powder Technology vol 254 pp 82ndash93 2014

[31] F Ayaz ldquoApplications of differential transform method to diff-erential-algebraic equationsrdquoAppliedMathematics and Compu-tation vol 152 no 3 pp 649ndash657 2004

[32] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004

[33] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[34] M Kurosaka ldquoThe oscillatory boundary layer growth over thetop and bottom plates of a rota ting channelrdquo Journal of FluidsEngineering Transactions of the ASME vol 95 no 1 pp 68ndash741973

[35] H S Takhar A J Chamkha and G Nath ldquoMHD flow overa moving plate in a rotating fluid with magnetic field hallcurrents and free stream velocityrdquo International Journal ofEngineering Science vol 40 no 13 pp 1511ndash1527 2002

[36] K Vajravelu and B V R Kumar ldquoAnalytical and numericalsolutions of a coupled non-linear system arising in a three-dimensional rotating flowrdquo International Journal of Non-LinearMechanics vol 39 no 1 pp 13ndash24 2004

[37] G Domairry and A Aziz ldquoApproximate analysis of MHDsqueeze flow between two parallel disks with suction or in-jection by homotopy perturbation methodrdquoMathematical Pro-blems in Engineering vol 2009 Article ID 603916 19 pages2009

[38] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4p 571 1952

[39] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled withnanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[40] M Sheikholeslami and D D Ganji ldquoHeat transfer of Cu-waternanofluid flow between parallel platesrdquo Powder Technology vol235 pp 873ndash879 2013

[41] M M Rashidi ldquoThe modified differential transform methodfor solving MHD boundary-layer equationsrdquo Computer PhysicsCommunications vol 180 no 11 pp 2210ndash2217 2009

[42] M M Rashidi and N Freidoonimehr ldquoSeries solutions forthe flow in the vicinity of the equator of an MHD boundary-layer over a porous rotating sphere with heat transferrdquoThermalScience vol 18 pp S527ndashS537 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0 02 04 06 08 1

0

03

06

09

12

minus03

120578

f998400 (120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 10 Effect ofΩ on the transverse velocity of1198911015840(120578)when120573 = 1120593 = 01 and119872 = 119908

0= 05

0 02 04 06 08 1

0

005

minus005

minus01

minus015

minus02

minus025

120578

g(120578

)

Ω = 1

Ω = 2

Ω = 3

Ω = 4

Ω = 5

Figure 11 Effect ofΩ on the transverse velocity of 119892(120578)when 120573 = 1120593 = 01 and119872 = 119908

0= 05

048

051

054

057

06

063

066

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1

f(120578

)

120578

Figure 12 Effect of119872 on the velocity component of 119891(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

0

03

06

09

12

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

minus03

120578

f998400 (120578

)

0 02 04 06 08 1

Figure 13 Effect of 119872 on the velocity component of 1198911015840(120578) when120573 = Ω = 1 120593 = 01 and 119908

0= 05

Mathematical Problems in Engineering 11

0

002

minus002

minus004

minus006

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1120578

g(120578

)

Figure 14 Effect of119872 on the velocity component of 119892(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

025

05

075

1

125

00 02 04 06 08 1

120578

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

f(120578

)

Figure 15 Effect of1199080on the velocity component of 119891(120578)when 120573 =

Ω = 1119872 = 05 and 120593 = 01

0

05

1

15

minus05

minus10 02 04 06 08 1

120578

f998400 (120578

)w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

Figure 16 Effect of 1199080on the velocity component of 1198911015840(120578) when

120573 = Ω = 1119872 = 05 and 120593 = 01

0

01

02

minus02

minus03

minus01

0 02 04 06 08 1

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

120578

g(120578

)

Figure 17 Effect of1199080on the velocity component of 119892(120578) when 120573 =

Ω = 1119872 = 05 and 120593 = 01

12 Mathematical Problems in Engineering

048

051

054

057

06

063

066

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1

f(120578

)

120578

Figure 18 Effect of nanoparticle types on the velocity componentof 119891(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0

03

06

09

12

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1120578

f998400 (120578

)

minus03

minus06

Figure 19 Effect of nanoparticle types on the velocity componentof 1198911015840(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0 02 04 06 08 1

0

002

minus006

minus004

minus002

AgCuCuO

TiO2Al2O3

120578

g(120578

)

Figure 20 Effect of nanoparticle types on the velocity componentof 119892(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

Nomenclature

119886 Stretching rate of the lower plate119861 External uniform magnetic field1198610 Constant magnetic flux density

119862119891 Skin friction coefficient

119891 119892 Self-similar velocitiesℎ(119905) Upper plane distance119869 Magnetic flux119901 Pressure119905 Time119879 Cauchy stress tensor119906 Velocity component in the x directionV Velocity component in the y direction119908 Velocity component in the z direction

Dimensionless Parameters

120573 Characteristic parameter of the flow (120574119886)

Ω Rotation parameter (120596119886)1198722 Magnetic parameter (1205901198612

0120588119891119886)

1199080 Suction parameter (119881

0119886ℎ)

Re119909 Local Reynolds number (120588

1198911198800ℎ120583119891)

Greek Symbols

120572 120575 and 120576 DTM constant parameters120590 Electrical conductivity120588 Density

Mathematical Problems in Engineering 13

120583 Dynamic viscosity120578 A scaled boundary-layer coordinate] Kinematic viscosity120574 Characteristic constant parameter120595 Stream function120596 Constant angular velocity120593 Nanoparticle volume fraction

Subscripts

119891 Fluid phase119899119891 Nanofluid119904 Solid phase

Conflict of Interests

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

Acknowledgment

EbrahimMomoniat acknowledges support from theNationalResearch foundation of South Africa under Grant number76868

References

[1] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquoComputers andMathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I Boundary layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 26ndash28 1961

[3] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer on a continuous flat surfacerdquoAIChEJournal vol 7 pp 221ndash225 1961

[4] L E Erickson L T Fan and V G Fox ldquoHeat and mass transferon a moving continuous flat plate with suction or injectionrdquoIndustrial and Engineering Chemistry Fundamentals vol 5 no1 pp 19ndash25 1966

[5] F K Tsou E M Sparrow and R J Goldstein ldquoFlow and heattransfer in the boundary layer on a continuousmoving surfacerdquoInternational Journal ofHeat andMass Transfer vol 10 no 2 pp219ndash235 1967

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[7] H I Andersson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquo Acta Mechanica vol 95 no 1ndash4 pp 227ndash230 1992

[8] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[9] A Mehmood and A Ali ldquoAnalytic solution of generalizedthree-dimensional flow and heat transfer over a stretching plane

wallrdquo International Communications in Heat andMass Transfervol 33 no 10 pp 1243ndash1252 2006

[10] A Mehmood and A Ali ldquoAnalytic homotopy solution ofgeneralized three-dimensional channel flow due to uniformstretching of the platerdquoActaMechanica Sinica vol 23 no 5 pp503ndash510 2007

[11] A K Borkakoti and A Bharali ldquoHydromagnetic flow and heattransfer between two horizontal plates the lower plate being astretching sheetrdquo Quarterly of Applied Mathematics vol 40 no4 pp 461ndash467 1983

[12] S Munawar A Mehmood and A Ali ldquoEffects of slip on flowbetween two stretchable disks using optimal homotopy analysismethodrdquo Canadian Journal of Applied Science vol 1 pp 50ndash682011

[13] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicropolar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 no 8 pp1119ndash1127 2002

[14] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[15] P D Ariel ldquoOn exact solutions of flow problems of a secondgrade fluid through two parallel porous wallsrdquo InternationalJournal of Engineering Science vol 40 no 8 pp 913ndash941 2002

[16] T Hayat and Z Abbas ldquoChannel flow of a Maxwell fluid withchemical reactionrdquo Zeitschrift fur angewandte Mathematik undPhysik vol 59 no 1 pp 124ndash144 2008

[17] G Domairry and A Aziz ldquoApproximate analysis of MHDdqueeze flow between two parallel disks with suction orinjection by homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2009 Article ID 603916 19 pages2009

[18] S U S Choi and J A Eastman ldquoEnhancing thermal conductiv-ity of fluids with nanoparticlesrdquo Materials Science vol 231 pp99ndash105 1995

[19] Y Xuan and Q Li ldquoInvestigation on convective heat transferand flow features of nanofluidsrdquo Journal of Heat Transfer vol125 no 1 pp 151ndash155 2003

[20] N Bachok A Ishak and I Pop ldquoFlow and heat transfer overa rotating porous disk in a nanofluidrdquo Physica B CondensedMatter vol 406 no 9 pp 1767ndash1772 2011

[21] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[22] M H Abolbashari N Freidoonimehr F Nazari and M MRashidi ldquoEntropy analysis for an unsteady MHD flow past astretching permeable surface in nano-fluidrdquoPowder Technologyvol 267 pp 256ndash267 2014

[23] O A Beg M M Rashidi M Akbari and A HosseinildquoComparative numerical study of single-phase and two-phasemodels for bio-nanofluid transport phenomenardquo Journal ofMechanics in Medicine and Biology vol 14 Article ID 145001131 pages 2014

[24] M M Rashidi A Hosseini I Pop S Kumar and N Freidoon-imehr ldquoComparative numerical study of single and two-phasemodels of nanofluid heat transfer in wavy channelrdquo AppliedMathematics and Mechanics vol 35 pp 831ndash848 2014

[25] E Abu-Nada Z Masoud H F Oztop and A Campo ldquoEffectof nanofluid variable properties on natural convection inenclosuresrdquo International Journal of Thermal Sciences vol 49no 3 pp 479ndash491 2010

14 Mathematical Problems in Engineering

[26] M M Rashidi S Abelman and N Freidoonimehr ldquoEntropygeneration in steady MHD flow due to a rotating porous diskin a nanofluidrdquo International Journal of Heat andMass Transfervol 62 no 1 pp 515ndash525 2013

[27] M Mustafa T Hayat I Pop S Asghar and S ObaidatldquoStagnation-point flow of a nanofluid towards a stretchingsheetrdquo International Journal of Heat and Mass Transfer vol 54no 25-26 pp 5588ndash5594 2011

[28] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica vol 49 pp 469ndash482 2014

[29] M Sheikholeslami and D D Ganji ldquoThree dimensional heatand mass transfer in a rotating system using nanofluidrdquo PowderTechnology vol 253 pp 789ndash796 2014

[30] M Sheikholeslami M Gorji-Bandpy and D D Ganji ldquoLatticeBoltzmann method for MHD natural convection heat transferusing nanofluidrdquo Powder Technology vol 254 pp 82ndash93 2014

[31] F Ayaz ldquoApplications of differential transform method to diff-erential-algebraic equationsrdquoAppliedMathematics and Compu-tation vol 152 no 3 pp 649ndash657 2004

[32] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004

[33] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[34] M Kurosaka ldquoThe oscillatory boundary layer growth over thetop and bottom plates of a rota ting channelrdquo Journal of FluidsEngineering Transactions of the ASME vol 95 no 1 pp 68ndash741973

[35] H S Takhar A J Chamkha and G Nath ldquoMHD flow overa moving plate in a rotating fluid with magnetic field hallcurrents and free stream velocityrdquo International Journal ofEngineering Science vol 40 no 13 pp 1511ndash1527 2002

[36] K Vajravelu and B V R Kumar ldquoAnalytical and numericalsolutions of a coupled non-linear system arising in a three-dimensional rotating flowrdquo International Journal of Non-LinearMechanics vol 39 no 1 pp 13ndash24 2004

[37] G Domairry and A Aziz ldquoApproximate analysis of MHDsqueeze flow between two parallel disks with suction or in-jection by homotopy perturbation methodrdquoMathematical Pro-blems in Engineering vol 2009 Article ID 603916 19 pages2009

[38] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4p 571 1952

[39] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled withnanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[40] M Sheikholeslami and D D Ganji ldquoHeat transfer of Cu-waternanofluid flow between parallel platesrdquo Powder Technology vol235 pp 873ndash879 2013

[41] M M Rashidi ldquoThe modified differential transform methodfor solving MHD boundary-layer equationsrdquo Computer PhysicsCommunications vol 180 no 11 pp 2210ndash2217 2009

[42] M M Rashidi and N Freidoonimehr ldquoSeries solutions forthe flow in the vicinity of the equator of an MHD boundary-layer over a porous rotating sphere with heat transferrdquoThermalScience vol 18 pp S527ndashS537 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0

002

minus002

minus004

minus006

M = 00M = 05M = 10M = 15

M = 20M = 25M = 30

0 02 04 06 08 1120578

g(120578

)

Figure 14 Effect of119872 on the velocity component of 119892(120578) when 120573 =

Ω = 1 120593 = 01 and 1199080= 05

025

05

075

1

125

00 02 04 06 08 1

120578

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

f(120578

)

Figure 15 Effect of1199080on the velocity component of 119891(120578)when 120573 =

Ω = 1119872 = 05 and 120593 = 01

0

05

1

15

minus05

minus10 02 04 06 08 1

120578

f998400 (120578

)w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

Figure 16 Effect of 1199080on the velocity component of 1198911015840(120578) when

120573 = Ω = 1119872 = 05 and 120593 = 01

0

01

02

minus02

minus03

minus01

0 02 04 06 08 1

w0 = 00

w0 = 01w0 = 02

w0 = 03

w0 = 05

w0 = 10

120578

g(120578

)

Figure 17 Effect of1199080on the velocity component of 119892(120578) when 120573 =

Ω = 1119872 = 05 and 120593 = 01

12 Mathematical Problems in Engineering

048

051

054

057

06

063

066

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1

f(120578

)

120578

Figure 18 Effect of nanoparticle types on the velocity componentof 119891(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0

03

06

09

12

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1120578

f998400 (120578

)

minus03

minus06

Figure 19 Effect of nanoparticle types on the velocity componentof 1198911015840(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0 02 04 06 08 1

0

002

minus006

minus004

minus002

AgCuCuO

TiO2Al2O3

120578

g(120578

)

Figure 20 Effect of nanoparticle types on the velocity componentof 119892(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

Nomenclature

119886 Stretching rate of the lower plate119861 External uniform magnetic field1198610 Constant magnetic flux density

119862119891 Skin friction coefficient

119891 119892 Self-similar velocitiesℎ(119905) Upper plane distance119869 Magnetic flux119901 Pressure119905 Time119879 Cauchy stress tensor119906 Velocity component in the x directionV Velocity component in the y direction119908 Velocity component in the z direction

Dimensionless Parameters

120573 Characteristic parameter of the flow (120574119886)

Ω Rotation parameter (120596119886)1198722 Magnetic parameter (1205901198612

0120588119891119886)

1199080 Suction parameter (119881

0119886ℎ)

Re119909 Local Reynolds number (120588

1198911198800ℎ120583119891)

Greek Symbols

120572 120575 and 120576 DTM constant parameters120590 Electrical conductivity120588 Density

Mathematical Problems in Engineering 13

120583 Dynamic viscosity120578 A scaled boundary-layer coordinate] Kinematic viscosity120574 Characteristic constant parameter120595 Stream function120596 Constant angular velocity120593 Nanoparticle volume fraction

Subscripts

119891 Fluid phase119899119891 Nanofluid119904 Solid phase

Conflict of Interests

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

Acknowledgment

EbrahimMomoniat acknowledges support from theNationalResearch foundation of South Africa under Grant number76868

References

[1] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquoComputers andMathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I Boundary layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 26ndash28 1961

[3] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer on a continuous flat surfacerdquoAIChEJournal vol 7 pp 221ndash225 1961

[4] L E Erickson L T Fan and V G Fox ldquoHeat and mass transferon a moving continuous flat plate with suction or injectionrdquoIndustrial and Engineering Chemistry Fundamentals vol 5 no1 pp 19ndash25 1966

[5] F K Tsou E M Sparrow and R J Goldstein ldquoFlow and heattransfer in the boundary layer on a continuousmoving surfacerdquoInternational Journal ofHeat andMass Transfer vol 10 no 2 pp219ndash235 1967

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[7] H I Andersson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquo Acta Mechanica vol 95 no 1ndash4 pp 227ndash230 1992

[8] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[9] A Mehmood and A Ali ldquoAnalytic solution of generalizedthree-dimensional flow and heat transfer over a stretching plane

wallrdquo International Communications in Heat andMass Transfervol 33 no 10 pp 1243ndash1252 2006

[10] A Mehmood and A Ali ldquoAnalytic homotopy solution ofgeneralized three-dimensional channel flow due to uniformstretching of the platerdquoActaMechanica Sinica vol 23 no 5 pp503ndash510 2007

[11] A K Borkakoti and A Bharali ldquoHydromagnetic flow and heattransfer between two horizontal plates the lower plate being astretching sheetrdquo Quarterly of Applied Mathematics vol 40 no4 pp 461ndash467 1983

[12] S Munawar A Mehmood and A Ali ldquoEffects of slip on flowbetween two stretchable disks using optimal homotopy analysismethodrdquo Canadian Journal of Applied Science vol 1 pp 50ndash682011

[13] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicropolar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 no 8 pp1119ndash1127 2002

[14] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[15] P D Ariel ldquoOn exact solutions of flow problems of a secondgrade fluid through two parallel porous wallsrdquo InternationalJournal of Engineering Science vol 40 no 8 pp 913ndash941 2002

[16] T Hayat and Z Abbas ldquoChannel flow of a Maxwell fluid withchemical reactionrdquo Zeitschrift fur angewandte Mathematik undPhysik vol 59 no 1 pp 124ndash144 2008

[17] G Domairry and A Aziz ldquoApproximate analysis of MHDdqueeze flow between two parallel disks with suction orinjection by homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2009 Article ID 603916 19 pages2009

[18] S U S Choi and J A Eastman ldquoEnhancing thermal conductiv-ity of fluids with nanoparticlesrdquo Materials Science vol 231 pp99ndash105 1995

[19] Y Xuan and Q Li ldquoInvestigation on convective heat transferand flow features of nanofluidsrdquo Journal of Heat Transfer vol125 no 1 pp 151ndash155 2003

[20] N Bachok A Ishak and I Pop ldquoFlow and heat transfer overa rotating porous disk in a nanofluidrdquo Physica B CondensedMatter vol 406 no 9 pp 1767ndash1772 2011

[21] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[22] M H Abolbashari N Freidoonimehr F Nazari and M MRashidi ldquoEntropy analysis for an unsteady MHD flow past astretching permeable surface in nano-fluidrdquoPowder Technologyvol 267 pp 256ndash267 2014

[23] O A Beg M M Rashidi M Akbari and A HosseinildquoComparative numerical study of single-phase and two-phasemodels for bio-nanofluid transport phenomenardquo Journal ofMechanics in Medicine and Biology vol 14 Article ID 145001131 pages 2014

[24] M M Rashidi A Hosseini I Pop S Kumar and N Freidoon-imehr ldquoComparative numerical study of single and two-phasemodels of nanofluid heat transfer in wavy channelrdquo AppliedMathematics and Mechanics vol 35 pp 831ndash848 2014

[25] E Abu-Nada Z Masoud H F Oztop and A Campo ldquoEffectof nanofluid variable properties on natural convection inenclosuresrdquo International Journal of Thermal Sciences vol 49no 3 pp 479ndash491 2010

14 Mathematical Problems in Engineering

[26] M M Rashidi S Abelman and N Freidoonimehr ldquoEntropygeneration in steady MHD flow due to a rotating porous diskin a nanofluidrdquo International Journal of Heat andMass Transfervol 62 no 1 pp 515ndash525 2013

[27] M Mustafa T Hayat I Pop S Asghar and S ObaidatldquoStagnation-point flow of a nanofluid towards a stretchingsheetrdquo International Journal of Heat and Mass Transfer vol 54no 25-26 pp 5588ndash5594 2011

[28] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica vol 49 pp 469ndash482 2014

[29] M Sheikholeslami and D D Ganji ldquoThree dimensional heatand mass transfer in a rotating system using nanofluidrdquo PowderTechnology vol 253 pp 789ndash796 2014

[30] M Sheikholeslami M Gorji-Bandpy and D D Ganji ldquoLatticeBoltzmann method for MHD natural convection heat transferusing nanofluidrdquo Powder Technology vol 254 pp 82ndash93 2014

[31] F Ayaz ldquoApplications of differential transform method to diff-erential-algebraic equationsrdquoAppliedMathematics and Compu-tation vol 152 no 3 pp 649ndash657 2004

[32] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004

[33] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[34] M Kurosaka ldquoThe oscillatory boundary layer growth over thetop and bottom plates of a rota ting channelrdquo Journal of FluidsEngineering Transactions of the ASME vol 95 no 1 pp 68ndash741973

[35] H S Takhar A J Chamkha and G Nath ldquoMHD flow overa moving plate in a rotating fluid with magnetic field hallcurrents and free stream velocityrdquo International Journal ofEngineering Science vol 40 no 13 pp 1511ndash1527 2002

[36] K Vajravelu and B V R Kumar ldquoAnalytical and numericalsolutions of a coupled non-linear system arising in a three-dimensional rotating flowrdquo International Journal of Non-LinearMechanics vol 39 no 1 pp 13ndash24 2004

[37] G Domairry and A Aziz ldquoApproximate analysis of MHDsqueeze flow between two parallel disks with suction or in-jection by homotopy perturbation methodrdquoMathematical Pro-blems in Engineering vol 2009 Article ID 603916 19 pages2009

[38] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4p 571 1952

[39] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled withnanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[40] M Sheikholeslami and D D Ganji ldquoHeat transfer of Cu-waternanofluid flow between parallel platesrdquo Powder Technology vol235 pp 873ndash879 2013

[41] M M Rashidi ldquoThe modified differential transform methodfor solving MHD boundary-layer equationsrdquo Computer PhysicsCommunications vol 180 no 11 pp 2210ndash2217 2009

[42] M M Rashidi and N Freidoonimehr ldquoSeries solutions forthe flow in the vicinity of the equator of an MHD boundary-layer over a porous rotating sphere with heat transferrdquoThermalScience vol 18 pp S527ndashS537 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

048

051

054

057

06

063

066

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1

f(120578

)

120578

Figure 18 Effect of nanoparticle types on the velocity componentof 119891(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0

03

06

09

12

AgCuCuO

TiO2Al2O3

0 02 04 06 08 1120578

f998400 (120578

)

minus03

minus06

Figure 19 Effect of nanoparticle types on the velocity componentof 1198911015840(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

0 02 04 06 08 1

0

002

minus006

minus004

minus002

AgCuCuO

TiO2Al2O3

120578

g(120578

)

Figure 20 Effect of nanoparticle types on the velocity componentof 119892(120578) when 120573 = Ω = 1119872 = 119908

0= 05 and 120593 = 01

Nomenclature

119886 Stretching rate of the lower plate119861 External uniform magnetic field1198610 Constant magnetic flux density

119862119891 Skin friction coefficient

119891 119892 Self-similar velocitiesℎ(119905) Upper plane distance119869 Magnetic flux119901 Pressure119905 Time119879 Cauchy stress tensor119906 Velocity component in the x directionV Velocity component in the y direction119908 Velocity component in the z direction

Dimensionless Parameters

120573 Characteristic parameter of the flow (120574119886)

Ω Rotation parameter (120596119886)1198722 Magnetic parameter (1205901198612

0120588119891119886)

1199080 Suction parameter (119881

0119886ℎ)

Re119909 Local Reynolds number (120588

1198911198800ℎ120583119891)

Greek Symbols

120572 120575 and 120576 DTM constant parameters120590 Electrical conductivity120588 Density

Mathematical Problems in Engineering 13

120583 Dynamic viscosity120578 A scaled boundary-layer coordinate] Kinematic viscosity120574 Characteristic constant parameter120595 Stream function120596 Constant angular velocity120593 Nanoparticle volume fraction

Subscripts

119891 Fluid phase119899119891 Nanofluid119904 Solid phase

Conflict of Interests

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

Acknowledgment

EbrahimMomoniat acknowledges support from theNationalResearch foundation of South Africa under Grant number76868

References

[1] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquoComputers andMathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I Boundary layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 26ndash28 1961

[3] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer on a continuous flat surfacerdquoAIChEJournal vol 7 pp 221ndash225 1961

[4] L E Erickson L T Fan and V G Fox ldquoHeat and mass transferon a moving continuous flat plate with suction or injectionrdquoIndustrial and Engineering Chemistry Fundamentals vol 5 no1 pp 19ndash25 1966

[5] F K Tsou E M Sparrow and R J Goldstein ldquoFlow and heattransfer in the boundary layer on a continuousmoving surfacerdquoInternational Journal ofHeat andMass Transfer vol 10 no 2 pp219ndash235 1967

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[7] H I Andersson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquo Acta Mechanica vol 95 no 1ndash4 pp 227ndash230 1992

[8] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[9] A Mehmood and A Ali ldquoAnalytic solution of generalizedthree-dimensional flow and heat transfer over a stretching plane

wallrdquo International Communications in Heat andMass Transfervol 33 no 10 pp 1243ndash1252 2006

[10] A Mehmood and A Ali ldquoAnalytic homotopy solution ofgeneralized three-dimensional channel flow due to uniformstretching of the platerdquoActaMechanica Sinica vol 23 no 5 pp503ndash510 2007

[11] A K Borkakoti and A Bharali ldquoHydromagnetic flow and heattransfer between two horizontal plates the lower plate being astretching sheetrdquo Quarterly of Applied Mathematics vol 40 no4 pp 461ndash467 1983

[12] S Munawar A Mehmood and A Ali ldquoEffects of slip on flowbetween two stretchable disks using optimal homotopy analysismethodrdquo Canadian Journal of Applied Science vol 1 pp 50ndash682011

[13] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicropolar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 no 8 pp1119ndash1127 2002

[14] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[15] P D Ariel ldquoOn exact solutions of flow problems of a secondgrade fluid through two parallel porous wallsrdquo InternationalJournal of Engineering Science vol 40 no 8 pp 913ndash941 2002

[16] T Hayat and Z Abbas ldquoChannel flow of a Maxwell fluid withchemical reactionrdquo Zeitschrift fur angewandte Mathematik undPhysik vol 59 no 1 pp 124ndash144 2008

[17] G Domairry and A Aziz ldquoApproximate analysis of MHDdqueeze flow between two parallel disks with suction orinjection by homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2009 Article ID 603916 19 pages2009

[18] S U S Choi and J A Eastman ldquoEnhancing thermal conductiv-ity of fluids with nanoparticlesrdquo Materials Science vol 231 pp99ndash105 1995

[19] Y Xuan and Q Li ldquoInvestigation on convective heat transferand flow features of nanofluidsrdquo Journal of Heat Transfer vol125 no 1 pp 151ndash155 2003

[20] N Bachok A Ishak and I Pop ldquoFlow and heat transfer overa rotating porous disk in a nanofluidrdquo Physica B CondensedMatter vol 406 no 9 pp 1767ndash1772 2011

[21] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[22] M H Abolbashari N Freidoonimehr F Nazari and M MRashidi ldquoEntropy analysis for an unsteady MHD flow past astretching permeable surface in nano-fluidrdquoPowder Technologyvol 267 pp 256ndash267 2014

[23] O A Beg M M Rashidi M Akbari and A HosseinildquoComparative numerical study of single-phase and two-phasemodels for bio-nanofluid transport phenomenardquo Journal ofMechanics in Medicine and Biology vol 14 Article ID 145001131 pages 2014

[24] M M Rashidi A Hosseini I Pop S Kumar and N Freidoon-imehr ldquoComparative numerical study of single and two-phasemodels of nanofluid heat transfer in wavy channelrdquo AppliedMathematics and Mechanics vol 35 pp 831ndash848 2014

[25] E Abu-Nada Z Masoud H F Oztop and A Campo ldquoEffectof nanofluid variable properties on natural convection inenclosuresrdquo International Journal of Thermal Sciences vol 49no 3 pp 479ndash491 2010

14 Mathematical Problems in Engineering

[26] M M Rashidi S Abelman and N Freidoonimehr ldquoEntropygeneration in steady MHD flow due to a rotating porous diskin a nanofluidrdquo International Journal of Heat andMass Transfervol 62 no 1 pp 515ndash525 2013

[27] M Mustafa T Hayat I Pop S Asghar and S ObaidatldquoStagnation-point flow of a nanofluid towards a stretchingsheetrdquo International Journal of Heat and Mass Transfer vol 54no 25-26 pp 5588ndash5594 2011

[28] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica vol 49 pp 469ndash482 2014

[29] M Sheikholeslami and D D Ganji ldquoThree dimensional heatand mass transfer in a rotating system using nanofluidrdquo PowderTechnology vol 253 pp 789ndash796 2014

[30] M Sheikholeslami M Gorji-Bandpy and D D Ganji ldquoLatticeBoltzmann method for MHD natural convection heat transferusing nanofluidrdquo Powder Technology vol 254 pp 82ndash93 2014

[31] F Ayaz ldquoApplications of differential transform method to diff-erential-algebraic equationsrdquoAppliedMathematics and Compu-tation vol 152 no 3 pp 649ndash657 2004

[32] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004

[33] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[34] M Kurosaka ldquoThe oscillatory boundary layer growth over thetop and bottom plates of a rota ting channelrdquo Journal of FluidsEngineering Transactions of the ASME vol 95 no 1 pp 68ndash741973

[35] H S Takhar A J Chamkha and G Nath ldquoMHD flow overa moving plate in a rotating fluid with magnetic field hallcurrents and free stream velocityrdquo International Journal ofEngineering Science vol 40 no 13 pp 1511ndash1527 2002

[36] K Vajravelu and B V R Kumar ldquoAnalytical and numericalsolutions of a coupled non-linear system arising in a three-dimensional rotating flowrdquo International Journal of Non-LinearMechanics vol 39 no 1 pp 13ndash24 2004

[37] G Domairry and A Aziz ldquoApproximate analysis of MHDsqueeze flow between two parallel disks with suction or in-jection by homotopy perturbation methodrdquoMathematical Pro-blems in Engineering vol 2009 Article ID 603916 19 pages2009

[38] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4p 571 1952

[39] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled withnanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[40] M Sheikholeslami and D D Ganji ldquoHeat transfer of Cu-waternanofluid flow between parallel platesrdquo Powder Technology vol235 pp 873ndash879 2013

[41] M M Rashidi ldquoThe modified differential transform methodfor solving MHD boundary-layer equationsrdquo Computer PhysicsCommunications vol 180 no 11 pp 2210ndash2217 2009

[42] M M Rashidi and N Freidoonimehr ldquoSeries solutions forthe flow in the vicinity of the equator of an MHD boundary-layer over a porous rotating sphere with heat transferrdquoThermalScience vol 18 pp S527ndashS537 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

120583 Dynamic viscosity120578 A scaled boundary-layer coordinate] Kinematic viscosity120574 Characteristic constant parameter120595 Stream function120596 Constant angular velocity120593 Nanoparticle volume fraction

Subscripts

119891 Fluid phase119899119891 Nanofluid119904 Solid phase

Conflict of Interests

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

Acknowledgment

EbrahimMomoniat acknowledges support from theNationalResearch foundation of South Africa under Grant number76868

References

[1] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquoComputers andMathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I Boundary layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 26ndash28 1961

[3] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer on a continuous flat surfacerdquoAIChEJournal vol 7 pp 221ndash225 1961

[4] L E Erickson L T Fan and V G Fox ldquoHeat and mass transferon a moving continuous flat plate with suction or injectionrdquoIndustrial and Engineering Chemistry Fundamentals vol 5 no1 pp 19ndash25 1966

[5] F K Tsou E M Sparrow and R J Goldstein ldquoFlow and heattransfer in the boundary layer on a continuousmoving surfacerdquoInternational Journal ofHeat andMass Transfer vol 10 no 2 pp219ndash235 1967

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[7] H I Andersson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquo Acta Mechanica vol 95 no 1ndash4 pp 227ndash230 1992

[8] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[9] A Mehmood and A Ali ldquoAnalytic solution of generalizedthree-dimensional flow and heat transfer over a stretching plane

wallrdquo International Communications in Heat andMass Transfervol 33 no 10 pp 1243ndash1252 2006

[10] A Mehmood and A Ali ldquoAnalytic homotopy solution ofgeneralized three-dimensional channel flow due to uniformstretching of the platerdquoActaMechanica Sinica vol 23 no 5 pp503ndash510 2007

[11] A K Borkakoti and A Bharali ldquoHydromagnetic flow and heattransfer between two horizontal plates the lower plate being astretching sheetrdquo Quarterly of Applied Mathematics vol 40 no4 pp 461ndash467 1983

[12] S Munawar A Mehmood and A Ali ldquoEffects of slip on flowbetween two stretchable disks using optimal homotopy analysismethodrdquo Canadian Journal of Applied Science vol 1 pp 50ndash682011

[13] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicropolar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 no 8 pp1119ndash1127 2002

[14] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[15] P D Ariel ldquoOn exact solutions of flow problems of a secondgrade fluid through two parallel porous wallsrdquo InternationalJournal of Engineering Science vol 40 no 8 pp 913ndash941 2002

[16] T Hayat and Z Abbas ldquoChannel flow of a Maxwell fluid withchemical reactionrdquo Zeitschrift fur angewandte Mathematik undPhysik vol 59 no 1 pp 124ndash144 2008

[17] G Domairry and A Aziz ldquoApproximate analysis of MHDdqueeze flow between two parallel disks with suction orinjection by homotopy perturbation methodrdquo MathematicalProblems in Engineering vol 2009 Article ID 603916 19 pages2009

[18] S U S Choi and J A Eastman ldquoEnhancing thermal conductiv-ity of fluids with nanoparticlesrdquo Materials Science vol 231 pp99ndash105 1995

[19] Y Xuan and Q Li ldquoInvestigation on convective heat transferand flow features of nanofluidsrdquo Journal of Heat Transfer vol125 no 1 pp 151ndash155 2003

[20] N Bachok A Ishak and I Pop ldquoFlow and heat transfer overa rotating porous disk in a nanofluidrdquo Physica B CondensedMatter vol 406 no 9 pp 1767ndash1772 2011

[21] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[22] M H Abolbashari N Freidoonimehr F Nazari and M MRashidi ldquoEntropy analysis for an unsteady MHD flow past astretching permeable surface in nano-fluidrdquoPowder Technologyvol 267 pp 256ndash267 2014

[23] O A Beg M M Rashidi M Akbari and A HosseinildquoComparative numerical study of single-phase and two-phasemodels for bio-nanofluid transport phenomenardquo Journal ofMechanics in Medicine and Biology vol 14 Article ID 145001131 pages 2014

[24] M M Rashidi A Hosseini I Pop S Kumar and N Freidoon-imehr ldquoComparative numerical study of single and two-phasemodels of nanofluid heat transfer in wavy channelrdquo AppliedMathematics and Mechanics vol 35 pp 831ndash848 2014

[25] E Abu-Nada Z Masoud H F Oztop and A Campo ldquoEffectof nanofluid variable properties on natural convection inenclosuresrdquo International Journal of Thermal Sciences vol 49no 3 pp 479ndash491 2010

14 Mathematical Problems in Engineering

[26] M M Rashidi S Abelman and N Freidoonimehr ldquoEntropygeneration in steady MHD flow due to a rotating porous diskin a nanofluidrdquo International Journal of Heat andMass Transfervol 62 no 1 pp 515ndash525 2013

[27] M Mustafa T Hayat I Pop S Asghar and S ObaidatldquoStagnation-point flow of a nanofluid towards a stretchingsheetrdquo International Journal of Heat and Mass Transfer vol 54no 25-26 pp 5588ndash5594 2011

[28] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica vol 49 pp 469ndash482 2014

[29] M Sheikholeslami and D D Ganji ldquoThree dimensional heatand mass transfer in a rotating system using nanofluidrdquo PowderTechnology vol 253 pp 789ndash796 2014

[30] M Sheikholeslami M Gorji-Bandpy and D D Ganji ldquoLatticeBoltzmann method for MHD natural convection heat transferusing nanofluidrdquo Powder Technology vol 254 pp 82ndash93 2014

[31] F Ayaz ldquoApplications of differential transform method to diff-erential-algebraic equationsrdquoAppliedMathematics and Compu-tation vol 152 no 3 pp 649ndash657 2004

[32] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004

[33] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[34] M Kurosaka ldquoThe oscillatory boundary layer growth over thetop and bottom plates of a rota ting channelrdquo Journal of FluidsEngineering Transactions of the ASME vol 95 no 1 pp 68ndash741973

[35] H S Takhar A J Chamkha and G Nath ldquoMHD flow overa moving plate in a rotating fluid with magnetic field hallcurrents and free stream velocityrdquo International Journal ofEngineering Science vol 40 no 13 pp 1511ndash1527 2002

[36] K Vajravelu and B V R Kumar ldquoAnalytical and numericalsolutions of a coupled non-linear system arising in a three-dimensional rotating flowrdquo International Journal of Non-LinearMechanics vol 39 no 1 pp 13ndash24 2004

[37] G Domairry and A Aziz ldquoApproximate analysis of MHDsqueeze flow between two parallel disks with suction or in-jection by homotopy perturbation methodrdquoMathematical Pro-blems in Engineering vol 2009 Article ID 603916 19 pages2009

[38] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4p 571 1952

[39] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled withnanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[40] M Sheikholeslami and D D Ganji ldquoHeat transfer of Cu-waternanofluid flow between parallel platesrdquo Powder Technology vol235 pp 873ndash879 2013

[41] M M Rashidi ldquoThe modified differential transform methodfor solving MHD boundary-layer equationsrdquo Computer PhysicsCommunications vol 180 no 11 pp 2210ndash2217 2009

[42] M M Rashidi and N Freidoonimehr ldquoSeries solutions forthe flow in the vicinity of the equator of an MHD boundary-layer over a porous rotating sphere with heat transferrdquoThermalScience vol 18 pp S527ndashS537 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Mathematical Problems in Engineering

[26] M M Rashidi S Abelman and N Freidoonimehr ldquoEntropygeneration in steady MHD flow due to a rotating porous diskin a nanofluidrdquo International Journal of Heat andMass Transfervol 62 no 1 pp 515ndash525 2013

[27] M Mustafa T Hayat I Pop S Asghar and S ObaidatldquoStagnation-point flow of a nanofluid towards a stretchingsheetrdquo International Journal of Heat and Mass Transfer vol 54no 25-26 pp 5588ndash5594 2011

[28] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica vol 49 pp 469ndash482 2014

[29] M Sheikholeslami and D D Ganji ldquoThree dimensional heatand mass transfer in a rotating system using nanofluidrdquo PowderTechnology vol 253 pp 789ndash796 2014

[30] M Sheikholeslami M Gorji-Bandpy and D D Ganji ldquoLatticeBoltzmann method for MHD natural convection heat transferusing nanofluidrdquo Powder Technology vol 254 pp 82ndash93 2014

[31] F Ayaz ldquoApplications of differential transform method to diff-erential-algebraic equationsrdquoAppliedMathematics and Compu-tation vol 152 no 3 pp 649ndash657 2004

[32] F Ayaz ldquoSolutions of the system of differential equationsby differential transform methodrdquo Applied Mathematics andComputation vol 147 no 2 pp 547ndash567 2004

[33] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[34] M Kurosaka ldquoThe oscillatory boundary layer growth over thetop and bottom plates of a rota ting channelrdquo Journal of FluidsEngineering Transactions of the ASME vol 95 no 1 pp 68ndash741973

[35] H S Takhar A J Chamkha and G Nath ldquoMHD flow overa moving plate in a rotating fluid with magnetic field hallcurrents and free stream velocityrdquo International Journal ofEngineering Science vol 40 no 13 pp 1511ndash1527 2002

[36] K Vajravelu and B V R Kumar ldquoAnalytical and numericalsolutions of a coupled non-linear system arising in a three-dimensional rotating flowrdquo International Journal of Non-LinearMechanics vol 39 no 1 pp 13ndash24 2004

[37] G Domairry and A Aziz ldquoApproximate analysis of MHDsqueeze flow between two parallel disks with suction or in-jection by homotopy perturbation methodrdquoMathematical Pro-blems in Engineering vol 2009 Article ID 603916 19 pages2009

[38] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4p 571 1952

[39] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled withnanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[40] M Sheikholeslami and D D Ganji ldquoHeat transfer of Cu-waternanofluid flow between parallel platesrdquo Powder Technology vol235 pp 873ndash879 2013

[41] M M Rashidi ldquoThe modified differential transform methodfor solving MHD boundary-layer equationsrdquo Computer PhysicsCommunications vol 180 no 11 pp 2210ndash2217 2009

[42] M M Rashidi and N Freidoonimehr ldquoSeries solutions forthe flow in the vicinity of the equator of an MHD boundary-layer over a porous rotating sphere with heat transferrdquoThermalScience vol 18 pp S527ndashS537 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of