Research Article Flows with Slip of Oldroyd-B Fluids over

Research ArticleFlows with Slip of Oldroyd-B Fluids over a Moving Plate

Abdul Shakeel1 Sohail Ahmad2 Hamid Khan1 Nehad Ali Shah3 and Sami Ul Haq4

1Department of Mathematics FAST-National University of Computer and Emerging Sciences PeshawarKhyber Pakhtunkhwa 25000 Pakistan2Department of Mathematics COMSATS Institute of Information Technology Attock Kamra Road Punjab 43600 Pakistan3Abdus Salam School of Mathematical Sciences Government College University Lahore Punjab 54000 Pakistan4Department of Mathematics Islamia College Peshawar Khyber Pakhtunkhwa 25000 Pakistan

Correspondence should be addressed to Sohail Ahmad drsohailahmadciit-attockedupk

Received 12 November 2015 Revised 3 March 2016 Accepted 13 March 2016

Academic Editor Luigi C Berselli

Copyright copy 2016 Abdul Shakeel et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A general investigation has been made and analytic solutions are provided corresponding to the flows of an Oldroyd-B fluid underthe consideration of slip condition at the boundaryThe fluid motion is generated by the flat plate which has a translational motionin its plane with a time-dependent velocityThe adequate integral transform approach is employed to find analytic solutions for thevelocity field Solutions for the flows corresponding to Maxwell fluid second-grade fluid and Newtonian fluid are also determinedin both cases namely flows with slip on the boundary and flows with no slip on the boundary respectively Some of our resultswere compared with other results from the literature The effects of several emerging dimensionless and pertinent parameters onthe fluid velocity have been studied theoretically as well as graphically in the paper

1 Introduction

Many materials in industry for instance grease polymermelts drillingmud clay coating suspensions certain oil anddifferent emulsions behave in such a way that we cannotdescribe mathematically through Navier-Stokes equationsFor this reason it is now generally accepted that non-New-tonian fluid models are more appropriate than Newtonianones and in practical applications the behavior of non-Newtonian fluids cannot be replaced with that of Newto-nian fluids Therefore the study of non-Newtonian fluidshas become very important due to their large number ofapplications in industry

The reader can see [1] for the latest and complete dis-cussion onOldroyd-B fluidmodels To the best of the authorsrsquoknowledge the first exact solutions corresponding to thesefluid models seem to be those obtained by Tanner [2] Fur-thermore some other useful as well as simpler solutions canbe found in [3] regarding the study of Oldroyd-B fluids

In the above studies the effect of fluid slippage is notconsideredTheflowof fluids induced by amotion of a plate iscalled Stokes flow Solution for some Stokes flows in different

geometries and under the assumption of no-slip boundarycondition can be found in [4ndash7]

However the no-slip condition is inadequate in severalsituations for instance mechanics of thin fluids problemshavingmultiple interfacesmicrochannel flows flows inwavytubes and flows of polymeric liquids with high molecularweight [8]

It is vital to study the effect of fluid slippage as it findsmany applications in industry When a surface moves theslip is mainly produced by the roughness of surface and rar-efaction of the fluid and the velocity on the surface Navier[9] proposed slip boundary condition in which it was statedthat the velocity of the fluid depends on the shear stressFor describing the slip that occurs at solid boundaries alarge number of models have been proposed [10] Otherstudies of slip at the boundary can be seen in [11ndash14] Somenon-Newtonian fluids such as polymer melts often exhibitmacroscopic wall slip which is generally described by anonlinear relation between wall slip velocity and the frictionat the wall

The aim of the present communication is to study Stokesflows of an Oldroyd-B fluid on a flat plate under the slip

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2016 Article ID 8619634 9 pageshttpdxdoiorg10115520168619634

2 Advances in Mathematical Physics

boundary conditions assumption between the plate and thefluid The motion of the plate is a rectilinear translation in itsplane Exact expressions for the velocity are determined bymeans of the Laplace transforms Expressions for the relativevelocity are also determined and the solutions correspondingto flows with no slip at the boundary are presented Theparticular case namely sine oscillations of thewall is studiedSome relevant properties of the velocity and comparisonsbetween solutions with slip and no slip at the boundary arepresented by using graphical illustrations generated by thesoftware Mathcad

2 Mathematical Formulation of the Problem

We consider an incompressible Oldroyd-B fluid occupyingthe space over an infinite plate which is situated in the (119909 119911)-plane of the Cartesian coordinate system with the positive 119910-axis in the upward direction Initially both the fluid and theplate are at rest At 119905 = 0

+ the fluid is set in motion by theplate which begins to move along the 119909-axis The velocity ofthe plate is assumed to be of the form119880

119900119891(119905) where119880

119900gt 0 is

a constant and 119891(119905) is a piecewise continuous dimensionlessfunction defined on [0infin) and 119891(0) = 0 Furthermore wesuppose that the Laplace transformof the function119891 exists Inthe case of parallel flow along the 119909-axis the velocity vector is^ = (119906(119910 119905) 0 0)whereas [15ndash17] lead us to the following gov-erning equation

120582

1205972119906 (119910 119905)

1205971199052

+

120597119906 (119910 119905)

120597119905

= ](1 + 120582119903

120597

120597119905

)

1205972119906 (119910 119905)

1205971199102

(1)

where 120582 is the relaxation time 120582119903is the retardation time ] =

120583120588 is the kinematic viscosity 120583 is the dynamic viscosity and120588 is the constant density of the fluid In this work we considerthe existence of slip at the wall and assume that the relativevelocity between the velocity of the fluid at the wall 119906(0 119905)and the speed of the wall is proportional to the shear rate atthewall [18 19]The adequate initial and boundary conditionsare given by

119906 (0 119905) minus 120573

120597119906 (0 119905)

120597119910

= 119880119900119891 (119905) 120573 ge 0 119905 gt 0

119906 (119910 0) =

120597119906 (119910 0)

120597119905

= 0 119910 ge 0

119906 (119910 119905) 997888rarr 0 for 119910 997888rarr infin

(2)

where 120573 is the slip coefficientBy using the characteristic time119879 introducing the follow-

ing nondimensional quantities to (1) and (2)

119905lowast

=

119905

119879

119879 gt 0

119910lowast

=

119910

119880119900119879

119906lowast

=

119906

119880119900

120582lowast

=

120582

119879

120582119903

lowast

=

120582119903

119879

120573lowast

=

120573

119880119900119879

119892 (119905lowast

) = 119891 (119879119905lowast

)

(3)

and dropping out the lowast notation we get the following non-dimensional problem

120582

1205972119906 (119910 119905)

1205971199052

+

120597119906 (119910 119905)

120597119905

=

1

Re(1 + 120582

119903

120597

120597119905

)

1205972119906 (119910 119905)

1205971199102

(4)

The nondimensional initial and boundary conditions are

119906 (0 119905) minus 120573

120597119906 (0 119905)

120597119910

= 119892 (119905) for 119905 gt 0 (5a)

119906 (119910 0) =

120597119906 (119910 0)

120597119905

= 0 for 119910 gt 0 (5b)

119906 (119910 119905) 997888rarr 0 for 119910 997888rarr infin (5c)

with Re = 119880119900

2119879] the Reynolds number

3 Calculations for Velocity Field

31 Oldroyd-B Fluid with Slip at the Wall Applying Laplacetransform to (4) (5a) and (5c) and using (5b) we obtain thetransformed problem

1205972119906 (119910 119902)

1205971199102

minus

Re (1205821199022 + 119902)(1 + 120582

119903119902)

119906 (119910 119902) = 0 (6)

119906 (0 119902) minus 120573

120597119906 (0 119902)

120597119910

= 119866 (119902) (7)

119906 (119910 119902) 997888rarr 0 119910 997888rarr infin (8)

The solution of the set of (6)ndash(8) is given by

119906 (119910 119902) = 119865 (119910 119902) sdot 119866 (119902) (9)

where

119865 (119910 119902) =

119887 exp (minus119886radic(1205821199022 + 119902) (120582119903119902 + 1))

119887 + radic(1205821199022+ 119902) (120582

119903119902 + 1)

119886 = 119910radicRe 119887 = 1

120573radicRe

(10)

and 119866(119902) = 119871119892(119905)

Advances in Mathematical Physics 3

In order to obtain the inverse Laplace transform of func-tion 119906(119910 119902) we consider the auxiliary functions

1198651(119910 119902) =

119887 exp (minus119886radic119902)119887 + radic119902

119908 (119902) =

1205821199022+ 119902

120582119903119902 + 1

(11)

Seeing that 119865(119910 119902) = 1198651[119910 119908(119902)] we have

119891 (119910 119905) = 119871minus1

119865 (119910 119902) = int

infin

0

1198911(119910 119909) ℎ (119909 119905) 119889119909 (12)

where

1198911(119910 119905) = 119871

minus1

1198651(119910 119902)

=

119890minus1199102Re4119905

120573radicRe120587119905

minus

1

1205732Re

119890119910120573+119905120573

2ReErfc(119910radicRe2radic119905

+

1

120573

radic119905

Re)

(13)

and ℎ(119909 119905) = 119871minus1exp(minus119909119908(119902)) is given by

ℎ (119909 119905) = exp (120574119909) 120575 (119905 minus 120582119909

120582119903

) minus radic

120574119909

120582119903119905 minus 120582119909

sdot 1198691(

2

120582119903

radic120574119909 (120582119903119905 minus 120582119909))

sdot exp((2120582 minus 120582

119903) 119909 minus 120582

119903119905

120582119903

2)

(14)

with 120574 = (120582 minus 120582119903)120582119903

2Now introducing 119891

1(119910 119905) and ℎ(119909 119905) into (12) we obtain

119891 (119910 119905) =

119887radic120582119903

radic120582120587119905

exp(minus120582Re1199102

4120582119903119905

+

120574120582119903

120582

119905) minus

1205821199031198872

120582

sdot exp(119910119887radicRe +1198872+ 120574

120582

120582119903119905)Erfc(

119910radic120582Re2radic120582119903119905

+ 119887radic120582119903119905

120582

) + int

infin

0

[

119887

radic120587119909

sdot exp(minus1199102Re4119909

+

(2120582 minus 120582119903) 119909 minus 120582

119903119905

120582119903

2)

minus 1198872exp(119910119887radicRe + 1198872119909 +

(2120582 minus 120582119903) 119909 minus 120582

119903119905

120582119903

2)

sdot Erfc(119910radicRe2radic119909

+ 119887radic119909)] times radic

120574119909

120582119903119905 minus 120582119909

sdot 1198691(

2

120582119903

sdot radic120574119909 (120582119903119905 minus 120582119909)) 119889119909

(15)

and the velocity field corresponding to the flow with slip atthe wall of an Oldroyd-B fluid is given by

119906119904(119910 119905) = (119891 lowast 119892) (119905) = int

119905

0

119891 (119910 119904) 119892 (119905 minus 119904) 119889119904 (16)

where 119891(119910 119905) is given by (15)

32 Oldroyd-B Fluid with No Slip at the Wall In this particu-lar case function 119865(119910 119902) given by (10) becomes

119865ns (119910 119902) = exp(minus119910radicReradic1205821199022+ 119902

120582119903119902 + 1

) (17)

By using the auxiliary functions

1198651ns (119910 119902) = exp (minus119910radicRe 119902)

1198911ns (119910 119905) =

119910radicRe2119905radic120587119905


)

(18)

together with the relation

119891ns (119910 119905) = 119871minus1

119865ns (119910 119902) = int

infin

0

1198911ns (119910 119909)

sdot ℎ (119909 119905) 119889119909 =

119910radicRe 1205822radic120587120582

119903

119905minus32exp(

120574120582119903

120582

119905

minus

1199102Re 1205824120582119903119905

) minus

119910radicRe 1205742radic120587

sdot int

infin

0

119890minus1199102Re4119909+((2120582minus120582

119903)119909minus120582119903119905)120582119903

2

sdot

1

119909radic120582119903119905 minus 120582119909

1198691(

2

120582119903

radic120574119909 (120582119903119905 minus 120582119909)) 119889119909

(19)

The velocity field corresponding to the flow with no-slipconditions of the Oldroyd-B fluid is given by

119906ns (119910 119905) = int

119905

0

119891ns (119910 119904) 119892 (119905 minus 119904) 119889119904 =2

radic120587

sdot int

infin

119910radicRe1205822radic120582119903119905

119890minus1199112+120574Re119910241199112

119892(119905 minus

120582Re1199102

41205821199031199112)119889119911

minus

119910radicRe 1205742radic120587

sdot int

infin

0

int

119905

0

119892 (119905 minus 119904)

119890minus1199102Re4119909+((2120582minus120582

119903)119909minus120582119903119904)120582119903

2

119909radic120582119903119904 minus 120582119909

times 1198691(

2

120582119903

radic120574119909 (120582119903119904 minus 120582119909)) 119889119904 119889119909

(20)


It is important to point out that the solution given by (20)is similar to that obtained recently by Fetecau et al [20 Eq(428)]

33 Maxwell Fluid with Slip Condition For the flows of Max-well fluids with slip condition at the wall (10) becomes

119865sM (119910 119902)

=

(119887radic2) exp(minus119910radic120582Reradic(119902 + 12120582)2 minus (12120582)2)

119887radic2 + radic(119902 + 12120582)2

minus (12120582)2

= 1198871exp

(minus119910radic120582Reradic(119902 + 1198861)2

minus 1198862

1)

1198871

+ radic(119902 + 1198861)2

minus 1198862

1= 1198651(119910radic119908

1(119902))

(21)

where 1198651(sdot sdot) is given by (11) and 119908

1(119902) = (119902 + 119886

1)2

minus 1198862

1 1198871=

119887radic120582 and 1198861= 12120582

The inverse Laplace transform of function (21) is

119891sm (119910 119905) = [1198911(119910 119905)

+ 1198861int

119905

0

1198911(1199101

radic1199052minus 1199062) 1198681(119886 119906) 119889119906] 119890

minus1198861119905

sdot

1198871119890minus1199102Re1205824119905minus119886

1119905

radic120587119905

minus 1198872

11198901199101198871radicRe120582+1198872

1119905minus1198861119905Erfc(

119910radicRe 1205822radic119905

+ 119887radic119905)

+ 1198861119890minus1198861119905

int

119905

0

[

1198871119890minus1199102Re1205824119911minus119886

1119911

radic120587119911

minus 1198872

11198901199101198871radicRe120582+1198872

1minus1198861119911Erfc(

119910radicRe 1205822radic119911

+ 1198871radic119911)]

sdot

119911

radic1199052minus 1199112

1198681(1198861

radic1199052minus 1199112) 119889119911

(22)

and the velocity field is given by

119906sM (119910 119905) = (119891 lowast 119892) (119905) = int

119905

0

119891SM (119910 119904) 119892 (119905 minus 119904) 119889119904 (23)

34 Maxwell Fluid with No Slip at the Wall Making 120573 rarr 0

and therefore 119887 rarr 0 into (21) or 120582119903rarr 0 into (17) we have

119865nsM (119910 119902) = exp(minus119910radicReradic1205821199022 + 119902)

= exp(minus119910radicRe 120582radic(119902 + 1198861)2

minus 1198862

1)

(24)

Using the formula

119871minus1

119890minus119910radicRe120582radic119902

= ℎ1(119910 119905) =

119910radicRe 120582119890minus1199102Re1205824119905

2119905radic120587119905

(25)

we obtain

119891nsM (119910 119905) = [ℎ1(119910 119905)

+ 1198861int

119905

0

ℎ1(119910 119911)

119911


1198681(1198861

radic1199052minus 1199112) 119889119911]

sdot 119890minus1198861119905

=

119910radicRe 1205822119905radic120587119905

119890minus1199102Re1205824119905minus119886

1119905

+ 1198861119890minus1198861119905

int

119905

0

119910radicRe 1205822119911radic120587119911

119890minus1199102Re1205824119911

sdot

119911


1198681(1198861

radic1199052minus 1199112) 119889119911

(26)

and the velocity field

119906nsM (119910 119905) = (119891 lowast 119892) (119905) = int

119905

0

119891nsM (119910 119904) 119892 (119905 minus 119904) 119889119904

=

2

radic120587

int

infin

119910radicRe1205822radic119905119890minus1199112minus1198861120582Re119910241199112

119892(119905

minus

120582Re1199102

41199112)119889119911 + 119886

1int

119905

0

int

119904

0

119892 (119905 minus 119904)

sdot

119910radicRe 1205822119911radic120587119911

119890minus1199102Re1205824119911minus119886

1119904

sdot

119911


1198681(1198861

radic1199042minus 1199112) 119889119904 119889119911

(27)

35 Second-Grade Fluid with Slip at the Wall Making 120582 rarr 0

into (10) and (14) and using the results

1198721=

119887

radic120587

int

infin

0

119890minus1199102Re4119909minus119909120582

119903119889119909

radic119909

=

119887

radic120587

int

infin

0

119890(minus1199102Re120582

119903)41199112minus1199112

2radic120582119903119889119911

= 119887radic120582119903

2

radic120587

int

infin

0

119890minus1199112minus(119910radicRe120582

119903)241199112

119889119911

= 119887radic120582119903

2

radic120587

sdot

radic120587

2

119890minus1sdot119910radicReradic120582

119903= 119887radic120582

119903119890minus119910radicReradic120582

119903

(28)


we obtain the velocity field of the second-grade fluid with slipat the wall in the equivalent forms

119906sSG (119910 119905) = 119887radic120582119903119890minus119910radicReradic120582

119903119892 (119905) minus 119887

2

119890119887119910radicRe

119892 (119905)

sdot int

infin

0

1198901198872119909minus119909120582

119903Erfc(119910radicRe2radic119909

+ 119887radic119909)119889119909 +

1

120582119903

sdot int

infin

0

[

119887119890minus1199102Re4119909

radic120587119909

minus 1198872

119890119910119887radicRe+1198872119909Erfc(

119910radicRe2radic119909

+ 119887radic119909)]

sdot 119890(minus119909+119905)120582

119903radic

119909

119905

1198681(

2

120582119903

radic119909119905)119889119909

(29)

or

119906sSG (119910 119905) = 119887 (1198872radic120582119903minus 119887 + radic120582

119903) 119890minus119910radicReradic120582

119903119892 (119905)

+

119890minus119905120582119903

radic120582119903119905

119892 (119905) times int

infin

0

[

119887119890minus1199102Re4119909minus119909120582

119903

radic120587119909

minus 1198872

119890119887119910radicRe+1198872119909minus119909120582


+ 119887radic119909)]

sdot radic1199091198681(

2

radic120582119903

radic119909119905)119889119909

(30)

36 Second-Grade FluidwithNo-SlipCondition Making120582 rarr0 into (17) and using (18) and (28) we have

119906nsSG (119910 119905) = 119890minus119910radicRe120582

119903119892 (119905) +

119910radicRe2120582119903radic120587

sdot int

infin

0

int

119905

0

119892 (119905 minus 119904)

119909radic119904

119890minus1199102Re4119909minus119909120582

119903minus1199041205821199031198681(

2

120582119903

sdot radic119909119904) 119889119904 119889119909

(31)

37 Newtonian Fluid withwithout Slip Condition Thesecases are obtained easily from (6) and (10) by making 120582 =

120582119903= 0 We obtain the following expressions for the velocity

field

119906sN (119910 119905) = int

119905

0

119892 (119905 minus 120591) VsN (119910 120591) 119889120591

119906nsN (119910 119905) = int

119905

0

119892 (119905 minus 120591) VnsN (119910 120591) 119889120591

(32)

where

VsN (119910 119905) =1

120573radicRe1

radic120587119905


) minus (

1

120573radicRe)

2

sdot exp(119910

120573

+

119905

1205732Re

)

sdot erfc(119910radicRe2radic119905

+

radic119905

120573radicRe)

(33)

for flows with slip on the boundary and

VnsN (119910 119905) =119910radicRe2119905radic120587119905


) (34)

for flows with no slip on the boundary respectivelyWemention that the results given by (33) and (34) are the

same as those obtained by Fetecau et al (see [21] Eq (29) for119872 = 0 120574 replaced by 120573radicRe and 119910 replaced by 119910radicRe)

Also if in our results we consider 119892(119905) = sin(120596119905) or 119892(119905) =cos(120596119905) (32)ndash(34) become equivalent to the results of Khaledand Vafai (see [22] Eqs (8) (9) (10) (16)) and to the resultsobtained by Hayat et al (see [23] Eqs (13) (14) with119872 = 0

and119870 rarrinfin)In the case of no slip on the boundary our solution (32)

together with (34) is identical with the result obtained byFetecau et al (see [24] Eq (31) with119870eff = 0)

4 Numerical Results and Conclusions

In this paper we have studied the flow of Oldroyd-B fluidsgenerated by a moving flat plate Using Laplace transformmethod we obtained analytical expressions of the velocityfor both cases of flows with slip at the boundary and withoutslip on the boundary The plate velocity was considered in ageneral form119880

0119891(119905) therefore solutions for several practical

problems can be obtained by choosing suitable forms of thefunction 119891(119905) From the dimensionless form of the studiedproblem it can be seen that only Reynolds number andnondimensional relaxation and retardation time influencethe fluid flows From this reason the numerical studies aremade for several values of Reynolds number and of the time119905 Graphs of velocity were plotted versus spatial coordinate 119910for the case of translation of the plate with constant velocitynamely for119891(119905) = 1 Velocity fields corresponding to flows ofMaxwell fluid second-grade fluid and Newtonian fluid werealso determined in both cases namely flows with slip on theboundary and flows with no slip on the boundary In orderto study the physical behavior of the fluid some numericalsimulations were made using the Mathcad software In Fig-ures 1 and 2 curves corresponding to velocity of the Oldroyd-B fluid for 120582 = 025 and 120582

119903= 015 and the dimensionless

friction coefficient 120573 = 075 are sketched In Figures 1 and2 the velocity curves for three values of time 119905 and for threevalues of Reynolds number Re in the case of flow with slip atthe plate and in the case of no slip at the plate are plotted


0 2 4 6 8

y

0 2 4 6 8

y

0 2 4 6 8

y

1

08

06

04

02

0

1

08

06

04

02

0

Velo

city

1

08

06

04

02

0

t = 5

t = 3

t = 15

minus02

Re = 125 slipRe = 175 slipRe = 225 slip

Re = 125 no slipRe = 175 no slipRe = 225 no slip





u(yt)

Velo

cityu(yt)

Velo

cityu(yt)

Figure 1 Velocity profiles of Oldroyd-B fluid for slip condition (16) with 120573 = 075 and no-slip condition (20) for 120582 = 025 120582119903= 015 and

different Reynolds number

For a fixed value of the time 119905 or for a fixed value of theReynolds number the issues whichmust be highlighted are asfollows

The fluid flows more slowly if slippage occurs on theboundary Increasing of the Reynolds number values leads

to slowing of the flows in both cases with or without slip Ifthe values of the time 119905 are increasing then it increases thethickness of the velocity boundary layer

Figure 3 was drawn in order to compare the velocityflows for Oldroyd-B Maxwell and Newtonian fluids It is


1

08

06

04

02

0

1

1

08

06

04

02

0

0

0 2 4 6 8

y

0 2 4 6 8

y

0 2 4 6 8

y

Re = 05

Re = 25

Re = 10

t = 05 no slipt = 20 no slipt = 50 no slip

t =

t =

t =


t =

t =

t =


t = 20 slipt = 50 slip

t = 05 slip

minus02

08

06

04

02

minus02

Velo

cityu(yt)

Velo

cityu(yt)

Velo

cityu(yt)

20 slip50 slip

05 slip

20 slip50 slip

05 slip


different values of the time 119905

important to note that for the flow with no slip on theboundary the velocity of Maxwell fluid has significant varia-tions in the area near plateThis no longer occurs if the flow iswith slip at the boundary Also the thickness of the boundary

layer of the Maxwell fluid is the smallest and the speed of thistype of fluid becomes zero more quickly than other fluids Ifthe values of the time 119905 are increasing then the differencesbetween velocities of the three fluids become insignificant


15

1

05

0

minus05

Velo

cityu(yt)

15

1

05

0

minus05

Velo

cityu(yt)

15

1

05

0

minus05

Velo

cityu(yt)

t = 05

t = 075

t = 155

0 1 2

y

3

0 1 2

y

3

0 1 2

y

3

Oldroyd-B fluid slipMaxwell fluid slipNewtonian fluid slip

Oldroyd-B fluid no slipMaxwell fluid no slipNewtonian fluid no slip





Figure 3 Comparison between velocity profiles of three fluid models with slip condition with 120573 = 055 and no-slip condition for Re = 065120582 = 055 and 120582

119903= 035


Competing Interests

The authors declare that they have no competing interests

References

[1] K R Rajagopal and A R Srinivasa ldquoA thermodynamic framework for rate type fluid modelsrdquo Journal of Non-NewtonianFluid Mechanics vol 88 no 3 pp 207ndash227 2000

[2] R I Tanner ldquoNote on the Rayleigh problem for a visco-elasticfluidrdquo Zeitschrift fur Angewandte Mathematik und Physik vol13 no 6 pp 573ndash580 1962

[3] K R Rajagopal and R K Bhatnagar ldquoExact solutions for somesimple flows of an Oldroyd-B fluidrdquo Acta Mechanica vol 113no 1 pp 233ndash239 1995

[4] N D Waters and M J King ldquoUnsteady flow of an elastico-viscous liquidrdquo Rheologica Acta vol 9 no 3 pp 345ndash355 1970

[5] P Puri and P K Kythe ldquoStokes first and second problems forRivlin-Ericksen fluids with neoclassical heat conditionrdquo ASMEJournal of Heat Transfer vol 120 pp 44ndash50 1996

[6] P M Jordan and A Puri ldquoRevisiting Stokesrsquo first problem forMaxwell fluidsrdquoTheQuarterly Journal ofMechanics and AppliedMathematics vol 58 no 2 pp 213ndash227 2005

[7] C Fetecau D Vieru and C Fetecau ldquoA note on the secondproblem of Stokes for Newtonian fluidsrdquo International Journalof Non-Linear Mechanics vol 43 no 5 pp 451ndash457 2008

[8] R I Tanner ldquoPartial wall slip in polymer flowrdquo Industrial andEngineering Chemistry Research vol 33 no 10 pp 2434ndash24361994

[9] M Navier ldquoMemoire sur les lois du mouvement des fluidsrdquoMemoires de lrsquoAcademie des Sciences de LrsquoInstitut de France vol6 pp 389ndash440 1823

[10] I J Rao and K R Rajagopal ldquoThe effect of the slip boundarycondition on the flow of fluids in a channelrdquo Acta Mechanicavol 135 no 3-4 pp 113ndash126 1999

[11] M Mooney ldquoExplicit formulas for slip and fluidityrdquo Journal ofRheology vol 2 no 2 pp 210ndash222 1931

[12] K B Migler H Hervet and L Leger ldquoSlip transition of a pol-ymer melt under shear stressrdquo Physical Review Letters vol 70no 3 pp 287ndash290 1993

[13] S Das S K Guchhait and R N Jana ldquoHall effects on unsteadyhydromagnetic flow past an accelerated porous plate in arotating systemrdquo Journal of Applied Fluid Mechanics vol 8 no3 pp 409ndash417 2015

[14] A S Chethan G N Sekhar and P G Siddheshwar ldquoFlow andheat transfer of an exponential stretching sheet in a viscoelasticliquid with navier slip boundary conditionrdquo Journal of AppliedFluid Mechanics vol 8 no 2 pp 223ndash229 2015

[15] M Khan R Malik and A Anjum ldquoExact solutions of MHDsecond Stokes flow of generalized Burgers fluidrdquo Applied Math-ematics and Mechanics vol 36 no 2 pp 211ndash224 2015

[16] I Khan F Ali and S Shafie ldquoStokesrsquo second problem for mag-netohydrodynamics flow in a Burgersrsquo fluid the cases 120574 = 120582

2

4

and 120574 gt 1205822

4rdquo PLoS ONE vol 8 no 5 Article ID e61531 2013[17] A M Siddiqui T Haroon M Zahid and A Shahzad ldquoEffect of

slip condition on unsteady flows of an oldroyd-B fluid betweenparallel platesrdquoWorld Applied Sciences Journal vol 13 no 11 pp2282ndash2287 2011

[18] DVieru andAA Zafar ldquoSomeCouette flows of aMaxwell fluidwith wall slip conditionrdquo Applied Mathematics amp InformationSciences vol 7 no 1 pp 209ndash219 2013

[19] D Vieru and A Rauf ldquoStokes flows of a Maxwell fluid with wallslip conditionrdquo Canadian Journal of Physics vol 89 no 10 pp1061ndash1071 2011

[20] C Fetecau N Nigar D Vieru and C Fetecau ldquoFirst generalsolution for unidirectional motions of rate type fluids over aninfinite platerdquoCommunications inNumerical Analysis vol 2 pp125ndash138 2015

[21] C Fetecau D Vieru C Fetecau and I Pop ldquoSlip effects on theunsteady radiative MHD free convection flow over a movingplate with mass diffusion and heat sourcerdquoThe European Phys-ical Journal Plus vol 130 article 6 2015

[22] A-R A Khaled and K Vafai ldquoThe effect of the slip conditionon Stokes and Couette flows due to an oscillating wall exactsolutionsrdquo International Journal of Non-Linear Mechanics vol39 no 5 pp 795ndash809 2004

[23] T Hayat M F Afzaal C Fetecau and A A Hendi ldquoSlip effectson the oscillatory flow in a porous mediumrdquo Journal of PorousMedia vol 14 no 6 pp 481ndash493 2011

[24] C Fetecau D Vieru C Fetecau and S Akhter ldquoGeneral solu-tions for magnetohydrodynamic natural convection flow withradiative heat transfer and slip condition over a moving platerdquoZeitschrift fur Naturforschung A vol 68 pp 659ndash667 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


boundary conditions assumption between the plate and thefluid The motion of the plate is a rectilinear translation in itsplane Exact expressions for the velocity are determined bymeans of the Laplace transforms Expressions for the relativevelocity are also determined and the solutions correspondingto flows with no slip at the boundary are presented Theparticular case namely sine oscillations of thewall is studiedSome relevant properties of the velocity and comparisonsbetween solutions with slip and no slip at the boundary arepresented by using graphical illustrations generated by thesoftware Mathcad

2 Mathematical Formulation of the Problem

We consider an incompressible Oldroyd-B fluid occupyingthe space over an infinite plate which is situated in the (119909 119911)-plane of the Cartesian coordinate system with the positive 119910-axis in the upward direction Initially both the fluid and theplate are at rest At 119905 = 0

+ the fluid is set in motion by theplate which begins to move along the 119909-axis The velocity ofthe plate is assumed to be of the form119880

119900119891(119905) where119880

119900gt 0 is

a constant and 119891(119905) is a piecewise continuous dimensionlessfunction defined on [0infin) and 119891(0) = 0 Furthermore wesuppose that the Laplace transformof the function119891 exists Inthe case of parallel flow along the 119909-axis the velocity vector is^ = (119906(119910 119905) 0 0)whereas [15ndash17] lead us to the following gov-erning equation

120582

1205972119906 (119910 119905)

1205971199052

+

120597119906 (119910 119905)

120597119905

= ](1 + 120582119903

120597

120597119905

)

1205972119906 (119910 119905)

1205971199102

(1)

where 120582 is the relaxation time 120582119903is the retardation time ] =

120583120588 is the kinematic viscosity 120583 is the dynamic viscosity and120588 is the constant density of the fluid In this work we considerthe existence of slip at the wall and assume that the relativevelocity between the velocity of the fluid at the wall 119906(0 119905)and the speed of the wall is proportional to the shear rate atthewall [18 19]The adequate initial and boundary conditionsare given by

119906 (0 119905) minus 120573

120597119906 (0 119905)

120597119910

= 119880119900119891 (119905) 120573 ge 0 119905 gt 0

119906 (119910 0) =

120597119906 (119910 0)

120597119905

= 0 119910 ge 0

119906 (119910 119905) 997888rarr 0 for 119910 997888rarr infin

(2)

where 120573 is the slip coefficientBy using the characteristic time119879 introducing the follow-

ing nondimensional quantities to (1) and (2)

119905lowast

=

119905

119879

119879 gt 0

119910lowast

=

119910

119880119900119879

119906lowast

=

119906

119880119900

120582lowast

=

120582

119879

120582119903

lowast

=

120582119903

119879

120573lowast

=

120573

119880119900119879

119892 (119905lowast

) = 119891 (119879119905lowast

)

(3)

and dropping out the lowast notation we get the following non-dimensional problem

120582

1205972119906 (119910 119905)

1205971199052

+

120597119906 (119910 119905)

120597119905

=

1

Re(1 + 120582

119903

120597

120597119905

)

1205972119906 (119910 119905)

1205971199102

(4)

The nondimensional initial and boundary conditions are

119906 (0 119905) minus 120573

120597119906 (0 119905)

120597119910

= 119892 (119905) for 119905 gt 0 (5a)

119906 (119910 0) =

120597119906 (119910 0)

120597119905

= 0 for 119910 gt 0 (5b)

119906 (119910 119905) 997888rarr 0 for 119910 997888rarr infin (5c)

with Re = 119880119900

2119879] the Reynolds number

3 Calculations for Velocity Field

31 Oldroyd-B Fluid with Slip at the Wall Applying Laplacetransform to (4) (5a) and (5c) and using (5b) we obtain thetransformed problem

1205972119906 (119910 119902)

1205971199102

minus

Re (1205821199022 + 119902)(1 + 120582

119903119902)

119906 (119910 119902) = 0 (6)

119906 (0 119902) minus 120573

120597119906 (0 119902)

120597119910

= 119866 (119902) (7)

119906 (119910 119902) 997888rarr 0 119910 997888rarr infin (8)

The solution of the set of (6)ndash(8) is given by

119906 (119910 119902) = 119865 (119910 119902) sdot 119866 (119902) (9)

where

119865 (119910 119902) =

119887 exp (minus119886radic(1205821199022 + 119902) (120582119903119902 + 1))

119887 + radic(1205821199022+ 119902) (120582

119903119902 + 1)

119886 = 119910radicRe 119887 = 1

120573radicRe

(10)

and 119866(119902) = 119871119892(119905)



1198651(119910 119902) =


119908 (119902) =

1205821199022+ 119902

120582119903119902 + 1

(11)


119891 (119910 119905) = 119871minus1

119865 (119910 119902) = int

infin

0

1198911(119910 119909) ℎ (119909 119905) 119889119909 (12)

where

1198911(119910 119905) = 119871

minus1

1198651(119910 119902)

=

119890minus1199102Re4119905

120573radicRe120587119905

minus

1

1205732Re

119890119910120573+119905120573


+

1

120573

radic119905

Re)

(13)


ℎ (119909 119905) = exp (120574119909) 120575 (119905 minus 120582119909

120582119903

) minus radic

120574119909

120582119903119905 minus 120582119909

sdot 1198691(

2

120582119903

radic120574119909 (120582119903119905 minus 120582119909))


119903) 119909 minus 120582

119903119905

120582119903

2)

(14)

with 120574 = (120582 minus 120582119903)120582119903



119891 (119910 119905) =

119887radic120582119903

radic120582120587119905


4120582119903119905

+

120574120582119903

120582

119905) minus

1205821199031198872

120582

sdot exp(119910119887radicRe +1198872+ 120574

120582

120582119903119905)Erfc(

119910radic120582Re2radic120582119903119905

+ 119887radic120582119903119905

120582

) + int

infin

0

[

119887

radic120587119909


+

(2120582 minus 120582119903) 119909 minus 120582

119903119905

120582119903

2)


(2120582 minus 120582119903) 119909 minus 120582

119903119905

120582119903

2)



120574119909

120582119903119905 minus 120582119909

sdot 1198691(

2

120582119903

sdot radic120574119909 (120582119903119905 minus 120582119909)) 119889119909

(15)


119906119904(119910 119905) = (119891 lowast 119892) (119905) = int

119905

0

119891 (119910 119904) 119892 (119905 minus 119904) 119889119904 (16)

where 119891(119910 119905) is given by (15)



120582119903119902 + 1

) (17)



1198911ns (119910 119905) =

119910radicRe2119905radic120587119905


)

(18)


119891ns (119910 119905) = 119871minus1

119865ns (119910 119902) = int

infin

0

1198911ns (119910 119909)

sdot ℎ (119909 119905) 119889119909 =

119910radicRe 1205822radic120587120582

119903

119905minus32exp(

120574120582119903

120582

119905

minus

1199102Re 1205824120582119903119905

) minus

119910radicRe 1205742radic120587

sdot int

infin

0

119890minus1199102Re4119909+((2120582minus120582

119903)119909minus120582119903119905)120582119903

2

sdot

1

119909radic120582119903119905 minus 120582119909

1198691(

2

120582119903

radic120574119909 (120582119903119905 minus 120582119909)) 119889119909

(19)


119906ns (119910 119905) = int

119905

0

119891ns (119910 119904) 119892 (119905 minus 119904) 119889119904 =2

radic120587

sdot int

infin

119910radicRe1205822radic120582119903119905

119890minus1199112+120574Re119910241199112

119892(119905 minus

120582Re1199102

41205821199031199112)119889119911

minus

119910radicRe 1205742radic120587

sdot int

infin

0

int

119905

0

119892 (119905 minus 119904)

119890minus1199102Re4119909+((2120582minus120582

119903)119909minus120582119903119904)120582119903

2

119909radic120582119903119904 minus 120582119909

times 1198691(

2

120582119903

radic120574119909 (120582119903119904 minus 120582119909)) 119889119904 119889119909

(20)




119865sM (119910 119902)

=


119887radic2 + radic(119902 + 12120582)2

minus (12120582)2

= 1198871exp


minus 1198862

1)

1198871

+ radic(119902 + 1198861)2

minus 1198862

1= 1198651(119910radic119908

1(119902))

(21)


1(119902) = (119902 + 119886

1)2

minus 1198862

1 1198871=

119887radic120582 and 1198861= 12120582


119891sm (119910 119905) = [1198911(119910 119905)

+ 1198861int

119905

0

1198911(1199101

radic1199052minus 1199062) 1198681(119886 119906) 119889119906] 119890

minus1198861119905

sdot

1198871119890minus1199102Re1205824119905minus119886

1119905

radic120587119905

minus 1198872

11198901199101198871radicRe120582+1198872

1119905minus1198861119905Erfc(

119910radicRe 1205822radic119905

+ 119887radic119905)

+ 1198861119890minus1198861119905

int

119905

0

[

1198871119890minus1199102Re1205824119911minus119886

1119911

radic120587119911

minus 1198872

11198901199101198871radicRe120582+1198872

1minus1198861119911Erfc(

119910radicRe 1205822radic119911

+ 1198871radic119911)]

sdot

119911


1198681(1198861

radic1199052minus 1199112) 119889119911

(22)


119906sM (119910 119905) = (119891 lowast 119892) (119905) = int

119905

0

119891SM (119910 119904) 119892 (119905 minus 119904) 119889119904 (23)





minus 1198862

1)

(24)

Using the formula

119871minus1


= ℎ1(119910 119905) =

119910radicRe 120582119890minus1199102Re1205824119905

2119905radic120587119905

(25)

we obtain

119891nsM (119910 119905) = [ℎ1(119910 119905)

+ 1198861int

119905

0

ℎ1(119910 119911)

119911


1198681(1198861

radic1199052minus 1199112) 119889119911]

sdot 119890minus1198861119905

=

119910radicRe 1205822119905radic120587119905

119890minus1199102Re1205824119905minus119886

1119905

+ 1198861119890minus1198861119905

int

119905

0

119910radicRe 1205822119911radic120587119911

119890minus1199102Re1205824119911

sdot

119911


1198681(1198861

radic1199052minus 1199112) 119889119911

(26)


119906nsM (119910 119905) = (119891 lowast 119892) (119905) = int

119905

0

119891nsM (119910 119904) 119892 (119905 minus 119904) 119889119904

=

2

radic120587

int

infin


119892(119905

minus

120582Re1199102

41199112)119889119911 + 119886

1int

119905

0

int

119904

0

119892 (119905 minus 119904)

sdot

119910radicRe 1205822119911radic120587119911

119890minus1199102Re1205824119911minus119886

1119904

sdot

119911


1198681(1198861

radic1199042minus 1199112) 119889119904 119889119911

(27)



1198721=

119887

radic120587

int

infin

0

119890minus1199102Re4119909minus119909120582

119903119889119909

radic119909

=

119887

radic120587

int

infin

0

119890(minus1199102Re120582

119903)41199112minus1199112

2radic120582119903119889119911

= 119887radic120582119903

2

radic120587

int

infin

0


119903)241199112

119889119911

= 119887radic120582119903

2

radic120587

sdot

radic120587

2


119903= 119887radic120582


119903

(28)




119903119892 (119905) minus 119887

2

119890119887119910radicRe

119892 (119905)

sdot int

infin

0

1198901198872119909minus119909120582


+ 119887radic119909)119889119909 +

1

120582119903

sdot int

infin

0

[

119887119890minus1199102Re4119909

radic120587119909

minus 1198872

119890119910119887radicRe+1198872119909Erfc(


+ 119887radic119909)]

sdot 119890(minus119909+119905)120582

119903radic

119909

119905

1198681(

2

120582119903

radic119909119905)119889119909

(29)

or



119903119892 (119905)

+

119890minus119905120582119903

radic120582119903119905

119892 (119905) times int

infin

0

[

119887119890minus1199102Re4119909minus119909120582

119903

radic120587119909

minus 1198872

119890119887119910radicRe+1198872119909minus119909120582


+ 119887radic119909)]

sdot radic1199091198681(

2

radic120582119903

radic119909119905)119889119909

(30)



119903119892 (119905) +

119910radicRe2120582119903radic120587

sdot int

infin

0

int

119905

0

119892 (119905 minus 119904)

119909radic119904

119890minus1199102Re4119909minus119909120582

119903minus1199041205821199031198681(

2

120582119903

sdot radic119909119904) 119889119904 119889119909

(31)



field

119906sN (119910 119905) = int

119905

0

119892 (119905 minus 120591) VsN (119910 120591) 119889120591

119906nsN (119910 119905) = int

119905

0

119892 (119905 minus 120591) VnsN (119910 120591) 119889120591

(32)

where

VsN (119910 119905) =1

120573radicRe1

radic120587119905


) minus (

1

120573radicRe)

2

sdot exp(119910

120573

+

119905

1205732Re

)


+

radic119905

120573radicRe)

(33)




) (34)













0 2 4 6 8

y

0 2 4 6 8

y

0 2 4 6 8

y

1

08

06

04

02

0

1

08

06

04

02

0

Velo

city

1

08

06

04

02

0

t = 5

t = 3

t = 15

minus02







u(yt)

Velo

cityu(yt)

Velo

cityu(yt)








1

08

06

04

02

0

1

1

08

06

04

02

0

0

0 2 4 6 8

y

0 2 4 6 8

y

0 2 4 6 8

y

Re = 05

Re = 25

Re = 10


t =

t =

t =


t =

t =

t =



t = 05 slip

minus02

08

06

04

02

minus02

Velo

cityu(yt)

Velo

cityu(yt)

Velo

cityu(yt)

20 slip50 slip

05 slip

20 slip50 slip

05 slip






15

1

05

0

minus05

Velo

cityu(yt)

15

1

05

0

minus05

Velo

cityu(yt)

15

1

05

0

minus05

Velo

cityu(yt)

t = 05

t = 075

t = 155

0 1 2

y

3

0 1 2

y

3

0 1 2

y

3








119903= 035


Competing Interests


References

















2

4

and 120574 gt 1205822

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198651(119910 119902) =


119908 (119902) =

1205821199022+ 119902

120582119903119902 + 1

(11)


119891 (119910 119905) = 119871minus1

119865 (119910 119902) = int

infin

0

1198911(119910 119909) ℎ (119909 119905) 119889119909 (12)

where

1198911(119910 119905) = 119871

minus1

1198651(119910 119902)

=

119890minus1199102Re4119905

120573radicRe120587119905

minus

1

1205732Re

119890119910120573+119905120573


+

1

120573

radic119905

Re)

(13)


ℎ (119909 119905) = exp (120574119909) 120575 (119905 minus 120582119909

120582119903

) minus radic

120574119909

120582119903119905 minus 120582119909

sdot 1198691(

2

120582119903

radic120574119909 (120582119903119905 minus 120582119909))


119903) 119909 minus 120582

119903119905

120582119903

2)

(14)

with 120574 = (120582 minus 120582119903)120582119903



119891 (119910 119905) =

119887radic120582119903

radic120582120587119905


4120582119903119905

+

120574120582119903

120582

119905) minus

1205821199031198872

120582

sdot exp(119910119887radicRe +1198872+ 120574

120582

120582119903119905)Erfc(

119910radic120582Re2radic120582119903119905

+ 119887radic120582119903119905

120582

) + int

infin

0

[

119887

radic120587119909


+

(2120582 minus 120582119903) 119909 minus 120582

119903119905

120582119903

2)


(2120582 minus 120582119903) 119909 minus 120582

119903119905

120582119903

2)



120574119909

120582119903119905 minus 120582119909

sdot 1198691(

2

120582119903

sdot radic120574119909 (120582119903119905 minus 120582119909)) 119889119909

(15)


119906119904(119910 119905) = (119891 lowast 119892) (119905) = int

119905

0

119891 (119910 119904) 119892 (119905 minus 119904) 119889119904 (16)

where 119891(119910 119905) is given by (15)



120582119903119902 + 1

) (17)



1198911ns (119910 119905) =

119910radicRe2119905radic120587119905


)

(18)


119891ns (119910 119905) = 119871minus1

119865ns (119910 119902) = int

infin

0

1198911ns (119910 119909)

sdot ℎ (119909 119905) 119889119909 =

119910radicRe 1205822radic120587120582

119903

119905minus32exp(

120574120582119903

120582

119905

minus

1199102Re 1205824120582119903119905

) minus

119910radicRe 1205742radic120587

sdot int

infin

0

119890minus1199102Re4119909+((2120582minus120582

119903)119909minus120582119903119905)120582119903

2

sdot

1

119909radic120582119903119905 minus 120582119909

1198691(

2

120582119903

radic120574119909 (120582119903119905 minus 120582119909)) 119889119909

(19)


119906ns (119910 119905) = int

119905

0

119891ns (119910 119904) 119892 (119905 minus 119904) 119889119904 =2

radic120587

sdot int

infin

119910radicRe1205822radic120582119903119905

119890minus1199112+120574Re119910241199112

119892(119905 minus

120582Re1199102

41205821199031199112)119889119911

minus

119910radicRe 1205742radic120587

sdot int

infin

0

int

119905

0

119892 (119905 minus 119904)

119890minus1199102Re4119909+((2120582minus120582

119903)119909minus120582119903119904)120582119903

2

119909radic120582119903119904 minus 120582119909

times 1198691(

2

120582119903

radic120574119909 (120582119903119904 minus 120582119909)) 119889119904 119889119909

(20)




119865sM (119910 119902)

=


119887radic2 + radic(119902 + 12120582)2

minus (12120582)2

= 1198871exp


minus 1198862

1)

1198871

+ radic(119902 + 1198861)2

minus 1198862

1= 1198651(119910radic119908

1(119902))

(21)


1(119902) = (119902 + 119886

1)2

minus 1198862

1 1198871=

119887radic120582 and 1198861= 12120582


119891sm (119910 119905) = [1198911(119910 119905)

+ 1198861int

119905

0

1198911(1199101

radic1199052minus 1199062) 1198681(119886 119906) 119889119906] 119890

minus1198861119905

sdot

1198871119890minus1199102Re1205824119905minus119886

1119905

radic120587119905

minus 1198872

11198901199101198871radicRe120582+1198872

1119905minus1198861119905Erfc(

119910radicRe 1205822radic119905

+ 119887radic119905)

+ 1198861119890minus1198861119905

int

119905

0

[

1198871119890minus1199102Re1205824119911minus119886

1119911

radic120587119911

minus 1198872

11198901199101198871radicRe120582+1198872

1minus1198861119911Erfc(

119910radicRe 1205822radic119911

+ 1198871radic119911)]

sdot

119911


1198681(1198861

radic1199052minus 1199112) 119889119911

(22)


119906sM (119910 119905) = (119891 lowast 119892) (119905) = int

119905

0

119891SM (119910 119904) 119892 (119905 minus 119904) 119889119904 (23)





minus 1198862

1)

(24)

Using the formula

119871minus1


= ℎ1(119910 119905) =

119910radicRe 120582119890minus1199102Re1205824119905

2119905radic120587119905

(25)

we obtain

119891nsM (119910 119905) = [ℎ1(119910 119905)

+ 1198861int

119905

0

ℎ1(119910 119911)

119911


1198681(1198861

radic1199052minus 1199112) 119889119911]

sdot 119890minus1198861119905

=

119910radicRe 1205822119905radic120587119905

119890minus1199102Re1205824119905minus119886

1119905

+ 1198861119890minus1198861119905

int

119905

0

119910radicRe 1205822119911radic120587119911

119890minus1199102Re1205824119911

sdot

119911


1198681(1198861

radic1199052minus 1199112) 119889119911

(26)


119906nsM (119910 119905) = (119891 lowast 119892) (119905) = int

119905

0

119891nsM (119910 119904) 119892 (119905 minus 119904) 119889119904

=

2

radic120587

int

infin


119892(119905

minus

120582Re1199102

41199112)119889119911 + 119886

1int

119905

0

int

119904

0

119892 (119905 minus 119904)

sdot

119910radicRe 1205822119911radic120587119911

119890minus1199102Re1205824119911minus119886

1119904

sdot

119911


1198681(1198861

radic1199042minus 1199112) 119889119904 119889119911

(27)



1198721=

119887

radic120587

int

infin

0

119890minus1199102Re4119909minus119909120582

119903119889119909

radic119909

=

119887

radic120587

int

infin

0

119890(minus1199102Re120582

119903)41199112minus1199112

2radic120582119903119889119911

= 119887radic120582119903

2

radic120587

int

infin

0


119903)241199112

119889119911

= 119887radic120582119903

2

radic120587

sdot

radic120587

2


119903= 119887radic120582


119903

(28)




119903119892 (119905) minus 119887

2

119890119887119910radicRe

119892 (119905)

sdot int

infin

0

1198901198872119909minus119909120582


+ 119887radic119909)119889119909 +

1

120582119903

sdot int

infin

0

[

119887119890minus1199102Re4119909

radic120587119909

minus 1198872

119890119910119887radicRe+1198872119909Erfc(


+ 119887radic119909)]

sdot 119890(minus119909+119905)120582

119903radic

119909

119905

1198681(

2

120582119903

radic119909119905)119889119909

(29)

or



119903119892 (119905)

+

119890minus119905120582119903

radic120582119903119905

119892 (119905) times int

infin

0

[

119887119890minus1199102Re4119909minus119909120582

119903

radic120587119909

minus 1198872

119890119887119910radicRe+1198872119909minus119909120582


+ 119887radic119909)]

sdot radic1199091198681(

2

radic120582119903

radic119909119905)119889119909

(30)



119903119892 (119905) +

119910radicRe2120582119903radic120587

sdot int

infin

0

int

119905

0

119892 (119905 minus 119904)

119909radic119904

119890minus1199102Re4119909minus119909120582

119903minus1199041205821199031198681(

2

120582119903

sdot radic119909119904) 119889119904 119889119909

(31)



field

119906sN (119910 119905) = int

119905

0

119892 (119905 minus 120591) VsN (119910 120591) 119889120591

119906nsN (119910 119905) = int

119905

0

119892 (119905 minus 120591) VnsN (119910 120591) 119889120591

(32)

where

VsN (119910 119905) =1

120573radicRe1

radic120587119905


) minus (

1

120573radicRe)

2

sdot exp(119910

120573

+

119905

1205732Re

)


+

radic119905

120573radicRe)

(33)




) (34)













0 2 4 6 8

y

0 2 4 6 8

y

0 2 4 6 8

y

1

08

06

04

02

0

1

08

06

04

02

0

Velo

city

1

08

06

04

02

0

t = 5

t = 3

t = 15

minus02







u(yt)

Velo

cityu(yt)

Velo

cityu(yt)








1

08

06

04

02

0

1

1

08

06

04

02

0

0

0 2 4 6 8

y

0 2 4 6 8

y

0 2 4 6 8

y

Re = 05

Re = 25

Re = 10


t =

t =

t =


t =

t =

t =



t = 05 slip

minus02

08

06

04

02

minus02

Velo

cityu(yt)

Velo

cityu(yt)

Velo

cityu(yt)

20 slip50 slip

05 slip

20 slip50 slip

05 slip






15

1

05

0

minus05

Velo

cityu(yt)

15

1

05

0

minus05

Velo

cityu(yt)

15

1

05

0

minus05

Velo

cityu(yt)

t = 05

t = 075

t = 155

0 1 2

y

3

0 1 2

y

3

0 1 2

y

3








119903= 035


Competing Interests


References

















2

4

and 120574 gt 1205822

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119865sM (119910 119902)

=


119887radic2 + radic(119902 + 12120582)2

minus (12120582)2

= 1198871exp


minus 1198862

1)

1198871

+ radic(119902 + 1198861)2

minus 1198862

1= 1198651(119910radic119908

1(119902))

(21)


1(119902) = (119902 + 119886

1)2

minus 1198862

1 1198871=

119887radic120582 and 1198861= 12120582


119891sm (119910 119905) = [1198911(119910 119905)

+ 1198861int

119905

0

1198911(1199101

radic1199052minus 1199062) 1198681(119886 119906) 119889119906] 119890

minus1198861119905

sdot

1198871119890minus1199102Re1205824119905minus119886

1119905

radic120587119905

minus 1198872

11198901199101198871radicRe120582+1198872

1119905minus1198861119905Erfc(

119910radicRe 1205822radic119905

+ 119887radic119905)

+ 1198861119890minus1198861119905

int

119905

0

[

1198871119890minus1199102Re1205824119911minus119886

1119911

radic120587119911

minus 1198872

11198901199101198871radicRe120582+1198872

1minus1198861119911Erfc(

119910radicRe 1205822radic119911

+ 1198871radic119911)]

sdot

119911


1198681(1198861

radic1199052minus 1199112) 119889119911

(22)


119906sM (119910 119905) = (119891 lowast 119892) (119905) = int

119905

0

119891SM (119910 119904) 119892 (119905 minus 119904) 119889119904 (23)





minus 1198862

1)

(24)

Using the formula

119871minus1


= ℎ1(119910 119905) =

119910radicRe 120582119890minus1199102Re1205824119905

2119905radic120587119905

(25)

we obtain

119891nsM (119910 119905) = [ℎ1(119910 119905)

+ 1198861int

119905

0

ℎ1(119910 119911)

119911


1198681(1198861

radic1199052minus 1199112) 119889119911]

sdot 119890minus1198861119905

=

119910radicRe 1205822119905radic120587119905

119890minus1199102Re1205824119905minus119886

1119905

+ 1198861119890minus1198861119905

int

119905

0

119910radicRe 1205822119911radic120587119911

119890minus1199102Re1205824119911

sdot

119911


1198681(1198861

radic1199052minus 1199112) 119889119911

(26)


119906nsM (119910 119905) = (119891 lowast 119892) (119905) = int

119905

0

119891nsM (119910 119904) 119892 (119905 minus 119904) 119889119904

=

2

radic120587

int

infin


119892(119905

minus

120582Re1199102

41199112)119889119911 + 119886

1int

119905

0

int

119904

0

119892 (119905 minus 119904)

sdot

119910radicRe 1205822119911radic120587119911

119890minus1199102Re1205824119911minus119886

1119904

sdot

119911


1198681(1198861

radic1199042minus 1199112) 119889119904 119889119911

(27)



1198721=

119887

radic120587

int

infin

0

119890minus1199102Re4119909minus119909120582

119903119889119909

radic119909

=

119887

radic120587

int

infin

0

119890(minus1199102Re120582

119903)41199112minus1199112

2radic120582119903119889119911

= 119887radic120582119903

2

radic120587

int

infin

0


119903)241199112

119889119911

= 119887radic120582119903

2

radic120587

sdot

radic120587

2


119903= 119887radic120582


119903

(28)




119903119892 (119905) minus 119887

2

119890119887119910radicRe

119892 (119905)

sdot int

infin

0

1198901198872119909minus119909120582


+ 119887radic119909)119889119909 +

1

120582119903

sdot int

infin

0

[

119887119890minus1199102Re4119909

radic120587119909

minus 1198872

119890119910119887radicRe+1198872119909Erfc(


+ 119887radic119909)]

sdot 119890(minus119909+119905)120582

119903radic

119909

119905

1198681(

2

120582119903

radic119909119905)119889119909

(29)

or



119903119892 (119905)

+

119890minus119905120582119903

radic120582119903119905

119892 (119905) times int

infin

0

[

119887119890minus1199102Re4119909minus119909120582

119903

radic120587119909

minus 1198872

119890119887119910radicRe+1198872119909minus119909120582


+ 119887radic119909)]

sdot radic1199091198681(

2

radic120582119903

radic119909119905)119889119909

(30)



119903119892 (119905) +

119910radicRe2120582119903radic120587

sdot int

infin

0

int

119905

0

119892 (119905 minus 119904)

119909radic119904

119890minus1199102Re4119909minus119909120582

119903minus1199041205821199031198681(

2

120582119903

sdot radic119909119904) 119889119904 119889119909

(31)



field

119906sN (119910 119905) = int

119905

0

119892 (119905 minus 120591) VsN (119910 120591) 119889120591

119906nsN (119910 119905) = int

119905

0

119892 (119905 minus 120591) VnsN (119910 120591) 119889120591

(32)

where

VsN (119910 119905) =1

120573radicRe1

radic120587119905


) minus (

1

120573radicRe)

2

sdot exp(119910

120573

+

119905

1205732Re

)


+

radic119905

120573radicRe)

(33)




) (34)













0 2 4 6 8

y

0 2 4 6 8

y

0 2 4 6 8

y

1

08

06

04

02

0

1

08

06

04

02

0

Velo

city

1

08

06

04

02

0

t = 5

t = 3

t = 15

minus02







u(yt)

Velo

cityu(yt)

Velo

cityu(yt)








1

08

06

04

02

0

1

1

08

06

04

02

0

0

0 2 4 6 8

y

0 2 4 6 8

y

0 2 4 6 8

y

Re = 05

Re = 25

Re = 10


t =

t =

t =


t =

t =

t =



t = 05 slip

minus02

08

06

04

02

minus02

Velo

cityu(yt)

Velo

cityu(yt)

Velo

cityu(yt)

20 slip50 slip

05 slip

20 slip50 slip

05 slip






15

1

05

0

minus05

Velo

cityu(yt)

15

1

05

0

minus05

Velo

cityu(yt)

15

1

05

0

minus05

Velo

cityu(yt)

t = 05

t = 075

t = 155

0 1 2

y

3

0 1 2

y

3

0 1 2

y

3








119903= 035


Competing Interests


References

















2

4

and 120574 gt 1205822

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119903119892 (119905) minus 119887

2

119890119887119910radicRe

119892 (119905)

sdot int

infin

0

1198901198872119909minus119909120582


+ 119887radic119909)119889119909 +

1

120582119903

sdot int

infin

0

[

119887119890minus1199102Re4119909

radic120587119909

minus 1198872

119890119910119887radicRe+1198872119909Erfc(


+ 119887radic119909)]

sdot 119890(minus119909+119905)120582

119903radic

119909

119905

1198681(

2

120582119903

radic119909119905)119889119909

(29)

or



119903119892 (119905)

+

119890minus119905120582119903

radic120582119903119905

119892 (119905) times int

infin

0

[

119887119890minus1199102Re4119909minus119909120582

119903

radic120587119909

minus 1198872

119890119887119910radicRe+1198872119909minus119909120582


+ 119887radic119909)]

sdot radic1199091198681(

2

radic120582119903

radic119909119905)119889119909

(30)



119903119892 (119905) +

119910radicRe2120582119903radic120587

sdot int

infin

0

int

119905

0

119892 (119905 minus 119904)

119909radic119904

119890minus1199102Re4119909minus119909120582

119903minus1199041205821199031198681(

2

120582119903

sdot radic119909119904) 119889119904 119889119909

(31)



field

119906sN (119910 119905) = int

119905

0

119892 (119905 minus 120591) VsN (119910 120591) 119889120591

119906nsN (119910 119905) = int

119905

0

119892 (119905 minus 120591) VnsN (119910 120591) 119889120591

(32)

where

VsN (119910 119905) =1

120573radicRe1

radic120587119905


) minus (

1

120573radicRe)

2

sdot exp(119910

120573

+

119905

1205732Re

)


+

radic119905

120573radicRe)

(33)




) (34)













0 2 4 6 8

y

0 2 4 6 8

y

0 2 4 6 8

y

1

08

06

04

02

0

1

08

06

04

02

0

Velo

city

1

08

06

04

02

0

t = 5

t = 3

t = 15

minus02







u(yt)

Velo

cityu(yt)

Velo

cityu(yt)








1

08

06

04

02

0

1

1

08

06

04

02

0

0

0 2 4 6 8

y

0 2 4 6 8

y

0 2 4 6 8

y

Re = 05

Re = 25

Re = 10


t =

t =

t =


t =

t =

t =



t = 05 slip

minus02

08

06

04

02

minus02

Velo

cityu(yt)

Velo

cityu(yt)

Velo

cityu(yt)

20 slip50 slip

05 slip

20 slip50 slip

05 slip






15

1

05

0

minus05

Velo

cityu(yt)

15

1

05

0

minus05

Velo

cityu(yt)

15

1

05

0

minus05

Velo

cityu(yt)

t = 05

t = 075

t = 155

0 1 2

y

3

0 1 2

y

3

0 1 2

y

3








119903= 035


Competing Interests


References

















2

4

and 120574 gt 1205822

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8

y

0 2 4 6 8

y

0 2 4 6 8

y

1

08

06

04

02

0

1

08

06

04

02

0

Velo

city

1

08

06

04

02

0

t = 5

t = 3

t = 15

minus02







u(yt)

Velo

cityu(yt)

Velo

cityu(yt)








1

08

06

04

02

0

1

1

08

06

04

02

0

0

0 2 4 6 8

y

0 2 4 6 8

y

0 2 4 6 8

y

Re = 05

Re = 25

Re = 10


t =

t =

t =


t =

t =

t =



t = 05 slip

minus02

08

06

04

02

minus02

Velo

cityu(yt)

Velo

cityu(yt)

Velo

cityu(yt)

20 slip50 slip

05 slip

20 slip50 slip

05 slip






15

1

05

0

minus05

Velo

cityu(yt)

15

1

05

0

minus05

Velo

cityu(yt)

15

1

05

0

minus05

Velo

cityu(yt)

t = 05

t = 075

t = 155

0 1 2

y

3

0 1 2

y

3

0 1 2

y

3








119903= 035


Competing Interests


References

















2

4

and 120574 gt 1205822

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

08

06

04

02

0

1

1

08

06

04

02

0

0

0 2 4 6 8

y

0 2 4 6 8

y

0 2 4 6 8

y

Re = 05

Re = 25

Re = 10


t =

t =

t =


t =

t =

t =



t = 05 slip

minus02

08

06

04

02

minus02

Velo

cityu(yt)

Velo

cityu(yt)

Velo

cityu(yt)

20 slip50 slip

05 slip

20 slip50 slip

05 slip






15

1

05

0

minus05

Velo

cityu(yt)

15

1

05

0

minus05

Velo

cityu(yt)

15

1

05

0

minus05

Velo

cityu(yt)

t = 05

t = 075

t = 155

0 1 2

y

3

0 1 2

y

3

0 1 2

y

3








119903= 035


Competing Interests


References

















2

4

and 120574 gt 1205822

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










15

1

05

0

minus05

Velo

cityu(yt)

15

1

05

0

minus05

Velo

cityu(yt)

15

1

05

0

minus05

Velo

cityu(yt)

t = 05

t = 075

t = 155

0 1 2

y

3

0 1 2

y

3

0 1 2

y

3








119903= 035


Competing Interests


References

















2

4

and 120574 gt 1205822

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Competing Interests


References

















2

4

and 120574 gt 1205822

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Flows with Slip of Oldroyd-B Fluids over