Nonlinear Analysis: Real World Applications 10 (2009) 3190–3195

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Nonlinear Analysis: Real World Applications

journal homepage: www.elsevier.com/locate/nonrwa

The Rayleigh–Stokes problem for an edge in a viscoelastic fluid with afractional derivative modelMasood KhanDepartment of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan

a r t i c l e i n f o

Article history:Received 13 February 2008Accepted 8 October 2008

Keywords:Rayleigh–Stokes problemGeneralized Oldroyd-B fluidExact solutionsFractional derivative model

a b s t r a c t

This paper investigates the exact analytic solutions for the Rayleigh–Stokes problem foran edge in a generalized Oldroyd-B fluid. This paper employs the fractional calculusapproach to study the flows in an Oldroyd-B fluid. The velocity field corresponding to anincompressible generalized Oldroyd-B fluid with a fractional derivative model within aninfinite edge is determined using Fourier sine and Laplace transforms. Two characteristicexamples: (i) flow due to an impulsive motion of edge, and (ii) flow due to a uniformlyaccelerated edge are considered. The solutions that have been obtained reduce to theknown solutions of an Oldroyd-B fluid by setting α = β = 1. Moreover, the similarsolutions for Maxwell and second grade fluids with fractional derivative models and thosefor the ordinary models appear as the limiting cases of the presented solutions.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

There are many fluids commonly used in industry such as polymer solutions, suspensions, blood, paints, certain oilsand greases exhibiting complex rheological behaviour. The use of viscoelastic fluids in different technological processeshas attracted considerable attention in recent years because of the possibility of a significant reduction in the turbulentfriction in flows of dilute solutions of polymers. The study of the combined effects of inertia and viscoelasticity is importantbecause of their significance in the phenomenon of turbulent drag reduction [1]. This phenomenon, due to a small amountof certain high molecular weight compounds, has been studied extensively, since its discovery by Toms [2]. Some recentinvestigations [3–7] involving viscoelastic fluids have been made. One of the most important classes for dilute solutions ofpolymers is the Oldroyd-B fluid. Thus, the flows of Oldroyd-B fluid have been studied in much detail, more than most othernon-Newtonian fluid models, and in complicated flow geometries [8–15]. Hence, the exact solutions for the flows of suchfluids are most welcome provided they correspond to physically realistic situations, as they serve a dual purpose. Firstly,they provide a solution to a flow that has technical relevance and secondly, these solutions can be used as tests to verifynumerical schemes that are developed for studyingmore complex flow problems. It is, therefore, desirable to seek the exactsolutions of the equations characterizing the flows of viscoelastic fluids.This article will investigate the exact solutions for the Rayleigh–Stokes problem for an edge in an Oldroyd-B fluid

with a fractional derivative model. For numerous fluids between elastic and viscous materials the fractional constitutiverelationship model has an advantage over the customary constitutive relationship model. The constitutive equations withfractional derivative have been proved to be a valuable tool for handling viscoelastic properties [16,17]. Very little effort [18–26] has so far been made to discuss the flows of viscoelastic fluids with a fractional derivative model. In this paper, theexplicit analytic expressions for the velocity field are constructed using Fourier sine and Laplace transforms. Some classicaland previous results can be regarded as particular cases of the presented results.

E-mail addresses:mkhan@qau.edu.pk, mkhan_21@yahoo.com.

1468-1218/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2008.10.002

M. Khan / Nonlinear Analysis: Real World Applications 10 (2009) 3190–3195 3191

2. Governing equations

The Cauchy stress tensor T for an incompressible Oldroyd-B fluid with a fractional derivative model is given by

T = −pI+ S,(1+ λDαt

)S = µ

(1+ λr D

βt

)A, (1)

where p is the scalar pressure, I the identity tensor, µ the dynamic viscosity, S the extra tensor, A = L + LT the firstRivlin–Ericksen tensor with L the velocity gradient, λ and λr (<λ) are the relaxation and retardation times, respectively,α and β are fractional calculus parameters such that 0 ≤ α ≤ β ≤ 1 and Dpt is the upper convected fractional timederivative defined by

Dpt S = Dpt S+ (V ·∇) S− LS− SLT, (2)

in which Dpt(= ∂

pt)is the fractional differentiation operator defined as [27]

Dpt [f (t)] =1

0(1− p)ddt

∫ t

0

f (τ )(t − τ)p

dτ , 0 ≤ p ≤ 1 (3)

when 0(·) denotes the Gamma function.For the following problems of two-dimensional flow, we consider the velocity field of the form

V = [u(y, z, t), 0, 0] , (4)

where u(y, z, t) is the velocity in the x-coordinate direction.Inserting Eq. (4), the constraint of incompressibility is identically satisfied and the balance of linear momentum, in the

absence of body forces and a pressure gradient in the flow direction, yields the following governing equation [28](1+ λDαt

) ∂u (y, z, t)∂t

= ν(1+ λrD

βt

) [ ∂2∂y2+∂2

∂z2

]u (y, z, t) ; y, z, t > 0 (5)

where ν = µ/ρ is the kinematic viscosity and ρ the constant density of the fluid.

3. Flow due to an impulsive motion of edge

Consider the flow of an incompressible generalized Oldroyd-B fluid occupying the space of the first quadrant of arectangular edge (∞ < x < ∞; y, z ≥ 0). Initially, the fluid and the edge are at rest. After time t = 0, the infinitelyextended edge is impulsively brought the constant velocity U in the x-direction and induces the motion to the fluid. Theflow is governed by Eq. (5) with the initial and the boundary conditions given by

u (y, z, 0) =∂u (y, z, 0)

∂t= 0; y, z > 0, (6)

u (0, z, t) = u (y, 0, t) = U; t > 0, (7)

u (y, z, t) ,∂u (y, z, t)

∂y,∂u (y, z, t)

∂z→ 0 as y2 + z2 →∞, t > 0. (8)

In order to solve the above problem, the Fourier sine transform and fractional Laplace transform are used. Consequently,multiplying Eq. (5) by (2/π) sin (ξy) sin (ηy) , integrating with respect to y and z from 0 to∞ and keeping in mind theconditions (7) and (8) we get[

1+ λDαt] ∂us (ξ , η, t)

∂t+ ν

(ξ 2 + η2

) [1+ λrD

βt

]us (ξ , η, t) =

2νVπ

(1+ λrD

βt

) ξ 2 + η2ξη

, (9)

where the double Fourier sine transform us (ξ , η, t) of u (y, z, t) have to satisfy the initial conditions

us (ξ , η, 0) =∂us (ξ , η, 0)

∂t= 0; ξ, η > 0. (10)

Denoting by

us (ξ , η, s) =∫∞

0us (ξ , η, t) e−stdt, (11)

the image function of us (ξ , η, t), applying the Laplace transform to Eq. (9), using the Laplace transform formula forsequential fractional derivative [27] subject to the conditions (10) we obtain

us (ξ , η, s) =2Vπ

ξ 2 + η2

ξη

ν + αrsign (1− β) sβ

s(s+ λsα+1 + ν

(ξ 2 + η2

)+ αr

(ξ 2 + η2

)sβ) , (12)

where αr = νλr and sign(·) is the signum function.

3192 M. Khan / Nonlinear Analysis: Real World Applications 10 (2009) 3190–3195

In order to findus (ξ , η, t) = L−1 {us (ξ , η, s)} and to avoid the burdensome calculations of residues and contour integrals,we apply the discrete Laplace transform method. First, we rewrite Eq. (12) in series form

us (ξ , η, s) =2Vπ

ξ 2 + η2

ξη

∞∑k=0

m+l=k∑m,l≥0

(−1)k λm

m!l!

[ν(ξ 2 + η2

)]l×

{ν

k!sm(α+1)−β(k+1)−1(s1−β + αr

(ξ 2 + η2

))k+1 + βr k!sm(α+1)−βk−1(s1−β + αr

(ξ 2 + η2

))k+1}, (13)

where βr = αrsign (1− β) .Now applying the discrete inverse Laplace transform to Eq. (13) to get

us (ξ , η, t) =2Vπ

ξ 2 + η2

ξη

∞∑k=0

m+l=k∑m,l≥0

(−1)k λm

m!l!

[ν(ξ 2 + η2

)]l×

{νtk+1−m(α+1)E(k)1−β,βk+2−m(α+1)

(−αr

(ξ 2 + η2

)t1−β

)+ βr tk+1−m(α+1)−βE

(k)1−β,β(k−1)+2−m(α+1)

(−αr

(ξ 2 + η2

)t1−β

)}, (14)

whence

Eλ,µ (z) =∞∑n=0

zn

0(λn+ µ), λ, µ > 0, (15)

represents the generalized Mittag-Leffler function with

E(k)λ,µ (z) =dk

dzkEλ,µ (z) =

∞∑n=0

(n+ k)!zn

n!0(λn+ λk+ µ), (16)

and 0(·) is the Gamma function.In obtaining Eq. (14) we used the following property of the generalized Mittag-Leffler function [21]

L−1{

k!sλ−µ(sλ ∓ c

)k+1}= tλk+µ−1E(k)λ,µ

(±ctλ

),(Re(s) > |c|1/λ

). (17)

Finally, inverting Eq. (14) by means of the Fourier sine formula [29], we get following expression for the velocity field

u (y, z, t) =4Vνπ2

∫∞

0

sin (ξy)ξ

∫∞

0

sin (ηy)η

∞∑k=0

m+l=k∑m,l≥0

(−1)k λm

m!l!

[ν(ξ 2 + η2

)]l+1×

{νtk+1−m(α+1)E(k)1−β,βk+2−m(α+1)

(−αr

(ξ 2 + η2

)t1−β

)+ βr tk+1−m(α+1)−βE

(k)1−β,β(k−1)+2−m(α+1)

(−αr

(ξ 2 + η2

)t1−β

)}dξdη. (18)

Looking at the velocity field (18), it seems that u (y, z, t) does not satisfy the boundary conditions (7). In order to do awaywith this inconvenience, we have also determined a new but equivalent expression for u (y, z, t) by writing us (ξ , η, s) fromEq. (12) under the suitable form

us (ξ , η, s) =2Vπξη

1s

{1−

s+ λsα+1 + αr(ξ 2 + η2

)sβ − βr

(ξ 2 + η2

)sβ

s+ λsα+1 + ν(ξ 2 + η2

)+ αr

(ξ 2 + η2

)sβ

}. (19)

Following the same way as before, of course, this solution clearly satisfies the boundary conditions (7) but it is large enoughto be included here.

M. Khan / Nonlinear Analysis: Real World Applications 10 (2009) 3190–3195 3193

4. Limiting cases

We will now consider the limiting cases wherein the solution reduces to that for specific fluids.

4.1 In a special case, when λr → 0, Eq. (18) reduces to

u (y, z, t) =4Vπ2

∫∞

0

sin (ξy)ξ

∫∞

0

sin (ηy)η

×

∞∑k=0

(−1)k

k!

[ν(ξ 2 + η2

)λ

]k+1t(k+1)(α+1)E(k)α,k+2+α

(−

(tλ

)α) dξdη, (20)

which represents the velocity field corresponding to the flow of a fractional Maxwell fluid performing the samemotion.4.2 For α = 1, Eq. (20) can be simplified to the similar solution for a Maxwell fluid

u (y, z, t) =4Vπ2

∫∞

0

sin (ξy)ξ

∫∞

0

sin (ηy)η

×

∞∑k=0

(−1)k

k!

[ν(ξ 2 + η2

)λ

]k+1t2k+2E(k)1,k+3

(−tλ

) dξdη. (21)

4.3 By letting now λ→ 0 in Eq. (18), we obtain the solution corresponding to a generalized second grade fluid

u (y, z, t) =4Vνπ2

∫∞

0

sin (ξy)ξ

∫∞

0

sin (ηy)η

∞∑k=0

(−1)k

k!

[ν(ξ 2 + η2

)]k+1×

{νtk+1E(k)1−β,βk+2

(−αr

(ξ 2 + η2

)t1−β

)+ βr tk+1−βE

(k)1−β,β(k−1)+2

(−αr

(ξ 2 + η2

)t1−β

)}dξdη. (22)

4.4 For β = 1, Eq. (22) can be reduced to

u (y, z, t) = V

{1−

4π2

∫∞

0

sin (ξy)ξ

∫∞

0

sin (ηy)η

exp

(−

ν(ξ 2 + η2

)1+ αr

(ξ 2 + η2

) t) dξdη} , (23)

which represents the solution corresponding to a second grade fluid, the velocity field (23) being identical with thatresulting from Eq. (24) in Ref. [30].

5. Flow due to a uniformly accelerated edge

Here the fluid and the rectangular edge are considered at rest for t ≤ 0 and the infinitely extended edge starts moving inthe x-direction at t > 0. The rectangular edge accelerate with a velocity of uniform acceleration A. The governing equation,the initial and a part of the boundary conditions are same. For uniform acceleration, instead of the conditions (7) we use thecondition

u (0, z, t) = u (y, 0, t) = At; t > 0. (24)

Adopting the same methodology as before the expression for the velocity field in case of uniformly accelerated edge isgiven by

u (y, z, t) =4Aνπ2

∫∞

0

sin (ξy)ξ

∫∞

0

sin (ηy)η

∞∑k=0

m+l=k∑m,l≥0

(−1)k λm

m!l!

[ν(ξ 2 + η2

)]l+1×

{νtk+2−m(α+1)E(k)1−β,βk+3−m(α+1)

(−αr

(ξ 2 + η2

)t1−β

)+ αr tk+2−m(α+1)−βE

(k)1−β,β(k−1)+3−m(α+1)

(−αr

(ξ 2 + η2

)t1−β

)}dξdη. (25)

3194 M. Khan / Nonlinear Analysis: Real World Applications 10 (2009) 3190–3195

6. Limiting cases

Let us deduce the limiting cases of the solution given be Eq. (25).

6.1 Making λr → 0 into Eq. (25), we find similar solution for a fractional Maxwell fluid given by

u (y, z, t) =4Aπ2

∫∞

0

sin (ξy)ξ

∫∞

0

sin (ηy)η

×

∞∑k=0

(−1)k

k!

[ν(ξ 2 + η2

)λ

]k+1t(k+1)(α+1)+1E(k)α,k+3+α

(−

(tλ

)α) dξdη. (26)

6.2 By setting α = 1, Eq. (26) becomes

u (y, z, t) =4Aπ2

∫∞

0

sin (ξy)ξ

∫∞

0

sin (ηy)η

×

∞∑k=0

(−1)k

k!

[ν(ξ 2 + η2

)λ

]k+1t2k+3E(k)1,k+4

(−tλ

) dξdη. (27)

This corresponds to the similar solution for a Maxwell fluid.6.3 Now letting λ→ 0 into Eq. (25) we find the similar solution for a generalized second grade fluid

u (y, z, t) =4Aνπ2

∫∞

0

sin (ξy)ξ

∫∞

0

sin (ηy)η

∞∑k=0

(−1)k

k!

[ν(ξ 2 + η2

)]k+1×

{νtk+2E(k)1−β,βk+3

(−αr

(ξ 2 + η2

)t1−β

)+ αr tk+2−βE

(k)1−β,β(k−1)+3

(−αr

(ξ 2 + η2

)t1−β

)}dξdη. (28)

6.4 Finally, lengthy but straight forward computations allow us to write Eq. (28) for β = 1 in the simpler form

u (y, z, t) = At −4Aνπ2

∫∞

0

sin (ξy)ξ

∫∞

0

sin (ηy)η

1ξ 2 + η2

{1− exp

(−

ν(ξ 2 + η2

)1+ αr

(ξ 2 + η2

) t)} dξdη, (29)

which corresponds to the solution for a second grade fluid and is in accordance with Eq. (4.1) from Ref. [31].

7. Concluding remarks

The aim of this note is to establish the closed form expressions for the velocity field to the unsteady flows of a generalizedOldroyd-B fluidwithin an edge. For the description of such a viscoelastic fluid, fractional calculus approach in the constitutiverelationship model is used. The motion of the fluid is due to an impulsive motion of edge and a uniformly accelerated edge.The expressions for the velocity field are obtained using Fourier sine transform and discrete Laplace transform of sequentialfractional derivative. The solutions that have been obtained are presented under integral and series form in terms of thederivatives of generalized Mittag-Leffler functions. The corresponding solutions for fractional Maxwell and generalizedsecond grade fluids as well as those for ordinary Oldroyd-B, Maxwell and second grade fluids can be obtained as limitingcases of the presented solutions. Moreover, a comparison of the analytical results with previously published results for somespecial cases is made and found to have excellent agreement.

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