Exact solutions for the unsteady axial flow of Non-Newtonian fluids through a circular cylinder

Commun Nonlinear Sci Numer Simulat 16 (2011) 226–238

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Commun Nonlinear Sci Numer Simulat

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Exact solutions for the unsteady axial flow of Non-Newtonian fluidsthrough a circular cylinder

Imran Siddique *, Zamra SajidDepartment of Mathematical Sciences, COMSATS Institute of Information Technology, Lahore, Pakistan

a r t i c l e i n f o

Article history:Received 26 November 2009Received in revised form 16 March 2010Accepted 16 March 2010Available online 21 March 2010

Keywords:Fractional calculusGeneralized Oldroyd-B fluidVelocity fieldExact solutionsLongitudinal shear stress

1007-5704/$ - see front matter � 2010 Elsevier B.Vdoi:10.1016/j.cnsns.2010.03.010

* Corresponding author. Tel.: +92 334 4140711.E-mail address: [email protected] (I. Sid

a b s t r a c t

The velocity field and the shear stress corresponding to the motion of a generalized Old-royd-B fluid due to an infinite circular cylinder subject to a longitudinal time-dependentshear stress are established by means of the Laplace and finite Hankel transforms. Theexact solutions, written under series form, can be easily specialized to give the similar solu-tions for generalized Maxwell and generalized second grade fluids as well as for ordinaryOldroyd-B, Maxwell, second grade and Newtonian fluids performing the same motion.Finally, some characteristics of the motion as well as the influence of the material param-eters on the behavior of the fluid are shown by graphical illustrations.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Non-Newtonian fluids, such as paints, grease, oils, blood, liquid polymers, glycerin etc., are frequently encountered inmany disciplinary fields including chemical engineering, foodstuff, biomedicine etc., and also are closely related to manyindustrial processes. Typical non-Newtonian characteristics include shear-thinning or shear-thickening, stress relaxation,normal stress difference and many more. Because of these complexities several models for non-Newtonian fluids have beenproposed in the literature. In the category of non-Newtonian fluids, the fluids of differential type have acquired special statusas well as much controversy [1]. These fluids cannot describe the influence of relaxation and retardation times. Amongstnon-Newtonian fluid models that are capable of describing these effects are the rate-type [2] models. One of the rate-typefluid models is the Oldroyd-B model [3]. Oldroyd-B fluid is one which stores energy like a linearized elastic solid, however itsdissipation depends upon two dissipative mechanisms meaning that they arise from a mixture of two viscous fluids. Oldroyddeveloped a systematic procedure for developing rate-type viscoelastic fluid models. He was careful to build into hisframework the invariance requirements that the models ought to meet, but did not concern itself with thermodynamicalissues. Rajagopal and Srinivasa [4] have elaborated a thermodynamic framework for systematically developing rate-typeviscoelastic fluid models. This framework has its basis on a proper choice of thermodynamicity, it means that body storesand dissipates energy at the same time. Within the context of such a theory, they developed a generalization of theOldroyd-B model and this reduces to the classical Oldroyd-B model when linearized appropriately. The Oldroyd-B modelis characterized by three material constants, which are viscosity, relaxation time and retardation time. Also, this model suc-cessfully describe the response of dilute polymeric fluids [5,6]. Some recent investigations describing the flows of Oldroyd-Bfluid can be found in [7–13].

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I. Siddique, Z. Sajid / Commun Nonlinear Sci Numer Simulat 16 (2011) 226–238 227

Much of the developments in the theory of viscoelastic flows are mainly restricted to the formulations of basic equationsand constitutive models. There are many fluids such as second grade, Maxwell, Oldroyd-B fluid etc. which lie in the categoryof viscoelastic fluid, and some recent attempts regarding the flow of a viscoelastic fluid through porous media are presentedin [14,15]. In order to describe the viscoelasticity [16], the fractional calculus approach is very important, which recently hasencountered much success in the description of varies problems of mechanics [16–22]. In particular, the constitutive equa-tions with fractional derivatives have been proved to be a valuable tool in handling viscoelastic properties. In general, theseequations are derived from well-known ordinary models via replacing time derivatives of stress and strain by derivatives offractional order. Thus, by introducing the fractional calculus approach into the constitutive relationship, a more appropriatemodel is presented for fluid materials between viscous and elastic. Moreover, a very good fit of the experimental data isachieved when the constitutive equation with fractional derivative is used. Song and Jiang [23] achieved satisfactory resultswhen applying the constitutive equation with fractional derivative to the experimental data of viscoelasticity. Tan et al.[24,25] applied fractional derivative to the constitutive relationship models of Maxwell viscoelastic and second grade fluidsand studied some unsteady flows. So far very little efforts [26–33] have been made to discuss the flows of viscoelastic fluidswith fractional calculus approach.

In this paper, we study the unsteady longitudinal flow of generalized Oldroyd-B fluid through an infinite circular cylinderof radius R. The flow of the fluid is generated by the cylinder that is pulled by a time-dependent shear stress along its axis.The velocity fields and the resulting shear stress are determined by means of Laplace and finite Hankel transforms and arepresented under integral and series forms in term of the generalized G-functions [34]. It is worthy to point out that solutionsthat have been obtained satisfy the governing differential equation and all imposed initial and boundary conditions as well.The solutions corresponding to generalized Maxwell and generalized second grade fluids, as well as to ordinary Oldroyd-B,Maxwell, second grade and Newtonian fluids, performing the same motion, are also determined as special cases of our gen-eral solutions.

2. Governing equations

For the problem under consideration we shall assume the velocity field v and the extra stress S of the form [8,13]

v ¼ vðr; tÞ ¼ uðr; tÞez; S ¼ Sðr; tÞ; ð1Þ

where ez is the unit vector in the z-direction of the cylindrical coordinates system r; h; and z. For such flows, the constraintof incompressibility is automatically satisfied. Furthermore, if the fluid is at rest upto the moment time t ¼ 0, then

vðr;0Þ ¼ 0; Sðr;0Þ ¼ 0: ð2Þ

The governing equations, corresponding to such motions for Oldroyd-B fluids, are [8,13,15]

1þ k@

@t

� �sðr; tÞ ¼ l 1þ kr

@

@t

� �@uðr; tÞ@r

; ð3Þ

1þ k@

@t

� �@uðr; tÞ@t

¼ m 1þ kr@

@t

� �@2

@r2 þ1r@

@r

!uðr; tÞ; ð4Þ

where sðr; tÞ ¼ Srzðr; tÞ is the shear stress that is different of zero, l is the dynamic viscosity of the fluid, m ¼ l=q is the kine-matic viscosity of the fluid (q being its constant density), k and kr are the relaxation and retardation times, respectively.

The governing equations corresponding to an incompressible generalized Oldroyd-B fluid, performing the same motion,are obtained from Eqs. (3) and (4) by replacing the inner time derivatives with fractional differential operators Da

t andDb

t ðb P aÞ, defined by [35–37]

Dpt f ðtÞ ¼

1Cð1�pÞ

ddt

R t0

f ðuÞðt�uÞp du; 0 < p < 1;

ddt f ðtÞ; p ¼ 1;

(ð5Þ

where Cð�Þ is the gamma function. Consequently, the governing equations corresponding to our problems are [27,31]

ð1þ kDat Þsðr; tÞ ¼ lð1þ krD

bt Þ@uðr; tÞ@r

; ð6Þ

ð1þ kDat Þ@uðr; tÞ@t

¼ mð1þ krDbt Þ

@2

@r2 þ1r@

@r

!uðr; tÞ; ð7Þ

where the new material constants k and kr have the dimensions of ta and tb respectively. In some recent papers (see [32], forinstance), the authors use ka and kb

r instead of k and kr into their constitutive equations. However, for simplicity, we keep thesame notations although these material constants have different significations in Eqs. (3), (4) and (6), (7), respectively.

228 I. Siddique, Z. Sajid / Commun Nonlinear Sci Numer Simulat 16 (2011) 226–238

3. Axial flow of an Oldroyd-B fluid with fractional derivatives through a circular cylinder

Let us consider an incompressible generalized Oldroyd-B fluid at rest in an infinite circular cylinder of radius R. At timet ¼ 0þ, the cylinder is suddenly pulled with a time-dependent shear stress. Due to the shear, the fluid is gradually moved. Itsvelocity is of the form (1)1, the governing equations are given by Eqs. (6) and (7) and the appropriate initial and boundaryconditions are

uðr;0Þ ¼ @uðr;0Þ@t

¼ 0; sðr; 0Þ ¼ 0; ; r 2 ½0;RÞ; ð8Þ

ð1þ kDat ÞsðR; tÞ ¼ lð1þ krD

bt Þ@uðr; tÞ@r

��r¼R

¼ fta; t > 0; ð9Þ

where f is a constant and a P 0.Eq. (9)1 is equivalent with sðR; tÞ ¼ f

k

R t0ðt � sÞaGa;0;1ð�1=k; sÞds, therefore a time-dependent shear stress on the boundary of

the cylinder, while Eq. (9)2 is equivalent with @uðR;tÞ@r ¼

flkr

R t0ðt � sÞaGb;0;1ð�1=kr ; sÞds, where Ga;b;cð�; tÞ is the generalized G-func-

tions [34]. Eqs. (6) and (7) with the conditions (8) and (9) can be solved by several methods. Integral transform technique canbe a systematic, efficient and powerful tool. For instance, Laplace transform can be used to eliminate the time variable andthe finite Hankel transform can be employed to eliminate the special variable.

Applying the Laplace transform to Eqs. (7) and (9)2, using (8)1,2 and the formulae [36,38]

LfDat f ðtÞg ¼ qaLff ðtÞg; Lftag ¼ Cðaþ 1Þ

qaþ1 ; a > �1; ð10Þ

we obtain the following problem with boundary condition

ðqþ kqaþ1Þ�uðr; qÞ ¼ mð1þ krqbÞ @2

@r2 þ1r@

@r

!�uðr; qÞ; ð11Þ

@�uðR; qÞ@r

¼ fl

Cðaþ 1Þqaþ1

1krqb þ 1

; ð12Þ

where �uðr; qÞ ¼R1

0 uðr; tÞe�qtdt is the Laplace transform of function uðr; tÞ.In the following, let us denote by [38]

�uHðrn; qÞ ¼Z R

0r�uðr; qÞJ0ðrrnÞdr; n ¼ 1;2;3 . . . ; ð13Þ

the finite Hankel transform of �uðr; qÞ, where rn are the positive roots of the transcendental equation J1ðRrÞ ¼ 0. In the aboverelations, Jmð�Þ is the first-kind, m-order Bessel function [39,40].

Applying the Hankel transform to Eq. (11) and taking into account Eqs. (12), (A.1) and (A.2) from Appendix A, we find that

�uHðrn; qÞ ¼ �u1Hðrn; qÞ þ �u2Hðrn; qÞ; ð14Þ

where

�u1Hðrn; qÞ ¼RfJ0ðRrnÞ

lr2n

Cðaþ 1Þqaþ1

1krqb þ 1

; ð15Þ

�u2Hðrn; qÞ ¼ �RfJ0ðRrnÞ

lr2n

Cðaþ 1Þqa

1krqb þ 1

kqa þ 1kqaþ1 þ qþ mkrr2

nqb þ mr2n: ð16Þ

The inverse Hankel transform of the function �uHðrn; qÞ is, see (A.3) from Appendix A,

�uðr; qÞ ¼ fr2

2lRCðaþ 1Þ

qaþ1

1krqb þ 1

�� 2flR

X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞCðaþ 1Þ

qa

1krqb þ 1

kqa þ 1kqaþ1 þ qþ mkrr2

nqb þ mr2n: ð17Þ

In order to determine the tension sðr; tÞ, we apply the Laplace transform to Eq. (6) with the initial condition (8)3 we obtain

�sðr; qÞ ¼ lkrqb þ 1kqa þ 1

@�uðr; qÞ@r

: ð18Þ

Differentiating Eq. (17) with respect to r and using the identity (A.1)1 we find that

�sðr; qÞ ¼ frR

Cðaþ 1Þqaþ1

1kqa þ 1

þ 2fR

X1n¼1

J1ðrrnÞrnJ0ðRrnÞ

Cðaþ 1Þqa

1kqaþ1 þ qþ mkrr2

nqb þ mr2n: ð19Þ

In order to determine the inverse Laplace transforms of functions �uðr; qÞ and �sðr; qÞ, we consider the following functions


BnðqÞ ¼1

kqaþ1 þ qþ mkrr2nqb þ mr2

n¼ 1

kqaþ1 þ snqb þ qþ mr2n � qb

; ð20Þ

CnðqÞ ¼kqa þ 1

kqaþ1 þ qþ mkrr2nqb þ mr2

n¼ kqa þ 1

kqaþ1 þ snqb þ qþ mr2n � qb

; ð21Þ

where

sn ¼ 1þ mkrr2n: ð22Þ

We rewrite functions BnðqÞ and CnðqÞ under the following forms

BnðqÞ ¼1

kqb q1þa�b þ snk

� �þ 1

k q1�b þ mr2nq�b � 1

� �� ¼X1k¼0

ð�1Þk

kqb

1k q1�b þ mr2

nq�b � 1� �� k

q1þa�b þ snk

� �kþ1

¼X1k¼0

ð�1Þk

kkþ1qb

Pbþcþd¼k

b;c;dP0k!qbð1�bÞ ðmr2

nq�bÞcð�1Þdb!c!d!

q1þa�b þ snk

� �kþ1 ¼X1k¼0

Xbþcþd¼k

b;c;dP0

ð�1Þkþdk!ðmr2nÞ

c

kkþ1b!c!d!

qb�ð1þbþcÞb

q1þa�b þ snk

� �kþ1 : ð23Þ

Similarly,

CnðqÞ ¼X1k¼0

Xbþcþd¼k

b;c;dP0


c

kkþ1b!c!d!

kqaþb�ð1þbþcÞb þ qb�ð1þbþcÞb

q1þa�b þ snk

� �kþ1 : ð24Þ

The inverse Laplace transforms of functions BnðqÞ and CnðqÞ are [41], see (A.4) from Appendix A,

bnðtÞ ¼ L�1fBnðqÞg ¼X1k¼0

Xbþcþd¼k

b;c;dP0


c

kkþ1b!c!d!G1þa�b;b�ð1þbþcÞb;kþ1 �

sn

k; t

; ð25Þ

cnðtÞ ¼ L�1fCnðqÞg ¼X1k¼0

Xbþcþd¼k

b;c;dP0


c

kkþ1b!c!d!kG1þa�b;aþb�ð1þbþcÞb;kþ1 �

sn

k; t

þ G1þa�b;b�ð1þbþcÞb;kþ1 �

sn

k; t

h i: ð26Þ

In order to avoid the tedious calculations of residues and contour integrals, we apply the discrete inverse Laplace transform[32,41] to Eqs. (17) and (19), using the convolution theorem, (25), (26), (A.5) and (A.6) form Appendix A, we find the expres-sions of velocity field uðr; tÞ and shear stress sðr; tÞ under the following forms:

If a > 0 then

uðr; tÞ ¼ fr2

2lRðf1 � f4ÞðtÞ �

2flR

X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞðf2 � f4 � cnÞðtÞ; ð27Þ

sðr; tÞ ¼ frRðf1 � f3ÞðtÞ þ

2fR

X1n¼1


ðf2 � bnÞðtÞ; ð28Þ

and if a ¼ 0 then

uðr; tÞ ¼ fr2

2lR1kr

Gb;�1;1 �1kr; t

� �� 2f

lR

X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞðf4 � cnÞðtÞ; ð29Þ

sðr; tÞ ¼ frR

1k

Ga;�1;1 �1k; t

� �þ 2f

R

X1n¼1


bnðtÞ; ð30Þ

4. Limiting cases

4.1. Oldroyd-B fluid as a particular case of Oldroyd-B fluid with fractional derivatives

Making a and b! 1 into Eqs. (27)–(30), and using (A.8)–(A.10), we obtain the velocity field and the associated shearstress under the forms

if a > 0 then

uðr; tÞ ¼ fr2

2lRðf1 � a2ÞðtÞ �

2flR

X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞðf2 � a2 � hnÞðtÞ; ð31Þ

sðr; tÞ ¼ frRðf1 � a1ÞðtÞ þ

2fR

X1n¼1


ðf2 � gnÞðtÞ; ð32Þ

and if a ¼ 0 then


uðr; tÞ ¼ fr2

2lR½1� expð�t=krÞ� �

2flR

X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞða2 � hnÞðtÞ; ð33Þ

sðr; tÞ ¼ frR½1� expð�t=kÞ� þ 2f

R

X1n¼1


gnðtÞ; ð34Þ

corresponding to the ordinary Oldroyd-B fluid, performing the same motion.

4.2. Maxwell fluid with fractional derivatives as a particular case of Oldroyd-B fluid with fractional derivatives

If in all equations and calculations of paragraph 3 we make kr ! 0, then we obtain the Maxwell fluid model with frac-tional derivatives which performing the same motion.

In this case we have sn ¼ 1; F4ðqÞ ¼ 1 (see (A.5)),

BnðqÞ ¼1

kqaþ1 þ qþ mr2n¼X1k¼0

1k� mr2

n

k

� �k q�k�1

ðqa þ 1k Þ

kþ1 ;

CnðqÞ ¼kqa þ 1

kqaþ1 þ qþ mr2n¼X1k¼0

1k� mr2

n

k

� �kkqa�k�1 þ q�k�1

ðqa þ 1k Þ

kþ1 ;

respectively,

bnðtÞ ¼ L�1fBnðqÞg ¼X1k¼0

1k� mr2

n

k

� �k

Ga;�k�1;kþ1 �1k; t

� �;

cnðtÞ ¼ L�1fCnðqÞg ¼X1k¼0

� mr2n

k

� �k

Ga;a�k�1;kþ1 �1k; t

� �þ 1

kGa;�k�1;kþ1 �

1k; t

� �� :

By means of the above expressions, from Eqs. (17) and (19) we obtain the following solutions:If a > 0 then

uðr; tÞ ¼ fr2

2lRta � 2af

lR

X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞX1k¼0

� mr2n

k

� �k

�Z t

0ðt � sÞa�1 Ga;a�k�1;kþ1 �

1k; s

� �þ 1


1k; s

� �� ds; ð35Þ

sðr; tÞ ¼ frkR

Z t

0ðt � sÞaGa;0;1 �

1k; s

� �dsþ 2af

kR

X1n¼1


X1k¼0

� mr2n

k

� �k Z t

0ðt � sÞa�1Ga;�k�1;kþ1 �

1k; s

� �ds; ð36Þ

and if a ¼ 0 then

uðr; tÞ ¼ fr2

2lR� 2f

lR

X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞcnðtÞ; ð37Þ

sðr; tÞ ¼ frR

1k

Ga;�1;1 �1k; t

� �þ 2f

R

X1n¼1


bnðtÞ: ð38Þ

For a ¼ 1 and using (A.6) into Eqs. (35) and (36), the known solutions [33, Eqs. (30) and (31)]

uðr; tÞ ¼ fr2

2lRt � 2f

lR

X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞX1k¼0

� mr2n

k

� �k

� Ga;a�k�2;kþ1 �1k; t

� �þ 1


1k; t

� �� ; ð39Þ

sðr; tÞ ¼ frkR

Ga;�2;1 �1k; t

� �þ 2f

kR

X1n¼1


Ga;�k�2;kþ1 �1k; t

� �; ð40Þ

corresponding to the similar flow of generalized Maxwell fluid are recovered.

4.3. Maxwell fluid as a particular case of Maxwell fluid with fractional derivatives

Making a! 1 into Eqs. (35)–(38), we get the velocity field and the shear stress under the formsIf a > 0 then

uðr; tÞ ¼ fr2

2lRta � 2af

lR

X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞ�X1k¼0

� mr2n

k

� �k Z t

0ðt � sÞa�1 G1;�k;kþ1 �

1k; s

� �þ 1

kG1;�k�1;kþ1 �

1k; s

� �� ds; ð41Þ

sðr; tÞ ¼ frkR

Z t

0ðt � sÞa expð�s=kÞdsþ 2af

kR

X1n¼1


X1k¼0

� mr2n

k

� �k Z t

0ðt � sÞa�1G1;�k�1;kþ1 �

1k; s

� �ds; ð42Þ


and if a ¼ 0 then

uðr; tÞ ¼ fr2

2lR� 2f

lR

X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞX1k¼0

� mr2n

k

� �k

� G1;�k;kþ1 �1k; t

� �þ 1

kG1;�k�1;kþ1 �

1k; t

� �� ; ð43Þ

sðr; tÞ ¼ frR

1� expð�t=kÞ½ � þ 2fkR

X1n¼1


X1k¼0

� mr2n

k

� �k

G1;�k�1;kþ1 �1k; t

� �; ð44Þ

corresponding to the ordinary Maxwell fluid, performing the same motion.For a ¼ 1 and using (A.6) into Eqs. (41) and (42), the known solutions

uðr; tÞ ¼ fr2

2lRt � 2f

lR

X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞ�X1k¼0

� mr2n

k

� �k

G1;�k�1;kþ1 �1k; t

� �þ 1

kG1;�k�2;kþ1 �

1k; t

� �� ; ð45Þ

sðr; tÞ ¼ frR

t � kð1� expð�t=kÞ½ � þ 2fkR

X1n¼1


X1k¼0

� mr2n

k

� �k

G1;�k�2;kþ1 �1k; t

� �; ð46Þ

corresponding to the similar flow of ordinary Maxwell fluid are recovered [45, Eqs. (17) and (20)].

4.4. Newtonian fluid as a particular case of Maxwell fluid

Making the limit k! 0 into Eqs. (41)–(44), the similar solutions [42] under the formsIf a > 0 then

uðr; tÞ ¼ fr2

2lRta � 2af

lR

X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞ

Z t

0ðt � sÞa�1 expð�mr2

nsÞds; ð47Þ

sðr; tÞ ¼ frR

ta þ 2afR

X1n¼1


Z t

0ðt � sÞa�1 exp �mr2

ns� �

ds; ð48Þ

and if a ¼ 0 then

uðr; tÞ ¼ fr2

2lR� 2f

lR

X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞexpð�mr2

ntÞ; ð49Þ

sðr; tÞ ¼ frRþ 2f

R

X1n¼1


expð�mr2ntÞ; ð50Þ

corresponding to the flow of Newtonian fluid are obtained.For a ¼ 1 into Eqs. (47) and (48), the known solutions for the flow of Newtonian fluids

uðr; tÞ ¼ fr2

2lRt � 2f

mlR

X1n¼1

J0ðrrnÞr4

nJ0ðRrnÞ½1� expð�mr2

ntÞ�; ð51Þ

sðr; tÞ ¼ frR

t þ 2fmR

X1n¼1

J1ðrrnÞr3

nJ0ðRrnÞ½1� expð�mr2

ntÞ�; ð52Þ

are recovered [12,13,33,42–46].

4.5. Second grade fluid with fractional derivatives as a particular case of Oldroyd-B fluid with fractional derivatives

Making k! 0 into Eqs. (17) and (19), we obtain the second grade fluid with fractional derivatives which performing thesame motion.

Using the following notations

lkr ¼ a1; a ¼ a1=q ¼ mkr ;

Eqs. (17) and (19) become

�uðr; qÞ ¼ fr2

2RCðaþ 1Þ

qaþ1

1a1qb þ l

� 2fR

X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞCðaþ 1Þ

qa

1a1qb þ l

1qþ ar2

nqb þ mr2n; ð53Þ

�sðr; qÞ ¼ frR

Cðaþ 1Þqaþ1 þ 2f

R

X1n¼1


Cðaþ 1Þqa

1qþ ar2

nqb þ mr2n: ð54Þ

Applying the discrete inverse Laplace transform [32,33] to Eqs. (53) and (54), using (A.4), (A.5), (A.11)–(A.13) and the con-volution theorem (A.7) form the Appendix A, we find the velocity field uðr; tÞ and the shear stress sðr; tÞ under the forms


If a > 0 then

uðr; tÞ ¼ fr2

2a1R

Z t

0ðt � sÞaGb;0;1 �

ma; s

dsþ far2

2lR

Z t

0ðt � sÞa�1Gb;b�1;1 �

ma; s

ds� 2af

lR

X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞX1k¼0

ð�mr2nÞ

k

�Z t

0ðt � sÞa�1½G1�b;�b�bk;kþ1ð�ar2

n; sÞ þ ar2nG1�b;�1�bk;kþ1ð�ar2

n; sÞ�ds; ð55Þ

sðr; tÞ ¼ frR

ta þ 2afR

X1n¼1


X1k¼0

ð�mr2nÞ

kZ t

0ðt � sÞa�1G1�b;�b�bk;kþ1ð�ar2

n; sÞds; ð56Þ

and if a ¼ 0 then

uðr; tÞ ¼ fr2

2a1RGb;�1;1 �

ma; t

þ fr2

2lRGb;b�1;1 �

ma; t

� 2f

lR

X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞX1k¼0

ð�mr2nÞ

k½G1�b;�bðkþ1Þ;kþ1ð�ar2n; tÞ

þ ar2nG1�b;�1�bk;kþ1ð�ar2

n; tÞ�; ð57Þ


R

X1n¼1


X1k¼0

ð�mr2nÞ

kG1�b;�bðkþ1Þ;kþ1ð�ar2n; tÞ: ð58Þ

For a ¼ 1, the known solutions

Fig. 1. Profiles of the velocity u(r, t) for a generalized Oldroyd-B fluid for different values of fractional coefficient.


uðr; tÞ ¼ fr2

2a1RGb;�2;1 �

ma; t

� 2f

a1R

X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞX1k¼0

ð�mr2nÞ

kZ t

0Gb;0;1 �m=a; sð Þ � G1�b;�bk�b�1;kþ1 �ar2

n; t � s� �

ds; ð59Þ

sðr; tÞ ¼ frR

t þ 2fR

X1n¼1


X1k¼0

ð�mr2nÞ

kG1�b;�bk�b�1;kþ1ð�ar2n; sÞds; ð60Þ

corresponding to the similar flow of generalized second grade fluids are recovered [12,33,46].

4.6. Second grade fluid as a particular case of second grade fluid with fractional derivatives

Making b! 1 into Eqs. (55)–(58), using (A.8)–(A.10), (A.14) and (A.15), we get the velocity field and the shear stress in theforms

if a > 0 then

uðr; tÞ ¼ fr2

2a1R

Z t

0ðt � sÞa expð�m=asÞdsþ far2

2lR

Z t

0ðt � sÞa�1 expð�m=asÞds� 2af

lR

X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞ

Z t

0ðt � sÞa�1

� exp � mr2n

1þ ar2n

s� �

ds; ð61Þ

sðr; tÞ ¼ frR

ta þ 2afR

X1n¼1


11þ ar2

n

Z t

0ðt � sÞa�1 exp � mr2

n

1þ ar2n

s� �

ds; ð62Þ

and if a ¼ 0 then

Fig. 2. Profiles of the velocity u(r, t) for a generalized Oldroyd-B fluid for different values of fractional coefficient a.


uðr; tÞ ¼ fr2

2lR� 2f

lR

X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞexp � mr2

n

1þ ar2n

t� �

; ð63Þ


R

X1n¼1


11þ ar2

nexp � mr2

n

1þ ar2n

t� �

; ð64Þ

corresponding to an ordinary second grade fluid, performing the same motion.For a ¼ 1, the expressions (61) and (62) can be written in the simplified forms

uðr; tÞ ¼ fr2

2lRðt � a=mÞ � 2f

mlR

X1n¼1

1� ð1þ ar2nÞ exp � mr2

n

1þ ar2n

t� ��

J0ðrrnÞr4

nJ0ðRrnÞ; ð65Þ

and

sðr; tÞ ¼ frR

t þ 2fmR

X1n¼1

1� exp � mr2n

1þ ar2n

t� ��

J1ðrrnÞr3

nJ0ðRrnÞ; ð66Þ

respectively, obtained in [13,44] by a different technique.

Fig. 3. Profiles of the velocity u(r) for a generalized Maxwell fluid for different values of fractional coefficient a.

Fig. 4. Profiles of the velocity u(r, t) for a generalized second grade fluid for different values of fractional coefficient.


For a! 0 into Eqs. (63)–(66), Eqs. (49)–(52) are recovered.

5. Conclusions

In this paper, the velocity fields and the resulting shear stress fields corresponding to the unsteady flow of a generalizedOldroyd-B fluid through an infinite circular cylinder have been determined, using Hankel and Laplace transforms. The mo-tion of the fluid is produced by the cylinder that after the initial moment is pulled by a time-dependent shear stress along itsaxis sðR; tÞ ¼ f

k

R t0ðt � sÞaGa;0;1ð�1=k; sÞds. The solutions that have been obtained, written under integral and series form in

terms of the generalized G-functions, satisfy all imposed initial and boundary conditions.Making a! 1 and b! 1 into Eqs. (27)–(30), the similar solutions for ordinary Oldroyd-B fluid are obtained. Making

kr ! 0 and b! 1 or kr ! 0; b! 1 and a! 1 into Eqs. (27)–(30), the solutions for generalized and ordinary Maxwell fluidsare obtained.

For k! 0, we have obtained the solutions corresponding to generalized second grade fluids, for k! 0 and b! 1, the sim-ilar solutions for ordinary second grade fluid are obtained, while for k! 0 and kr ! 0 the solutions for Newtonian fluids arealso determined.

In Figs. 1 and 2 are plotted the velocity diagrams corresponding to generalized Oldroyd-B fluids as functions of the radiusr for two values of time t. It is clearly seen that the diagrams of u(r) corresponding to generalized Oldroyd-B fluids ðb – 1Þtend to the diagram of the Oldroyd-B fluid ðb ¼ 1Þ. If the fractional coefficient a is fixed, then the velocity increases in thecentral area of the cylinder for increasing b, while near the boundary of the cylinder the velocity decreases (Fig. 1). The effect


of the fractional coefficient a is opposite. For fixed b and increasing a, the velocity decreases in the central area of the cyl-inder and increases near the wall (Fig. 2).

The velocity field corresponding to generalized Maxwell fluids decreases when the fractional coefficient a increases(Fig. 3), while the generalized second grade fluids have similar flows with the generalized Oldroyd-B fluids (Fig. 4).

In all figures we used R ¼ 1; f ¼ 0:5; k ¼ 6; l ¼ 32; m ¼ 0:0357; a ¼ 0:143 and the roots rn have been approximated byð4n�1Þp

4R [45].

Acknowledgements

The authors would like to express their sincere thanks to the referees for their fruitful comments and suggestions regard-ing an earlier version of this paper.

Appendix A

� �
ddr
J0½uðrÞ� ¼ �J1½uðrÞ�u0ðrÞ;ddr

J1½uðrÞ� ¼ J0½uðrÞ� �1

uðrÞ J1½uðrÞ� u0ðrÞ: ðA:1ÞZ R

0r

@2

@r2 þ1r@

@r

!�uðr; qÞJ0ðrrnÞdr ¼ �r2

n�uHðrn; qÞ þ RJ0ðRrnÞ

@�uðR; qÞ@r

: ðA:2Þ

�u1ðr; qÞ ¼fr2

2lRCðaþ 1Þ

qaþ1

1krqb þ 1

; �u2ðr; qÞ ¼2R2

X1n¼1

J0ðrrnÞJ2

0ðRrnÞ�u2Hðrn; qÞ: ðA:3Þ

L�1 qc

ðqb � pÞd

( )¼ Gb;c;dðp; tÞ; Reðbd� cÞ > 0; ReðqÞ > 0;

dqb

�� < 1; ðA:4Þ

where
4
Gb;c;dðp; tÞ ¼X1j¼0

Cðdþ jÞpj

CðdÞCðjþ 1ÞtðdþjÞb�c�1

C½ðdþ jÞb� c� :

F1ðqÞ ¼Cðaþ 1Þ

qaþ1 ; f 1ðtÞ ¼ L�1fF1ðqÞg ¼ ta; a P 0;

F2ðqÞ ¼Cðaþ 1Þ

qa ; f 2ðtÞ ¼ L�1fF2ðqÞg ¼ ata�1; a > 0; ðA:5Þ

F3ðqÞ ¼1

kqa þ 1; f 3ðtÞ ¼ L�1fF3ðqÞg ¼

1k

Ga;0;1 �1k; t

� �;

F4ðqÞ ¼1

krqb þ 1; f 4ðtÞ ¼ L�1fF4ðqÞg ¼

1kr

Gb;0;1 �1kr; t

� �;

Z t

0Gb;c;dðp; sÞds ¼ Gb;c�1;dðp; tÞ: ðA:6Þ

If u1ðtÞ ¼ L�1f�u1ðqÞg and u2ðtÞ ¼ L�1f�u2ðqÞg then

L�1f�u1ðqÞ�u2ðqÞg ¼ ðu1 � u2ÞðtÞ ¼Z t

0u1ðt � sÞu2ðsÞds ¼

Z t

0u1ðsÞu2ðt � sÞds; ðA:7Þ

f3ðtÞ ¼1k

G1;0;1 �1k; t

� �¼ 1

k

X1j¼0

1j!� t

k

� �j

¼ 1k

expð�t=kÞ ¼ a1ðtÞ;

f4ðtÞ ¼1kr

expð�t=krÞ ¼ a2ðtÞ; ðA:8Þ

G1;�1;1 �1k; t

� �¼ k 1� e�

tk

h i; ðA:9Þ

bnðtÞ ¼X1k¼0

1k� mr2

n

k

� �k

G1;�k�1;kþ1 �sn

k; t

¼ gnðtÞ;

cnðtÞ ¼X1k¼0

� mr2n

k

� �k

G1;�k;kþ1 �sn

k; t

þX1k¼0

1k� mr2

n

k

� �k

G1;�k�1;kþ1 �sn

k; t

¼ hnðtÞ; ðA:10Þ


1a1qb þ l

1qþ ar2

nqb þ mr2n¼ 1

lX1k¼0

ð�mr2nÞ

k q�b�bk þ ar2nq�1�bk

ðq1�b þ ar2nÞ

kþ1 � qb�1

qb þ ma

" #; ðA:11Þ

1qþ ar2

nqb þ mr2n¼ q�b

q1�b þ ar2n þ mr2

nq�b¼X1k¼0

ð�mr2nÞ

k q�b�bk

q1�b þ ar2n

� �kþ1 ; ðA:12Þ

r2

4¼X1n¼1

J0ðrrnÞr2

nJ0ðRrnÞ; ðA:13Þ

fr2

2a1RG1;�1;1 �

ma; t

þ fr2

2lRG1;0;1 �

ma; t

¼ fr2

2R1a1

amð1� e�m=atÞ þ 1

le�m=at

� �¼ fr2

2lR; ðA:14Þ

X1k¼0

ð�mr2nÞ

kð1þ ar2nÞG0;�k�1;kþ1ð�ar2

n; tÞ ¼X1k¼0

ð�mr2nÞ

kð1þ ar2nÞX1j¼0

ð�ar2nÞ

j Cðkþ jþ 1ÞCðkþ 1ÞCðjþ 1Þ �

tk

Cðkþ 1Þ

¼X1k¼0

ð�mr2nÞ

kð1þ ar2nÞ

Cðkþ 1Þ tk 1

ð1þ ar2nÞ

kþ1 ¼X1k¼0

1k!� mr2

n

1þ ar2n

t� �k

¼ exp � mr2n

1þ ar2n

t� �

: ðA:15Þ

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Exact solutions for the unsteady axial flow of Non-Newtonian fluids through a circular cylinder

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