flow through an oscillating rectangular duct for generalized maxwell fluid with fractional...

$: Flow through an oscillating rectangular duct for generalized Maxwell fluid with fractional derivatives$
Commun Nonlinear Sci Numer Simulat 17 (2012) 3219–3234

Contents lists available at SciVerse ScienceDirect

Commun Nonlinear Sci Numer Simulat

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

Flow through an oscillating rectangular duct for generalized Maxwellfluid with fractional derivatives

Mudassar Nazar ⇑, Muhammad Zulqarnain, Muhammad Saeed Akram, Muhammad AsifCentre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan

a r t i c l e i n f o a b s t r a c t

Article history:Received 6 April 2011Received in revised form 9 September 2011Accepted 2 October 2011Available online 14 January 2012

Keywords:Generalized Maxwell fluidOscillating rectangular ductVelocity field

1007-5704/$ - see front matter � 2011 Elsevier B.Vdoi:10.1016/j.cnsns.2011.10.002

⇑ Corresponding author.E-mail address: [email protected] (M. N

The velocity field and the associated shear stresses corresponding to the unsteady flow ofgeneralized Maxwell fluid on oscillating rectangular duct have been determined by meansof double finite Fourier sine and Laplace transforms. These solutions are also presented as asum of the steady-state and transient solutions. The solutions corresponding to Maxwellfluids, performing the same motion, appear as limiting cases of the solutions obtained here.In the absence of w, namely the frequency, and making a ? 1, all solutions that have beendetermined reduce to those corresponding to the Rayleigh Stokes problem on oscillatingrectangular duct for Maxwell fluids. Finally, some graphical representations confirm theabove assertions.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

How to predict the behavior of non-Newtonian fluids is still a considerable problem in fluid mechanics. The main reasonfor this is probably that non-Newtonian fluids help us to understand the wide variety of fluids that exist in the physicalworld. A constitutive equation relates a response to the perturbation associated with the response. Amongst the many mod-els like differential, rate and integral types which have been treated as non-Newtonian behavior, the fluids of rate type [1]have received special attention. Among the viscoelastic rate type model, which is used widely, is the Maxwell model [2].

Fetecau et al. [3] studied the velocity field and the adequate shear stress corresponding to the unsteady flow of a general-ized Maxwell fluid using Fourier sine and Laplace transforms. Zheng [4] obtained analytical solutions for generalized Max-well fluid flow due to oscillatory and constantly accelerating plate using the fractional calculus approach established theconstitutive relationship of fluid model. Vieru et al. [5] discussed the flow of a generalized Maxwell fluid induced by a con-stantly accelerating plate between two side walls.

Fractional calculus has encountered much success in the description of viscoelasticity and complex dynamical systems[6,7]. The starting point of the fractional derivative model of viscoelastic fluids is usually a classical differential equationwhich is modified by replacing the time derivative of an integer order by the Riemann–Liouville fractional calculus operator.This generalization allows one to define precisely non-integer order integrals or derivatives. A very good agreementis achieved with experimental data when the Maxwell model is used with its first order derivatives replaced by thefractional-order derivatives [8].

Nadeem et al. [9] discussed the Rayleigh Stokes problem for rectangular pipe in Maxwell and second grade fluids. Oscil-lating viscoelastic rectilinear flows in straight ducts due to an oscillating pressure gradient have been studied analytically byBroer [10] and Thurston [11,12] for a linear Maxwell fluid, by Jones and Walters [13,14] and Peev et al. [15] for a linearWalter B fluid, by Bhatnagar [16] for a Jeffrey fluid (linear Oldroyd), by Ramkissoon et al. [17] for a short memory Walters

. All rights reserved.

azar).

http://dx.doi.org/10.1016/j.cnsns.2011.10.002

mailto:[email protected]

http://dx.doi.org/10.1016/j.cnsns.2011.10.002

http://www.sciencedirect.com/science/journal/10075704

http://www.elsevier.com/locate/cnsns

3220 M. Nazar et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3219–3234

liquid B, and by Rahaman and Ramkissoon [18] for an upper convected Maxwell fluid. Furthermore, oscillating tube flow forviscoelastic fluids, mimicking human blood, has been considered by Walitza et al. [19].

The aim of this paper is to establish new exact solutions for generalised Maxwell fluid on oscillating rectangular duct bymeans of double finite Fourier sine and Laplace transforms for fractional calculus approach. The governing equation of mo-tion is a fractional order partial differential equation and obtain the velocity field and the shear stress. To solve such a prob-lem one has to use one additional initial condition, the first time derivative of the velocity is zero at time t = 0 [20,21].

2. Governing equations

Consider an incompressible generalized Maxwell fluid at rest in a duct of rectangular cross-section whose sides are atx = 0, x = d, y = 0 and y = h. At time t = 0+ the duct begin to oscillate along z-axis.

The velocity field is

vðx; y; tÞ ¼ wðx; y; tÞk ð1Þ

and the governing equation is

ð1þ kaDat Þ@wðx; y; tÞ

@t¼ m

@2wðx; y; tÞ@x2 þ @

2wðx; y; tÞ@y2

" #; ð2Þ

where

Dat f ðtÞ ¼ 1

Cð1� aÞ

Z t

0

f 0ðsÞðt � sÞa

ds;

is the Caputo’s fractional operator and C is the gamma function.We consider the following initial and boundary conditions

wðx; y;0Þ ¼ @wðx; y;0Þ@t

¼ 0; ð3aÞ

wð0; y; tÞ ¼ wðd; y; tÞ ¼ wðx;0; tÞ ¼ wðx; h; tÞ ¼ U cosðxtÞ ð3bÞ

or

wðx; y;0Þ ¼ @wðx; y;0Þ@t

¼ 0; ð4aÞ

wð0; y; tÞ ¼ wðd; y; tÞ ¼ wðx;0; tÞ ¼ wðx; h; tÞ ¼ U sinðxtÞ; ð4bÞ

We denote by u(x,y, t) the solution of problem (2), (3a), (3b) and by v(x,y, t) the solution of problem (2), (4a), (4b) and definethe complex velocity field

Fðx; y; tÞ ¼ uðx; y; tÞ þ ivðx; y; tÞ; ð5Þ

which is the solution of the following problem:

1þ kaDat

� � @Fðx; y; tÞ@t

¼ m@2Fðx; y; tÞ

@x2 þ @2Fðx; y; tÞ@y2

" #; ð6Þ

Fðx; y;0Þ ¼ @Fðx; y;0Þ@t

¼ 0; ð7Þ

Fð0; y; tÞ ¼ Fðd; y; tÞ ¼ Fðx;0; tÞ ¼ Fðx;h; tÞ ¼ Ueixt : ð8Þ

The fractional differential Eq. (6) with the initial and boundary conditions (7) and (8) will be solved by means of the Fouriersine and Laplace transforms.

3. Calculation of velocity field

Multiplying both sides of Eq. (6) by sin(amx) sin(bny), integrating with respect to xand y over [0,d] � [0,h] and usingEq. (8), we find that

1þ kaDat

� � @FmnðtÞ@t

þ m a2m þ b2

n

� �FmnðtÞ ¼ m½1� ð�1Þm�½1� ð�1Þn�a

2m þ b2

n

ambnUeixt ; ð9Þ

where am ¼ mpd ; bn ¼ np

h ; and the double Fourier sine transforms

FmnðtÞ ¼Z d

0

Z h

0Fðx; y; tÞ sinðamxÞ sinðbnyÞdxdy; m;n ¼ 1;2;3 . . . ð10Þ

M. Nazar et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3219–3234 3221

of F(x,y, t) have to satisfy the initial conditions

Fmnð0Þ ¼ 0;@Fmnð0Þ@t

¼ 0; m;n ¼ 1;2;3 . . . ð11Þ

By applying the Laplace transforms to Eq. (9) with the initial conditions (11), we obtain

qþ kaqaþ1 þ m a2m þ b2

n

� �� FmnðqÞ ¼ m½1� ð�1Þm�½1� ð�1Þn�a

2m þ b2

n

ambnU

1q�xi

; ð12Þ

which can be simplified as

FmnðqÞ ¼ mU½1� ð�1Þm�½1� ð�1Þn�a2m þ b2

n

ambn

1q�xi

1qþ kaqaþ1 þ m a2

m þ b2n

� �¼ amn

Uq�xi

� amnU 1þ ix1

q� ix

� �kaqa þ 1

kaqaþ1 þ qþ mkmn; ð13Þ

where

amn ¼1

ambn½1� ð�1Þm�½1� ð�1Þn�; kmn ¼ a2

m þ b2n; ð14Þ

and FmnðqÞ ¼R1

0 FmnðtÞe�qtdt is the Laplace transform of Fmn(t). In order to obtain the inverse Laplace transformFmnðtÞ ¼ L�1fFmnðqÞg, we consider the function

HmnðqÞ ¼kaqa þ 1

kaqaþ1 þ qþ mkmn; ð15Þ

which can be written in the following equivalent forms:

HmnðqÞ ¼kaðqa þ k�aÞ

kaq½ðqa þ k�aÞ þ mk�akmnq�1� ¼qa�1 þ k�aq�1

ðqa þ k�aÞ þ mk�akmnq�1¼X1k¼0

ð�1Þkðmk�akmnq�1Þkðqa�1 þ k�aq�1Þðqa þ k�aÞkþ1

¼X1k¼0

�mkmn

ka

� k qa�k�1 þ k�aq�k�1

ðqa þ k�aÞkþ1 : ð16Þ

Applying the double inverse Fourier sine transform to Eq. (13), we obtain

Fðx; y; qÞ ¼ 4Udh

1q�xi

X1m;n¼1

amn sinðamxÞ sinðbnyÞ � 4Udh

X1m;n¼1

amn 1þ ixq�xi

� �HmnðqÞ � sinðamxÞ sinðbnyÞ ð17Þ

Using the formula [22]

1 ¼ 4dh

X1m;n¼1

amn sinðamxÞ sinðbnyÞ;

we obtain for Fðx; y; qÞ the expression

Fðx; y; qÞ ¼ Uq�xi

� 16Udh

X1m;n¼1

sinðaMxÞaM

sinðbNyÞbN

1þ xiq�xi

� �HMNðqÞ; ð18Þ

where M ¼ 2m� 1; N ¼ 2n� 1; aM ¼ ð2m�1Þpd and bN ¼ ð2n�1Þp

h .By applying the inverse Laplace transform to Eq. (18), using (16) and the formula

L�1 qb

ðqa � dÞc �

¼ Ga;b;cðd; tÞ; Reðac � bÞ > 0; ReðqÞ > 0; jqaj > jdj;

we obtain for the complex velocity field F(x,y, t), the following expression:

Fðx; y; tÞ ¼ Ueixt � 16Udh

X1m;n¼1

sinðaMxÞaM

sinðbNyÞbN

X1k¼0

� mkMN

ka

� k

�(

Ga;a�k�1;kþ1ð�k�a; tÞ þ k�aGa;�k�1;kþ1ð�k�a; tÞ� �

þxiX1k¼0

� mkMN

ka

� k

�Z t

0eixðt�sÞ½Ga;a�k�1;kþ1ð�k�a; sÞ þ k�aGa;�k�1;kþ1ð�k�a; sÞ�ds

�: ð19Þ

In the above relations the generalized G-functions Ga,b,c(d, t) are defined by

Ga;b;cðd; tÞ ¼X1j¼0

djCðc þ jÞCðcÞCðjþ 1Þ

tðcþjÞa�b�1

C½ðc þ jÞa� b� : ð20Þ


Setting d = 2a, h = 2b and changing the origin of the coordinate system (taking x = x⁄ + a, y = y⁄ + b and dropping out the starnotation), the complex velocity can be written in the form

Fðx; y; tÞ ¼ Ueixt � 4Uab

X1m;n¼1

ð�1Þmþn cosðaMxÞaM

cosðbNyÞbN

X1k¼0

� mkMN

ka

� k

�(

½Ga;a�k�1;kþ1ð�k�a; tÞ

þ k�aGa;�k�1;kþ1ð�k�a; tÞ� þxiX1k¼0

� mkMN

ka

� k

�Z t

0eixðt�sÞ Ga;a�k�1;kþ1ð�k�a; sÞ þ k�aGa;�k�1;kþ1ð�k�a; sÞ

� �ds�: ð21Þ

The velocity field corresponding to the cosine oscillations of the duct, respectively to the sine oscillations of the duct are gi-ven by

uðx; y; tÞ ¼ Re½Fðx; y; tÞ�

¼ U cosðxtÞ þ 4Uab

X1m;n¼1


cosðbNyÞbN

X1k¼0

� mkMN

ka

� k

�(


þ k�aGa;�k�1;kþ1ð�k�a; tÞ� �xX1k¼0

� mkMN

ka

� k

�Z t

0sinxðt � sÞ½Ga;a�k�1;kþ1ð�k�a; sÞ þ k�aGa;�k�1;kþ1ð�k�a; sÞ�ds

�; ð22Þ

vðx; y; tÞ ¼ Im½Fðx; y; tÞ�

¼ U sinðxtÞ þ 4Uab

X1m;n¼1


cosðbNyÞbN

X1k¼0

� mkMN

ka

� k

�(


þ k�aGa;�k�1;kþ1ð�k�a; tÞ� þxX1k¼0

� mkMN

ka

� k

�Z t

0cos xðt � sÞ½Ga;a�k�1;kþ1ð�k�a; sÞ þ k�aGa;�k�1;kþ1ð�k�a; sÞ�ds

�: ð23Þ

4. Calculation of shear stresses

In the considered problem Sxx = Sxy = Syy = 0,

1þ kaDat

� �s1ðx; y; tÞ ¼ l @xðx; y; tÞ

@x; s1 ¼ Sxz; ð24Þ

1þ kaDat

� �s2ðx; y; tÞ ¼ l @xðx; y; tÞ

@y; s2 ¼ Syz; ð25Þ

1þ kaDat

� �rðx; y; tÞ ¼ 2k s1

@x@xþ s2

@x@y

� �; r ¼ Szz: ð26Þ

We denote by s1c, s2c the tensions for cosine oscillations, respectively s1s, s2s the tensions for sine oscillations of the duct.If we introduce

T1ðx; y; tÞ ¼ s1cðx; y; tÞ þ is1sðx; y; tÞ; ð27ÞT2ðx; y; tÞ ¼ s2cðx; y; tÞ þ is2sðx; y; tÞ; ð28Þ

we obtain

1þ kaDat

� �T1ðx; y; tÞ ¼ l @Fðx; y; tÞ

@x; ð29Þ

1þ kaDat

� �T2ðx; y; tÞ ¼ l @Fðx; y; tÞ

@y; ð30Þ

with F(x,y, t) given by (5).The fluid begins from rest at time t = 0, we have the initial conditions

T1ðx; y;0Þ ¼ T2ðx; y; 0Þ ¼ 0: ð31Þ
Applying the Laplace transforms to Eqs. (29) and (30) with the initial conditions (31), we obtain
T1ðx; y; qÞ ¼l

kaqa þ 1@Fðx; y; qÞ

@x; ð32Þ

T2ðx; y; qÞ ¼l

kaqa þ 1@Fðx; y; qÞ

@y; ð33Þ


where

Fig. 1.parame

Fðx; y; qÞ ¼ Uq�xi

� 16Udh

X1m;n¼1

sinðaMxÞaM

sinðbNyÞbN

qq�xi

kaqa þ 1kaqaþ1 þ qþ mkMN

: ð34Þ

Introducing Eq. (34) into Eqs. (32) and (33), we get

T1ðx; y; qÞ ¼ �16lU

dh

X1m;n¼1

cosðaMxÞ sin bNybN

qðq�xiÞðkaqaþ1 þ qþ mkMNÞ

; ð35Þ

T2ðx; y; qÞ ¼ �16lU

dh

X1m;n¼1

sinðaMxÞaM

cosðbNyÞ qðq�xiÞðkaqaþ1 þ qþ mkMNÞ

: ð36Þ

Profiles of velocity u(x,y, t) given by Eq. (22) for x = 2, m = 0.0012, U = 0.001, a = 0.025, b = 0.02, k = 0.5 and different values of y, t and the fractionalter a.


Let us take

Fig. 2.parame

GmnðqÞ ¼q

ðq�xiÞðkaqaþ1 þ qþ mkmnÞ¼ 1

q�xiq

kaq½ðqa þ k�aÞ þ mkmnk�aq�1�

¼ 1q�xi

1ka

X1k¼0

� mkmn

ka

� k q�k

ðqa þ k�aÞkþ1 : ð37Þ

The inverse Laplace transform of the function Gmn(q) is

gmnðtÞ ¼1ka

X1k¼0

� mkmn

ka

� k Z t

0eixðt�sÞGa;�k;kþ1ð�k�a; sÞds: ð38Þ

Profiles of velocity u(x,y, t) given by Eq. (22) for x = 2, m = 0.0012, U = 0.001, a = 0.025, b = 0.02, k = 1.0 and different values of y, t and the fractionalter a.


Now we obtain the following expressions of the complex shear stresses:

Fig. 3.parame

T1ðx; y; tÞ ¼ �16lU

dh

X1m;n¼1

cosðaMxÞ sinðbNÞybN

1ka

X1k¼0

� mkMN

ka

� k Z t

0eixðt�sÞGa;�k;kþ1ð�k�a; sÞds; ð39Þ

T2ðx; y; tÞ ¼ �16lU

dh

X1m;n¼1

sinðaMxÞaM

cosðbNyÞ 1ka

X1k¼0

� mkMN

ka

� k Z t

0eixðt�sÞGa;�k;kþ1ð�k�a; sÞds: ð40Þ

The real tensions are given by

Profiles of velocity v(x,y, t) given by Eq. (23) for x = 2, m = 0.0012, U = 0.001, a = 0.025, b = 0.02, k = 0.5 and different values of y, t and the fractionalter a.

Fig. 4.parame


s1cðx; y; tÞ ¼ Re½T1ðx; y; tÞ�

¼ �16lUdh

X1m;n¼1

X1k¼0

cosðaMxÞka

sinðbNyÞbN

� mkMN

ka

� k

�Z t

0cos xðt � sÞGa;�k;kþ1ð�k�a; sÞds; ð41Þ

s2cðx; y; tÞ ¼ Re½T2ðx; y; tÞ�

¼ �16lUdh

X1m;n¼1

X1k¼0

sinðaMxÞaM

cosðbNyÞka � mkMN

ka

� k

�Z t

0cos xðt � sÞGa;�k;kþ1ð�k�a; sÞds; ð42Þ

Profiles of velocity v(x,y, t) given by Eq. (23) for x = 2, m = 0.0012, U = 0.001, a = 0.025, b = 0.02, k = 1.0 and different values of y, t and the fractionalter a.


for cosine oscillations of the duct, respectively

Fig. 5.fraction

s1sðx; y; tÞ ¼ Im½T1ðx; y; tÞ�

¼ �16lUdh

X1m;n¼1

X1k¼0

cosðaMxÞka

sinðbNyÞbN

� mkMN

ka

� k

�Z t

0sinxðt � sÞGa;�k;kþ1ð�k�a; sÞds; ð43Þ

s2sðx; y; tÞ ¼ Im½T2ðx; y; tÞ�

¼ �16lUdh

X1m;n¼1

X1k¼0

sinðaMxÞaM

cosðbNyÞka � mkMN

ka

� k

�Z t

0sinxðt � sÞGa;�k;kþ1ð�k�a; sÞds; ð44Þ

for sine oscillations of the duct.

5. Limiting cases

5.1. Maxwell fluids as a particular case of the generalized Maxwell fluids

Making a = 1 into Eq. (21) and using (A5) from the Appendix, we obtain the complex velocity field

Absolute values of the velocity u(x,y, t) given by Eq. (22) for x = 2, m = 0.0012, U = 0.001, a = 0.025, b = 0.02, k = 0.5, and different values of theal parameter a.

Fig. 6.fraction



X1m;n¼1


cosðbNyÞbN

� coshumnt2k

� þ 1

umnsinh

umnt2k

� � �e�

t2k þxi

Z t

0eixðt�sÞ cosh

umns2k

� þ 1

umnsinh

umns2k

� � �e�

s2kds

�; ð45Þ

corresponding to an ordinary Maxwell fluid performing the same motion, where

umn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4kmkmn

p: ð46Þ

Integrating by parts, after a straightforward calculation, function F(x,y, t) can be written in the simpler form


X1m;n¼1


cosðbNyÞbN

� AMNeixt þ ð1� AMNÞ coshUMNt

2k

� þ 1þ 2kxi

UMNsinh

UMNt2k

� � �e�

t2k

�; ð47Þ

where

Amn ¼x2½1� kðmkmn � kx2Þ� þxmkmni

ðmkmn � kx2Þ2 þx2: ð48Þ

Absolute values of the velocity u(x,y, t) given by Eq. (23) for x = 2, m = 0.0012, U = 0.001, a = 0.025, b = 0.02, k = 0.5, and different values of theal parameter a.


We denote by

Fig. 7.

cmn ¼ ReðAmnÞ ¼x2½1� kðmkmn � kx2Þ�ðmkmn � kx2Þ2 þx2

; dmn ¼ ImðAmnÞ ¼xmkmn

ðmkmn � kx2Þ2 þx2; ð49Þ

and, from Eq. (47) we obtain

uðx; y; tÞ ¼ U cosðxtÞ � 4Uab

X1m;n¼1


� cosðbNyÞbN

cMN cosðxtÞ � dMN sinðxtÞ þ e�t

2k ð1� cMNÞ coshUMNt

2k

� þ 1� cMN þ 2kxdMN

UMNsinh

UMNt2k

� � � �;

ð50Þ

vðx; y; tÞ ¼ U sinðxtÞ � 4Uab

X1m;n¼1


� cosðbNyÞbN

cMN sinðxtÞ � dMN cosðxtÞ � e�t

2k dMN coshUMNt

2k

� þ dMN � 2kxð1� cMNÞ

UMNsinh

UMNt2k

� � � �:

ð51Þ

The shear stresses corresponding to the flow of a Maxwell fluid are obtained from Eqs. (39) and (40) for a = 1. Using (A9)(from Appendix) we obtain the following forms of the shear stresses:

Tension Sxz given by Eq. (52) for x = 2, m = 0.0012, l = 1.48, U = 0.001, a = 0.025, b = 0.02, k = 0.5, and different values of the fractional parameter a.

Fig. 8.


s1cðx; y; tÞ ¼ Re½T1ðx; y; tÞ� ¼ �16lU

dh

X1m;n¼1

cosðaMxÞ sinðbNyÞbN

eMN cosðxtÞ � fMN sinðxtÞf

þ e�t

2kxeMN þ 2mkMNfMN

xUMNsinh

UMNt2k

� � eMN cosh

UMNt2k

� � ��; ð52Þ

s2cðx; y; tÞ ¼ Re½T2ðx; y; tÞ� ¼ �16lU

dh

X1m;n¼1

cosðbNyÞ sinðaMxÞaM

eMN cosðxtÞ � fMN sinðxtÞf

þ e�t

2kxeMN þ 2mkMNfMN

xUMNsinh

UMNt2k

� � eMN cosh

UMNt2k

� � ��; ð53Þ

s1sðx; y; tÞ ¼ Im½T1ðx; y; tÞ� ¼ �16lU

dh

X1m;n¼1

cosðaMxÞ sinðbNyÞbN

fMN cosðxtÞ þ eMN sinðxtÞf

þ e�t

2kxfMN � 2mkMNeMN

xUMNsinh

UMNt2k

� � fMN cosh

UMNt2k

� � ��; ð54Þ

s2sðx; y; tÞ ¼ Im½T2ðx; y; tÞ� ¼ �16lU

dh

X1m;n¼1

cosðbNyÞ sinðaMxÞaM

fMN cosðxtÞ þ eMN sinðxtÞf

þ e�t

2kxfMN � 2mkMNeMN

xUMNsinh

UMNt2k

� � fMN cosh

UMNt2k

� � ��: ð55Þ

Tension Syz given by Eq. (53) for x = 2, m = 0.0012, l = 1.48, U = 0.001, a = 0.025, b = 0.02, k = 0.5, and different values of the fractional parameter a.


where

Fig. 9.

emn ¼x2

ðmkmn � kx2Þ2 þx2; f mn ¼

xðmkmn � kx2Þðmkmn � kx2Þ2 þx2

ð56Þ

6. Concluding remarks

Here we established exact solutions for the velocity field and the associated shear stresses corresponding to flows of aMaxwell fluid with fractional derivatives within an oscillating rectangular duct. Both cases of the cosine and sine oscillationsof the duct have been analyzed and the solutions have been determined by means of the Laplace and double finite sine Fou-rier transforms. The solutions corresponding to Maxwell fluids have been determined as particular case. Finally, some phys-ical aspects are revealed by graphical illustrations.

Figs. 1 and 2 are sketched to show the profiles of the velocity field, Eq. (22), for a Maxwell fluid with fractional derivatives,versus x 2 [�0.025,025] and for different values of the spatial coordinate y, fractional coefficient a and time t. From thesefigures, it is obvious to note that the velocity field is an increasing function of spatial variable y, and for large values of timet, the velocity is an increasing function of the fractional parameter a. Consequently, the fractional Maxwell fluid flows slowerin comparison with the arbitrary Maxwell fluid; the velocity field is a decreasing function for increasing relaxation time k.

Figs. 3 and 4 are sketched to show the profiles of the velocity field of a Maxwell fluid with fractional derivatives in the caseof sine oscillations of the duct, Eq. (23). From these figures, one can easily see that the velocity field has the similar behavioras in previous case.

Tension Sxz given by Eq. (54) for x = 2, m = 0.0012, l = 1.48, U = 0.001, a = 0.025, b = 0.02, k = 0.5, and different values of the fractional parameter a.

Fig. 10. Tension Syz given by Eq. (55) for x = 2, m = 0.0012, l = 1.48, U = 0.001, a = 0.025, b = 0.02, k = 0.5, and different values of the fractional parameter a.


In Figs. 5 and 6, we plotted the absolute values of the velocity field versus t for x = 0.01 m, y = 0 m, for both cosine and sineoscillations of the duct and for small and large values of time t. These diagrams are plotted for three values of the fractionalcoefficient a. It is clear that for small values of the time t, the velocity field is not monotonous function of the fractional coef-ficient a, while for large values of the time t, the velocity field is increasing for increasing a.

In Figs. 7–10, we drew diagrams of the shear stresses Sxz, Syz, versus t, for both cases of the oscillations of the duct and forthree values of fractional coefficient a. From these figures, it clearly results that for small values of time t, the tensions Sxz, Syz

change their monotony with respect to a, while for large values of time t these functions are increasing functions of a.

Acknowledgement

This article is fully supported by Higher Education Commission of Pakistan under the project entitled ‘‘Flow of Rate TypeFluid in Oscillating Rectangular Duct’’.

Appendix A

For a = 1, Eq. (15) becomes

HmnðqÞ ¼kqþ 1

kq2 þ qþ mkmn¼X1k¼0

� mkmn

k

� k q�k þ k�1q�k�1

ðqþ k�1Þkþ1 ; ðA1Þ

with the inverse Laplace transform

hmnðtÞ ¼X1k¼0

� mkmn

k

� k

G1;�k;kþ1 �1k; t

� þ 1

kG1;�k�1;kþ1 �

1k; t

� � �: ðA2Þ


On the other hand the function Hmn(q) can be written as

HmnðqÞ ¼kðqþ 1

kÞ

k qþ 1k

� �2 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�4kmkmn

p2k

� 2" # ¼ qþ 1

2k

� �þ 1

2k

qþ 12k

� �2 � umn2k

� �2 ¼qþ 1

2k

� �qþ 1

2k

� �2 � un2k

� �2 þ1

umn

umn2k

qþ 12k

� �2 � umn2k

� �2 ;

umn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4kmkmn

p; ðA3Þ


hmnðtÞ ¼ coshumn

2kt

� þ 1

umnsinh

umn

2kt

� � �e�

t2k: ðA4Þ

Comparing (A2) and (A4), we obtain that

X1k¼0

� mkmn

k

� k

G1;�k;kþ1 �1k; t

� þ 1

kG1;�k�1;kþ1 �

1k; t

� � �¼ cosh

umn

2kt

� þ 1

umnsinh

umn

2kt

� � �e�

t2k: ðA5Þ

Let us

GmnðqÞ ¼q

ðq�wiÞðkq2 þ qþ mkmnÞ¼ 1

q�wi1k

X1k¼0

� mkmn

k

� k q�k

ðqþ k�1Þkþ1 ; ðA6Þ


gmnðtÞ ¼1k

X1k¼0

� mkmn

k

� k Z t

0eiwðt�sÞG1;�k;kþ1 �

1k; s

� ds: ðA7Þ

On the other hand

GmnðqÞ ¼1

q�wiq

k qþ 12k

� �2 � umn2k

� �2h i ¼ 1

k1

q�wiqþ 1

2k

� �qþ 1

2k

� �2 � umn2k

� �2 �1

umn

umn2k

qþ 12k

� �2 � umn2k

� �2

" #; ðA8Þ


gmnðtÞ ¼1k

Z t

0eiwðt�sÞ cosh

umns2k

� � 1

umnsinh

umns2k

� � �e�

s2kds

¼ 1k

eiwtZ t

0e� iwþ 1

2kð Þs coshumns2k

� ds� 1

kumneiwt

Z t

0e� iwþ 1

2kð Þs sinhumns2k

� ds: ðA9Þ

References

[1] Fetecau C, Fetecau Corina. Decay of a potential vortex in a Maxwell fluid. Int J Non Linear Mech 2003;38:985–90.[2] Akhtar W, Fetecau Corina, Tigoiu V, Fetecau C. Flow of a Maxwell fluid between two side walls induced by a constantly accelerating plate. Z Angew

Math Phys 2009;60:498–510.[3] Fetecau Corina, Athar M, Fetecau C. Unsteady flow of a generalized Maxwell fluid with fractional derivative due to a constantly accelerating plate.

Comput Math Appl 2009;57:596–603.[4] Zheng L. Exact solutions for generalized Maxwell fluid flow due to oscillatory and constantly accelerating plate. Nonlinear Anal Real World Appl

2010;11:3744–51.[5] Vieru D, Fetecau Corina, Athar M, Fetecau C. Flow of a generalized Maxwell fluid induced by a constantly accelerating plate between two side walls. Z

Angew Math Phys 2009;60:498–510.[6] Palade LI, Attane P, Huilgol RR, Mena B. Anomalous stability behavior of a properly invariant constitutive equation which generalizes fractional

derivative models. Int J Eng Sci 1999;37:315–29.[7] Rossikhin YA, Shitikova MV. A new method for solving dynamic problems of fractional derivative viscoelasticity. Int J Eng Sci 2001;39:149–76.[8] Makris N, Constantinou MC. Fractional derivative model for viscous dampers. J Struct Eng ASCE 1991;117:2708–24.[9] Nadeem S, Asghar S, Hayat T, Hussain Mazhar. The Rayleigh Stokes problem for rectangular pipe in Maxwell and second grade fluid. Meccanica

2008;43:495–504.[10] Broer LJF. On the hydrodynamics of viscoelastic fluids. Appl Sci Res 1956–1957;A6:226–36.[11] Thurston GR. Theory of oscillation of a viscoelastic medium between parallel planes. J Appl Phys 1959;30(12):1855–60.[12] Thurston GR. Theory of oscillation of a viscoelastic fluid in a circular tube. JASA 1960;32(2):210–3.[13] Jones JR, Walters TS. Flow of elastico-viscous liquids in channels under the influence of a periodic pressure gradient Part I. Rheol Acta 1967;6(3):240–5.[14] Jones JR, Walters TS. Flow of elastico-viscous liquids in channels under the influence of a periodic pressure gradient Part II. Rheol Acta

1967;6(4):330–8.[15] Peev G, Elenkov D, Kunev I. On the problem of oscillatory laminar flow of elastico-viscous liquids in channels. Rheol Acta 1970;9:506–8.[16] Bhatnagar RK. Flow of an Oldroyd fluid in a circular pipe with time dependent pressure gradient. Appl Sci Res 1975;30:241–67.[17] Ramkissoon H, Eswaran CV, Majumdar SR. Unsteady flow of an elastico-viscous fluid in tubes of uniform cross-section. Int J Non Linear Mech

1989;24(6):585–97.[18] Rahaman KD, Ramkissoon H. Unsteady axial viscoelastic pipe flows. J Non Newton Fluid Mech 1995;57:27–38.[19] Walitza E, Maisch E, Chmiel H, Andrade I. Experimental and numerical analysis of oscillatory tube flow of viscoelastic fluids represented at the example

of human blood. Rheol Acta 1979;18:116–21.


[20] Bohme G. Stromungsmechanik nicht-newtonscher fluide. Stuttgart: B.G. Teubner; 1981.[21] Wood WP. Transient viscoelastic helical flows of circular and annular cross-section. J Non Newton Fluid Mech 2001;100:115–26.[22] Debnath L, Bhatta D. Integral Transforms and their Applications. Boca Raton, London, New York: Chapman & Hall, CRC; 2007.

flow through an oscillating rectangular duct for generalized maxwell fluid with fractional...

Documents