Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel

Applied Mathematics and Computation 215 (2010) 3645–3654

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Applied Mathematics and Computation

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

Peristaltic flow of viscoelastic fluid with fractional Maxwell modelthrough a channel

D. Tripathi, S.K. Pandey, S. Das *

Department of Applied Mathematics, Institute of Technology, Banaras Hindu University, Varanasi 221 005, India

a r t i c l e i n f o

Keywords:PeristalsisViscoelastic fluidFractional Maxwell modelHomotopy perturbation methodAdomian decomposition methodFriction forceMittag–Leffler function

0096-3003/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.amc.2009.11.002

* Corresponding author.E-mail addresses: [email protected], subir_da

a b s t r a c t

The paper presents the transportation of viscoelastic fluid with fractional Maxwell modelby peristalsis through a channel under long wavelength and low Reynolds number approx-imations. The propagation of wall of channel is taken as sinusoidal wave propagation(contraction and relaxation). Homotopy perturbation method (HPM) and Adomian decom-position method (ADM) are used to obtain the analytical approximate solutions of theproblem. The expressions of axial velocity, volume flow rate and pressure gradient areobtained. The effects of fractional parameters ðaÞ, relaxation time ðk1Þ and amplitude ð/Þon the pressure difference and friction force across one wavelength are calculated numer-ically for different particular cases and depicted through graphs.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

The continuous wavelike muscle contraction and relaxation of the physiological vessels such as oesophagus, stomach,intestines, sometimes in the ureters, and blood vessels (arteries, veins, capillaries etc.) and other hollow tubes is knownas peristalsis. The transportation of fluids by peristalsis is known as peristaltic transport. The waves can be short, local re-flexes or long, continuous contractions along the length of the organ. In the esophagus, peristaltic waves push food into thestomach. In the stomach, they help mix stomach contents and propel food to the small intestine, where they expose food tothe intestinal wall for absorption and move it forward. Peristalsis in the large intestine pushes waste towards the anal canaland is important in removing gas and dislodging potential bacterial colonies. The flows of blood through the blood vesselsand urine through ureters are the peristaltic flow.

Shapiro et al. [1] studied the peristaltic transport of Newtonian fluid with long wavelength and low Reynolds numberapproximation. They discussed the pressure, mechanical efficiency, reflux limit and trapping limit in both two-dimensionaland axi-symmetric cases by assuming infinite length of vessels. Bohme and Friedrich [2] investigated the peristaltic flow ofviscoelastic liquids. Recently, many investigators are interested to investigate the important applications of viscoelastic fluidin physiology, engineering and medical science. Tsiklauri and Beresnev [3], El-Shehawy et al. [4] and Hayat et al. [5] haveconsidered the linear viscoelastic model as Maxwell model. They discussed the effect of relaxation time on the peristaltictransport. While Hayat et al. [6–10] have investigated the peristaltic transport of viscoelastic fluid by taking Jeffrey modeland they have also discussed the effect of relaxation and retardation time on the peristaltic transport. Blood as well as foodmaterials such as bread, fruit jam and some non-vegetarian foods (cf., [11,12]) are found to be viscoelastic in nature. It is aninteresting fact that almost all edible semi-solids bear both viscous and elastic properties.

In order to describe the viscoelastic properties of materials in various fields, recently the constitutive equations with or-dinary and fractional time derivatives have been introduced. Rheological models with fractional time derivatives have

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[email protected] (S. Das).

3646 D. Tripathi et al. / Applied Mathematics and Computation 215 (2010) 3645–3654

played an important role to study the valuable tool of viscoelastic properties. In general, fractional Maxwell model is derivedfrom known Maxwell model by replacing the ordinary derivatives of stress–strain by derivatives of fractional order. Friedrich[13] has developed a model to determine the relaxation and retardation time with different fractional time derivatives in thestress-strain relation by using the Riemann–Liouville definition. Tan et al. [14] have investigated the unsteady flow of vis-coelastic fluid with fractional Maxwell model between two parallel plates and Qi and Jin [15] have discussed unsteady flowsbetween coaxial cylinders, while Qi and Xu [16] have studied in the channel and solutions are obtained by using Laplacetransform, Fourier transform and Weber transform. Hayat et al. [17] have constructed periodic unidirectional flows of a vis-coelastic fluid with the fractional Maxwell model and solutions are solved by Fourier transform. Khan et al. [18] have dis-cussed the decay of potential vertex for viscoelastic fluid with fractional Maxwell model and analytical solutions areobtained by Hankel transform and discrete Laplace transform.

In all these previous investigations authors have discussed the velocity field, but they did not study the effect of fractionalparameters on pressure and friction force. In this paper we consider the peristaltic flow of viscoelastic fluid with fractionalmodel under the assumption of long wavelength and low Reynolds number. We have discussed the effect of relaxation time,fractional parameters and amplitude on the pressure difference and friction force across one wavelength. This evolutionmodel is first of its kind. In this article two powerful mathematical tools Homotopy perturbation method and Adomiandecomposition method are used to obtain approximate analytical solutions of the problem and the numerical results ofthe problem for different particular cases are depicted graphically.

HPM is an approach for finding the approximate analytical solution of linear and nonlinear problems. The method wasfirst proposed by He [19,20] and was successfully applied to solve nonlinear wave equations by He [21–23], linear partialdifferential equations of fractional order by He [24], Momani and Odibat [25], Ganji et al. [26], Das and Gupta [27], Daset al. [28], etc.

ADM was first proposed by Adomian [29–32] and used to solve a wide class of nonlinear and partial differential equations[33–39].

2. Mathematical model

The constitutive equation of shear stress–strain relationship of viscoelastic fluid with fractional Maxwell model is givenby [13,16]

1þ ~ka1@a

@~ta

� �~s ¼ l _c; ð1Þ

where ~k1; a; ~t; ~s; l; _c are the relaxation time, fractional parameter, time, shear stress, viscosity, rate of shear strain.The governing equations of motion for incompressible fluids in two-dimensional case are

q @@~tþ ~u @

@~nþ ~v @

@~g

� �~u ¼ � @~p

@~nþr � ~s;

q @@~tþ ~u @

@~nþ ~v @

@~g

� �~v ¼ � @~p

@~gþr � ~s;

9>=>; ð2Þ

where k1; a; ~t; ~s; l; _c are the fluid density, velocity, axial coordinate, transverse velocity, transverse coordinate andpressure.

The physical parameters are non-dimensionalized as follows:

n ¼ ~nk ; g ¼ ~g

a ; t ¼ c~tk ; k1 ¼ c~k1

k ; u ¼ ~uc ; v ¼ ~v

cd ; h ¼ ~ha ;

/ ¼ ~/a ; p ¼ ~pa2

lck ; Q ¼ ~Qac ; s ¼ a~s

lc ; Re ¼ qcadl ; d ¼ a

k ;

9=; ð3Þ

where ~h; ~/; ~Q are transverse displacement of the walls, amplitude of the wave, volume flow rate and their counterpartswithout � are the corresponding parameters in the dimensionless form. The parameters k; a; c symbolize the wavelength,the semi-width of the channel and the wave velocity respectively. Re stands for the Reynolds number while d is defined asthe wave number.

Using Eq. (1) in Eq. (2), in view of non-dimensionalisation and low Reynolds number approximation, reduce to

1þ ka1@a

@ta� �

@p@n ¼ @2u

@g2 ;

@p@g ¼ 0:

9=; ð4Þ

Boundary conditions are given by

@uðn;0; tÞ@g

¼ 0; uðn; h; tÞ ¼ 0;@p@n

����t¼0¼ 0: ð5Þ

D. Tripathi et al. / Applied Mathematics and Computation 215 (2010) 3645–3654 3647

Integrating Eq. (4) with respect to g, and using first condition of Eq. (5), the velocity gradient is obtained as

@u@g¼ 1þ ka

1@a

@ta

� �@p@n

g: ð6Þ

Further integrating Eq. (5) from h to g, we get the axial velocity as

u ¼ 12

1þ ka1@a

@ta

� �@p@nðg2 � h2Þ: ð7Þ

The volume flow rate is defined as Q ¼R h

0 udg, which, by virtue of Eq. (7), reduces to

Q ¼ �h3

31þ ka

1@a

@ta

� �@p@n: ð8Þ

The transformations between the wave and the laboratory frames, in the dimensionless form, are given by Shapiro et al. [1]

f ¼ n� t; 1 ¼ g; U ¼ u� 1; V ¼ v; h ¼ Q � h; ð9Þ

where the left side parameters are in the wave frame and the right side parameters are in the laboratory frame.We further assume that the wall undergoes contraction and relaxation is given by

h ¼ 1� / cos2ðpnÞ: ð10Þ

The following are the existing relations between the averaged flow rate, the flow rate in the wave frame and that in the lab-oratory frame [1]:

�Q ¼ hþ 1� /2¼ Q � hþ 1� /

2: ð11Þ

Eq. (8), in view of Eq. (11) gives

@a

@ta@p@n

� �þ 1

ka1

@p@n¼ � 3

ka1

�Q þ h� 1þ ð/=2Þh3

� �: ð12Þ

3. Solution of the problem by homotopy perturbation method

Eq. (12) can be written as

Dat f þ 1

ka1

f ¼ � k2

ka1; ð13Þ

where f ðn; tÞ ¼ @p@n and k2 ¼ 3

�Qþh�1þð/=2Þh3

� �with the initial condition

f ðn;0Þ ¼ 0: ð14Þ

According to the homotopy perturbation method, we construct the following homotopy:

Dat f ¼ �q

1ka

1f þ k2

ka1

: ð15Þ

In view of homotopy perturbation method, we use the homotopy parameter ‘q’ to expand the solution:

f ¼ f0 þ qf1 þ q2f2 þ q3f3 þ q4f4 þ � � � ð16Þ

When q! 1, Eq. (15) corresponds Eq. (13) and Eq. (16) becomes the approximate solution of Eq. (13). Substituting Eq. (16) inEq. (15) and comparing the like powers of q, we obtain the following set of linear differential equations:

q0 : Dat f0 ¼ 0; ð17Þ

q1 : Dat f1 ¼ �

1ka

1f0 �

k2

ka1; ð18Þ

q2 : Dat f2 ¼ �

1ka

1f1; ð19Þ

q3 : Dat f3 ¼ �

1ka

1f2; ð20Þ

q4 : Dat f4 ¼ �

1ka

1f3 ð21Þ

and so on.

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The method is based on applying the operator Jat (the inverse operator of Caputo derivative Dat ) on both sides of the Eqs.

(17)–(21), we obtain

f0 ¼ 0; ð22Þ

f1 ¼ �k2

ka1

ta

Cðaþ 1Þ ; ð23Þ

f2 ¼k2

k2a1

t2a

Cð2aþ 1Þ ; ð24Þ

f3 ¼ �k2

k3a1

t3a

Cð3aþ 1Þ ; ð25Þ

f4 ¼k2

k4a1

t4a

Cð4aþ 1Þ : ð26Þ

Thus the exact solution may be obtained as

f ðn; tÞ ¼X1r¼0

fr

¼X1r¼0

½wðaÞ�r tra

Cðraþ 1Þ ; where ½wðaÞ�r ¼ð�1Þr k2

kra1; r P 1

0; r ¼ 0

(

¼ EaðwðaÞtaÞ; ð27Þ

where EaðtÞ ¼P1

r¼0tr

Cðraþ1Þ ; ða > 0Þ is the Mittag–Leffler function in one parameter.

4. Solution of the problem by Adomian decomposition method

We consider Eq. (13) as

@f@t¼ � @1�a

@t1�a1ka

1f þ k2

ka1

� �;

Ltf ¼ �@1�a

@t1�a1ka

1f þ k2

ka1

� �;

ð28Þ

where Lt � @@t symbolizes the linear differential operation.

Applying the integration inverse operator L�1t ¼

R t0ð�Þdt to Eq. (28) and using Eq. (14), we get

f ðn; tÞ ¼ �L�1t

@1�a

@t1�a1ka

1f þ k2

ka1

� �: ð29Þ

The Adomian decomposition method (Adomian [30,31]) assumes an infinite series solutions for unknown function f ðn; tÞ gi-ven by

f ðn; tÞ ¼X1n¼0

fnðn; tÞ; ð30Þ

where the components f0; f1; f2; . . . are usually determined recursively by

f0 ¼ 0

f1 ¼ �L�1t

@1�a

@t1�a1ka

1f0 þ

k2

ka1

� �;

f2 ¼ �L�1t

@1�a

@t1�a1ka

1f1

� �;

fnþ1 ¼ �L�1t

@1�a

@t1�a1ka

1fn

� �; n P 1

ð31Þ

Finally the expression of f ðn; tÞ obtained from Eq. (30) will be same as the expression obtained by homotopy perturbationmethod.

The pressure difference Dp and friction force F across one wavelength can be calculated by using formulae [40]

Dp ¼Z 1

0

@p@n

dn; ð32Þ

F ¼Z 1

0h � @p

@n

� �dn: ð33Þ

Fig. 1. Pressure vs. time at / ¼ 0:5; �Q ¼ 0:5; k1 ¼ 1:0 for various values of a.

Fig. 2. Pressure vs. time at / ¼ 0:5, �Q ¼ 0:5; a ¼ 1=2 for various values of k1.

Fig. 3. Pressure vs. time at a ¼ 1=2, �Q ¼ 0:5; k1 ¼ 1:0 for various values of /.

D. Tripathi et al. / Applied Mathematics and Computation 215 (2010) 3645–3654 3649

Fig. 4. Friction force vs. time at / ¼ 0:5, �Q ¼ 0:5; k1 ¼ 1:0 for various values of a.

Fig. 5. Friction force vs. time at / ¼ 0:5; �Q ¼ 0:5; a ¼ 1=2 for various values of k1.

Fig. 6. Friction force vs. time at a ¼ 1=2; �Q ¼ 0:5; k1 ¼ 1:0 for various values of /.

3650 D. Tripathi et al. / Applied Mathematics and Computation 215 (2010) 3645–3654

D. Tripathi et al. / Applied Mathematics and Computation 215 (2010) 3645–3654 3651

All the results can be reduced to the classical Maxwell model if a ¼ 1 and it also reduces to Newtonian fluid if k1 ! 0.

5. Numerical results and discussion

In this section, we present the salient features of fractional time derivative and viscoelastic characteristics on pressurerise per wavelength ðDpÞ, friction force (F) through Figs. 1–12. In the present analysis we have considered F to be acting

Fig. 7. Pressure vs. averaged flow rate at / ¼ 0:5; t ¼ 1:0; k1 ¼ 1:0 for various values of a.

Fig. 8. Pressure vs. averaged flow rate at / ¼ 0:5; t ¼ 1:0; a ¼ 1=2 for various values of k1.

Fig. 9. Pressure vs. averaged flow rate at a ¼ 1=2; t ¼ 1:0; k1 ¼ 1:0 for various values of /.

3652 D. Tripathi et al. / Applied Mathematics and Computation 215 (2010) 3645–3654

in the opposite direction. The first three figures are prepared for the pressure vs. time and next three figures represent fric-tion force vs. time for fixed average flow rate �Q ¼ 0:5. Figs. 7–9 illustrates pressure vs. averaged flow rate and Figs. 10–12show friction force vs. averaged flow rate for fixed time t ¼ 1:0. The influences of different physical parameters such as relax-ation time ðk1Þ, amplitude ð/Þ and fractional parameter ðaÞ are particularly emphasized throughout the analysis. All the com-putations and graphs are made by using MATHEMATICA.

Fig. 10. Friction force vs. averaged flow rate at / ¼ 0:5; t ¼ 1:0; k1 ¼ 0:5 for various values of a.

Fig. 11. Pressure vs. averaged flow rate at / ¼ 0:5; t ¼ 1:0; a ¼ 1=2 for various values of k1.

Fig. 12. Friction force vs. averaged flow rate at / ¼ 0:5; t ¼ 1:0; k1 ¼ 0:5 for various values of /.

D. Tripathi et al. / Applied Mathematics and Computation 215 (2010) 3645–3654 3653

In Fig. 1, the pressure rise per wavelength Dp is plotted against the time t at different fractional Brownian motionsa ¼ 1=3; 1=2; 2=3 and at standard motion a ¼ 1:0 for fixed / ¼ 0:5 and k1 ¼ 1:0. In this case, pressure decreases with in-crease in fractional parameter a. Fig. 2 shows pressure rise per wavelength Dp vs. the time t at various value of relaxationtime k1 ¼ 0:4; 0:6; 0:8; 1:0 for fixed a ¼ 0:5 and / ¼ 0:5 and it is observed that pressure diminishes with relaxation timek1. Fig. 3 represents that pressure rise per wavelength Dp vs. the time t at various value of amplitude/ ¼ 0:4; 0:5; 0:6; 0:7 for fixed a ¼ 0:5 and k1 ¼ 1:0 and it is found that pressure increases with increasing amplitude (/Þ.As reveled by Figs. 4–6 the qualitative nature of variation in friction force F and pressure difference Dp w.r.to time are similarthough their quantitative behavior is very much different.

In Figs. 7–9, we plot graphs between pressure rise per wavelength Dp and averaged flow rate at different fractionalBrownian motions a ¼ 1=3; 1=2; 2=3 and at standard motion a ¼ 1:0 for relaxation time ðk1 ¼ 0:4; 0:6; 0:8; 1:0Þ and ampli-tude (/ ¼ 0:4; 0:5; 0:6; 0:7Þ respectively. It is seen that the relation between pressure and averaged flow rate is linear. Theaveraged flow rate is maximum at zero pressure, i.e., for Dp ¼ 0. It is revealed that pressure decreases with increase of boththe parameters a and k1, but increases with the increase in amplitude /. When a ¼ 1, the expressions reduce to those of clas-sical Maxwell model. The graphs (Figs. 7 and 9) depicting the relationship between pressure and averaged flow rate are ob-served similar in nature to those of Figs. 5 and 7, drawn by Hayat et al. [41]. But a quantitative difference is observed in thiscase due to the fact that they have taken peristaltic flow of viscoelastic fluid with classical Maxwell model in an asymmetricchannel and that wall equation is also different. For k1 ! 0, all the expressions reduce to that of Newtonian fluid. The illus-trations (Figs. 7–9) are also qualitatively similar to Fig. 7 of Shapiro et al. [1], who studied the peristaltic transport of New-tonian fluid under the assumption of long wavelength and low Reynolds number.

Figs. 10–12 depict that the results of friction force F are qualitatively similar to that of Dp, but quantitatively different.Here Figs. 10–12 are also similar to Fig. 5 (friction force vs. averaged flow rate) sketched by Mekheimer [40], where he inves-tigated the peristaltic flow of couple stress fluid under the effect of magnetic field and with different wall equation.

6. Conclusion

Fractional Maxwell model plays an important role to study the description of viscoelastic properties. Comparing profilesof the pressure for generalized Maxwell model (i.e., 0 < a < 1Þ and classical Maxwell model ða ¼ 1Þ, we may conclude thatthe pressure is minimum for classical model. When we go from classical to fractional, i.e., generalized Maxwell model, thepressure increases. Further comparing with the profiles of pressure for the generalized Maxwell model with k1 > 0 and New-tonian fluid with k1 ! 0, it is observed that the pressure is maximum for the second one. When we go from Newtonian to thegeneralized one, pressure diminishes. Similar effects are observed for friction force. The authors strongly believe that thepresent study of flow of viscoelastic fluid with fractional Maxwell model constitutes a significant change from the classicalapproach and thus will considerably benefit the engineers working in this field.

Acknowledgement

The authors of this article express their sincere thanks to the reviewers for their valuable suggestions for the improve-ment of the article.

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