Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 329639, 24 pages

doi:10.1155/2012/329639

Research Article

Peristaltic Flow of Carreau Fluid in a Rectangular Duct through a Porous Medium

R. Ellahi,¹ Arshad Riaz,¹ S. Nadeem,² and M. Ali³

¹Department of Mathematics and Statistics, FBAS, IIU, Islamabad 44000, Pakistan

²Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan

³Mechanical Engineering Department, College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia

Received 27 January 2012; Revised 21 March 2012; Accepted 21 March 2012

Academic Editor: Anuar Ishak

Copyright © 2012 R. Ellahi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We have examined the peristaltic flow of Carreau fluid in a rectangular channel through a porous medium. The governing equations of motion are simplified by applying the long wavelength and low Reynolds number approximations. The reduced highly nonlinear partial differential equations are solved jointly by homotopy perturbation and Eigen function expansion methods. The expression for pressure rise is computed numerically by evaluating the numerical integration. The physical features of pertinent parameters have been discussed by plotting graphs of velocity, pressure rise, pressure gradient, and stream functions.

1. Introduction

Investigation of flow through a porous medium has many applications in various branches of science and technology. The applications in which flow through a porous medium is mostly prominent are filtration of fluids, seepage of water in river beds, movement of underground water and oils, limestone, rye bread, wood, the human lung, bile duct, gallbladder with stones, and small blood vessels which are few examples of flow through porous medium [1]. Peristaltic mechanism is another important phenomenon which has exploited the attention of many researchers due to its physiological and industrial applications. A large amount of literature is available on the peristalsis involving Newtonian and non-Newtonian fluids with different flow geometries [2–12]. The peristaltic flow through a porous medium has been also discussed by number of researchers. Pandey and Chaube [13] have examined the peristaltic flow of micropolar fluid through a porous medium in the presence of external magnetic field. They pointed out that the maximum pressure is strongly dependent on permeability of porous medium. Interaction of peristaltic flow with pulsatile fluid through a porous medium has been investigated by Afifi and Gad [14]. Rao and Mishra [15] have examined the peristaltic transport of a power law fluid in a porous tube. In another paper Mishra and Rao [16] have studied the peristaltic transport in a channel with a porous peripheral layer. Some other papers on this topic are given in the references [17–20]. The mathematical model of peristaltic flow in a two-dimensional symmetric and asymmetric channel was discussed by Eytan and Elad [17] because of its application in interuterine fluid flow in a nonpregnant uterus. A number of researchers have examined the peristaltic flow of two-dimensional flow in an asymmetric channel [18–25]. However Reddy et al. [26] have recently given the idea that the sagittal cross-section of the uterus may be better approximated by a tube of rectangular cross-section than a two-dimensional channel and presented the influence of lateral walls on peristaltic flow in a rectangular duct. Only few papers (two or three) have been given in literature which discuss the peristaltic flows in a rectangular channel [26, 27].

Motivated by the previous studies, in the present investigation, we have examined the peristaltic flow of Carreau fluid in a rectangular symmetric channel through a porous medium. In the laboratory frame under the assumptions of long wavelength and low Reynolds number, the solutions of the governing equations of Carreau fluid in a rectangular duct have been found by using homotopy perturbation method. The physical features of the pertinent parameters are discussed by plotting pressure rise, velocity, pressure gradient, and stream functions.

2. Mathematical Formulation

Consider the peristaltic flow of an incompressible Carreau fluid in a duct of rectangular cross-section through a porous medium having the channel width and height . We are considering the Cartesian coordinate system in such a way that -axis is taken along the axial direction, -axis is taken along the lateral direction, and -axis is along the vertical direction of rectangular channel.

The peristaltic waves on the walls are represented as

where and are the amplitudes of the waves, is the wavelength, is the velocity of the propagation, is the time, and is the direction of wave propagation. The walls parallel to -plane remain undisturbed and are not subject to any peristaltic wave motion. We assume that the lateral velocity is zero as there is no change in lateral direction of the duct cross-section. Let be the velocity for a rectangular duct. The governing equations for the flow problem are stated as follows.

Continuity equation:

-momentum equation:

-momentum equation:

-momentum equation:

in which is the density, is the pressure, and is the stress tensor for Carreau fluid, which is defined as

Let us define a wave frame moving with the velocity away from the fixed frame by the transformation

Define the following nondimensional quantities:

Using the previous nondimensional quantities in (2.2) to (2.5), the resulting equations (after dropping the bars) can be written as

where

Under the assumption of long wavelength and low Reynolds number , (2.10), take the form

The corresponding boundary conditions are

where .

3. Solution of the Problem

The solution of the previously mentioned nonlinear partial differential equation has been calculated by homotopy perturbation method (HPM), which is defined as [28–44]

in which is embedding parameter which has the range . For our convenience we have taken as the linear operator. We choose the following initial guess:

Define

Substituting (3.3) into (3.1) and then comparing the like powers of , one obtains the following problems with the corresponding boundary conditions:

From (3.4) we have

With the help of (3.8), (3) can be written as

in which

The solution of (3.9) is calculated by eigenfunction expansion method and is defined as

where and other constants are defined in the appendix section. The HPM solution up to first iteration is finally defined as

where and are defined in (3.8) and (3.11). The volumetric flow rate is calculated as

The instantaneous flux is given by

The average volume flow rate over one period of the peristaltic wave is defined as

The pressure gradient is obtained after solving (3.13) and (3.15). The pressure rise is evaluated by using the following expression:

4. Results and Discussions

In this section, we have discussed the homotopy perturbation solution given in (3.12) through plotting the graphs for velocity, pressure gradient, pressure rise, and stream lines. In Figures 4(a) and 4(b), the velocity field is plotted for different values of flow rate , both for two dimensional and three-dimensional flows, respectively. It is observed that with the increase of flow rate , the velocity field decreases. The variation of velocity field for different values of is displayed in Figures 5(a) and 5(b). It is seen that with the increase in , the velocity field decreases, and the maximum velocity is at the centre of the channel for small values of . In Figures 6(a) and 6(b), the velocity field increases with the increase in . However, with the increase in , the velocity field decreases near the channel wall and increases in the middle (see Figures 7(a) and 7(b)). The velocity field distributions for different values of and are shown in Figures 8 and 9. It is seen that the magnitude of velocity increases for different values of , while decreasing for . The pressure gradient graphs for various values of , and are plotted in Figures 10, 11, 12, and 13. It is observed that with the increase in and , the pressure gradient decreases but increases with and , and the maximum pressure gradient occurs in the middle of the channel for small values of the parameters. It means that flow can easily pass in the centre of the channel. The plots for pressure rise for different values of , and are portrayed in Figures 14, 15, and 16. It is seen that pressure rise increases with the increase in in the peristaltic pumping region; that is, when and in the augmented pumping region , the pressure rise decreases (see Figure 14). The pressure rise increases with the increase in in the region and gives opposite behavior in the region [0.6,3] (see Figure 15). In Figure 16, it is depicted that with the increase in , pressure rise decreases for small and increases for large . The trapping phenomenon is presented by plotting stream lines and is shown in Figures 17, 18, and 19. The stream lines for various values of are shown in Figure 17. It is seen that the trapping bolus decreases with the increase in , while size of the bolus is reduced with . The stream lines for different values of and are sketched in Figures 18 and 19. It is depicted that with the increase in both parameters, the numbers of trapping bolus increase. It is also noted that bolus becomes small in size for but enlarges for the parameter .

Tables 1 and 2 are placed to compare the present work with [26]. Figures 2 and 3 are drawn according to the data given in Tables 1 and 2. From these figures it is clear to see that the analytical solution is in good agreement with the exact solution. Also, it is observed that velocity is reduced in the case of non-Newtonian (Carreau) fluid (see Figure 1). From Figure 2, it is seen that pressure rise is increased for the present fluid model.

Table 1: Velocity for various values of for fixed , .

For Newtonian fluid, when ,
For Carreau fluid when ,

for
for

Exact solution [26]
Analytical solution
Analytical solution

−1.6
−1.0000
−1.0000
−1.0000

−1.2
0.2108
0.2115
−0.0938

−0.8
1.1161
1.1173
0.1677

−0.4
1.6746
1.6758
0.2445

0.0
1.8633
1.8645
0.2613

0.4
1.6746
1.6758
0.2445

0.8
1.1161
1.1173
0.1677

1.2
0.2108
0.2115
−0.0938

1.6
−1.0000
−1.0000
−1.0000

Table 2: Pressure rise for various values of .

For Newtonian fluid, when ,
For Carreau fluid when ,

for ,
for ,

Exact solution [26]
Analytical solution
Analytical solution

−0.1
12.514
12.7087
12.8287

0.0
9.4799
9.4399
9.8599

0.1
6.4457
6.4712
6.8912

0.2
3.4115
3.4025
3.9225

0.3
0.3773
0.3338
0.9538

0.4
−2.6569
−2.6350
−2.0149

0.5
−5.6911
−5.6037
−4.9837

0.6
−8.7254
−8.7724
−7.9524

0.7
−11.7596
−11.7412
−10.9211

0.8
−14.7938
−14.7099
−13.8899

Figure 1: Schematic diagram for peristaltic flow in a rectangular duct.

Figure 2: Velocity for various values of for fixed .

Figure 3: Pressure rise for various values of , when .

(a)

(b)

Figure 4: Velocity profile for different values of for fixed . (a) For 2-dimensional and (b) for -dimensional.

(a)

(b)

Figure 5: Velocity profile for different values of for fixed . (a) For 2-dimensional and (b) For 3-dimensional.

(a)

(b)

Figure 6: Velocity profile for different values of for fixed . (a) For -dimensional and (b) for -dimensional.

(a)

(b)

Figure 7: Velocity profile for different values of for fixed . (a) For -dimensional and (b) for -dimensional.

(a)

(b)

Figure 8: Velocity profile for different values of for fixed . (a) For -dimensional and (b) for -dimensional.

(a)

(b)

Figure 9: Velocity profile for different values of for fixed . (a) For -dimensional and (b) for -dimensional.

Figure 10: Variation of with for different values of at .

Figure 11: Variation of with for different values of at .

Figure 12: Variation of with for different values of at .

Figure 13: Variation of with for different values of at .

Figure 14: Variation of with for different values of at .

Figure 15: Variation of with for different values of at .

Figure 16: Variation of with for different values of at ,  .

(a)

(b)

(c)

(d)

Figure 17: Stream lines for different values of . (a) For , (b) for , (c) for , and (d) for . The other parameters are .

(a)

(b)

(c)

(d)

Figure 18: Stream lines for different values of . (a) For , (b) for , (c) for , and (d) for . The other parameters are .

(a)

(b)

(c)

(d)

Figure 19: Stream lines for different values of . (a) for , (b) for , (c) for , and (d) for . The other parameters are .

Appendix

Consider

while the other constants are defined as

Acknowledgment

The author M. Ali extends his appreciation to the deanship of Scientific Research at king Saud University for funding this work through the research group project no. RGP-VPP-080.

References

E. F. Elshehawey, N. T. Eldabe, E. M. Elghazy, and A. Ebaid, “Peristaltic transport in an asymmetric channel through a porous medium,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 140–150, 2006.

S. Nadeem and N. S. Akbar, “Influence of heat transfer on peristaltic transport of a Johnson-Segalman fluid in an inclined asymmetric channel,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 10, pp. 2860–2877, 2010.

S. Nadeem and S. Akram, “Influence of inclined magnetic field on peristaltic flow of a Williamson fluid model in an inclined symmetric or asymmetric channel,” Mathematical and Computer Modelling, vol. 52, no. 1-2, pp. 107–119, 2010.

S. Nadeem and S. Akram, “Heat transfer in a peristaltic flow of MHD fluid with partial slip,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 2, pp. 312–321, 2010.

N. S. Akbar, T. Hayat, S. Nadeem, and A. A. Hendi, “Effects of slip and heat transfer on the peristaltic flow of a third order fluid in an inclined asymmetric channel,” International Journal of Heat and Mass Transfer, vol. 54, no. 7-8, pp. 1654–1664, 2011.

S. Nadeem and S. Akram, “Peristaltic transport of a hyperbolic tangent fluid model in an asymmetric channel,” Zeitschrift fur Naturforschung, vol. 64, no. 9-10, pp. 559–567, 2009.

S. Nadeem and S. Akram, “Slip effects on the peristaltic flow of a Jeffrey fluid in an asymmetric channel under the effect of induced magnetic field,” International Journal for Numerical Methods in Fluids, vol. 63, no. 3, pp. 374–394, 2010.

S. Nadeem and S. Akram, “Peristaltic flow of a couple stress fluid under the effect of induced magnetic field in an asymmetric channel,” Archive of Applied Mechanics, vol. 81, no. 1, pp. 97–109, 2011.

S. Nadeem and N. S. Akbar, “Effects of heat transfer on the peristaltic transport of MHD Newtonian fluid with variable viscosity: application of Adomian decomposition method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 11, pp. 3844–3855, 2009.

S. Nadeem, T. Hayat, N. S. Akbar, and M. Y. Malik, “On the influence of heat transfer in peristalsis with variable viscosity,” International Journal of Heat and Mass Transfer, vol. 52, no. 21-22, pp. 4722–4730, 2009.

S. Nadeem and N. S. Akbar, “Influence of heat transfer on a peristaltic transport of Herschel-Bulkley fluid in a non-uniform inclined tube,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 12, pp. 4100–4113, 2009.

S. Nadeem and S. Akram, “Heat transfer in a peristaltic flow of MHD fluid with partial slip,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 2, pp. 312–321, 2010.

S. K. Pandey and M. K. Chaube, “Peristaltic flow of a micropolar fluid through a porous medium in the presence of an external magnetic field,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 9, pp. 3591–3601, 2011.

N. A. S. Afifi and N. S. Gad, “Interaction of peristaltic flow with pulsatile fluid through a porous medium,” Applied Mathematics and Computation, vol. 142, no. 1, pp. 167–176, 2003.

A. R. Rao and M. Mishra, “Peristaltic transport of a power-law fluid in a porous tube,” Journal of Non-Newtonian Fluid Mechanics, vol. 121, no. 2-3, pp. 163–174, 2004.

M. Mishra and A. R. Rao, “Peristaltic transport in a channel with a porous peripheral layer: model of a flow in gastrointestinal tract,” Journal of Biomechanics, vol. 38, no. 4, pp. 779–789, 2005.

O. Eytan and D. Elad, “Analysis of intra-uterine fluid motion induced by uterine contractions,” Bulletin of Mathematical Biology, vol. 61, no. 2, pp. 221–238, 1999.

S. Nadeem and S. Akram, “Peristaltic flow of a Williamson fluid in an asymmetric channel,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 7, pp. 1705–1716, 2010.

S. Srinivas and M. Kothandapani, “Peristaltic transport in an asymmetric channel with heat transfer—a note,” International Communications in Heat and Mass Transfer, vol. 35, no. 4, pp. 514–522, 2008.

S. Srinivas and R. Gayathri, “Peristaltic transport of a Newtonian fluid in a vertical asymmetric channel with heat transfer and porous medium,” Applied Mathematics and Computation, vol. 215, no. 1, pp. 185–196, 2009.

K. Vajravelu, S. Sreenadh, and V. R. Babu, “Peristaltic pumping of a Herschel-Bulkley fluid in a channel,” Applied Mathematics and Computation, vol. 169, no. 1, pp. 726–735, 2005.

S. Nadeem and N. S. Akbar, “Effects of temperature dependent viscosity on peristaltic flow of a Jeffrey-six constant fluid in a non-uniform vertical tube,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp. 3950–3964, 2010.

D. Srinivasacharya, M. Mishra, and A. R. Rao, “Peristaltic pumping of a micropolar fluid in a tube,” Acta Mechanica, vol. 161, no. 3-4, pp. 165–178, 2003.

M. Elshahed and M. H. Haroun, “Peristaltic transport of Johnson-Segalman fluid under effect of a magnetic field,” Mathematical Problems in Engineering, vol. 2005, no. 6, pp. 663–677, 2005.

M. H. Haroun, “Non-linear peristaltic flow of a fourth grade fluid in an inclined asymmetric channel,” Computational Materials Science, vol. 39, no. 2, pp. 324–333, 2007.

M. V. S. Reddy, M. Mishra, S. Sreenadh, and A. R. Rao, “Influence of lateral walls on peristaltic flow in a rectangular duct,” Journal of Fluids Engineering, vol. 127, no. 4, pp. 824–827, 2005.

S. Nadeem and S. Akram, “Peristaltic flow of a Jeffrey fluid in a rectangular duct,” Nonlinear Analysis, vol. 11, no. 5, pp. 4238–4247, 2010.

J. H. He, “A note on the homotopy perturbation method,” Thermal Science, vol. 14, no. 2, pp. 565–568, 2010.

J. H. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters A, vol. 350, no. 1-2, pp. 87–88, 2006.

J. H. He, “An elementary introduction to the homotopy perturbation method,” Computers and Mathematics with Applications, vol. 57, no. 3, pp. 410–412, 2009.

A. Yldrm, “Applying He's variational iteration method for solving differential-difference equation,” Mathematical Problems in Engineering, vol. 2008, Article ID 869614, 7 pages, 2008.

S. T. Mohyud-Din and A. Yildirim, “Solving nonlinear boundary value problems using He's polynomials and Padé approximants,” Mathematical Problems in Engineering, vol. 2009, Article ID 690547, 17 pages, 2009.

S. T. Mohyud-Din, A. Yildirim, S. A. Sezer, and M. Usman, “Modified variational iteration method for free-convective boundary-layer equation using padé approximation,” Mathematical Problems in Engineering, vol. 2010, Article ID 318298, 11 pages, 2010.

G. Domairry, M. Ahangari, and M. Jamshidi, “Exact and analytical solution for nonlinear dispersive K (m, p) equations using homotopy perturbation method,” Physics Letters A, vol. 368, no. 3-4, pp. 266–270, 2007.

S. S. Ganji, A. Barari, M. G. Sfahani, G. Domairry, and P. T. Baboli, “Consideration of transient stream/aquifer interaction with the nonlinear Boussinesq equation using HPM,” Journal of King Saud University, vol. 23, no. 2, pp. 211–219, 2011.

M. Jalaal, D. D. Ganji, and G. Ahmadi, “Analytical investigation on acceleration motion of a vertically falling spherical particle in incompressible Newtonian media,” Advanced Powder Technology, vol. 21, no. 3, pp. 298–304, 2010.

M. Jalaal and D. D. Ganji, “On unsteady rolling motion of spheres in inclined tubes filled with incompressible Newtonian fluids,” Advanced Powder Technology, vol. 22, no. 1, pp. 58–67, 2011.

M. Jalaal and D. D. Ganji, “An analytical study on motion of a sphere rolling down an inclined plane submerged in a Newtonian fluid,” Powder Technology, vol. 198, no. 1, pp. 82–92, 2010.

M. Jalaal, D. D. Ganji, and G. Ahmadi, “An analytical study on settling of non-spherical particles,” Asia-Pacific Journal of Chemical Engineering, vol. 7, pp. 63–72, 2012.

D. D. Ganji and A. Sadighi, “Application of He's homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 4, pp. 411–418, 2006.

H. Tari, D. D. Ganji, and H. Babazadeh, “The application of He's variational iteration method to nonlinear equations arising in heat transfer,” Physics Letters A, vol. 363, no. 3, pp. 213–217, 2007.

D. D. Ganji and A. Rajabi, “Assessment of homotopy-perturbation and perturbation methods in heat radiation equations,” International Communications in Heat and Mass Transfer, vol. 33, no. 3, pp. 391–400, 2006.

D. D. Ganji, M. Rafei, A. Sadighi, and Z. Z. Ganji, “A comparative comparison of He's method with perturbation and numerical methods for nonlinear vibrations equations,” International Journal of Nonlinear Dynamics in Engineering and Sciences, vol. 1, no. 1, pp. 1–20, 2009.

Z. Z. Ganji, D. D. Ganji, and E. Esmaeilpour, “Study on nonlinear Jeffery-Hamel flow by He's semi-analytical methods and comparison with numerical results,” Computers and Mathematics with Applications, vol. 58, no. 11-12, pp. 2107–2116, 2009.