Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel. (English) Zbl 1352.76131
Summary: The paper presents the transportation of viscoelastic fluid with fractional Maxwell model by peristalsis through a channel under long wavelength and low Reynolds number approximations. The propagation of wall of channel is taken as sinusoidal wave propagation (contraction and relaxation). Homotopy perturbation method (HPM) and Adomian decomposition method (ADM) are used to obtain the analytical approximate solutions of the problem. The expressions of axial velocity, volume flow rate and pressure gradient are obtained. The effects of fractional parameters \((\alpha )\), relaxation time \((\lambda_{1})\) and amplitude (\(\phi\)) on the pressure difference and friction force across one wavelength are calculated numerically for different particular cases and depicted through graphs.
MSC:
| 76Z05 | Physiological flows |
| 76A10 | Viscoelastic fluids |
| 76M25 | Other numerical methods (fluid mechanics) (MSC2010) |
Keywords:
peristalsis; viscoelastic fluid; fractional Maxwell model; homotopy perturbation method; Adomian decomposition method; friction force; Mittag-Leffler functionReferences:
| [1] | Shapiro, A. H.; Jafferin, M. Y.; Weinberg, S. L., Peristaltic pumping with long wavelengths at low Reynolds number, J. Fluid Mech., 35, 669-675 (1969) |
| [2] | Bohme, G.; Friedrich, R., Peristaltic flow of viscoelastic liquids, J. Fluid Mech., 128, 109-122 (1983) · Zbl 0515.76131 |
| [3] | Tsiklauri, D.; Beresnev, I., Non-Newtonian effects in the peristaltic flow of a Maxwell fluid, Phys. Rev. E, 64, 036303 (2001) |
| [4] | El-Shehawy, E. F.; El-Dabe, N. T.; El-Desoki, I. M., Slip effects on the peristaltic flow of a non-Newtonian Maxwellian fluid, Acta Mech., 186, 141-159 (2006) · Zbl 1106.76009 |
| [5] | Hayat, T.; Ali, N.; Asghar, S., Hall effects on the peristaltic flow of a Maxwell fluid in a porous medium, Phys. Lett. A, 363, 397-403 (2007) · Zbl 1197.76126 |
| [6] | Hayat, T.; Ali, N., Peristaltic motion of a Jeffrey fluid under the effect of a magnetic field in a tube, Commun. Nonlinear Sci. Numer. Simul., 13, 1343-1352 (2008) · Zbl 1221.76016 |
| [7] | Hayat, T.; Ali, N.; Asghar, S.; Siddiqui, A. M., Exact peristaltic flow in tubes with an endoscope, Appl. Math. Comput., 182 (2006) · Zbl 1103.76073 |
| [8] | Hayat, T.; Ahmad, N.; Ali, N., Effects of endoscope and magnetic field on the peristalsis involving Jeffrey fluid, Commun. Nonlinear Sci. Numer. Simul., 13, 1581-1591 (2008) · Zbl 1221.76078 |
| [9] | Hayat, T.; Ali, N.; Asghar, S., An analysis of peristaltic transport for flow of a Jeffrey fluid, Acta Mech., 193, 101-112 (2007) · Zbl 1120.76002 |
| [10] | Hayat, T.; Javed, M.; Ali, N., MHD peristaltic transport of a Jeffrey fluid in a channel with compliant walls and porous space, Trans. Porous Med. (2007) |
| [11] | Barnes, H. A.; Hutton, J. F.; Walters, K., An introduction to Rhyology (1989), Elsevier: Elsevier Amsterdam · Zbl 0729.76001 |
| [12] | Takahashi, T.; Ogoshi, H.; Miyamoto, K.; Yao, M. L., Viscoelastic properties of commercial plain yoghurts and trial foods for swallowing disorders, Rheology, 27, 169-172 (1999) |
| [13] | Freidrich, C., Relaxation and retardation functions of the Maxwell model with fractional derivatives, Rheol. Acta, 30, 151-158 (1991) |
| [14] | Tan, W.; Pan, W.; Xu, M., A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates, Int. J. Non-Linear Mech., 38, 645-650 (2003) · Zbl 1346.76009 |
| [15] | Qi, H.; Jin, H., Unsteady rotating flows of a viscoelastic fluid with the fractional Maxwell model between coaxial cylinders, Acta Mech. Sinica, 22, 301-305 (2006) · Zbl 1202.76016 |
| [16] | Qi, H.; Xu, M., Unsteady flow of viscoelastic fluid with fractional Maxwell model in channel, Mech. Res. Commun., 34, 210-212 (2007) · Zbl 1192.76008 |
| [17] | Hayat, T.; Nadeem, S.; Asghar, S., Periodic unidirectional flows of a viscoelastic fluid with the fractional Maxwell model, Appl. Math. Comput., 151, 153-161 (2004) · Zbl 1161.76436 |
| [18] | Khan, M.; Ali, S. H.; Fetecau, C.; Qi, H., Decay of potential vortex for a viscoelastic fluid with fractional Maxwell model, Appl. Math. Model., 33, 2526-2533 (2009) · Zbl 1185.76523 |
| [19] | He, J. H., Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng., 178, 257-262 (1999) · Zbl 0956.70017 |
| [20] | He, J. H., A coupling method of homotopy technique and perturbation technique for non-linear problems, Int. J. Nonlinear Mech., 35, 37-43 (2000) · Zbl 1068.74618 |
| [21] | He, J. H., Periodic solutions and bifurcations of delay-differential equations, Phys. Lett. A, 347, 228-230 (2005) · Zbl 1195.34116 |
| [22] | He, J. H., Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons Fract., 26, 695-700 (2005) · Zbl 1072.35502 |
| [23] | He, J. H., Homotopy Perturbation method for bifurcation of nonlinear problems, Int. J. Nonlinear Sci. Numer. Simul., 6, 207-208 (2005) · Zbl 1401.65085 |
| [24] | He, J. H., Homotopy perturbation method for solving boundary value problems, Phys. Lett. A, 350, 87-88 (2006) · Zbl 1195.65207 |
| [25] | Momani, S.; Odibat, Z., Comparison between the HPM and the VIM for linear fractional partial differential equations, Comput. Math. Appl., 54, 910-919 (2007) · Zbl 1141.65398 |
| [26] | Ganji, Z. Z.; Ganji, D. D.; Jafari, H.; Rostamian, M., Application of the homotopy perturbation method to coupled system of partial differential equations with time fractional derivatives, Topolog. Methods Nonlinear Anal., 31, 341-348 (2008) · Zbl 1163.35312 |
| [28] | Das, S.; Gupta, P. K.; Rajeev, A fractional predator-prey model and its solution, Int. J. Nonlinear Sci. Numer. Simul., 10, 873-876 (2009) |
| [29] | Adomian, G., Nonlinear stochastic differential equations, J. Math. Anal. Appl., 55, 441-452 (1976) · Zbl 0351.60053 |
| [30] | Adomian, G., A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135, 501-544 (1988) · Zbl 0671.34053 |
| [31] | Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Boston and London · Zbl 0802.65122 |
| [32] | Adomian, G.; Rach, R., On the solution of algebraic equations by the decomposition method, J. Math. Anal. Appl., 105, 141-166 (1985) · Zbl 0552.60060 |
| [33] | Abbaoui, K.; Cherruault, Y., Convergence of Adomian’s method applied to nonlinear equations, Math. Comput. Model., 20, 69-73 (1994) · Zbl 0822.65027 |
| [34] | Abbaoui, K.; Cherruault, Y., New ideas for proving convergence of decomposition methods, Comput. Math. Appl., 29, 103-108 (1995) · Zbl 0832.47051 |
| [35] | Abbaoui, K.; Cherruault, Y.; Seng, V., Practical formulae for the calculus of multivariable Adomian polynomials, Math. Comput. Model., 22, 89-93 (1995) · Zbl 0830.65010 |
| [36] | Cherruault, Y., Convergence of Adomian’s method, Kyberbetes, 8, 31-38 (1988) · Zbl 0697.65051 |
| [37] | Cherruault, Y.; Adomian, G., Decomposition method: a new proof of convergence, Math. Comput. Model., 18, 103-106 (1993) · Zbl 0805.65057 |
| [38] | Cherruault, Y.; Adomian, G.; Abbaoui, K.; Rach, R., Further remarks on convergence of decomposition method, Int. J. Bio-Medical Comput., 38, 89-93 (1995) |
| [39] | Gabet, L., The theoretical foundation of the Adomian method, Comput. Math. Appl., 27, 41-52 (1994) · Zbl 0805.65056 |
| [40] | Mekheimer, Kh. S., Effect of the induced magnetic field on the peristaltic flow of a couple stress fluid, Phys. Lett. A, 372, 4271-4278 (2008) · Zbl 1221.76231 |
| [41] | Hayat, T.; Alvi, N.; Ali, N., Peristaltic mechanism of a Maxwell fluid in an asymmetric channel, Nonlinear Anal.: Real World Appl., 9, 1474-1490 (2008) · Zbl 1154.74334 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
