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Travelling Wave Solutions for the Unsteady Flow of a Third Grade Fluid Induced Due to Impulsive Motion of Flat Porous Plate Embedded in a Porous Medium

Published online by Cambridge University Press:  13 March 2014

T. Aziz ,
F. M. Mahomed ,
A. Shahzad  and
R. Ali
T. Aziz*
Affiliation:
Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, South Africa
F. M. Mahomed
Affiliation:
Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, South Africa
A. Shahzad
Affiliation:
Department of Mathematics, Quaid-i-Azam University, 45320 Islamabad, Pakistan
R. Ali
Affiliation:
Department of Applied Mathematics, Technical University Dortmund LS-III, Vogelpothsweg 87, Germany
*
*tahaaziz77@yahoo.com
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Abstract

This work describes the time-dependent flow of an incompressible third grade fluid filling the porous half space over an infinite porous plate. The flow is induced due to the motion of the porous plate in its own plane with an arbitrary velocity V(t). Translational type symmetries are employed to perform the travelling wave reduction into an ordinary differential equation of the governing nonlinear partial differential equation which arises from the laws of mass and momentum. The reduced ordinary differential equation is solved exactly, for a particular case, as well as by using the homotopy analysis method (HAM). The better solution from the physical point of view is argued to be the HAM solution. The essentials features of the various emerging parameters of the flow problem are presented and discussed.

Keywords

Third grade fluidSuction/InjectionTravelling wave solutionsHomotopy analysis method (HAM)
Type
Research Article
Information
Journal of Mechanics , Volume 30 , Issue 5 , October 2014 , pp. 527 - 535
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2014 

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References

1.Rivlin, R. S. and Ericksen, J. L., “Stress Deformation Relation for Isotropic Materials,” Journal of Rational Mechanics and Analysis, 4, pp. 323–329 (1955).Google Scholar
2.Vieru, D., Siddique, I., Kamran, M. and Fetecau, C., “Energetic Balance for the Flow of a Second-Grade Fluid Due to a Plate Subject to a Shear Stress,” Computers & Mathematics with Applications, 56, pp. 1128–1137 (2008).Google Scholar
3.Cortell, R., “MHD Flow and Mass Transfer of an Electrically Conducting Fluid of Second Grade in a Porous Medium Over a Stretching Sheet with Chemically Reactive Species,” Chemical Engineering Processing, 46, pp. 721–728 (2007).Google Scholar
4.Vajravelu, K. and Rollins, D., “Hydromagnetic Flow of a Second Grade Fluid Over a Stretching Sheet,” Applied Mathematics and Computation, 148, pp. 783–791 (2004).Google Scholar
5.Emin, M., Erdogen, C. and Erdem Imrak, “On Unsteady Unidirectional Flows of a Second Grade Fluid,” International Journal of Non-Linear Mechanics, 40, pp. 1238–1251 (2005).Google Scholar
6.Fosdick, R. L. and Rajagopal, K. R., “Thermodynamics and Stability of Fluids of Third Grade,” Proceedings of the Royal Society of London, Series A, 339, pp. 351–377 (1980).Google Scholar
7.Ariel, P. D., “Flow of a Third Grade Fluid Through a Porous Flat Channel,” International Journal of Engineering Science, 41, pp. 1267–1285 (2003).Google Scholar
8.Hayat, T., Mamboundou, H. M. and Mahomed, F. M., “Unsteady Solutions in a Third Grade Fluid Filling the Porous Space,” Mathematics Problems in Engineering, 2008, Article ID 139560 (2008).Google Scholar
9.Hayat, T., Kara, A. H. and Momoniat, E., “Exact Flow of a Third-Grade Fluid on a Porous Wall,” International Journal of Non-Linear Mechanics, 38, pp. 1533–1537 (2003).CrossRefGoogle Scholar
10.Szeri, A. Z. and Rajagopal, K. R., “Flow of a Non-Newtonian Fluid Between Heated Parallel Plates,” International Journal of Non-Linear Mechanics, 20, pp. 91–101 (1985).Google Scholar
11.Akyildiz, F. T., Bellout, H. and Vajravelu, K., “Exact Solutions of Nonlinear Differential Equation Arising in Third Grade Fluid Flows,” International Journal of Non-Linear Mechanics, 39, pp. 1571–1578 (2004).Google Scholar
12.Yurusay, M. and Pakdemirli, M., “Approximate Analytical Solutions for the Flow of a Third Grade Fluid in a Pipe,” International Journal of NonLinear Mechanics, 37, pp. 187–195 (2002).Google Scholar
13.Rajagopal, K. R., Szeri, A. Z. and Troy, W., “An Existence Theorem for the Flow of a Non-Newtonian Fluid Past an Infinite Porous Plate,” International Journal of Non-Linear Mechanics, 21, pp. 279–289 (1986).Google Scholar
14.Makinde, O. D., “Thermal Criticality for a Reactive Gravity Driven Thin Film Flow of a Third-Grade Fluid with Adiabatic Free Surface Down an Inclined Plane,” Applied Mathematics and Mechanics, English Edition, 30, pp. 373–380 (2009).Google Scholar
15.Hayat, T. and Kara, A. H., “Couette Flow of a Third-Grade Fluid with Variable Magnetic Field,” Mathematical and Computer Modelling, 43, pp. 132–137 (2006).Google Scholar
16.Vafai, K., Handbook of Porous Media, Taylor & Francis, New York, 2005.CrossRefGoogle Scholar
17.Bear, J., Dynamics of Fluids in Porous Media, Dover, New York (1972).Google Scholar
18.Liu, S. and Masliyah, J. H., “On Non-Newtonian Fluid Flow in Ducts and Porous Media,” Chemical Engineering Sciences, 53, pp. 1175–1201 (1998).CrossRefGoogle Scholar
19.Nield, D. A. and Bejan, A., Convection in Porous Media, 2nd Edition, Springer, Berlin. 1999.Google Scholar
20.Shah, C. B. and Yortsos, Y. C., “Aspects of Flow of Power Law Fluids in Porous Media,” Journal of American Institute of Chemical Engineering, 41, pp. 1099–1112 (1995).Google Scholar
21.Tan, W. C. and Masuoka, T., “Stokes' First Problem for a Second Grade Fluid in a Porous Half Space with Heated Boundary,” International Journal of Non-Linear Mechanics, 40, pp. 515–522 (2005).Google Scholar
22.Tan, W. C. and Masuoka, T., “Stokes' First Problem for an Oldroyd-B Fluid in a Porous Half Space,” Physics Fluids, 17, pp. 23101–23107 (2005).Google Scholar
23.Liao, S. J., “On the Proposed Homotopy Analysis Technique for Nonlinear Problems and Its Applications,” Ph. D Dissertation, Shanghai Jiao Tong University (1992).Google Scholar
24.Liao, S. J. and Campo, A., “Analytic Solutions of the Temperature Distribution in Blasius Viscous Flow Problems,” Journal of Fluid Mechanics, 25, pp. 411–453 (2002).Google Scholar
25.Ziabakhsh, Z. and Domairry, G., “Analytic Solution of Natural Convection Flow of a Non-Newtonian Fluid Between Two Vertical Flat Plates Using Ho-motopy Analysis Method,” Communications in Nonlinear Science and Numerical Simulation, 14, pp. 1868–1880 (2009).Google Scholar
26.Liao, J. S., “On the Homotopy Analysis Method for Non-Linear Problems,” Applied Mathematics and Computation, 47, pp. 499–513 (2004).Google Scholar
27.Esmaeilpour, M., Domairry, G., Sadoughi, N. and Davodi, A. G., “Homotopy Analysis Method for the Heat Transfer of a Non-Newtonian Fluid Flow in a Axisymmetric Channel with a Porous Wall,” Communications in Nonlinear Science and Numerical Simulation, 15, pp. 2424–2430 (2010).Google Scholar
28.Wang, C., and Pop, I., “Analysis of the Flow of a Power-Law Fluid Flow on an Unsteady Stretching Surface by Means of Homotopy Analysis Method,” Journal of Non-Newtonian Fluid Mechanics, 138, pp. 161–172 (2006).Google Scholar
29.Sajid, M. and Hayat, T., “The Application of Homotopy Analysis Method to Thin Film Flows of a Third Grade Fluid,” Chaos Solitons and Fractals, 38, pp. 506–515 (2008).Google Scholar
30.Hayat, T. and Sajid, M., “On Analytic Solution for Thin Film Flow of a Forth Grade Fluid Down a Vertical Cylinder,” Physics Letters A, 361, pp. 316–322 (2007).Google Scholar
31.Hayat, T., Khan, M. and Sajid, M., Asghar, S., “Rotating Flow of a Third Grade Fluid in a Porous Space With Hall Current,” Nonlinear Dynamics, 49, pp. 83–91 (2007).CrossRefGoogle Scholar
32.Abbasbandy, S., “The Application of the Homotopy Analysis Method to Nonlinear Equations Arising in Heat Transfer,” Physics Letters A, 360, pp. 109–113 (2006).Google Scholar
33.Abbasbandy, S., “Homotopy Analysis Method for Heat Radiation Equations,” International Communications Heat and Mass Transfer, 34, pp. 380–387 (2007).Google Scholar
34.Fakhar, K., Kara, A. H., Khan, I. and Sajid, M., “On the Computation of Analytical Solutions of an Unsteady Magnetohydrodynamic Flow of a Third Grade Fluid with Hall Effects,” Communications Mathematical Applied, 61, pp. 980–987 (2011).Google Scholar
35.Fakhar, K., Xu, Z. and Yi, C., “Exact Solutions of a Third Grade Fluid Flow on a Porous Plate,” Applied Mathematics and Computation, 202, pp. 376–382 (2008).Google Scholar