Jeffery Hamel Flow of a non-Newtonian Fluid

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan

2 HITEC University Taxila Cantt Pakistan

Abstract

This paper presents the Jeffery Hamel flow of a non-Newtonian fluid namely Casson fluid. Suitable similarity transform is applied to reduce governing nonlinear partial differential equations to a much simpler ordinary differential equation. Variation of Parameters Method (VPM) is then employed to solve resulting equation. Same problem is solved numerical by using Runge-Kutta order 4 method. A comparison between both the solutions is carried out to check the efficiency of VPM. Effects of emerging parameters are demonstrated both for diverging and converging channels using graphical simulation.

Keywords

Main Subjects

[1] G. Hamel, Spiralförmige Bewgungen Zäher Flüssigkeiten, Jahresber. Deutsch.Math. Verein., 25 (1916) 34-60.
G. Hamel, SpiralförmigeBewgungenZäherFlüssigkeiten, Jahresber. Deutsch.Math.  Verein., 25 (1916) 34-60.
[2] L. Rosenhead, The steady two-dimensional radial flow of viscous fluid between two inclined plane walls, Proc. R. Soc. A 175 (1940) 436-467.
[3] K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967.
[4] R. Sadri., Channel entrance flow, PhD thesis, Dept. Mechanical Engineering, the University of Western Ontario, 1997.
[5] M. Hamadiche, J. Scott,D. Jeandel.Temporal stability of Jeffery–Hamel flow. J Fluid Mech, 268 (1994) 71–88.
[6] E. Fraenkel, Laminar flow in symmetrical channels with slightly curved walls. I:On the Jeffery-Hamel solutions for flow between plane walls, Proc. R. Soc. Lond, A 267 (1962) 119-138.
[7] Hermann Schlichting, Boundary-layer Theory, McGraw-Hill Press, New York, 2000.
[8] E. W. Mrill, A. M. Benis, E. R. Gilliland, , T. K. Sherwood, E. W. Salzman, Pressure flow relations of human blood hollow fibers at low flow rates. Journal of Applied Physiology, 20 (1965), 954–967.
[9] D. A. McDonald, Blood Flows in Arteries, 2nd ed., Arnold, London (1974).
[10] S. Nadeem, R.Ul Haq, C. Lee, MHD flow of a Casson fluid over an exponentially shrinking sheet, Scientia Iranica, 19 (2012) 1150-1553.
[11] N. Ahmed, U. Khan, S. I. U. Khan, Y. X. Jun, Z. A. Zaidi, S. T. Mohyud-Din, Magneto hydrodynamic (MHD) Squeezing Flow of a Casson Fluid between Parallel Disks, International Journal of Physical Sciences, 8 (2013) 1788-1799.
[12] S. Nadeem, R. Ul Haq, N. S. Akbar, Z. H. Khan, MHD three-dimensional Casson fluid flow past a porous linearly stretching sheet, Alexandria Engineering Journal, In Press.
[13] S. Abbasbandy, A new application of He's variational iteration method for quadratic Riccati differential equation by using Adomian's polynomials, Journal of Computational and Applied Mathematics 207 (2007), 59-63.
[14] S. Abbasbandy, Numerical solutions of nonlinear Klein-Gordon equation by variational iteration method, International Journal of Numerical Methods in Engineering, 70 (2007), 876-881.
[15] M. A. Abdou and A. A. Soliman, Variational iteration method for solving Burger's and coupled Burger's equations, Journal of Computational and Applied Mathematics 181 (2005), 245-251.
[16] M. A. Noor and S. T. Mohyud-Din, Variational iteration technique for solving higher order boundary value problems, Applied Mathematics and Computation, 189 (2007) 1929—1942.
[17] M. A. Abdou and A. A. Soliman, New applications of variational iteration method, Physica D, 211 (1-2) (2005), 1-8.
[18] M. Asadullah, U. Khan, N. Ahmed, R. Manzoor, S.T. Mohyud-Din, International Journal of Modern Mathematical Sciences, 6 (2013), 92-106.
[19] R. Ellahi, The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: Analytical solutions, Applied Mathematical Modeling, 37 (2013) 1451-1467.
[20] R. Ellahi, M. Raza, K. Vafai, Series solutions of non-Newtonian nanofluids with Reynolds’ model and Vogel’s model by means of the homotopy analysis method, Mathematical and Computer Modelling, 55 (2012) 1876–1891.
[21] M. A. Noor, S. T. Mohyud-Din and A. Waheed, Variation of parameters method for solving fifth-order boundary value problems. Applied Mathematics and Information Sciences, 2 (2008), 135 -141.
[22] S. T. Mohyud-Din, M. A. Noor, A. Waheed, Variation of parameter method for solving sixth-order boundary value problems, Communication of the Korean Mathematical Society, (2009), 24, 605-615.
[23] S. T. Mohyud-Din, M. A. Noor, A. Waheed, Variation of parameter method for initial and boundary value problems, World Applied Sciences Journal, 11 (2010) 622-639.
[24] T. Mohyud-Din, M. A. Noor, A. Waheed, Modified Variation of Parameters Method for Second-order Integro-differential Equations and Coupled Systems, World Applied Sciences Journal, 6 (2009) 1139-1146.
[25] U. Khan, N. Ahmed, Z. A. Zaidi, S. U. Jan, S. T. Mohyud-Din, On Jeffery-Hamel Flows, International Journal of Modern Mathematical Sciences, 7 (2013), 236-247.
[26] U. Khan, N. Ahmed, Z. A. Zaidi, M. Asadullah, S. T. Mohyud-Din, MHD Squeezing Flow between Two Infinite Plates, Ain Shams Engineering Journal, In Press.