Analytical Expression for the Hydrodynamic Fluid Flow through a Porous Medium

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Analytical Expression for the Hydrodynamic Fluid Flow through a Porous Medium V.Ananthaswamy1, S.Uma Maheswari2

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Department of Mathematics, The Madura College (Autonomous), Maduri, Tamil Nadu, India

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M. Phil., Mathematics, The Madura College (Autonomous), Maduri, Tamil Nadu, India

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ananthu9777@rediffmail.com; 2 umashanmugam1992@gmail.com

Abstract

In this research article the effect of variable viscosity on the temporal development of small disturbances in a pressure-driven fluid flow through a channel saturated with porous medium is investigated. The approximate analytical solution of the second order boundary value problem for the dimensionless velocity is derived by using the Homotopy analysis method. This method can be easily extended to solve the non-liner initial and boundary value problems in physical, chemical in engineering and sciences. Keywords

Variable Viscosity; Porous Medium; Hydrodynamic Flow; Boundary Value Problem; Homotopy Analysis Method Introduction In the last few years, studies related to hydrodynamic stability of a moving viscous fluid through a channel filled with saturated porous medium played a key role in transport process, petrochemical engineering and geo-physical flows. It was because of that the study provided useful information on the sequence of fluids from lamina to turbulent flows. Turbulent flow has been used in some real life application. For example, in arterial blood flow with multiple stenosis, shipping over deep seas and in aviation industry and many more. Currently, several work has been done in this area of research for example, Makinde [2003] reported the linear stability of hydromagnetic plane-Poiseuille flow at high Reynolds numbers by using the multideck asymptotic approach. Makinde and Mhone [2007] investigated the temporal development of small disturbances in magneto hydrodynamic Jeffery–Hamel flows through a convergent-divergent channel. Furthermore Makinde [2009] examined the temporal development of small disturbances in a pressure-driven fluid flow through a channel filled with a saturated porous medium by using the Brinkman flow model. In all the above mentioned studies, the fluid viscosity has been studied constant. Viscosity is a very sensitive fluid property that varies with temperature, pressure or both in some cases. Therefore, as suggested in [Liao, (1992 & 1995)] the effect of stenosis can be captured in the model by taking the artery as a porous medium. Motivated by the results in [Protter (1984), Liao., (1992 & 1995) ], the specific objective of this paper is to investigate the effects of variations in viscosity and porous permeability on the flow stability which has not been accounted for in the previous models in literature. Mathematical Formulations of the Problem Consider the flow of a variable viscous, incompressible fluid through a channel filled with saturated porous materials. The dynamic viscosity of the fluid is assumed to vary with the channel width. If we employed a Cartesian coordinate system such that the x-axis corresponds to the flow direction and the y-axis is normal to it, then in 2-dimensions, the flow governing equations can be written as: u v  0 x y

International Journal of Automation and Control Engineering, Vol. 4, No. 2—October 2015 2325-7407/15/02 067-10 © 2015 DEStech Publications, Inc. doi:10.12783/ijace.2015.0402.02

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