Unsteady Mhd Flow of a Viscous Fluid Through Porous Medium

Investigating the Unsteady MHD Flow of Viscous Fluid through Porous Media under Transverse Magnetic Field

by Mr. Satish Kumar*, Dr. Bhagvat Swaroop,

- Published in Journal of Advances in Science and Technology, E-ISSN: 2230-9659

Volume 3, Issue No. 6, Aug 2012, Pages 0 - 0 (0)

Published by: Ignited Minds Journals

ABSTRACT

Thispaper analyzed the exact solution of the unsteady motion of an electricallyconducting; incompressible viscous fluid through porous medium under the actionof transverse of magnetic field is obtained. The velocity and temperatureprofiles are obtained analytically and used to compute the wall shear stressand rate of heat Transfer at the channel walls. On the basis of certainsimplifying assumption, and the fluid equations of continuity momentum andenergy are obtained.

KEYWORD

unsteady MHD flow, viscous fluid, porous medium, exact solution, electrically conducting, incompressible, transverse magnetic field, velocity profile, temperature profile, wall shear stress, rate of heat transfer, channel walls, simplifying assumption, fluid equations, continuity momentum, energy

INTRODUCTION

The study of flow of an electrically conducting fluid has many applications in engineering problems such as MHD generator plasma studies, nuclear reactor, geothermal energy extraction, and boundary layer control in the field of aerodynamic. The study of the motion of Newtonian fluids in the presence of a magnetic field has applications in many areas, including the handling of biological fluids and the flow of nuclear fuel slurries, liquid metals and alloys, plasma, mercury amalgams, and blood. This happens because the inertial effects become important. Recently, Ahamadi and manvi (1971) derived a general equation of motion and applied the results obtained to some basic flow problems. Raptis (1983) studied the free convective flow through porous medium bounded by an infinite vertical plate with oscillating plate temperature and constant suction. Attia (1999) discussed the Transient MHD flow and heat transfer between two parallel plates with temperature dependent viscosity. Kunugi et al (2005) gave MHD effect on flow structures and heat transfer characteristics of liquid metal–gas annular flow in a vertical pipe. Abbas et al (2008) studied the Hydromegnatic flow in a viscoelastic fluid due to the oscillatory stretching surface. Samadi et al (2009) discussed the Analytic solution for heat transfer of a third grade viscoelastic fluid in non-Darcy porous media with thermo physical effects. In the present paper, we investigate the combined effects of a transverse magnetic field on unsteady flow of a conducting optically thin fluid through a channel filled with saturated porous medium and non-uniform walls temperature. In the following sections, the problem is formulated, solved and the pertinent results are discussed.

MATHEMATICAL FORMULATION

We consider the unsteady of an electrically conducting, incompressible and viscous fluid through a porous medium in the present of transverse magnetic field. Take a Cartesian coordinate system (x, y) where ox lies along the centre of channel, y is the distance measured in the normal suction. The governing equations of motion are



22002

upuuBugTTtxyK …. (1)

10p

y

 …. (2)

0u x

 …. (3)

With the initial boundary conditions

…. (4)

Where u is the axial velocity, t the time, T the fluid temperature, g the gravitational force, q the radiative heat flux, the coefficient of volume expansion, K the permeability of porous medium, 00eBH

2

permeability, 0 the intensity of magnetic field, ethe conductivity of the fluid,  is the fluid density, is the kinematics viscosity coefficient, T0 and Tw is the wall temperature. Let us introduce the following non-dimensional variable



0

02220202

Re,,,,,

,,,

w we

TTUaxyuxyuddUTT gTTddBdPKHPDaGrUaU

 





 

Where Gr the grashoff number, U is the flow mean velocity, H the Hartmann number, Re the Reynolds number, Da is the Darcy number and s =1/Da is the porous medium shape factor parameter. Using the non-dimensional variable the equations (1), (2) and (3) becomes



2222ReupuusHuGrtxyK

 …. (5)

0p y

 

 …. (6)

0u x

 

 …. (7)

With the initial boundary conditions

0,0,1, 0,0,0,

uy uy

 



 .... (8)

From equation (6) it is clear that P* is independent of y*, solving the equation (5) for purely oscillatory flow. Let

0,,ititpeuytuyex

, …. (9)

oscillation, substituting the equation (9) in (5) we obtain the following form

220002

dumuGrdy

…. (10)

With the boundary conditions