American Journal of Computational and Applied M athematics 2013, 3(4): 220-224 DOI: 10.5923/j.ajcam.20130304.04
On the Steady MHD Poiseuille Flow between Two Infinite Parallel Porous Plates in an Inclined Magnetic Field Alfred W. Manyonge 1,* , Jacob K. Bitok2 , Dionysius W. Kiema3 1
Department of Pure and Applied M athematics, M aseno University, M aseno, P.O. BOX 333-40105, Kenya Department of M athematics and Computer Science, Chepkoilel University College, Eldoret, P.O. BOX 1125-30100, Kenya 3 Department of M athematics, Pwani University College, M ombasa, P.O. Kilifi, Kenya
2
Abstract In this paper, we examine the motion of a two dimensional steady flow of a v iscous, electrically conducting, incompressible flu id flowing between two infinite parallel p lates one of which is porous and under the in fluence of a transverse magnetic field and constant pressure gradient. The lower plate is assumed porous while the upper plate is not. The resulting coupled governing equation of motion is solved analytically by an analytical approach. Analytical expression for th e flu id velocity obtained is expressed in terms of Hart mann number. The effects of the magnetic inclinations, Hart mann number, suction/injection and pressure gradient to the velocity are discussed graphically.
Keywords Magnetohydrodynamic(M HD) Flow, Magnetic Field, Navier-Stokes Equations, Porous Plate, Analytic Solution
1. Introduction Magnetohydrodynamics (MHD) is the flu id mechanics of electrically conducting fluids. So me of these flu ids include liquid metals (such as mercury, molten iron) and ionised gases known by Physicists as plasma, an examp le being the solar atmosphere. The subject of MHD is largely perceived to have been initiated by Swedish electrical engineer Hannes Alfve`n[1] in 1942. The description is as follows: If an electrically conducting flu id is placed in a constant magnetic field, the motion of the fluid induces currents which create forces on the fluid. The production of these currents has led to their use such as in the design of among other devices the MHD generators for electricity production. The equations that describe MHD flow are a co mb ination of continuity equation, Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism.
2. Literature Review The governing equations are differential equations that have to be solved either analytically or nu merically. Shercliff[2] studied the steady motion of an electrically conducting flu id in pipes under transverse magnetic fields. Drake[3] cons idered flo w in a channel due to p erio d ic pressure g rad ient and so lved the resu lt ing equ at ion by * Corresponding author:
[email protected] (Al fred W. Manyonge) Published online at http://journal.sapub.org/ajcam Copyright © 2013 Scientific & Academic Publishing. All Rights Reserved
separation of variables method. Singh & Ram[4] considered laminar flo w of an electrically conducting fluid through a channel in the presence of a transverse magnetic field under the influence of a periodic pressure gradient and solved the resulting differential equation by the method of Lap lace transform. Ram et al. [5] have analysed hall effects on heat and mass transfer flo w through porous med ia. Shimo mura[6] discussed magnetohydrodynamics turbulent channel flow under a uniform transverse magnetic field. Singh[7] considered steady magnetohydrodynamic flu id flow between two parallel p lates. Kazuyuki[8] discussed inertia effects in two dimensional MHD channel flow and Al-Hadhrami[9] considered flo w of fluids through horizontal channels of porous materials and obtained velocity expressions in terms of the Reynolds number. Ganesh[10] studied unsteady MHD Stokes flow of a viscous fluid between two parallel porous plates. They considered fluid being withdrawn through both walls of the channel at the same rate.
3. Motivation In this paper we consider two dimensional Po iseuille flow of an electrically conducting flu id between two infin ite parallel plates under the influence of transverse magnetic field under a constant pressure gradient and assess the effect to velocity if the lower p late is porous. The resulting differential equation is solved by an analytical approach and the solution expressed in terms of Hart mann number. The solution of this equation is important, for example in the design of MHD power generators.
American Journal of Computational and Applied M athematics 2013, 3(4): 220-224
4. Mathematical Formulation
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formed by the two plates as the x- axis and assume that flow is in this direction. Thus v=0 and u=u(y). Also v=0 imp lies that continuity equation collapses to
The MHD phenomenon can simply be described as follows: Consider an electrically conducting fluid moving u 0 with velocity V. At right angles to this flow, we apply a x magnetic field, the field strength of which is represented by Hence equations (5) and (6) beco me: the vector B. We shall assume that the flu id has attained 1 1 p 2u steady state conditions i.e. flow variab les are independent of 0 f Bx (7) x y 2 the time t. This condition is purely for analytic reasons so 1 1 p that no macroscopic charge density is being built up at any 0 f (8) place in the system as well as all currents are constant in time. By y Because of the interaction of the two fields, namely, velocity Let the pressure gradient, and magnetic fields, an electric field vector denoted E is p induced at right angles to both V and B. This electric field is s x given by Now since p=p(x), E VXB........................................ .......... .......... ..(1) Equation (8) co llapses and (1) p dp where X stands for cross product of the two vectors V and B. s.............................. (9) .............. x dx If we assume that the conducting fluid is isotropic in spite of Co mbin ing (7) and (9) we find the magnetic field, we can denote the electrical conductivity of the fluid by a scalar σ. By Oh m's law[11], the density of 1 s 2u 0 f Bx .................... .............(10 (10) the current induced in the conducting fluid denoted J is given y 2 by The x-co.......( mponent J (VXB)............................. .......... 2) of the Lorentz force in Equation (10) (2).......... can be expressed as follows Simu ltaneously occurring with the induced current is the 2 1 B0 Lorentz force F given by (11)...(11) f B x [(uiXjB0 ) XjB0 ui.......... F JXB..............................(3) .............................. ..(3) This force occurs because, as an electric generator, the where B0 is the magnetic field strength component assumed conducting fluid cuts the lines of the magnetic field. The to be applied to a direction perpendicular to flu id motion . vector F is the vector cross product of both J and B and is a Equation (11) is achieved when we use expressions of force vector perpendicular to the plane of both J and B. This and induced current in Equations (2) and (3) together with induced force is parallel to V but in opposite direction. the fact that for any three vectors A, B and C it can be shown Laminar flow through a channel under uniform t ransverse that (AXB)XC=(B.A)C-(B.C)A. In Equation (11) i and j magnetic field is important because of some of the uses such refer to unit rectangular vectors i and j. Hence Equation (10) as the MHD generator, MHD pu mp and electro magnetic becomes: flow meter. d 2u 2 s We now consider an electrically conducting, viscous, B0 u 0.............................. (12) ....(12) 2 dy steady, incompressible fluid moving between two infin ite parallel plates both kept at a constant distance 2h between We now want to investigate how magnetic inclination to them. Both plates of the channel are fixed with no motion. the velocity influences the fluid flow. Th is behaviour is This is plane Poiseuille flow driven by a constant pressure modelled by introducing an angle of inclination to the second gradient. The equations of mot ion are the continuity equation term of Equation (12) as fo llo ws
u v d 2u 2 s 0.............................. (4)..........................(4) B0 u sin 2 0.......... (13) .............. 2 x y dy
and the Navier-Stokes equations
u u 1 1 p 2u 2u u v f Bx ( )....( 5) (5) x y x x 2 y 2 v v 1 1 p 2v 2v u v f By ( )....( 6) (6) x y y x 2 y 2
where ρ is the flu id density, fBx, fBy , u, v are the co mponents of the body force per unit mass of the fluid and the velocity in x and y directions respectively, µ is the fluid v iscosity and p is the pressure acting on the fluid. This flow is practically horizontal, in wh ich we choose the axis of the channel
where θ is the angle between V and B . In Equation (13), we assume that the two fields are inclined to each other at an angle θ lying in the range 0
