A finite element method is given to obtain the solution in terms of velocity and induced magnetic field for the steady M H D (magnetohydrodynamic) flow through a rectangular pipe having arbitrarily conducting walls. Linear and then quadratic approximations have been taken for both velocity and magnetic field for comparison and it is found that with the quadratic approximation it is possible to increase the conductivity and Hartmann number M ( M < 100). A special solution procedure has been used for the resulting block tridiagonal system of equations. Computations have been carried out for several values of Hartmann number
( 5 < M < 100) and wall conductivity. It is also found that, if the wall conductivity increases, the flux decreases. The same is the effect of increasing the Hartmann number. Selected graphs are given showing the behaviour of the velocity field and induced magnetic field.
TNTRODIJCTION
The problem of magnetohydrodynamic flow through channels has become important because of several engineering applications such as designing of the cooling system for a nuclear reactor, M HD flowmeters, M HD generators, blood flow measurements, etc. In general, the problems of MHD flow are extremely complex owing to the coupling of the equations of fluid mechanics and electrodynamics, and analytic solutions are out of the question. The exact solutions are, therefore, available only for some simple geometries subject to simple boundary conditions.'. In most of the studies, the walls have been taken as either non-conducting or perfectly conducting, or a combination of the two.39 Recently, Singh and Lal"' have obtained numerical solutions of steady-state MHD flows through pipes of various cross-sections using either a finite difference or finite element method (FEM). But with linear approximation in the finite element method they could obtain results at most up to M = 5.
The present paper is an extension of the above studies to the case of arbitrary wall conductivity, high Hartmann number (up to 100) by using the FEM with linear and then quadratic approximations for the velocity and magnetic fields. A variational principle for the problem has been obtained and then the Ritz FEM has been applied taking linear and quadratic elements. The results are obtained for various values of wall conductivity and Hartmann number. The flux has also been calculated for each case.
BASIC EQUATIONS
The fluid is taken as viscous, incompressible and having uniform electrical conductivity. It is driven down a rectangular pipe, with arbitrary wall conductivity, by means of a constant applied