A stabilized finite element method using the residual-free bubble functions (RFB) is proposed for solving the governing equations of steady magnetohydrodynamic duct flow. A distinguished feature of the RFB method is the resolving capability of high gradients near the layer regions without refining mesh. We show that the RFB method is stable by proving that the numerical method is coercive even not only at low values but also at moderate and high values of the Hartmann number. Numerical results confirming theoretical findings are presented for several configurations of interest. The approximate solution obtained by the RFB method is also compared with the analytical solution of ShercliffÕs problem.

SUMMARYA 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. TNTRODIJCTIONThe 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 EQUATIONSThe 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

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The magnetohydrodynamic (MHD) flow in a rectangular duct is investigated for the case when the flow is driven by the current produced by electrodes, placed one in each of the walls of the duct where the applied magnetic field is perpendicular. The flow is steady, laminar and the fluid is incompressible, viscous and electrically conducting. A stabilized finite element with the residual-free bubble (RFB) functions is used for solving the governing equations. The finite element method employing the RFB functions is capable of resolving high gradients near the layer regions without refining the mesh. Thus, it is possible to obtain solutions consistent with the physical configuration of the problem even for high values of the Hartmann number. Before employing the bubble functions in the global problem, we have to find them inside each element by means of a local problem. This is achieved by approximating the bubble functions by a nonstandard finite element method based on the local problem. Equivelocity and current lines are drawn to show the well-known behaviours of the MHD flow. Those are the boundary layer formation close to the insulated walls for increasing values of the Hartmann number and the layers emanating from the endpoints of the electrodes. The changes in direction and intensity with respect to the values of wall inductance are also depicted in terms of level curves for both the velocity and the induced magnetic field

SUMMARYThe magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in a rectangular duct with an external magnetic field applied transverse to the flow has been investigated. The walls parallel to the applied magnetic field are conducting while the other two walls which are perpendicular to the field are insulators. The boundary element method (BEM) with constant elements has been used to cast the problem into the form of an integral equation over the boundary and to obtain a system of algebraic equations for the boundary unknown values only. The solution of this integral equation presents no problem as encountered in the solution of the singular integral equations for interior methods.Computations have been carried out for several values of the Hartmann number (1 < M < 10). It is found that as M increases, boundary layers are formed close to the insulated boundaries for both the velocity and the induced magnetic field and in the central part their behaviours are uniform. Selected graphs are given showing the behaviours of the velocity and the induced magnetic field.