Magnetohydrodynamic flow on a half‐plane

Abstract

SUMMARYWe investigate the magnetohydrodynamic flow (MHD) on the upper half of a non-conducting plane for the case when the flow is driven by the current produced by an electrode placed in the middle of the plane. The applied magnetic field is perpendicular to the plane, the flow is laminar, uniform, steady and incompressible. An analytical solution has been developed for the velocity field and the induced magnetic field by reducing the problem to the solution of a Fredholm's integral equation of the second kind, which has been solved numerically. Infinite integrals occurring in the kernel of the integral equation and in the velocity and magnetic field were approximated for large Hartmann numbers by using Bessel functions. As the Hartmann number M increases, boundary layers are formed near the non-conducting boundaries and a parabolic boundary layer is developed in the interface region. Some graphs are given to show examples of this behaviour. KEY WORDS MHD Flow Half-plane FORMULATION OF THE PROBLEMWe consider the steady flow of an incompressible fluid with uniform prroperties driven by the interaction of imposed electric currents and a uniform transverse magnetic field. Imposed currents enter the fluid at l = f a , through external circuits and move up on the plane. We assume that all the physical variables, including pressure, and the boundary conditions are functions of < and q only. The pressure gradient is zero. There is only one component of velocity and of magnetic field (in the z-direction). The equations describing such flows are the same as those of MHD duct flow problems when pressure gradient is taken as zero. Figure 1 shows the geometry of the problem.A uniform magnetic field of strength H , is directed along the axis of q. The wall is electrically insulated except for a length 2a, in the middle, where a perfectly conducting electrode is placed so that this part is conducting. Thus the partial differential equations describing the flow (in nondimensionalized form) are', VzV+M(aB/ay)=O,V 2 B + M ( a v/aq) = 0,

Finite element method solution of electrically driven magnetohydrodynamic flow

“…For the induced magnetic field, we have a layer formation, emanating from the points of discontinuity (∓l, ∓1) on the boundary. The structure of these layers shows similarity to the solution of the MHD flow on the half plane y > 0 with a conducting portion [13,9]. There, the layers propagating from the endpoints of the conducting part of the boundary is of parabolic type and have thickness O(1/ √ M).…”