Natural convection of an electrically conducting fluid in the presence of a magnetic field

Abstract

The case of a vertical hot plate surrounded by an electrically conducting fluid is examined in the presence of a magnetic field acting in a direction perpendicular to the induced movement caused by the buoyant forces. It is found that similarity solutions exist, provided that the intensity of the magnetic field changes with the inverse fourth root of the distance measured in the direction of the flow. The resulting differential equations of motion and energy have solutions depending on the Prandtl number, the Grashof number and a non-dimensional third number (say Z) representing the ratio of the ponderomotive force over the buoyant force. Theoretical asymptotic solutions have been obtained for constant wall temperature in the following cases: 1. (a) Very high Prandtl and small Z numbers. In this case the inertia forces may be neglected. 2. (b) Very high Z numbers regardless of Prandtl numbers. 3. (c) Zero and small Prandtl numbers. Exact solutions obtained by an analogue computer are also reported. It is found that the action of the magnetic field is to decelerate the flow thus decreasing the Nusselt number. For a constant Prandtl number the rate of decrease of the heat transfer coefficient with increasing values of the non-dimensional number Z is higher for smaller values of Z; on the other hand for the same value of the parameter Z, the rate of decrease of the same coefficient is higher for lower Prandtl numbers. The case of non-similarity solutions is also investigated; the basic differential equations for a constant transverse magnetic field and fields depending on a power of the vertical distance are given. It is found that experiments in the laboratory are feasible since the parameter Z is of the order of one to ten for liquid metals.