Fluid flow in a hemispherical container induced by a distributed source of current and a superimposed uniform magnetic field

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Abstract
In arc welding electromagnetic forces are thought to be the major cause of motion in the weldpool, and it has recently been found by experimentalists that the application to the workpiece of a uniform magnetic field normal to the weldpool surface appears to stabilize the welding process. In this paper we investigate motion in a hemisphere due to a stationary distributed current source with a superimposed uniform magnetic field applied parallel to the axis of symmetry. When inertial effects are ignored and a simple source-sink model of the current source is introduced, we obtain an analytic solution for the fluid flow valid for low currents and low applied fields. A numerical scheme is then developed to solve the full nonlinear-flow problem and results are obtained for the source-sink model of the current source and for a more realistic numerically constructed distributed current source. For values of the externally applied magnetic field and the current flowing through the hemisphere typical of those occurring in practical welding situations we discover that the flow in an axial section through the pool is radially outwards on the free surface. This is in the opposite direction to that generally predicted by the theory when there is no superimposed magnetic field.