Fluid flow in a hemisphere induced by a distributed source of current

One of the main problems in welding is to produce consistent weld profiles. Simple heat-flow models of the weldpool, which are currently used to predict the shape of the solid-liquid boundary, do not take account of fluid motion which is observed in practice and the effect of such motion could be significant. Electromagnetic j × B forces due to the welding arc have been proposed as a major cause of the motion and we attempt here to develop existing flow models towards more practical welding situations. We consider the steady-state flow of an incompressible viscous conducting fluid in a hemispherical container due to various axisymmetric representations of the distributed current sources which can arise in arc welding. A solution is found for sufficiently small currents that inertial effects may be ignored and no singularities appear in the velocity field. We discover that varying the current distribution can lead to qualitatively different flow patterns, i.e. poloidal flows in opposite directions and breakup into two distinct counter-rotating loops.