Nonlinear steady hydromagnetic convection in mushy layers during alloy solidification

We consider the problem of nonlinear steady convection in a horizontal mushy layer, which is subjected to a vertical magnetic field, during alloy solidification. Under near-eutectic approximation and the limit of large far-field temperature, we determine the finite-amplitude solutions to the weakly nonlinear problem by using a perturbation technique. We obtain specific information about convective flow solutions in the form of rolls, squares, rectangles, down-hexagons with down-flow at the cells' centers and up-flow at the cells' boundaries and up-hexagons with up-flow at the cells' centers and down-flow at the cells' boundaries. The stability of these solutions is also investigated with respect to arbitrary disturbances. We found, in particular, that only up-hexagons or supercritical rolls can be stable. No subcritical non-hexagonal pattern convection was found to exist. The subcritical domain of stable hexagons and the supercritical domain of stable rolls were found to be enhanced by the magnetic field as well as by the non-uniformity of the permeability of the medium, but the magnetic field narrows down the supercritical domain for the stable hexagons. The magnetic field appears to reduce the tendency for chimney formation at the nodes of up-hexagons, while such a tendency is enhanced by the magnetic field at the nodes of down-hexagons.