Instabilities of dynamic thermocapillary liquid layers with magnetic fields

This paper presents a linear-stability analysis for the transition from a steady, two-dimensional thermocapillary convection in a liquid-metal layer to a periodic, three-dimensional flow involving hydrothermal waves which propagate in the direction normal to the plane of the base how. There is a uniform magnetic field applied parallel to the free surface in the plane of the base flow and there is a linear temperature gradient along the free surface in the base how. The ratio of the layer's length to its depth, 2L, is large. The magnetic Reynolds number is small. A key parameter is lambda, the ratio of the large Hartmann number based on depth to L. The value of lambda, increases as either the magnetic field strength is increased or L is decreased. The results for very small values of lambda, agree with the results of a previous treatment of this instability without a magnetic field. As lambda is increased, the critical Marangoni number and the wavenumber for the hydrothermal rolls both increase. For large values of lambda, the base flow and the hydrothermal waves are confined to a free-surface layer with O(lambda(-1/2)) dimensionless thickness.