Instabilities of dynamic thermocapillary liquid layers with magnetic fields

Abstract

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 flow. 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 flow. The ratio of the layer's length to its depth, 2L, is large. The magnetic Reynolds number is small. A key parameter is λ, the ratio of the large Hartmann number based on depth to L. The value of λ increases as either the magnetic field strength is increased or L is decreased. The results for very small values of λ agree with the results of a previous treatment of this instability without a magnetic field. As λ is increased, the critical Marangoni number and the wavenumber for the hydrothermal rolls both increase. For large values of λ, the base flow and the hydrothermal waves are confined to a free-surface layer with O(λ-1/2) dimensionless thickness.