Theoretical Aspects of Thermocapillary Convection in Liquid Metals under Magnetic Fields Influence

Magnetic fields provide the possibility to influence the convective flow pattern in an predefined contactless way, in particular, to suppress and control oscillatory instabilities. Therefore, there is increasing interest now in the use of magnetic fields in a variety of materials processing technologies like crystal growth from the melt or directional solidification.
Available knowledge is very limited on the influence of a steady magnetic field (described by the Hartmann number Ha) on convective motions, described by the Grashoff number Gr (buoyancy), and by the Marangoni number Ma (thermocapillarity at free surfaces) in low Prandtl number fluids like liquid metals. Therefore, in the present paper we present theoretical results on the prototype problem of an infinitely extended fluid layer heated either from the side or from the bottom, and exposed to a steady external magnetic field.
In the first part we consider the stability of a unidirectional thermocapillary flow set up by a temperature gradient parallel to the free surface. The magnetic field is supposed to be parallel to the free surface. In this case the magnetic field has no influence on the basic flow, the stability of which is well known without magnetic field. The restriction to disturbances propagating in the same direction as the magnetic field allows to obtain an analytical solution of the stability problem and an explicit expression for the dispersion relation. The hydrothermal wave instability is suppressed by the magnetic field, the critical Marangoni number grows proportionally to Ha. The most unstable wave pattern is stretched along the field lines. The wave number of the most unstable mode decreases as Ha-4/5 for Ha >> 1.
The second part of this communication is devoted to the analysis of the thermocapillary instability in a layer with both temperature gradient and magnetic field being perpendicular to the free surface. Although this problem has been extensively treated in the past, the ...