Weakly nonlinear Marangoni instability in the presence of a magnetic field: effect of the boundary conditions and magnetic Prandtl number

The onset of Marangoni convection in a horizontal layer of conducting liquid in the presence of a vertical magnetic field is considered. The layer is placed on a heated rigid wall and is open to an ambient gas above. The wall and gas phase are assumed to be both thermally and electrically either conducting or insulating. The influence of the boundary conditions and magnetic Prandtl number on the Marangoni instability is studied using a weakly nonlinear analysis. It is shown that the stability conditions are influenced by the electromagnetic properties of the wall when the magnetic field strength is small or moderate, while the properties of the gas are more important under strong magnetic fields. For both conducting and insulating wall, the effect of the magnetic Prandtl number is found to be strong for moderate magnetic fields when the ambient gas is nonconducting, and weak for a conducting gas.