The effect of a uniform magnetic field on the onset of steady B�nard-Marangoni convection in a layer of conducting fluid

Abstract

In this paper we use a combination of analytical and numerical techniques to analyse the effect of a uniform vertical magnetic field on the onset of steady Benard-Marangoni convection in a horizontal layer of quiescent, electrically conducting fluid subject to a uniform vertical temperature gradient. The critical values of the Rayleigh and Marangoni numbers for the onset of steady convection are calculated and the latter is found to be critically dependent on the non-dimensional Crispation and Bond numbers. The stability of the layer to long wavelength disturbances is analysed and the two different asymptotic limits of strong surface tension (small Crispation number) and strong magnetic field (large Chandrasekhar number) are investigated. In the latter case analytical results for the critical Rayleigh and Marangoni numbers are obtained and are found to be in excellent agreement with the results of numerical calculations. We conclude that the presence of the magnetic field always has a stabilising effect on the layer. Treating the Marangoni number as the critical parameter we show that if the free surface is non-deformable then any particular disturbance can be stabilised with a sufficiently strong magnetic field, but if the free surface is deformable and gravity waves are excluded then the layer is always unstable to infinitely long wavelength disturbances with or without a magnetic field. Including gravity has a stabilising effect on the long wavelength modes, but not all disturbances can be stabilised no matter now strong the magnetic field is.