Onset of Benard-Marangoni convection in a horizontal layer of fluid

In this paper we use classical linear stability theory to investigate the onset of Benard-Marangoni convection in a planar horizontal layer of fluid heated from below in the most physically-relevant case when the non-dimensional Rayleigh number and Marangoni number are linearly dependent. We use a combination of analytical and numerical techniques to obtain for the first time a detailed description of the marginal stability curves for the onset of both steady and overstable convection which substantially extends the numerical results of earlier authors. We perform a comprehensive asymptotic analysis of the marginal curves in the limits of both long and short wavelength disturbances and present the results of numerical calculations which illustrate the effects of varying the problem parameters on the marginal curves. In particular, we give an example of a situation in which there is competition between a steady and an overstable mode at the onset of convection and investigate the onset of steady convection in the limit of weak free surface deformation.