The onset of Benard-Marangoni convection in a horizontal layer of fluid

In this paper we use classical linear stability theory to investigate the o nset of Benard-Marangoni convection in a planar horizontal layer of fluid h eated from below in the most physically-relevant case when the non-dimensio nal Rayleigh number and Marangoni number are linearly dependent. We use a c ombination of analytical and numerical techniques to obtain for the first t ime a detailed description of the marginal stability curves for the onset o f both steady and overstable convection which substantially extends the num erical results of earlier authors. We perform a comprehensive asymptotic an alysis of the marginal curves in the limits of both long and short waveleng th disturbances and present the results of numerical calculations which ill ustrate the effects of varying the problem parameters on the marginal curve s. In particular, we give an example of a situation in which there is compe tition between a steady and an overstable mode at the onset of convection a nd investigate the onset of steady convection in the limit of weak free sur face deformation. (C) 1999 Elsevier Science Ltd. All rights reserved.