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Onset of stationary Bénard - Marangoni convection in a fluid layer with variable surface tension and viscosity

The onset of stationary Bénard - Marangoni convective instabilities in a fluid layer with thermally dependent surface tension and viscosity is studied by means of linear stability analysis. The dependence of viscosity and surface tension of the fluid on temperature is assumed to be exponential and linear respectively. The upper surface is free and is subject to a general thermal condition, while the lower boundary is rigid and is fixed at a constant temperature or a constant heat flux. For the latter case, the analytically asymptotic solution of long wavelength is obtained. For pure Bénard convection, the system becomes more stable when the Biot number Bi increases. For both the upper and lower boundaries fixed at a constant heat flux, the critical Rayleigh number decreases monotonically with the physical viscosity parameter B, and the corresponding critical wavenumber vanishes. The Biot number Bi, affecting the system, depends strongly on the parameter . The critical Rayleigh number decreases with and a jump in the critical wavenumber for large and B and small Bi exists. The critical conditions and , for various values of or Bi, approach constant values as the viscosity parameter B becomes large.