Effect of thermal boundary conditions on benard--marangoni convection with temperature dependent viscosity and density

The effect of thermal boundary conditions on in a horizontal fluid layer with temperature dependent viscosity and density is investigated. The lower boundary is considered to be rigid which is either at fixed temperature or at fixed heat flux condition, while the temperature dependence of the surface tension at the upper free boundary and subject to a general thermal boundary condition. The problem is solved numerically using the Galerkin technique. It is found that the critical thermal Rayleigh number or critical Marangoni number making the onset of Bénard-Marangoni convection is greatest for fixed temperature and least for fixed heat flux condition. The effect of increasing in Biot number, temperature dependent viscosity and density is to delay the onset of convection, while increasing in the Marangoni number due to surface tension effect at the free boundary is to hasten the convection. Some known results are recovered as special cases.