Benard-Marangoni instability in a liquid layer with temperature-dependent viscosity

Surface-tension driven instability in a thin horizontal layer of variable-viscosity fluid, bounded by a heated rigid plate from below and a free surface from above, is studied. The viscosity is assumed to decrease exponentially with increasing the temperature. The linear hydrodynamic stability analysis is applied. The corresponding eigenvalue problem is solved analytically. The critical values of the Marangoni number, based on the mean of the liquid viscosities at the layer boundaries, depend on a viscosity parameter presenting a logarithm of the ratio of the maximum viscosity at the free surface to the minimum value at the plate. For a given liquid, the viscosity variation has a small stabilizing effect in very thin layers and a considerable destabilizing influence in relatively thick ones.