Coupled double-diffusive thermocapillary instability: Linear and nonlinear analysis

The onset of Marangoni instability of the quiescent equilibrium in a binary liquid layer open to the atmosphere at its nondeformable interface in the no-gravity environment and subjected to the simultaneous presence of the normal temperature gradient and of the tangential temperature and solute concentration gradients is studied. The no-flow equilibrium is possible for specially chosen values of the imposed tangential gradients only. Linear stability analysis shows that the instability is longwave for very small values of the parameter gamma that specifies the ratio between the tangential and normal temperature gradients. For higher values of gamma the instability is shortwave. It is found in the latter case that the instability is always oscillatory for nonzero gamma and for any value of the inverse Lewis number L-1 not equal 1. Weakly nonlinear analysis in the regime of small gamma is carried out to derive the nonlinear evolution equation describing a wave propagation along the layer. The primary bifurcation from the equilibrium state is found to be supercritical for very small values of gamma and becomes subcritical thereafter.