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 γ that specifies the ratio between the tangential and normal temperature gradients. For higher values of γ the instability is shortwave. It is found in the latter case that the instability is always oscillatory for nonzero γ and for any value of the inverse Lewis number L-1 ≠ 1. Weakly nonlinear analysis in the regime of small γ 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 γ and becomes subcritical thereafter.