Thermo-capillary effects along a deformable singular interface between two immiscible fluids

In this paper, we study the onset of instabilities triggered by the thermo-capillary effect along an isotropic material deformable interface between two one-component immiscible fluids described using the method of Bedeaux, Albano and Mazur (BAM). The BAM method allows a complete description of the non-equilibrium thermodynamics of the interface between the two fluids. In our analysis, all physical variables have a singular contribution along the interface. We begin with the derivation of the conservation laws, the entropy production and the thermodynamic fluxes for the bulk phases and the interface. Then, these equations are used to analyze the interface between two superposed immiscible fluids. The system is delimited by two horizontal conducting rigid plates kept at different constant temperatures. We apply normal mode analysis and study the linearized equations using long-wave expansions. The temperature and velocity profiles are obtained for the mechanical equilibrium and the linear perturbations. Finally, we obtain the stationary and oscillatory instability boundaries. The instability boundaries are functions of the ratios of transport coefficients of the bulk phases and dimensionless numbers (Marangoni, Prandl and Galileo numbers) and they also elicit dependencies in transport coefficients specific to the singular material interface.