Rayleigh-Taylor instability of a miscible interface in a confined domain

Abstract

On the basis of the phase-field approach, we investigate the simultaneous diffusive and convective evolution of an isothermal binary mixture of two slowly miscible liquids that are confined in a horizontal plane layer. We assume that two miscible liquids are brought into contact, so the binary system is thermodynamically unstable and the heavier liquid is placed on top of the lighter liquid, so the system is gravitationally unstable. Our model takes into account the non-Fickian nature of the interfacial diffusion and the dynamic interfacial stresses at a boundary separating two miscible liquids. The numerical results demonstrate that the classical growth rates that characterise the initial development of the Rayleigh-Taylor instability can be retrieved in the limit of the higher Peclet numbers (weaker diffusion) and thinner interfaces. The further nonlinear development of the Rayleigh-Taylor instability, characterised, e.g., by the size of the mixing zone, is however limited by the height of the plane layer. On a longer time scale, the binary system approaches the state of thermodynamic and hydrodynamic equilibrium. In addition, a novel effect is found. It is commonly accepted that the interface between the miscible liquids slowly smears in time due to diffusion. We however found that when the binary system is subject to hydrodynamic transformations the interface boundary stretches, so its thickness changes (the interface becomes thinner) on a much faster convective time scale. The thickness of the interface is inversely proportional to the surface tension, and the stronger surface tension limits the development of the Rayleigh-Taylor instability.