On the Two-Phase Navier-Stokes Equations with Boussinesq-Scriven Surface

Abstract. Two-phase ows with interface modeled as a Boussinesq- Scriven surface uid are analysed concerning their fundamental mathematical properties. This extended form of the common sharp-interface model for two-phase ows includes both surface tension and surface viscosity. For this system of partial differential equations with free interface it is shown that the energy serves as a strict Ljapunov functional, where the equilibria of the model without boundary contact consist of zero velocity and spheres for the dispersed phase. The linearizations of the problem are derived formally, showing that equilibria are linearly stable, but nonzero velocities may lead to problems which linearly are not wellposed. This phenomenon does not occur in absence of surface viscosity. The present paper aims at initiating a rigorous mathematical study of two-phase ows with surface viscosity