Stability of Equilibria for Two-Phase Flows with Soluble Surfactant

The presence of surfactants, ubiquitous at most fluid-liquid interfaces, has a pronounced effect on the surface tension, hence on the stress balance at the phase boundary: local variations of the capillary forces induce transport of momentum along the interface-so-called Marangoni effects. The mathematical model governing the dynamics of such systems for the case in which the surfactant is soluble in one of the adjacent bulk phases has been discussed in a recent paper of the authors. This leads to the two-phase balances of mass and momentum, complemented by a species equation for both the interface and the relevant bulk phase. Within the model, the motions of the surfactant and of the adjacent bulk fluids are coupled by means of an interfacial momentum source term that represents Marangoni stresses. In this paper, we develop a strict Lyapunov functional for the problem and identify and study the equilibria of the system. We show that they are linearly stable, which will allow us to prove that they are also stable for the nonlinear problem.