Modeling and numerical simulation of surfactant systems with incompressible fluid flows on surfaces

Abstract

In this paper, we consider the mathematical modeling and numerical approximation for the fluid-surfactant phase field model coupled with the Navier–Stokes equations on surfaces. Taking account of the effect of the curvature and fluid flows, we first propose the incompressible Navier–Stokes fluid-surfactant phase field model on surfaces. Then, by introducing two power-type scalar auxiliary variables in the phase field equations for the potential energy terms and an exponential-type one in the momentum equation with respect to the inertial term, we transform the proposed model to an equivalent form. Furthermore, with the help of the derived three scalar auxiliary variables (3-SAV) equations, via utilizing the pressure correction method and implicit–explicit techniques in a proper sequence to eliminate the influence of the skew-symmetric loss in the surface finite element method, and applying stabilized methods for the convective and diffusion terms in the phase field and momentum equations according to the feature of the velocity field on surfaces respectively, we construct a linearized and decoupled fully discrete scheme, in which only some linear equations need to be solved at each time step. The unconditional energy stability of the fully discrete scheme is also proved. Finally, we show some numerical examples to verify the rationality of the model and the efficiency of the scheme, and discuss the influence of the geometric curvature and fluid flows on the system.