Numerical approximation of a phase-field surfactant model with fluid flow

Modelling interfacial dynamics with soluble surfactants in a multiphase system is a challenging task. Here, we consider the numerical approximation of a phase-field surfactant model with fluid flow. The nonlinearly coupled model consists of two Cahn-Hilliard-type equations and incompressible Navier-Stokes equation. With the introduction of two auxiliary variables, the governing system is transformed into an equivalent form, which allows the nonlinear potentials to be treated efficiently and semi-explicitly. By certain subtle explicit-implicit treatments to stress and convective terms, we construct first and second-order time marching schemes, which are extremely efficient and easy-to-implement, for the transformed governing system. At each time step, the schemes involve solving only a sequence of linear elliptic equations, and computations of phase-field variables, velocity and pressure are fully decoupled. We further establish a rigorous proof of unconditional energy stability for the first-order scheme. Numerical results in both two and three dimensions are obtained, which demonstrate that the proposed schemes are accurate, efficient and unconditionally energy stable. Using our schemes, we investigate the effect of surfactants on droplet deformation and collision under a shear flow, where the increase of surfactant concentration can enhance droplet deformation and inhibit droplet coalescence

On the use of a friction model in a Volume of Fluid solver for the simulation of dynamic contact lines

We consider the implementation of a friction contact angle model in a Navier-Stokes VoF-CSF solver for the simulation of moving contact lines at the nano-scale. A liquid-liquid interface confined in a Couette flow generated by two solid walls moving at the same velocity in opposite directions is considered to discuss the relevance of the friction model. The simulations are compared with a reference case obtained using MD simulations by Qian et al. [46]. We show that the Navier Stokes simulations are able to reproduce the MD simulations for both the interface shape and the velocity field. The appropriate contact line friction is found to be grid convergent and of the same order as the friction measured in MD simulations. A detailed investigation of the interface shape has revealed an auto-similar linear profile in the center of the channel. Close to the wall the interface shape follows the classical Log evolution given by the Cox relation despite the wall confinement