Numerical investigation of the sharp-interface limit of the Navier-Stokes-Cahn-Hilliard equations

In this article, we study the behavior of the Abels-Garcke-Gr\"un Navier-Stokes-Cahn-Hilliard diffuse-interface model for binary-fluid flows, as the diffuse-interface thickness passes to zero. We consider this so-called sharp-interface limit in the setting of the classical oscillating-droplet problem. To provide reference limit solutions, we derive new analytical expressions for small-amplitude oscillations of a viscous droplet in a viscous ambient fluid in two dimensions. We probe the sharp-interface limit of the Navier-Stokes-Cahn-Hilliard equations by means of an adaptive finite-element method, in which the refinements are guided by an a-posteriori error-estimation procedure. The adaptive-refinement procedure enables us to consider diffuse-interface thicknesses that are significantly smaller than other relevant length scales in the droplet-oscillation problem, allowing an exploration of the asymptotic regime. For two distinct modes of oscillation, we determine the optimal scaling relation between the diffuse-interface thickness parameter and the mobility parameter. Additionally, we examine the effect of deviations from the optimal scaling of the mobility parameter on the approach of the diffuse-interface solution to the sharp-interface solution.