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

In this article, we study the behaviour of the Abels-Garcke-Grun Navier-Stokes-Cahn-Hilliard diffuse-interface model for binary-fluid flows, as the diffuse-interface thickness passes to zero. For the diffuse-interface model to approach a classical sharp-interface model in the limit epsilon -> +0, the so-called mobility parameter m in the diffuse-interface model must scale appropriately with the interface-thickness parameter epsilon. In the literature various scaling relations in the range o(1) to O(epsilon(3)) have been proposed, but the optimal order to pass to the limit has not been explored previously. Our primary objective is to elucidate this optimal order of the m-epsilon scaling relation in terms of the rate of convergence of the diffuse-interface solution to the sharp-interface solution. Additionally, we examine how the convergence rate is affected by a sub-optimal parameter scaling. We centre our investigation around the case of an oscillating droplet. 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. For two distinct modes of oscillation, we probe the sharp-interface limit of the Navier-Stokes-Cahn-Hilliard equations by means of an adaptive finite-element method. 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.

Automated summary

In this article, the behavior of the Abels-Garcke-Grun Navier-Stokes-Cahn-Hilliard diffuse-interface model for binary-fluid flows as the diffuse-interface thickness approaches zero is studied. The optimal order of the m-epsilon scaling relation and its impact on the convergence rate of the diffuse-interface solution to the sharp-interface solution are elucidated. The case of an oscillating droplet is investigated, and new analytical expressions for small-amplitude oscillations are derived. The sharp-interface limit of the Navier-Stokes-Cahn-Hilliard equations is probed using an adaptive finite-element method.