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

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In this article, we study the behaviour of the Abels–Garcke–Grün 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 $\varepsilon \to +0$ , the so-called mobility parameter $m$ in the diffuse-interface model must scale appropriately with the interface-thickness parameter $\varepsilon$ . In the literature various scaling relations in the range $o(1)$ to $O(\varepsilon ^3)$ 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 $m$ – $\varepsilon$ 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.