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In this article, we study the behavior of the Abels-Garcke-Gr\"unNavier-Stokes-Cahn-Hilliard diffuse-interface model for binary-fluid flows, asthe diffuse-interface thickness passes to zero. We consider this so-calledsharp-interface limit in the setting of the classical oscillating-dropletproblem. To provide reference limit solutions, we derive new analyticalexpressions for small-amplitude oscillations of a viscous droplet in a viscousambient fluid in two dimensions. We probe the sharp-interface limit of theNavier-Stokes-Cahn-Hilliard equations by means of an adaptive finite-elementmethod, in which the refinements are guided by an a-posteriori error-estimationprocedure. The adaptive-refinement procedure enables us to considerdiffuse-interface thicknesses that are significantly smaller than otherrelevant length scales in the droplet-oscillation problem, allowing anexploration of the asymptotic regime. For two distinct modes of oscillation, wedetermine the optimal scaling relation between the diffuse-interface thicknessparameter and the mobility parameter. Additionally, we examine the effect ofdeviations from the optimal scaling of the mobility parameter on the approachof the diffuse-interface solution to the sharp-interface solution.

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