Gradient flow formulation and second order numerical method for motion by mean curvature and contact line dynamics on rough surface
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We study the dynamics of a droplet placing on an inclined rough surface, which can be described by gradient flows on a Hilbert manifold. We propose unconditionally stable first/second order numeric schemes to simulate the geometric motion of the droplet described using motion by mean curvature with moving contact lines. The schemes base on (i) explicit moving boundaries, which decouple the dynamics of the contact lines and the capillary surface, (ii) a semi-Lagrangian method on moving grids and (iii) a predictor-corrector method with an inexact nonlinear elliptic solver. To demonstrate the accuracy and long-time validation of the proposed schemes, several challenging computational examples - including breathing droplets, droplets on inhomogeneous rough surfaces and quasi-static Kelvin pendant droplets - are constructed and compared with exact solutions to quasi-static dynamics obtained by desingularized differential-algebraic system of equations (DAEs).
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