A 2D sharp and conservative VOF method for modeling the contact line dynamics with hysteresis on complex boundary

A sharp and conservative numerical method is proposed for studying the 2D contact line dynamics along complex geometrical boundaries, while a hybrid volume-of-fluid and embedded boundary method is designed to model the liquid/gas and fluid/solid interfaces, respectively. Unlike diffusive numerical methods that artificially thicken the interface to enhance stability, our method devises a sharp and precise scheme for discretizing the advection term of the VOF equation, with special attention to arbitrary solid boundaries within the same discretized interfacial cell. This scheme conserves the volume fraction exactly and maintains numerical stability even in small and irregular cells cut by the solid boundary. Another novel aspect of our contribution is the precise imposition of the contact angle condition at the contact line. A special height function method is designed and implemented for cells cut by the solid boundary. Furthermore, the contact angle condition, whether static or dynamic during contact line motion, is extended to more general cases by considering hysteresis phenomena. The code is released on the Basilisk website, enabling the first implementation of a sharp geometrical VOF method capable of accurately simulating contact line dynamics on complex solid substrates in 2D.