Liquid withdrawal from capillary tubes: Explicit and implicit analytical solution for constant and dynamic contact angle

In a preceding paper, we derived analytical solutions for the displacement of a gas by a liquid in horizontal and inclined capillary tubes where the tube inlet is connected to a liquid reservoir of constant pressure. We considered quite general models for the dynamic contact angle and were able to derive implicit equations for the velocity of the gas-liquid interface. These solutions allowed us to identify five different flow scenarios for liquid withdrawal that differed in the direction of flow and the sign of the acceleration of the gas-liquid interface. In this paper, we consider the special case where the dynamic contact angle is determined by a nonequilibrium Young force that depends linearly on the capillary number. Thus we can derive explicit and the more traditional implicit analytical solutions for both the position and the velocity of the gas-liquid interface. We also construct diagrams that allow us to predict which of the five flow scenarios will occur depending on the nondimensional parameters that define the problem. The diagrams can be combined with diagrams previously obtained for infiltration and the entire parameter space subdivided into regions that are associated with either liquid withdrawal, liquid infiltration, or metastable and stable equilibrium states. Our solutions are also valid within the limit where the contact angle is constant.