Effects of dynamic contact angle on liquid withdrawal from capillary tubes: (semi)-analytical solutions

The displacement of a gas by a liquid in both horizontal and inclined capillary tubes where the tube inlet is connected to a liquid reservoir of constant pressure can be described by the Lucas-Washburn theory. One can also use the Lucas-Washburn theory to model the reverse flow, that is, liquid withdrawal, even though the latter case has received relatively little attention. In this paper, we derive analytical solutions for the travel time of the gas-liquid interface as a function of interface velocity. The interface position can be obtained by numerically integrating the numerically inverted interface velocity. Therefore we refer to these solutions as (semi)-analytical. We neglect inertial forces. However, we account for a dynamic contact angle where the nondimensional non-equilibrium Young force depends on the capillary number in the form of either a power law or a power series. We explore the entire nondimensional parameter space. The analytical solutions allow us to show that five different liquid withdrawal scenarios may occur that differ in the direction of flow and the sign of the acceleration of the gas-liquid interface: horizontal, upward, steady-state downward, accelerating downward, and decelerating downward flow. In the last case, the liquid is withdrawn from the tube either completely or partially. The (semi)-analytical solutions are also valid within the limit where the contact angle is constant.