Effects of dynamic contact angle on liquid infiltration into horizontal capillary tubes: (semi)-analytical solutions

We generalize Washburn's analytical solution for capillary flow in a horizontally oriented tube by accounting for a dynamic contact angle. We consider two general models for dynamic contact angle: the uncompensated Young force on the contact line depends on the capillary number in the form of either (1) a power law with exponent beta or (2) a power series. By considering the ordinary differential equation (ODE) for the velocity of the gas-liquid interface instead of the ODE for the interface position, we are able to derive new analytical solutions. For both dynamic contact angle models, we derive analytical solutions for the travel time of the gas-liquid interface as a function of interface velocity. The interface position as a function of time can be obtained through numerical integration. For the power law and beta=1 (an approximation of Cox's model for dynamic contact angle), we obtain an analytical solution for both interface position and velocity as a function of time. For the power law and beta=3, we can express the interface velocity as a function of time.