On numerical modelling of contact lines in fluid flows

We study numerically a reduced model proposed by Benilov and Vynnycky (J. Fluid Mech. {\bf 718} (2013), 481), who examined the behavior of a contact line with a $180^{\circ}$ contact angle between liquid and a moving plate, in the context of a two-dimensional Couette flow. The model is given by a linear fourth-order advection-diffusion equation with an unknown velocity, which is to be determined dynamically from an additional boundary condition at the contact line. The main claim of Benilov and Vynnycky is that for any physically relevant initial condition, there is a finite positive time at which the velocity of the contact line tends to negative infinity, whereas the profile of the fluid flow remains regular. Additionally, it is claimed that the velocity behaves as the logarithmic function of time near the blow-up time. Compared to the previous computations based on COMSOL built-on algorithms, we use MATLAB software package and develop a direct finite-difference method to study dynamics of the reduced model under different initial conditions. We confirm the first claim but also show that the blow-up behavior is better approximated by a power function, compared with the logarithmic function. This numerical result suggests a simple explanation of the blow-up behavior of contact lines.