D.E. Pelinovsky and C. Xu
On numerical modelling and the blow-up behavior of contact lines with a 180-degree contact angle
Journal of Engineering Mathematics 92 (2015), 31-44
We study numerically a reduced model proposed by Benilov and Vynnycky [J. Fluid Mech. 718 (2013), 481], who examined
the behavior of a contact line with a 180-degree 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.
linear advection-diffusion equation, variable speed, fluid flows, finite-difference method, blow-up rate.