  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.
