Dynamics of viscous fluids it typically described by Navier-Stokes equations. However, in the context of modeling thin films, one can use the fact that the in plane dimensions of the flow (parallel to the solid surface) are much longer than the perpendicular one. One typically averages over the short dimension (lubrication approximation), effectively reducing the dimension of the system by one. Even though this procedure simplifies significantly the mathematical formulation, there is still a fourth order partial differential equation in two space and one time dimension to be solved (the fourth order term appears as the result of the capillary forces).

An important aspect of modeling thin films flows is the behavior of the fluid at the very front (so called contact line), where three phases - solid, liquid, and gas, meet. Typically assumed "no-slip" boundary conditions for the fluid is problematic, since it eliminates the tangential component of fluid velocity, and effectively prevents the contact line from moving. There are a couple of approaches to this problem. One is to relax the no-slip boundary condition, so that to allow for the fluid motion at the contact line. Another one (which is predominantly used for the simulations presented here) is to assume the presence of a thin layer (precursor) of fluid in front of the apparent contact line. In our work from 2001, Global models for moving contact lines, J. Diez, L. Kondic, and A. L. Bertozzi, Phys. Rev. E 63, 011208 (2001), we show that these two approaches are closely related and lead to very similar results.

Through the years, this basic mathematical model has been substantially extended and applied to a number of different problems, as listed here.