PHYSICAL JOURNAL E Numerical solutions of thin-film equations for polymer flows

on the numerical implementation of thin-film equations that describe the capillary-driven evolution of viscous films, in two-dimensional configurations. After recalling the general forms and features of these equations, we focus on two particular cases inspired by experiments: the leveling of a step at the free surface of a polymer film, and the leveling of a polymer droplet over an identical film. In each case, we first discuss the long-term self-similar regime reached by the numerical solution before comparing it to the experimental profile. The agreement between theory and experiment is excellent, thus providing a versatile probe for nanorheology of viscous liquids in thin-film geometries. Even though hydrodynamics is a well-established field, several limits and assumptions remain strongly debated. For instance, at which length scale and why does the the-ory break down? Can one generalize the concept of vis-cosity at small scales? What are the details of flow at the liquid/solid boundary? How does a droplet spread on a given wet or liquid substrate, and what is the role of the so-called precursor film