The capillarity–driven spreading of a thin droplet of a viscous liquid on a solid plane is modelled by the lubrication approximation, an evolution equation for the film height h. However, as a consequence of the no–slip boundary condition for the liquid at the solid plane, logarithmic divergences in the viscous dissipation rate occur if the support of h changes.
This well–known singularity is removed by relaxing the no–slip condition, thereby introducing a microscopic lengthscale b. Matched asymptotics suggests a relationship (Tanner’s law) between the speed of the contact line (the boundary of the support of h) and the macroscopic contact angle (the slope of h near the boundary of its support), modulo a logarithm involving b. This dynamic contact angle condition, which balances viscous forces and surface tension, is quite different from the static contact angle condition (Young’s law), which balances just the surface tensions.
Tanner’s law predicts a specific scaling for the spreading of the droplet. In a joint work with L. Giacomelli, we rigorously derive a bound on the spreading, which is consistent with the one predicted based on Tanner’s law, including the logarithmic terms. Mathematically speaking, this amounts to estimates of appropriate integral quantities of the evolution equation, which comes in form of a nonlinear parabolic equation of fourth order.