*Wissenschaftliches Programm* *Liste der Vortragenden*

**Sektion 2**

Dienstag, 19.09.2000,
14.00–14.45 Uhr, POT 51

#### Felix Otto, Universität Bonn

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.