Advancing fronts for the thin-film equation with null slip and repulsive potentials: the case of partial wetting

Abstract

For negative values of the spreading coefficient (that is, in the so-called ``partial wetting'' regime), we prove that the thin-film equation with zero slip and repulsive potentials $P$ of the form $P(h)\approx h^{1-m}$ as $h\to 0$, $m>1$, admits for any positive speed a one-parameter family of travelling-wave solutions with a contact line and (as in standard slippage models) a logarithmically-corrected linear behaviour as $h\to +\infty$. These waves have locally finite rate of dissipation for any $m>1$ and locally finite energy for any $m\in (1,3)$. The result thus confirms that mildly repulsive potentials effectively resolve the no-slip paradox. The family is parametrized by a thermodynamically consistent contact-line condition which reduces to the classical fixed microscopic contact-angle one if $P\equiv 0$.