Advancing Fronts for the Thin-Film Equation with Null Slip and Repulsive Potentials: The Case of Partial Wetting

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)≈h1-m as h→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→+∞. Locally around the contact-line, these waves have finite rate of dissipation for any m>1 and finite energy for any m∈(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≡0.