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 \(h\rightarrow 0\) , \(m>1\) , 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 \(h\rightarrow +\infty \) . Locally around the contact-line, these waves have finite rate of dissipation for any \(m>1\) and 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 \(P\equiv 0\) .
