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 P ( h ) ≈ h 1 - m as h→ 0 h → 0 , $$m>1$$ 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→ +∞ h → + ∞ . Locally around the contact-line, these waves have finite rate of dissipation for any $$m>1$$ m > 1 and finite energy for any m∈ (1,3) 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 P ≡ 0 .