Thin-Film Equations with Singular Potentials: An Alternative Solution to the Contact-Line Paradox

Abstract

AbstractIn the regime of lubrication approximation, we look at spreading phenomena under the action of singular potentials of the form $$P(h)\approx h^{1-m}$$ P ( h ) ≈ h 1 - m as $$h\rightarrow 0^+$$ h → 0 + with $$m>1$$ m > 1 , modeling repulsion between the liquid–gas interface and the substrate. We assume zero slippage at the contact line. Based on formal analysis arguments, we report that for any $$m>1$$ m > 1 and any value of the speed (both positive and negative) there exists a three-parameter, hence generic, family of fronts (i.e., traveling-wave solutions with a contact line). A two-parameter family of advancing “linear-log” fronts also exists, having a logarithmically corrected linear behavior in the liquid bulk. All these fronts have finite rate of dissipation, indicating that singular potentials stand as an alternative solution to the contact-line paradox. In agreement with steady states, fronts have microscopic contact angle equal to $$\pi /2$$ π / 2 for all $$m>1$$ m > 1 and finite energy for all $$m<3$$ m < 3 . We also propose a selection criterion for the fronts, based on thermodynamically consistent contact-line conditions modeling friction at the contact line. So as contact-angle conditions do in the case of slippage models, this criterion selects a unique (up to translation) linear-log front for each positive speed. Numerical evidence suggests that, fixed the speed and the frictional coefficient, its shape depends on the spreading coefficient, with steeper fronts in partial wetting and a more prominent precursor region in dry complete wetting.