The rapidly advancing contact line Part-1: Navier slip and microscale inertial effects

Abstract

Curtain coating, in which a moving plate is coated by a falling liquid sheet, sustains advancing contact lines at large capillary numbers Ca ~ O(1), based on plate speed. Steady states exist up to a critical capillary number, beyond which wetting failure occurs through air-bubble entrainment. In the steady regime, experiments report that velocity along the fluid-fluid interface accelerates as the contact line is approached, down to tens of micrometres; this has been interpreted as evidence against the Navier slip model. We ask whether this acceleration is compatible with slip models, and show that it is. Although Navier slip implies a vanishing velocity at the contact line, the experimentally accessible microscale region lies outside the slip region. The curtain-coating setup is revealing because the local Reynolds number, based on distance from the contact line r ~ 10 microns, is order unity, so the observable flow is governed by local inertia. Our two-phase Navier-Stokes Volume-of-Fluid simulations with quadtree adaptive mesh refinement resolve the smallest scales and study the flow with a Navier slip boundary condition and fixed contact angle. The simulations reproduce the non-monotonic dependence of the critical capillary number on global Reynolds number, based on feed-flow velocity, and the variation of the macroscopic contact angle at the inflection point, in agreement with Liu et al (2016). The interfacial velocity in the microscale region is well described by an inertially corrected wedge flow solution whose wedge angle is set by the inflection-point value, with agreement improving as slip length is reduced; at larger scales, interface bending follows the Benney solution. These inertial effects, absent from pure Stokes flow, are essential in the experimental region. Thus qualitative microscale observations do not decisively invalidate slip models for advancing contact lines.