The rapidly advancing contact line: testing the limits of numerical resolution*

A rapidly advancing contact line can be obtained by letting a feed flow of velocity V fall from a height H on a plate moving horizontally at velocity U . For given fluids and a given geometry the problem has two control parameters, a Reynolds Re number based on V and a capillary number Ca based on U . Beyond a critical condition wetting failure happens and no steady-state solutions are found. This "curtain coating" or "hydrodynamic assist" setup is extremely challenging to simulate in the conditions of the experiments because of the centimetre-to-nanometer length scale ratio and an advancing Ca of order 1. We attempt it using the Volume-of-Fluid (VoF) method with quadtree adaptive mesh refinement with capillary forces computed using the Continuous Surface Force method and Height Functions for the curvature calculation. We use a fixed contact angle and a Navier slip boundary condition, focusing on the apparent contact angle and the critical conditions at which the steady-state solution disappears. Critical parameters of the stability window are first validated against the previous computations of Liu et al. (2016). Finally, we go at the limit of our resolution, that is a hundred nanometer and compare against the experimental parameters of the Blake et al. (1999). The stability window is seen moving towards the experimental window as the slip length is decreased, making this a remarkable comparison. Finally, we answer a long-standing physical question on the origin of accelerating flow around the rapidly advancing contact line, observed in the experiments (Clarke 1994) but not in the Stokes flow simulation (Wilson, Summers and Shikhmurzaev 2001). We observe the accelerating flow around the contact line in simulations, which is theoretically attributed to inertia (Varma et al. 2021). We propose an inertially corrected wedge theory, with the wedge angle based on the inflection point angle of the liquid curtain to describe the flow at tens of microns. It is only inside the cut-off length scale, the slip length, the velocity starts to decrease as expected from Stokes flow solution.