Stability of dynamic contact lines in three dimensions

The wetting or dewetting of a solid surface by a viscous fluid is a pervasive phenomenon in nature, and also one of industrial importance for a wide range of coating applications. The macroscopic flow is strongly influenced by the wetting dynamics of the three-phase contact line which separates the wet and dry regions of the substrate. It is well known (Snoeijer 2013) that there is a critical capillary number, Ca_crit above which the contact line is linearly unstable to shape perturbations; in the wetting case (an advancing contact line) this leads to air entrainment, and in the dewetting case (a receeding contact line) this leads to thin-film formation. It has been recently shown (Keeler 2022) that, in two-dimensions, a fold bifurcation occurs at Ca_crit where the branches of the stable and unstable steady states, which coexist below Ca_crit, meet. In this study we extend this work into three dimensions, using, as a starting point, a "Landau-Levich" geometry in which a vertical plate is withdrawn or inserted into a bath of viscous fluid. We explore the dynamic wetting transition and its bifurcation structure through a combination of a semi-analytic linear stability analysis and full-scale direct numerical simulation, and investigate the formation dynamics of three-dimensional structures at and behind the receding contact line.