A time-dependent formulation for nonlinear gravity-capillary surface waves with viscosity

We develop a time-dependent conformal method to study the effect of viscosity on steep surface waves. When the effect of surface tension is included, numerical solutions are found that contain highly oscillatory parasitic capillary ripples. These small amplitude ripples are associated with the high curvature at the crest of the underlying viscous-gravity wave, and display asymmetry about the wave crest. Previous inviscid studies of steep surface waves have calculated intricate bifurcation structures that appear for small surface tension. We show numerically that viscosity suppresses these. While the discrete solution branches still appear, they collapse to form a single smooth branch in limit of small surface tension. Temporal stability is also studied, and these solutions are seen to be stable when time-evolved from different initial conditions. Our work provides a convenient method for the numerical computation and analysis of water waves with viscosity, without evaluating the free-boundary problem for the full Navier-Stokes equations.