Non-instantaneous dynamics in viscid gravity-capillary flows

Gravity-capillary waves and free-surface flows are present in a wide range of physical systems and at different spatial scales. Irrotational free-surface flows can be described in terms of dynamical equations exclusively for degrees of freedom at the surface, by taking advantage of the potential nature of flow in the bulk of the fluid. However, viscosity and no-slip boundaries generate rotational flows that go unaddressed in such a description. In this work, we derive dynamics for the surface degrees of freedom of gravity-capillary free-surface flow, by showing that, in a weakly viscid flow regime, bulk flows translate to non-instantaneous dynamics at the surface. Our analysis consists of a non-perturbative analytical treatment of viscid effects in linearized (i.e., wave-like) dynamics, supported by direct numerical simulations. In particular, the present work highlights the importance of bulk diffusive modes in viscid dynamics of a free surface.