Interfacial vorticity dynamics for Navier–Stokes–Korteweg system: General theory and application to two-dimensional near-wall cavitation bubble

Abstract

A theory of interfacial vorticity dynamics is developed for the single-component liquid–vapor flows described by the Navier–Stokes–Korteweg equations. The non-ideal equation of state and surface capillarity are collectively considered via the chemical potential gradient force. Explicit formulas of the interfacial enstrophy and chemical potential fluxes are derived to elucidate their generation mechanisms from the interface, which are expressed by using the fundamental surface physical quantities. Then, the transport equation of the Lamb dilatation (namely, the Lamb vector divergence) is rigorously derived, generalizing the previous results of the Lamb-vector dynamics for single-phase incompressible viscous flows. The derived results are evaluated in a near-wall cavitation bubble combined with the van der Waals equation of state. The roles of surface shear stress and vorticity, surface chemical potential gradient, surface acceleration, surface stretching-shrinking motion and dilatational waves are emphasized in generating the interfacial enstrophy and chemical potential fluxes, featured by the strong vorticity concentration, interface deformation and inward-propagating microjet due to the longitudinal pressure difference. These distinct dynamical features observed near the bubble interface are well captured by the spatial distribution of the Lamb dilatation. Analyzing the dynamics of the Lamb dilatation based on the transport equation demonstrates the significance of the high-curvature rotating surface region and its vicinity in view of their dominance to the unsteady diffusion effect of the Lamb dilatation. The present exposition could provide new physical insights into liquid–vapor flows from the perspective of interfacial vorticity dynamics.