The generation and conservation of vorticity: deforming interfaces and boundaries in two-dimensional flows

Abstract

This article presents a revised formulation of the generation and transport of vorticity at generalised fluid–fluid interfaces, substantially extending the work of Brons et al. (J. Fluid Mech., vol. 758, 2014, pp. 63–93). Importantly, the formulation is effectively expressed in terms of the conservation of vorticity, and the latter is shown to hold for arbitrary deformation and normal motion of the interface; previously, vorticity conservation had only been demonstrated for stationary interfaces. The present formulation also affords a simple physical description of the generation of vorticity in incompressible, Newtonian flows: the only mechanism by which vorticity may be generated on an interface is the inviscid relative acceleration of fluid elements on each side of the interface, due to pressure gradients or body forces. Viscous forces act to transfer circulation between the vortex sheet representing the interface slip velocity, and the fluid interior, but do not create vorticity on the interface. Several representative example flows are considered and interpreted under the proposed framework, illustrating the generation, transport and, importantly, the conservation of vorticity within these flows.