The intrinsic decomposition of vorticity dynamics on an arbitrarily moving and deforming boundary

Abstract

Boundary vorticity dynamics provides a rigorous theoretical foundation for understanding vorticity creation at boundaries, vorticity-boundary interactions, as well as the rational design of effective boundary flow control strategies. It cornerstone is the boundary vorticity flux (BVF), first introduced by Lighthill in 1963, which quantities the local rate of vorticity production at a boundary, and thereby serves as a mathematical measure of distributed vorticity source strength. By adopting a differential-geometric approach, we develop a general theory of the intrinsic decomposition of BVF for compressible Newtonian fluid interacting with an arbitrarily moving and deforming boundary surface. The analyses are further extended to the decomposition of boundary enstrophy dynamics, centered on the boundary enstrophy flux (BEF). Beyond the existing literature, the new theory explicitly identifies a complete set of boundary sources for the rigid-rotation and spin modes, as well as for various enstrophy constituents, arising from the interplay among external force, surface geometry and kinematics, and both longitudinal and transverse physical processes on a deformable boundary. It is noteworthy that introducing a conjugate curvature tensor pair consistently yields compact mathematical representations for all source terms, manifesting as bilinear (or quadratic-form-type) couplings between fundamental vortcity modes and the surface curvature tensors, irrespective of the complexity or generality of the boundary kinematics.