Instantaneous drag for flow through general curvilinear channels in terms of vortex dynamics: the Josephson-Anderson relation*

The detailed Josephson-Anderson (JA) relation for classical fluids [1] equates instantaneous work by pressure drop over any streamwise segment of a general channel to wall-normal flux of spanwise vorticity, spatially integrated over that section. The potential flow with the same mass flux, as in Kelvin’s minimum energy theorem, appears as background field and incorporates information about channel geometry. We have generalized this result to streamwise periodic channels for convenient use in numerical simulations. Whereas the usual Neumann b.c. [1] create an unphysical vortex sheet in a periodic channel, we exploit instead Dirichlet b.c. to define the background potential. We show that the minimum energy theorem still holds and the JA relation again equates work by pressure drop to integrated flux of spanwise vorticity. The result holds for both Newtonian and non-Newtonian fluids and for general curvilinear walls.  We present some numerical results for our new formula. The relation arose in quantum superfluid theory and it holds also for external flows around solid bodies [2], related to works of Burgers, Lighthill, etc. Drag and dissipation are thus related very generally to vorticity structure and dynamics locally in space and time, with many applications to drag-reduction, e.g. by polymers, and calculation of drag, e.g. in rough-wall channels.

[1] E. R. Huggins, Phys. Rev. A 1, 332-337 (1970) 

[2] G. L. Eyink, Phys. Rev. X 11, 031054 (2021)