The Josephson-Anderson Relation and the Classical D'Alembert Paradox

Generalizing prior work of P. W. Anderson and E. R. Huggins, we show that a "detailed Josephson-Anderson relation" holds for drag on a finite body held at rest in a classical incompressible fluid flowing with velocity ${\bf V}.$ The relation asserts an exact equality between the instantaneous power consumption by the drag, $-{\bf F}\cdot{\bf V},$ and the vorticity flux across the potential mass current, $-(1/2)\int dJ\int \epsilon_{ijk}\Sigma_{ij}\,d\ell_k.$ Here $\Sigma_{ij}$ is the flux in the $i$th coordinate direction of the conserved $j$th component of vorticity and the line-integrals over $\ell$ are taken along streamlines of the potential flow solution ${\bf u}_\phi=\nabla\phi$ of the ideal Euler equation, carrying mass flux $dJ=\rho\,{\bf u}_\phi\cdot d{\bf A}.$ The results generalize the theories of M. J. Lighthill for flow past a body and, in particular, the steady-state relation $(1/2)\epsilon_{ijk}\langle\Sigma_{jk}\rangle =\partial_i\langle h\rangle,$ where $h=p+(1/2)|{\bf u}|^2$ is the generalized enthalpy or total pressure, extends Lighthill's theory of vorticity generation at solid walls into the interior of the flow. We use these results to explain drag on the body in terms of vortex dynamics, unifying the theories for classical fluids and for quantum superfluids. The results offer a new solution to the "D'Alembert paradox" at infinite Reynolds numbers and imply the necessary conditions for turbulent drag reduction.