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