  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.
