Josephson-Anderson Relation and the Classical D’Alembert Paradox

Abstract

Generalizing the 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 V. The relation asserts an exact equality between the instantaneous power consumption by the drag -F·V and the vorticity flux across the potential mass current -(1/2)∫dJ∫ε_{ijk}Σ_{ij}dℓ_{k}. Here, Σ_{ij} is the flux in the ith coordinate direction of the conserved jth component of vorticity, and the line integrals over ℓ are taken along streamlines of the potential-flow solution u_{ϕ}=∇ϕ of the ideal Euler equation, carrying mass flux dJ=ρu_{ϕ}·dA. Drag and dissipation are thus associated with the motion of vorticity relative to this background ideal potential flow solving Euler’s equation. The results generalize the theories of M. J. Lighthill for flow past a body and, in particular, the steady-state relation (1/2)ε_{ijk}⟨Σ_{jk}⟩=∂_{i}⟨h⟩, where h=p+(1/2)|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, provide an explanation for a long-standing puzzle about the experimental conditions required for anomalous turbulent energy dissipation, and imply the necessary and sufficient conditions for turbulent drag reduction.