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

Abstract

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 $\mathbf{V}.$ The relation asserts an exact equality between the instantaneous power consumption by the drag, $-\mathbf{F}\cdot \mathbf{V},$ 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 ${\mathbf{u}}_{\varphi }=\nabla \varphi$ of the ideal Euler equation, carrying mass flux $dJ=\rho {\mathbf{u}}_{\varphi }\cdot d\mathbf{A}.$ The results generalize the theories of M. J. Lighthill for flow past a body and, in particular, the steady-state relation $(1/2){\epsilon }_{ijk}⟨{\Sigma }_{jk}⟩={\partial }_i⟨h⟩,$ where $h=p+(1/2)|\mathbf{u}{|}^2$ 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.

arXiv comment18 pages, 5 figures. A few minor typos have been corrected

References in corpus (25)

• Potentially Singular Solutions of the 3D Incompressible Euler Equations
• Von Kármán vortex street in a Bose-Einstein condensate
• Pressure drag in linear and nonlinear quantum fluids
• On the triggering of the Ultimate Regime of convection
• A stochastic Lagrangian representation of the 3-dimensional incompressible Navier-Stokes equations
• Vortex shedding and drag in dilute Bose-Einstein condensates
• Onsager's Conjecture for the Incompressible Euler Equations in Bounded Domains
• Onsager's Conjecture with Physical Boundaries and an Application to the Vanishing Viscosity Limit
• Generation and Decay of Two-Dimensional Quantum Turbulence in a Trapped Bose-Einstein Condensate
• Turbulent Flow in Pipes and Channels as Cross-Stream "Inverse Cascades" of Vorticity
• Onsager's conjecture and anomalous dissipation on domains with boundary
• Anderson's considerations on the flow of superfluid helium: some offshoots
• Quantum Analogs of Classical Wakes in Bose-Einstein Condensates
• A stochastic-Lagrangian approach to the Navier--Stokes equations in domains with boundary
• A superfluid boundary layer
• Remarks on the emergence of weak Euler solutions in the vanishing viscosity limit
• Motion of an object through a dilute quantum fluid
• Boundary layers and emitted excitations in nonlinear Schrodinger superflow past a disk
• Is electromagnetic gauge invariance spontaneously violated in superconductors?
• Genesis of d'Alembert's paradox and analytical elaboration of the drag problem
• Flying in a superfluid: starting flow past an airfoil
• Vorticity Determines the Force on Bodies Immersed in Active Fluids
• Stochastic Lagrangian Dynamics of Vorticity. II. Channel-Flow Turbulence
• Theoretical analysis of quantum turbulence using the Onsager "ideal turbulence" theory
• Spectral Flow, the Magnus Force, and the Josephson-Anderson Relation

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• Scaling of global properties of fluctuating streamwise velocities in pipe flow: Impact of the viscous term
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• First observation of the Josephson-Anderson relation in experiments on hydrodynamic drag