Inertial Momentum Dissipation for Viscosity Solutions of Euler Equations: External Flow Around a Smooth Body

TL;DR

This paper investigates the momentum transfer and surface forces in inviscid flow around a smooth body, linking viscous effects to Euler solutions in the zero-viscosity limit, and validating classical theories of vorticity and drag.

Contribution

It introduces a novel framework for understanding wall friction and pressure in Euler solutions derived from viscous flows, connecting these to classical theories and providing new insights into inviscid limit behavior.

Findings

Viscous skin friction and wall pressure persist as distributions in the inviscid limit.

Wall friction vanishes under bounded velocity and pressure conditions near the wall.

Pressure forces account for all drag when wall-normal velocity diminishes near the wall.

Abstract

We study the local balance of momentum for weak solutions of incompressible Euler equations obtained from the zero-viscosity limit in the presence of solid boundaries, taking as an example flow around a finite, smooth body. We show that both viscous skin friction and wall pressure exist in the inviscid limit as distributions on the body surface. We define a nonlinear spatial flux of momentum toward the wall for the Euler solution, and show that wall friction and pressure are obtained from this momentum flux in the limit of vanishing distance to the wall, for the wall-parallel and wall-normal components, respectively. We show furthermore that the skin friction describing anomalous momentum transfer to the wall will vanish if velocity and pressure are bounded in a neighborhood of the wall and if also the essential supremum of wall-normal velocity within a small distance of the wall vanishes with this distance (a precise form of the vanishing wall-normal velocity condition). In the latter case, all of the limiting drag arises from pressure forces acting on the body and the pressure at the body surface can be obtained as the limit approaching the wall of the interior pressure for the Euler solution. As one application of this result, we show that Lighthill's theory of vorticity generation at the wall is valid for the Euler solutions obtained in the inviscid limit. Further, in a companion work, we show that the Josephson-Anderson relation for the drag, recently derived for strong Navier-Stokes solutions, is valid for weak Euler solutions obtained in their inviscid limit.