Inertial Momentum Dissipation for Viscosity Solutions of Euler Equations. I. Flow Around a Smooth Body
Abstract
We study the local balance of momentum for weak solutions of incompressible Euler equations obtained from the zeroviscosity 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 wallparallel and wallnormal 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 wallnormal velocity within a small distance of the wall vanishes with this distance (a precise form of the nonflowthrough 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 JosephsonAnderson relation for the drag, recently derived for strong NavierStokes solutions, is valid for weak Euler solutions obtained in their inviscid limit.
 Publication:

arXiv eprints
 Pub Date:
 June 2022
 arXiv:
 arXiv:2206.05325
 Bibcode:
 2022arXiv220605325Q
 Keywords:

 Mathematical Physics;
 Mathematics  Analysis of PDEs;
 Physics  Fluid Dynamics
 EPrint:
 24 pages, 1 figure. The revision contains a new remark about strongweak uniqueness