English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this article we examine the interaction of incompressible 2D flows withcompact material boundaries. Our focus is the dynamic behavior of thecirculation of velocity around boundary components and the possible exchangebetween flow vorticity and boundary circulation in flows with vortex sheetinitial data We begin by showing that the velocity can be uniquelyreconstructed from the vorticity and boundary component circulations, whichallows to recast 2D Euler evolution using vorticity and the circulations asdynamic variables. The weak form of this vortex dynamics formulation of theequations is called the weak vorticity formulation. The main result in thisarticle is the equivalence between the weak velocity and weak vorticityformulations, without sign assumptions. Next, we focus on weak solutionsobtained by mollifying initial data and passing to the limit, with the portionof vorticity singular with respect to the Lebesgue measure assumed to benonnegative. For these solutions we prove that the circulations around eachboundary component cannot be smaller than the initial data circulation, so thatnonnegative vorticity may be absorbed by the boundary, but not produced by theboundary. In addition, we prove that if the weak solution conserves circulationat the boundary components it is a boundary coupled weak solution, a strongerversion of the weak vorticity formulation. We prove existence of a weaksolution which conserves circulation at the boundary components if the initialvorticity is integrable. In addition, we discuss the definition of themechanical force which the flow exerts on material boundary components and itsrelation with conservation of circulation. Finally, we describe thecorresponding results for a bounded domain with holes, and the adaptationsrequired in the proofs.

Weak vorticity formulation for the incompressible Euler equations in domains with boundary