Exact results with the J-integral applied to free-boundary flows

Abstract

We apply the J-integral to free-boundary flows in a channel geometry such as viscous fingering or blob injection in Hele-Shaw cells, void propagation in electromigration, and injection of air bubbles into inviscid liquids. The theory of that and related conservation integrals, developed in elasticity, is outlined in a way that is applicable to fluid mechanics problems. Depending on the boundary conditions, for infinite bubbles in Laplacian fields we are able to use the J-integral to predict finger width if such solutions exist or to predict that there are no solutions. For finite sized bubbles, bounds can sometimes be derived. In the case of Hele-Shaw flows, in which solutions appear as a continuum, finger width cannot be constrained, but we do obtain a new derivation and generalization of Richardson moment conservation. Applications to vortex motion are also outlined briefly.