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

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. 1. Introduction We present here the technique of the path-independent J-integral (Rice 1968a, b) for interfacial fluid instabilities. The J-integral is commonly used in elasticity for cracks and notches in static and steady-state dynamic cases. It is useful both for its path independence, which we exploit here, and its property of giving the configurational force on an elastic singularity Rice 1968b). It allows an easy determination of the asymptotic stress and strain in the close vicinity of a crack, for example. It is not restricted to the linear elasticity case where the field in the sample is biharmonic (Eshelby 1970; Knowles & Sternberg 1972), and can also take into account plasticity for non-growing cracks, within the approximation of 'deformation theory' (Hutchinson 1968; Rice & Rosengren 1968). In this paper we present some applications of this technique when the field in the flow is Laplacian. We treat successively Darcy's law in the Hele-Shaw context, electromigration, and Euler flow for inviscid fluids, in the strip geometry. We consider infinite or finite bubbles in two or three dimensions but with axisymmetry. Since the boundary conditions which constrains the dynamics, and so the existence of the solution, concern derivatives of the field φ in the normal and the tangential directions, locally defined on the interface and on the cell boundaries, the J-integral involving φ seems to be especially convenient. The application is very simple and for an infinite finger-shaped solution, we are in some cases able to predict the asymptotic relative width of the finger compared to the cell size when it is unique