Global geometry and coarse-grained formulation of the evolution of pointwise intermaterial interface measure in chaotic flows

Abstract

The article develops a coarse-grained model for the evolution of the intermaterial contact interface measure (length in 2 d, area in 3 d) in chaotic flows, starting from the global geometric properties characterizing these flows. The model reduces to a finite-volume formulation for the balance equations expressing the evolution in time and space of the interface measure, where its local growth rate within chaotic regions is controlled by the average 〈 D( x, t): e u ( x, t) e u ( x, t)〉, D( x, t) being the deformation tensor and e u ( x, t) the unit vector spanning the unstable invariant subspace at the point x. The analysis is developed for two- and three-dimensional flows, and is useful not only for short-cut analysis of interface measures but also as a tool in the development of coarse-grained models of reaction/diffusion kinetics in chaotic flows accounting for the complex lamellar structure generated in such systems.