Mixing by chaotic advection in a class of spatially periodic flows

Abstract

The paths followed by individual fluid particles can be extremely complicated even in smooth laminar flows. Such chaotic advection causes mixing of the fluid. This phenomenon is studied analytically for a class of spatially periodic flows comprising a basic flow of two-dimensional (or axisymmetric) counter-rotating vortices in a layer of fluid, and modulated by a perturbation which is periodic in time and/or space. Examples of this type of flow include Bénard convection just above the point of instability of two-dimensional roll cells, and Taylor vortex flow between concentric rotating cylinders. The transport of chaotically advected particles is modelled as a Markov process. This predicts diffusion-like mixing, and provides an expression for the diffusion coefficient. This expression explains some features of experimental results reported by Solomon & Gollub (1988): its accuracy is investigated through a detailed comparison with numerical results from a model of wavy Taylor vortex flow. The approximations used in the analysis are equivalent to those used to obtain the quasi-linear result for diffusion in the standard map.