Determination of the local micromixing structure in laminar flows

In rapidly mixing flows with low diffusion rates and/or fast reactions, accurate calculation of the local concentration distribution becomes impractical. One approach is to consider mixing segregated fluids and follow the interface between the two fluids as it stretches in the flow. For a rapid mixing flow such as a chaotic or turbulent flow, the area of the interface between segregated fluids grows exponentially, so that determination of the location of the interface rapidly becomes unfeasible. In this study, a method for calculating the local lamellar structure (i.e. the orientation and thicknesses of the lamellae) for any advection time and at any location in the flow is developed for two-dimensional flows. Locally the lamellar structure is assumed to be one-dimensional, and the orientation of the lamellae can be determined from the direction corresponding to the maximum stretching rate for fluid arriving at the location. The lamellar structure can be readily determined by mapping a short finite line, oriented perpendicular to the orientation of the lamellae, backwards in time to the initial conditions. Once the line is returned to its initial location the intersection of the line with the initial interface between the segregated fluids can be used to accurately determine the detailed lamellar structure, in particular the thicknesses of the lamellae. The accuracy of the method is demonstrated for two deterministic chaotic flows, the sine flow and a piecewise linear flow, the saw-tooth flow. The use of the piecewise linear flow enables accurate testing of the method as precise simulations of the full interface can be obtained for relatively long advection times, where the initial interface is a polygon.