Distributive mixing in creeping flow : coherent structures, quality measures and reversibility

The objective of this paper is to show the advantages of an analytical solution to study the problem of mixing of highly viscous materials, i.e. distributive mixing with no diffusion. This study is based on the construction of Eulerian and Lagrangian representations of the mixing process, the determination of the periodic points, and the quantification of the mixed state. It addresses problems: how to organize steady or periodic distributive mixing in a two-dimensional wedge cavity (a prototype for an extruder mixer) and where to put a blob (or blobs) in order to achieve the best result of mixing in finite time. The central question is what determine a coherent structures of mixing patterns. Three criteria for estimating the quality of mixing are developed. The idea of our approach traces back to the fundamental work of Gibbs (1902), Danckwerts (1952) and Welander (1955) and consists of using the concept of a `coarse grain' density of the mixed component. Numerical data are presented showing the change in time of the statistical values of the square density, the entropy and the intensity of segregation. These data permit to estimate the dynamics of mixing and deviation from a uniform state at any moment in time. Time reversibility is an important issue in numerical simulations, it concerns the reversibility both of individual points and quality measures. We have concluded that computation of the measures shows a complete reversibility in spite of irreversibility of some individual points