Quantifying fluid mixing with the Shannon entropy

We introduce a methodology to quantify the quality of mixing in various systems, including polymeric ones, by adapting the Shannon information entropy. For illustrative purposes we use particle advection of two species in a two-dimensional cavity flow. We compute the entropy by using the probability of finding a suitable chosen group/complex of particles of a given species, at a given location. By choosing the size of the group to be in direct proportion to the overall concentration of the components in the mixture we ensure that the entropic measure is maximized for the case of perfect mixing, that is, when at each location the component concentration is equal to the corresponding overall component concentrations. The scale of observation role in evaluating mixing is analyzed using the entropic methodology. We also illustrate the effect of initial conditions on mixing in a laminar system, typical in operations involving polymers.