Straining flow of a weakly interacting polymer-surfactant solution

In this paper, we consider the straining flow of a weakly interacting polymer-surfactant solution below a free surface, with the bulk surfactant concentration above the critical micelle concentration. We formulate a set of coupled differential equations describing the concentration of monomers, micelles, polymer, and polymer-micelle aggregates in the flow. We analyse the model in several asymptotic limits, and make predictions about the distribution of each of the species. In particular, in the large-reaction-rate limit we find that the model predicts a region near the free surface where no micelles or aggregates are present, and beneath this a region where the concentration of surfactant is constant, across which the concentration of aggregates increases until all the free polymer is consumed. For certain parameter regimes, a maximum in the concentration of the polymer-micelle complex occurs within the bulk fluid. In the finite-reaction-rate limit, micelles, and aggregates are present right up to the free surface, and the plateau in the concentration of surfactant in the bulk is no longer present. Results from the asymptotic theory compare favorably with full numerical solutions.