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.

The hybrid numerical-asymptotic (HNA) approach aims to reduce the computational cost of conventional numerical methods for high-frequency wave scattering problems by enriching the numerical approximation space with oscillatory basis functions, chosen based on partial knowledge of the high-frequency solution asymptotics. In this paper, we propose a new methodology for the treatment of shadow boundary effects in HNA boundary element methods, using the classical geometrical theory of diffraction phase functions combined with mesh refinement. We develop our methodology in the context of scattering by a class of sound-soft non-convex polygons, presenting a rigorous numerical analysis (supported by numerical results) which proves the effectiveness of our HNA approximation space at high frequencies. Our analysis is based on a study of certain approximation properties of the Fresnel integral and related functions, which govern the shadow boundary behaviour.

The theoretical investigation here of a three-dimensional array of jets of fluid (air guns) and their interference is motivated by applications to the food sorting industry especially. Three-dimensional motion without symmetry is addressed for arbitrary jet cross-sections and incident velocity profiles. Asymptotic analysis based on the comparatively long axial length scale of the configuration leads to a reduced longitudinal vortex system providing a slender flow model for the complete array response. Analytical and numerical studies, along with comparisons and asymptotic limits or checks, are presented for various cross-sectional shapes of nozzle and velocity inputs. The influences of swirl and of unsteady jets are examined. Substantial cross-flows are found to occur due to the interference. The flow solution is non-periodic in the cross-plane even if the nozzle array itself is periodic. The analysis shows that in general the bulk of the three-dimensional motion can be described simply in a cross-plane problem but the induced flow in the cross-plane is sensitively controlled by edge effects and incident conditions, a feature which applies to any of the array configurations examined. Interference readily alters the cross-flow direction and misdirects the jets. Design considerations centre on target positioning and jet swirling.