# Mathematics Colloquia and Seminars

Wormlike micellar solutions consist of long self-assembled micellar aggregates (wormlike micelles') which entangle in solution, thus exhibiting viscoelastic properties; in addition these aggregates break and reform. These solutions are used as thickeners in many home care products (e.g. shampoo) and as fracturing agents in oil recovery.  In wall-driven shearing flows it is known that many wormlike micellar solutions exhibit a shear-banding transition resulting in regions of high and low shear-rate.  The VCM model (Vasquez, McKinley and Cook, Journal of Non-Newtonian Fluid Mechanics, 2007) describes the coupled evolution of two micellar species: long chains which can break in half to form shorter chains, which can themselves recombine to form a long worm.  In this presentation, predictions of the VCM model are examined in pressure-driven flow through straight and wavy channels and in time-dependent filament stretching through use of asymptotics, adaptive spectral methods, linear stability analysis and domain perturbation analysis.  In straight channel flow, the velocity profile exhibits a complex spatial structure including an apparent slip boundary layer at the walls and an interior layer connecting shear bands.  An adaptive Chebyshev collocation method is developed to track and resolve the spatial and temporal evolution.  Linear stability analysis shows that at banding an interfacial instability can arise resulting in a 2D sinuous (snake-like’) perturbation flow in the flow/gradient plane, with local fluctuations along the interface between bands.   Finally, extensional flow is investigated in a converging/diverging channel and in filament stretching.