Constitutive relations for modelling shear banding in complex fluids

Shear banding is observed in a wide range of complex fluids, including colloidal suspensions, wormlike micellar solutions, polymeric fluids, foams, and emulsions. These materials often exhibit a non-monotonic steady-state relationship between shear stress and shear rate in viscometric flows. In fluids where this non-monotonic response is observed with respect to shear rate, shear bands typically develop in the flow-gradient direction, and such fluids tend to exhibit "spurt" phenomena in pressure-driven flows. Conversely, in fluids where the non-monotonicity is observed with respect to shear stress, banding tends to occur in the vorticity direction. To model these phenomena, thermodynamically consistent rate-type constitutive relations have been developed by specifying two scalar potentials: a non-convex rate of dissipation potential and a convex free energy potential. From the class of admissible constitutive relations, a choice has been made by requiring that the rate of entropy production be non-negative and maximal. The resulting constitutive relations have been studied under simple flows and show good agreement with experimental observations reported in the literature.