A differential velocities-based study of diffusion effects in shear banding micellar solutions

By using the generalized bracket approach of non-equilibrium thermodynamics, we recently developed a multi-fluid approach to account for the Fickian diffusion and stress-induced migration in wormlike micellar solutions. In this work, we propose a modification to this framework where differential velocities are not treated as intermediate variables anymore. As the differential velocities are now used as state variables, additional boundary conditions arising from the presence of the diffusive derivative terms can be directly imposed with respect to these variables. The advantage of this modification is that no microstructural information is required for the specification of the boundary conditions. Sample calculations on this new model performed for transient cylindrical Couette flow revealed that the diffusional processes greatly affect the temporal evolution of the rheology and microstructure of the wormlike micelles. In the nonlinear viscoelastic regime, the stress-induced migration dominates over the Fickian diffusion. The smaller the diffusivity constant, the more time is required to reach steady state. Furthermore, the steady-state solution was found to be independent of the (nonzero) value of the diffusivity constant. To smooth the kink between the shear bands and to obtain a unique solution, we believe that nonlocal and surface effects would have to be accounted for. The proposed multi-fluid description of diffusion is general and could be applied to other complex fluids such as polymer solutions.