Wormlike Micellar Solutions: III. VCM Model Predictions in Steady and Transient Shearing Flows

The two species, scission/reforming Vasquez-Cook-McKinley (VCM) model was formulated to describe the coupling between the viscoelastic fluid rheology and the kinetics of wormlike micellar assembly and deformation-induced rupture. The model self-consistently captures the nonlocal effects of stress-induced diffusion and has been studied in various limits for a number of canonical flow fields including Large Amplitude Oscillatory Shear (LAOS), steady and transient extensional flow as well as steady pressure-driven channel flow. However, a complete study of the spatiotemporal model predictions, both with (and without) inertia, and with (or without) the stress-concentration diffusive coupling, has not yet been reported. In this paper we present a comprehensive investigation of the full VCM model in steady and transient shearing flow including inertial and diffusive (non-local) effects. The consequences Preprint submitted to Journal of Non-Newtonian Fluid Mechanics 20 April 2014 of varying the model parameters, the effect of the start-up ramp rate, and the role of geometry on the steady state flow curve are each investigated. As a result of the onset of shear-banding and nonlocal effects in the velocity, stress and concentration profiles, we show that the measured rheological properties in a wormlike micellar solution described by the VCM model can depend on the initial ramp rate as well as specific details of the geometry such as the length scale of the rheometric fixture chosen and its curvature. The complete time evolution of the rheological response at high Deborah numbers is examined, from the initial formation of inertial waves through nonlinear overshoots in the viscoelastic stresses, shear band formation (and elastic recoil in the local velocity), to the long time diffusion-mediated approach to a final steady state.