Multiple Shear-Banding Transitions for a Model of Wormlike Micellar Solutions

Abstract

Wormlike micelles are long wormy cylindrical aggregates of surfactants, self-assembled within a solvent, which entangle and continuously break and reform at thermal equilibrium. Rheological characterization and flow visualization experiments with micellar solutions show that under steady state shearing flow the deformation field may not remain homogeneous but instead spatially localize, resulting in the formation of pronounced shear bands. Models which capture this banding behavior generally display a nonmonotonic constitutive response or “flow curve” (of the shear stress resulting from the imposed shear rate). Homogeneous steady state solutions along the decreasing portion of this constitutive curve are unstable and, under shear rate control, the solution in this regime bifurcates to a spatially inhomogeneous flow with two shear rates selected from the positive slope portions of the curve that coexist at identical values of the stress. Tracking of the spatio-temporal development of the banded solution structure shows a strong elastic recoil in the local fluid velocity profile at short times (earlier than the effective relaxation time of the entangled chains). At longer times the velocity profile approaches its steady banded state. These predictions agree with experimental observations by Miller and Rothstein [J. Non-Newtonian Fluid Mech., 143 (2007), pp. 22--37]. In this paper the interplay of the competing roles of inertia, the imposed shear rate, and the transient dynamics of the start up in the flow are examined using the VCM (Vasquez--Cook--McKinley) model. This constitutive model is a scission/reforming network model developed to capture the essential physics of the deformable micellar microstructure and its coupling to the macroscopic flow field. The addition of inertia into the coupled set of nonlinear partial differential equations describing the material response changes the type of the equation set, introducing a transient damped (diffusive and dispersive) inertio-elastic shear wave following the imposition of flow. Depending on the relative time scales associated with the damping, the shear wave speed, the start-up ramp speed, and the imposed shear rate, the reflections of the damped transient wave from the boundaries can interfere with the microscopic mechanisms leading to elastic recoil and the localization of the shear that leads to formation of a shear band. The result of this interference is the establishment of a transient velocity profile with a varying number of (two, three, or four) shear bands. When there is no stress diffusion in the model the multiple-banded profile exists to steady state, and the resulting macroscopic flow is thus not uniquely specified by the imposed shear rate alone.