Rate-type constitutive relations for modelling the viscoelastic response of fluids that exhibit a non-monotonic steady-state response.

Many complex fluids, such as colloidal suspensions, wormlike micellar solutions, polymeric fluids, foams, and emulsions, undergo distinct shear-induced microstructural changes which are often manifested as a non-monotonic steady-state relationship between shear stress and shear rate in viscometric flows where either stress or strain rate becomes multi-valued. Literature suggests that non-monotonicity in the steady-state flow curve is a sufficient condition for observing “shear banding”- where the fluid separates into distinct zones of differing shear rates. Whether it is a necessary condition remains an open question. In fluids where this non-monotonic response is observed with respect to shear rate, shear bands typically develop in the flow-gradient direction, which is also referred to as “gradient banding”, and such fluids tend to experience a sudden change in flow rate in pressure-driven flows, even when the pressure is varied gradually. This is often referred to as “spurt”. Conversely, in fluids where the non-monotonicity is observed with respect to shear stress, banding tends to occur in the vorticity direction and is therefore called “vorticity banding”.

In order to efficiently model this phenomenon, we have developed two classes of thermodynamically consistent rate-type models that are capable of capturing the viscoelastic response of such fluids by specifying two scalar potentials: a non-convex rate of dissipation potential, given either as a function of stress or the kinematic variables, and Gibbs or Helmholtz free energy potential that is convex. 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. We studied the response of the models under simple viscometric flows, and they show good agreement with the experimental observations. We also studied the creep and stress relaxation responses of these models.