Shear-banding instabilities

Gradient-banding and vorticity-banding instabilities, as well as a shear-induced instability due to shear-gradient induced mass transport will be discussed. Various scenarios that underly these instabilities are addressed and simple constitutive relations that allow for a (semi-) quantitative analysis are proposed. A relatively simple constitutive equation that has been proposed some time ago is reviewed, which captures a number of the experimentally observed gradient-banding phenomena. This constitutive equation is based on the usual formal expansion of the stress tensor with respect to gradients in the flow velocity, but now including the second order term. The second order term is necessary to describe the relatively large spatial gradients within the interface between the two bands. The resulting simple constitutive equation is shown to give rise to stationary gradient-banded states, where the shear rates within the bands are constant, it describes stress selection under controlled rate conditions and explains why banding can not occur under controlled stress conditions. The simple constitutive equation does not include coupling to concentration, which may give rise to banding also under controlled stress conditions. Two examples of mechanisms that lead to the strong shear thinning that is necessary for gradient banding are discussed: (i) transient forces due to entanglements in polymer systems, and (ii) critical slowing down. The latter mechanism is shown to be important for a worm-like micellar system. The mechanism that leads to vorticity banding is still under debate. Vorticity banding of fd-virus suspensions within the two-phase isotropic-nematic coexistence will be discussed. Experiments on the kinetics of banding and particle-tracking experiments lead to a recently proposed mechanism for the vorticity-banding instability, where the instability is identified as an elastic instability similar to the polymer-Weissenberg effect. The role of polymer chains in the classic Weissenberg effect is now played by inhomogeneities formed during the initial stages of phase separation. For other systems than fd-virus suspensions that exhibit vorticity banding, the inhomogeneities general have a different origin, like in weakly aggregated colloids and worm-like micellar systems where the inhomogeneities are the colloidal aggregates and the worms, respectively. An instability that has been discovered some time ago, which is an instability due to shear-gradient induced mass transport is also discussed. The coupling between shear-gradients and mass transport has been formally introduced through a shear-rate dependent chemical potential, of which the microscopic origin was not explained. It will be shown that the microscopic origin of this coupling is related to the shear-induced distortion of the pair-correlation function. Contrary to the stationary gradient-banded and vorticity-banded state, it is not yet known what the stationary state is when this shear-concentration-coupling instability occurs