A constitutive relation describing the shear-banding transition.

An additional contribution to the standard expression for the shear stress must be considered in order to describe shear banding. A possible extension of the standard constitutive relation is proposed. Its physical, purely hydrodynamic origin is discussed. The corresponding Navier-Stokes equation is analyzed for the two-plate geometry, where flow gradients are assumed to exist only in the direction perpendicular to the two plates. The linearized Navier-Stokes equation is shown to be very similar to the Cahn-Hilliard equation for spinodal decomposition, with a similar term that stabilizes rapid spatial variations. Only slowly varying flow gradients are unstable. Just as in the initial stage of spinodal decomposition there is a most rapidly growing wavelength in the initial stage of the shear-banding transition, leading to a predictable number of shear bands. A modified Maxwell equal area construction is derived, which dictates the stress and the shear rates in the bands under controlled shear conditions, and which shows that under controlled stress conditions no true shear bands can coexist. The kinetics of the shear-banding transition is studied numerically. For the two-plate geometry it is found that there exist multiple stationary states under controlled shear conditions, depending on the initial state of the flow profile. Shear banding occurs not only when the system is initially unstable, but can also be induced outside the unstable region when the amplitude of the initial perturbation is large enough. The shear-banding transition can thus proceed via "spinodal demixing" (from an unstable initial state) or via "condensation." Under controlled stress conditions no stationary state is found. Here, coupling with flow gradients extending in other directions, not perpendicular to the two plates, should probably be taken into account.