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 hy drodynamic origin is discussed. The responding Navier-Stokes equation is an alyzed for the two-plate geometry, where flow gradients are assumed to exis t only in the direction perpendicular to the two plates. The linearized Nav ier-Stokes equation is shown to be very similar to the Cahn-Hilliard equati on for spinodal decomposition, with a similar term. that: stabilizes rapid spatial variations. Only slowly varying flow gradients are unstable. Just a s in the initial stage of spinodal decomposition there is a most rapidly gr owing wavelength in the Initial stage of the shear-banding transition, lead ing to a predictable number of shear bands. A modified Maxwell equal area c onstruction is derived, which dictates the stress and the shear rates in th e bands under controlled shear conditions, and which shows that under contr olled stress conditions no true shear bands can coexist. The kinetics of th e shear-banding transition is studied numerically. I;or the two-plate geome try it is found that there exist multiple stationary states under controlle d shear conditions, depending on the initial state of the flow profile. She ar banding occurs not only when the system is initially unstable, but can a lso be induced outside the unstable region when the amplitude of the initia l perturbation is large enough. The shear-banding transition can thus proce ed via "spinodal demixing" (from an unstable initial state) or via "condens ation." Under controlled stress conditions no stationary state is found. He re, coupling with flow gradients extending in other directions, not perpend icular to the two plates, should probably be taken into account. [S1063-651 X(99)18310-7].