OFFPRINT Criterion for purely elastic Taylor-Couette instability in the flows of shear-banding fluids

In the past twenty years, shear-banding flows have been probed by various techniques, such as rheometry, velocimetry and flow birefringence. In micellar solutions, many of the data collected exhibit unexplained spatio-temporal fluctuations. Recently, it has been suggested that those fluctuations originate from a purely elastic instability of the flow. In cylindrical Couette geometry, the instability is reminiscent of the Taylor-like instability observed in viscoelastic polymer solutions. In this letter, we describe how the criterion for purely elastic Taylor-Couette instability should be adapted to shear-banding flows. We derive three categories of shear-banding flows with curved streamlines, depending on their stability. Copyright c © EPLA, 2011 “The stability of viscous liquids contained between two rotating cylinders” of radii Ri and Ro —or Taylor-Couette (TC) flow— is the benchmark problem for instability of flows with curved streamlines. It was the title of a seminal paper by Taylor in 1923 [1], wherein the author showed that the purely annular flow eventually becomes unstable. Above a critical rotation speed, a secondary vortex flow sets in, with periodicity λ∼ d along the vorticity direction, where d≡Ro−Ri. The original study by Taylor concerned simple incompressible Newtonian fluids. But many fluids are non-Newtonian and exhibit viscoelastic contributions to the stress [2]. In 1990, Larson, Shaqfeh and Muller showed that the TC flow of polymer solutions could also become unstable to a Taylor-like instability [3]. The kinematics of the unsteady flow are roughly similar to those of the Newtonian case, i.e. after a critical threshold, Taylor vortices appear, but the destabilizing mechanisms are very different, depending on two different kinds of non-linearities. It is well known that Newtonian fluids can exhibit increasingly unstable flows for large values of the Reynolds number. When only the inner cylinder is rotating, the Reynolds number depends on the rotation rate of the (a)E-mail: [email protected] inner cylinder Ωi, such that Re≡ ΩiRid ν , where ν is the kinematic viscosity of the fluid [4]. In a simple Newtonian fluid, the constitutive relation is the simple linear relation between stress and shear rate. Then, the only non-linearity in the equations of motion comes from the advective term on the velocity (v ·∇)v, in the equation of motion to ensure consistency between Eulerian and Lagrangian descriptions of fluid motion. The Reynolds number Re is linked to the relative magnitude of this term with respect to the dissipation terms [4]. In polymer solutions, and in many non-Newtonian fluids, the primary non-linearity usually comes from the constitutive relation rather than from the momentum balance. The constitutive equation is dynamical, i.e. it relates to the stress relaxation dynamics and typically includes a convected derivative on the stress T [2]. In this convected derivative, consistency between Eulerian and Lagrangian descriptions requires a convective term, now applied on the stress (v ·∇)T, and material frame independence requires additional terms of similar dimensionality ∇v ·T [2,3]. The dimensionless group linked to the magnitude of those new non-linear terms is the Weissenberg number Wi≡ τ γ̇, where γ̇ ≡ ΩiRi d is the typical shear rate in the flow and τ is the stress relaxation time [5]. The similarity between Wi and Re is more apparent by