Absolute instability in viscoelastic mixing layers

Abstract

The spatiotemporal linear stability of viscoelastic planar mixing layers is investigated. A one-parameter family of velocity profiles is used as the base state with the parameter, S, controlling the amount of shear and backflow. The influence of viscoelasticity in dilute polymer solutions is modeled with the Oldroyd-B and FENE-P constitutive equations. Both models require the specification of the ratio of the polymer-relaxation and convective time scales (the Weissenberg number, We) and the ratio of solvent and solution viscosities (β). The maximum polymer extensibility, L, must also be specified for the FENE-P model. We examine how the variation of these parameters along with the Reynolds number, Re, affects the minimum value of S at which the flow becomes locally absolutely unstable. With the Oldroyd-B model, the influence of viscoelasticity is shown to be almost fully captured by the elasticity, \documentclass[12pt]{minimal}\begin{document}$E^* \equiv \frac{(1-\beta ) We}{Re}$\end{document}E*≡(1−β)WeRe, and Scrit decreases as elasticity is increased, i.e., elasticity is destabilizing. A simple approximate dispersion relation obtained via long-wave asymptotic analysis is shown to accurately capture this destabilizing influence. Results obtained with the FENE-P model exhibit a rich variety of behavior. At large values of the extensibility, L, results are similar to those for the Oldroyd-B fluid as expected. However, when the extensibility is reduced to more realistic values (L ≈ 100), one must consider the scaled shear rate, \documentclass[12pt]{minimal}\begin{document}$\eta _c \equiv \frac{We S}{2L}$\end{document}ηc≡WeS2L, in addition to the elasticity. When ηc is large, the base-state polymer stress obtained by the FENE-P model is reduced, and there is a corresponding reduction in the overall influence of viscoelasticity on stability. Additionally, elasticity exhibits a stabilizing effect which is driven by the streamwise-normal perturbation polymer stress. As ηc is reduced, the base-state and perturbation normal polymer stresses predicted by the FENE-P model move towards the Oldroyd-B values, and the destabilizing influence of elasticity observed with the Oldroyd-B model is again present.