An introductory essay on subcritical instabilities and the transition to turbulence in visco-elastic parallel shear flows

摘要

This paper is an pedagogical essay on the scenario of the instabilities and the transition to turbulence in visco-elastic polymer flows. When polymers are long, they get easily stretched by the shear present in flows, and the viscosity of the solution or melt is large. As a result, inertial effects are usually negligible as the Reynolds numbers are small but the fluid is strongly nonNewtonian due to the shear-induced elasticity and anistropy, and the slow relaxation effects. The dimensionless number governing these nonNewtonian effects is the Weissenberg number Wi. From a number of precise experiments and theoretical investigations in the last fifteen years, it has become clear that as the Weissenberg number increases, visco-elastic fluids exhibit flow instabilities driven by the anisotropy of the normal stress components and the curvature of the streamlines. The combination of these normal stress effects that drive laminar curved flow unstable and the possibilty of the elastic effects to store energy in high shear regions and to dump it elsewhere in less sheared regions, appears to be strongly self-enhancing: Instabilities and the transition to a turbulent regime driven by these elastic forces, are often found to be hysteretic and strongly subcritical (nonlinear). There are two main underlying themes of this introductory essay. First of all, that it is profitable to let one be motivated by transition scenarios in Newtonian fluids as a function of Reynolds number, when investigating possible transition scenarios in visco-elastic fluids as a function of Weissenberg number. Secondly, that the self-enhancing effects of polymer stretching will also cause subcritical instabilities in visco-elastic parallel shear flows. The aim of this paper is to introduce and discuss these issues at a pictorial level which is accessible for a nonexpert. After introducing some of the basic ingredients of polymer rheology we follow a number of the important theoretical and experimental developments of the last fifteen years and discuss the picture that emerges from it. We then turn to a discussion of recent theoretical and numerical approaches aimed at establishing whether visco-elastic parallel shear flows indeed also exhibit a subcritical transition to elastic turbulence. We show how a simple extension of the well-known condition of Pakdel and McKinley for the instability threshold of curved flows, can be extended to the nonlinear (subcritical) instability scenario of parallel visco-elastic shear flows. This extension predicts the critical amplitude for the nonlinear instability to decrease as 1/Wi~2 and to be independent of the wavenumber k of the perturbations. The fact that the threshold is k-independent over a large range of k's suggest that many modes will be excited at the same time, and hence that the instability will generally drive the flow turbulent. Implications of these results and an outlook for the future are discussed as well.