Transition in inertialess flows of viscoelastic fluids: the role of uncertainty
Wednesday, October 21, 2009 - 11:15am - 12:15pm
In this talk a system theoretic approach is used to model and analyze the early stages of transition in inertialess channel flows of viscoelastic fluids. We argue that modeling of uncertainty, such as the approximate nature of polymer constitutive equations, is central to understanding the dynamics of viscoelastic fluids. Robustness with respect to uncertainty is quantified by induced norms from spatio-temporal body forces to components of velocity and polymer stress fluctuations. This input-output approach has strong connections to the analysis of pseudospectra of linear operators in Hilbert space, and it exhibits the importance of streamwise elongated flow patterns in viscoelastic fluids. For streamwise independent fluctuations, we establish an explicit unfavorable scaling of the L2-induced norms with the Weissenberg number. This suggests that small amount of modeling uncertainty can destabilize nominally stable dynamics and promote transition to elastic turbulence. We also demonstrate that small stability margins originate from the stretching of polymer stress fluctuations by a background shear and identify the spatial structure of the most amplified fluctuations. One of the main messages of this talk is that, at the level of velocity fluctuation dynamics, polymer stretching and the Weissenberg number in elasticity-dominated flows of viscoelastic fluids effectively play the role of vortex tilting and the Reynolds number in inertia-dominated flows of Newtonian fluids.