Irreversibility of mechanical and hydrodynamic instabilities

Abstract

The literature on dynamical systems has, for the most part, considered self-oscillators (i.e., systems capable of generating and maintaining a periodic motion at the expense of an external energy source with no corresponding periodicity) either as applications of the concepts of limit cycle and Hopf bifurcation in the theory of differential equations, or else as instability problems in feedback control systems. Here we outline a complementary approach, based on physical considerations of work extraction and thermodynamic irreversibility. We illustrate the power of this method with two concrete examples: the mechanical instability of rotors that spin at super-critical speeds, and the hydrodynamic Kelvin-Helmholtz instability of the interface between fluid layers with different tangential velocities. Our treatment clarifies the necessary role of frictional or viscous dissipation (and therefore of irreversibility), while revealing an underlying unity to the physics of many irreversible processes that generate mechanical work and an autonomous temporal structure (periodic, quasi-periodic, or chaotic) in the presence of an out-of-equilibrium background.