Linear and Nonlinear Stability of Fluid Motion

Abstract

In this chapter, we investigate some qualitative and quantitative properties of solutions of fluid dynamics equations (mainly Navier equations), give some results for the stability of various flows, and discuss some instabilities. First, Sect. 9.1 is devoted to a general discussion of different (qualitative) aspects of the linear and nonlinear stability theory of fluid motion, and in Sect. 9.2, the fundamental ideas concerning the linear and nonlinear stability of fluid motion are briefly presented. Using a technique of the Lyapunov-Schmidt type derived from bifurcation theory and perturbation methods, coupled with multiple scales technique, we present in Sect. 9.3, a unified approach to nonlinear hydrodynamic stability. In Sect. 9.4, we consider some facets of the Rayleigh-Bénard (RB) convective instability (buoyancy effect) problem and the Bénard-Marangoni (BM) thermocapillary instability (deformable free-surface effect) problem. In Sect. 9.5, we discuss some features of nonlinear Couette-Taylor viscous flow between two rotating concentric (infinite) cylinders. Finally, in Sect. 9.6, some concluding remarks are given relative to the stability of viscous fluid flows.