Hydrodynamic Instabilities

Hydrodynamic instabilities occur when a control parameter (vertical temperature difference in Rayleigh‒Bénard convection) exceeds a certain threshold. The generic Landau model of bifurcations may be applied to some instabilities. In the simplest case, the amplitude increases continuously and reversibly above threshold: in the Rayleigh‒Bénard case, it is determined by a balance between the driving forces due to the density gradient and the diffusive effects opposing it. This chapter then discusses instabilities governed by centrifugal forces (Taylor‒Couette) or surface-tension gradients (Bénard‒Marangoni). This last case leads to the concept of sub-critical instabilities. The Kelvin‒Helmholtz instability for parallel flows at different velocities is an example of open flows; the influence of the shape of the velocity profiles of these flows is described. Finally the stability of Couette and Poiseuille flows is discussed.