Chapter 6 An Introduction to Hydrodynamic Stability Anubhab Roy and Rama Govindarajan Abstract In this chapter, our objective is twofold: (1) to describe common physical mechanisms which cause flows to become unstable, and (2) to introduce recent viewpoints on the subject. In the former, we present some well-known instabili- ties, and also discuss how surface tension and viscosity can act as both stabilisers and destabilisers. The field has gone through a somewhat large upheaval over the last two decades, with the understanding of algebraic growth of disturbances, and of absolute instability. In the latter part we touch upon these aspects. 6.1 Introduction “... not every solution of the equations of motion, even if it is exact, can actually occur in Nature. The flows that occur in Nature must not only obey the equations of fluid dynamics, but also be stable.” – Landau & Lifshitz Flow stability has preoccupied fluid dynamicists for centuries. The ubiquitous nature of turbulence, and the inability to offer a universal theory of it, led many to try tackling what was perceived as a simpler problem: of transition from a laminar state to a turbulent one. In the last two centuries, such efforts many-a-times have offered remarkable insight and helped make successful predictions. The description has undergone major reforms from time to time, and the complete process is not fully understood. The transition to turbulence begins usually with an instability of the laminar state, which is the subject of this brief review. An effort is made to address the basic tenets of hydrodynamic stability with a focus on a few recent viewpoints on the subject. R. Govindarajan (  ) Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India e-mail: rama@jncasr.ac.in J.M. Krishnan et al. (eds.), Rheology of Complex Fluids, DOI 10.1007/978-1-4419-6494-6 6, © Springer Science+Business Media, LLC 2010 131