Introduction

In this chapter we discuss how various gravitational, capillary, and thermal phenomena can initiate an instability in a fluid initially at rest. In such a motionless fluid, advection of momentum plays a negligible role in the small amplitude theory, unlike the situations that will be studied in later chapters. We also present the basic techniques for studying linear stability: derivation of the equations for small perturbations of a base state, linearization, and determination of the normal modes and the dispersion relation.

An important part of this chapter is devoted to the analysis of problems in terms of ratios of characteristic scales. The approach we take is to simplify the problem by evaluating the order of magnitude of the involved phenomena before embarking on long analytic or numerical calculations. This allows us to retain only the most important effects and to elucidate mechanisms. This dimensional analysis, which is essential in both fundamental and applied research, is often sufficient for determining the scaling laws governing the problem. It allows the choice of a set of suitable reference scales to recast the problem in dimensionless form, or, in other words, to recast the problem in a system of units composed of scales that are intrinsic to the problem. This modeling approach then guides the later calculations, for example, by revealing a small parameter which suggests an asymptotic expansion. It also serves to justify or reject a posteriori certain hypotheses, and leads to a better understanding of the physics of the problem.