Hydrodynamics plays a ubiquitous role in our current environment. In the preamble of his lectures notes on fluid dynamics, K. Moffatt (1973) gives a wheel-shaped picture of the world as conceived by an egocentric hydrodynamicist. His picture strikingly illustrates the interrelations between this scientific domain and others, from the most basic fields such as pure mathematics to applied engineering or medicine, all organized in concentric circles. In our Fig. 1, we display an adaptation of his viewpoint, focused more on the material contained in this book, namely topics involving the innermost levels relating physics and mathematics to fluid mechanics. But we hope that this will not hide the fact that enrichments are often motivated by problems raised in fields possibly remote from the “center.” This Overview chapter is therefore a long digression about this figure, intended to provide the reader with a documented guide on how the material in the rest of the book is organized.

As implied by the first part of its title, the book begins with a general introduction to hydrodynamics under the leadership of B. Castaing. From a physical point of view, hydrodynamics accounts for the low-frequency, long-wavelength properties of matter, i.e., its macroscopic behavior as related to conserved or quasi-conserved quantities (cf. Martin et al., 1972). The usual understanding is more restrictively centered on continuous media that “flow,” the description of which will be introduced in C1-1. To be more specific, let us consider the simplest case of a one-component Newtonian fluid.