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The study of low-Mach-number flows occurs in the general context of asymptotic modelling of fluid flows, and in a recent book (Zeytounian 2004, Chap. 6) an argument was given for the value of asymptotic modelling of flows at low Mach number as a effective way of building consistent models amenable to numerical simulation, because it is well known that time-dependent compressible-flow schemes often become ineffective at low Mach numbers.

It has long been known (Janzen 1913 and Rayleigh 1916) that asymptotic methods provide a sound way of describing incompressible aerodynamics (hydrodynamics) in terms of a low-Mach-number ( ≪ 1) flow, and a comprehensive discussion of this topic was given by Imai (1957), as far as steady flows of an inviscid fluid are concerned.

In this chapter, we describe an initial (rather ‘naive’) approach to the asymptotics of low-Mach-number flows, based mainly on consideration of the equations, and show for various cases why the initial and boundary conditions (linked to a physically realistic fluid flow problem) are very important in the framework of a consistent theory for the asymptotic modelling of low- Mach-number flows.

In this chapter, we consider first, in Sect. 4.1, a detailed derivation of the Navier–Fourier initial–boundary–value problem for the Navier velocity vector , the pseudo–pressure and the perturbation of the temperature , which are the solutions of the system of the Navier equations (3.55) and an approximate equation for , derived from the full NSF equation (3.48c), for the temperature .

In the present chapter, the main problem concerns low-Mach-number flow affected by acoustic effects in a confined gas over a long time.

First, in Sect. 6.1, we discuss in detail various facets of the applicability of the Boussinesq approximation to slow atmospheric motion. In Sect. 6.2, we give an asymptotic derivation of the Boussinesq lee-wave-model inviscid equations, from the system of Euler equations (1.40) derived in Sect. 1.7.1.

In this last chapter, we consider, first, in Sect. 7.1, mainly the asymptotic derivation of the KZK equation of nonlinear acoustics, which generalizes the well-known Burgers' unsteady one-dimensional dissipative model equation (Burgers 1948) to an equation with a diffraction and parabolic effect.