A. Basic concepts and definitions

A fluid is considered a continuous isotropic substance, the individual elements of which continue to deform as the result of applied tangential surface stresses. Fluids comprise both liquids and gases.

Although fluid matter, whether liquid or gaseous, is discrete on a microscopic — i.e., molecular-scale, it is convenient — and will be done in the following — to consider small fluid elements to contain many molecules, and to work with the average statistical properties of such large numbers of molecules; that is, the detailed molecular structure is not taken into account but is replaced by a continuous model of the fluid, assuming that the smallest relevant length scale considered is still very large compared with the average intermolecular distance (continuum model of fluids). ^†

^† The continuum model is valid for gases as long as the Knudsen number is small, i.e., Kn = l ₀ /l < 0.1, where l ₀ is the mean free path for the molecules, and l is the characteristic length of the flow field; see Schaaf and Chambré (1958); Schaaf (1958).

The physical behavior of a fluid can be described by its thermodynamic and transport properties. The density ρ is such a continuum property of the fluid, defined by the following limiting process, which is typical in continuum mechanics: