What is a Fluid?

By definition, a solid material is rigid. Although a rigid material tends to shatter when subjected to very large stresses, it can withstand a moderate shear stress (i.e., a stress that tends to deform the material by changing its shape, without necessarily changing its volume) for an indefinite period. To be more exact, when a shear stress is first applied to a rigid material it deforms slightly, but then springs back to its original shape when the stress is relieved.

A plastic material, such as clay, also possess some degree of rigidity. However, the critical shear stress above which it yields is relatively small, and once this stress is exceeded the material deforms continuously and irreversibly, and does not recover its original shape when the stress is relieved.

By definition, a fluid material possesses no rigidity at all. In other words, a small fluid element is unable to withstand any tendency of an applied shear stress to change its shape. This does not preclude the possibility that such an element may offer resistance to shear stress. However, any resistance must be incapable of preventing the change in shape from eventually occurring, which implies that the force of resistance vanishes with the rate of deformation. An obvious corollary is that the shear stress must be zero everywhere inside a fluid that is in mechanical equilibrium.

Fluids are conventionally classified as either liquids or gases. The most important difference between these two types of fluid lies in their relative compressibility. To be more exact, gases can be compressed much more easily than liquids. Consequently, any motion that involves significant pressure variations is generally accompanied by much larger changes in mass density in the case of a gas than in the case of a liquid.

A macroscopic fluid ultimately consists of a huge number of individual molecules. However, most practical applications of fluid mechanics are concerned with behavior on lengthscales that are far longer than the typical intermolecular spacing. Under these circumstances, it is reasonable to suppose that the bulk properties of a given fluid are the same as if it were completely continuous in structure. A corollary of this assumption is that when, in the following, we talk about infinitesimal volume elements, we really mean elements that are sufficiently small that the bulk fluid properties (such as mass density, pressure, and velocity) are approximately constant across them, but are still sufficiently large that they contain a very great number of molecules (which implies that we can safely neglect any statistical variations in the bulk properties). The continuum hypothesis also requires infinitesimal volume elements to be much larger than the molecular mean-free-path between collisions.

In addition to the continuum hypothesis, our study of fluid mechanics is premised on three major assumptions:

1. Fluids are isotropic media. In other words, there is no preferred direction in a fluid.
2. Fluids are Newtonian. In other words, there is a linear relationship between the local shear stress and the local rate of strain, as first postulated by Isaac Newton (1642-1727). It is also assumed that there is a linear relationship between the local heat flux density and the local temperature gradient.
3. Fluids are classical. In other words, the macroscopic motion of ordinary fluids is well described by Newtonian dynamics, and both quantum and relativistic effects can be safely ignored.

It should be noted that the previous assumptions are not valid for all fluid types (e.g., certain liquid polymers, which are non-isotropic; thixotropic fluids, such as jelly or paint, which are non-Newtonian; and quantum fluids, such as liquid helium, which exhibit non-classical effects on macroscopic lengthscales). However, most practical applications of fluid mechanics involve the equilibrium and motion of bodies of water or air, extending over macroscopic lengthscales, and situated relatively close to the Earth's surface. Such bodies are very well described as isotropic, Newtonian, classical fluids.