This course will deal with a mathematical idealization of common fluids such as air or water. The main idealization is embodied in the notion of a continuum and our “fluids” will generally be identified with a certain connected set of points in RN, where we will consider dimension N to be 1,2, or 3. Of course the fluids will move, so basically our subject is that of a moving continuum.

This description is an idealization which neglects the molecular structure of real fluids. Liquids are fluids characterized by random motions of molecules on the scale of 10−7 − 10−8 cm, and by a substantial resistance to compression.

Gases consist of molecules moving over much larger distances, with mean free paths of the order of 10−3 cm, and are readily compressed. Both liquids and gases will fall within the scope of the theory of fluid motion which we will develop below. The theory will deal with observable properties such as velocity, density, and pressure. These properties must be understood as averages over volumes which contains many molecules but are small enough to be “infinitesimal” with respect to the length scale of variation of the property. We shall use the term fluid parcel to indicate such a small volume. The notion of a particle of fluid will also be used, but should not be confused with a molecule. For example, the time rate of change of position of a fluid particle will be the fluid velocity, which is an average velocity taken over a parcel and is distinct from molecular velocities. The continuum theory has wide applicability to the natural world, but there are certain situations where it is not satisfactory. Usually these will involve small domains where the molecular structure becomes important, such as shock waves or fluid interfaces.