Continuum Fluid Mechanics and the Navier-Stokes Equations

The Navier-Stokes (NS) equations provide us with a nearly all-encompassing, highly accurate physical theory that can predict practically all phenomena of interest in aerodynamics, including “aerodynamic” flows of liquids such as water. In Section 3.1, we briefly consider the general way in which these equations represent the physics, the assumptions that had to be made to arrive at them, and their range of validity. Then in the sections after that, we delve into the specifics of the equations and what they mean.

3.1 The Continuum Formulation and Its Range of Validity

In the NS formulation, the fluid is treated as a continuous material, or continuum, with local physical properties that can be represented by continuous functions of space and time. These continuum properties, of course, depend on the properties of the molecules that make up the gas or liquid and on the lower level physics of their motions and interactions. However, the continuum properties represent only the integrated effects of the lower level physics, not the details. As I noted in Chapter 2, this provides a representation that is not merely adequate, but highly accurate over a wide range of conditions.

The early historic development of the NS formulation followed an ad hoc approach, assuming continuum behavior a priori and developing a model for the effects of viscosity based on experiments in very simple flows. Much of the hard work involved in this ...

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