Mathematical Principles of Classical Fluid Mechanics

TLDR: In this article, the authors consider the problem of describing the physical behavior of a fluid in terms of a system of differential equations, and their derivation from fundamental axioms, and the various forms in which they take when more or less special assumptions concerning the fluid or the fluid motion are made.

Abstract: Classical fluid mechanics is a branch of continuum mechanics; that is, it proceeds on the assumption that a fluid is practically continuous and homogeneous in structure. The fundamental property which distinguishes a fluid from other continuous media is that it cannot be in equilibrium in a state of stress such that the mutual action between two adjacent parts is oblique to the common surface. Though this property is the basis of hydrostatics and hydrodynamics, it is by itself insufficient for the description of fluid motion. In order to characterize the physical behavior of a fluid the property must be extended, given suitable analytical form, and introduced into the equations of motion of a general continuous medium, this leading ultimately to a system of differential equations which are to be satisfied by the, velocity, density, pressure, etc. of an arbitrary fluid motion. In this article we shall consider these differential equations, their derivation from fundamental axioms, and the various forms which they take when more or less special assumptions concerning the fluid or the fluid motion are made.