Mathematical Theory of Compressible Fluid Flow

The theory of fluid flow (for an incompressible or compressible fluid, whether liquid or gas) is based on the Newtonian mechanics of a small solid body. The essential part of Newton's Principle can be formulated into the following statements: (a) To each small solid body can be assigned a positive number ra, invariant in time and called its mass; and (b) The body moves in such a way that at each moment the product of its acceleration vector by ra is equal to the sum of certain other vectors, called forces, which are determined by the circumstances under which the motion takes place (Newton's Second Law).1 For example, if a bullet moves through the atmosphere, one force is gravity rag, directed vertically downward (g = 32.17 ft/sec2 at latitude 45°N); another is the air resistance, or drag, opposite in direction to the velocity vector, with magnitude depending upon that of the velocity, etc. For example, if a bullet moves through the atmosphere, one force is gravity rag, directed vertically downward (g = 32.17 ft/sec2 at latitude 45°N); another is the air resistance, or drag, opposite in direction to the velocity vector, with magnitude depending upon that of the velocity, etc. By means of a limiting process, this principle can be adapted to the case of a continuum in which a velocity vector q and an acceleration vector dq/dt exist at each point. Let Ρ be a point with coordinates (x, y, z), or position vector r, and dV a volume element in the neighborhood of P; to this volume element will be assigned a mass pdV, where ρ is the density, or mass per unit volume. Density will be measured in slugs per cubic foot. For air under standard conditions (temperature 59°F, pressure 29.92 in. Hg, or 2116 lb/ft2), ρ = 0.002378 slug/ft3, as compared with ρ = 1.94 slug/ft3 for water.