Stokes-Navier Equations and the Fundamental Equations of Flow Noise

Abstract

For ideal fluids, pressure may be considered to represent a function of the density that is defined by the state equation and the Euler equations. For viscous flow, variable pressure has to be replaced by the stress tensor. A scalar “pressure” no longer exists. The only scalar point functions that can still be retained for a viscous fluid are density and the trace of stress tensor (the average of the normal stresses in the three coordinated directions). The trace of the stress tensor and the density are, therefore, necessarily functions of one another. This conclusion makes it possible to expand (as Stokes has) the concept of pressure for viscous flow by defining it as the average value of the normal stresses. This generalized pressure is a function of the density and measures not the true stress in the fluid but the average of the stresses over all possible directions. In the classical theory of internal friction, the relation λ+23u=0 is wrongly impressed on the volume viscosity λ. In modern derivations, the volume viscosity λ is retained in its general form, but the pressure is no longer identified with a third of the exact value of the trace of the stress tensor. This situation makes it necessary to rederive the fundamental equations of flow and of flow noise; the derivations to be given are exact; the only assumptions that will be introduced are those of an isotropic fluid and of ideal fluid friction; the friction forces depend on the velocities only, and not on their higher-order time derivatives.