On the limits of the Navier-Stokes equations

Abstract

Heuristic derivations of the Navier-Stokes equations are unable to reveal the applicability limits of these equations. In this paper we rederive the Navier-Stokes equations from kinetic theory, using a method that affords a step by step insight into the required simplifying assumptions. The major task on this way is to find the conditions needed to truncate the resulting infinite system of transport equations at a finite level. The minimal obtainable closed set comprises three equations, for the particle number density $\mathit{N} $, the macroscopic velocity $\mathbf{v}$, and the temperature $\mathit{T}$. It is verified that this 3-equation system conserves the total energy, i.e., the sum of kinetic and internal energy. As a consequence, the energy is not conserved if the integrity of this closed system is violated, as for instance in the case of the so-called incompressible Navier-Stokes equations where the equation for $\mathbf{v}$ is the only one kept, and the other two discarded and jointly replaced by the incompressibility condition $\nabla \cdot \mathbf{v} = 0$. It is shown that a second viscosity, found in parts of the literature, does not exist as long as the Navier-Stokes equation is applied inside the range of its validity. Outside this range, a second viscosity builds up, however not as a matter constant. The Navier-Stokes equation in its known form rests upon the tacit assumption that the particles are points without volume and without collective forces between them, whereby dense gases and liquids are excluded, and the applicability limited to ideal gases. In the final section of this paper, an attempt is made to generalize the equations for applicability to real fluids.