To avoid misunderstandings of what the Navier-Stokes equations does and does not model, as well as I begin with a discussion on its framework,
This text is not a scientific one and I have chosen not provide references. These can be found in my scientific papers on the new model. A very good paper for understanding the underpinnings of the Navier-Stokes equations is of course this one by Stokes.
The compressible Navier-Stokes equations
The (modern) compressible Navier-Stokes equations formulated in an Eulerian (fixed-in-space) frame are:
These equations describe the conservation of mass, momentum and energy. rho is the density; v the velocity; p is pressure; S the stress tensor; E the total energy; T temperature; kappa is the heat conductivity coefficient; mu the viscosity coefficient; R is the gas constant.
These equations are hyperbolic-parabolic and should not be confused with the incompressible Navier-Stokes equations that is used for liquids. Here, only gas flows are discussed.
The hyperbolic continuity equation and its consequences
Perhaps, the most eye-catching feature of the (compressible) Navier-Stokes equations, is the zero on the right-hand of the continuity (mass) equation. The other equations, conservation of momentum and energy, have viscous, and heat conducitve, terms on the right-hand side and the zero breaks this symmetry.
The zero in the continuity equation has hitherto prevented any attempt to prove well-posedness. For obvious physical reasons, solutions to the Navier-Stokes equations must have non-negative density, temperature and pressure. (This is often referred to as «positivity».) Mathematically, the equations break down should any thermodynamic variable become negative. However, due to the zero in the continuity equation positivity can not be proven to hold. That is, the equations do not guarantee that, when given reasonable input data, the solution will not evolve into an unphysical one that violates positivity.
This is not only a mathematical nuisance. In fact, a negative density or pressure is the most common reason for a CFD code to crash which has spawned research into numerical fixes that prevent this. (See next paragraph.) Nevertheless, it may be argued that a model is not designed to be convenient for mathematical and numerical analysts; it is supposed model the physical reality. However, a negative denisty is arguable unphysical. More generally, unless a model is well-posed, predictive simulations are impossible to carry out. Despite the fact that well-posedness of the Navier-Stokes equations is largely unknown, numerical computations are routinely carried out. However, for non-linear PDEs it is perfectly possible that a consistent numerical approximation of a non-linear conservation law is stable, and yet it produces a numerical solution that is utterly incorrect! Stability does not guarantee that the code produces an approximate solution.
Due to positivity problems one may even argue that it is not the Navier-Stokes equations that are solved in CFD codes. (Unless the flow is subsonic, laminar and dynamically stable. Then there is some mathematical theory supporting numerical schemes.) Typically, a substantial amount of artificial diffusion is needed to stabilise the computations. In particular, artificial diffusion is needed in the continuity equation in order to prevent negative thermodynamic variables. Furthermore, since it is usually impossible to fully resolve realistic flows, these artificial diffusion terms are often larger than the physical diffusion. Hence, it is not the Navier-Stokes equations that are effectively approximated but rather some homogenised Euler system.
There are in fact a number of physically strange properties of the Navier-Stokes equations, that all emanate from the zero in the continuity equation. Namely, there may be a non-zero entropy and energy gradient through an adiabatic wall; relaxation to thermodynamic equilibrium requires convective transport even at the diffusive scale; remarkably complicated (and physically inexplicable) models are required for multi-component flows in order to not violate the non-diffusive continuity equation of the Navier-Stokes equation; based on the constitutive law for Newtonian fluids (Force in x-direction = mu*du/dy) and resistance to compression/expansion (mu*du/dx), it is impossible to interpret all viscous terms that appear as boundary terms on an Eulerian control volume as forces in the direction of the momentum.