In the derivation of the diffusive compressible Euler (dcE) model and the Navier-Stokes-Fourier (NSF) equations, there are a few differences in the modelling choices that result in the two different sets of equations.
Gases and liquids are fundamentally different. In liquids the mean free path is short. Typically less than the diameter of the molecules themselves. This implies that the molecules are almost «rubbing» against each other. In addition there are strong intermolecular forces. Momentum and internal energy (heat) is thus dissipated through collisions whereby the molecules do not exchange positions. (This process is called conduction.) That is, there is virtually no mass diffusion. Hence, a mass element is a good starting point for modelling the macroscopic dynamics. This leads to the (incompressible) Navier-Stokes equations. (The compressible Navier-Stokes are not used for liquids for at least two reasons: 1) The sound waves do not affect the dynamics of the fluid and can be modelled separately once the flow field is known. 2) When solving the compressible Navier-Stokes numerically, the time step is restricted by the speed of sound; the higher the speed of sound is, the smaller the time step. In a liquid the speed of sound is very large making the compressible equations very expensive to solve without any added benefits in view of 1.)
On the other hand, in gases the mean free path is large in comparison with the size of the molecules. This means that they are travelling past each other before they collide with another particle. In this view the random transport of momentum and internal energy is caused by the movement of a molecule across a distance. Of course, collisions will disperse momentum and heat in the new location. (This process is called diffusion.) However, since the random transport is caused by the movement of molecules, mass is also transported and diffused. Since this is the case, it makes little sense to use a mass element as the basis for a macroscopic model. That is why I propose to use Eulerian control volume where the convective and diffusive transport across its boundaries are modelled. That is, conservation is the governing principle. Once this view has been adopted, it makes no sense to interpret the model from a mass element or Lagrangian viewpoint.
Further to the last point, since mass elements are not employed there is no force balance appearing in the derivation of the diffusive compressible Euler model. There is no need to model viscous forces and there is no stress tensor. (Note that viscosity is a model and the viscosity coefficient is a model coefficient, not a fundamental constant of nature.) It is fundamental misconception to try to interpret anything in the diffusive compressible Euler model as a stress tensor because a stress tensor acts on mass elements, but a mass element has no meaning when mass is diffused.
Remark: In the original paper, I did rewrite the equations in a Lagrangian form in order to deduce a value for the diffusion coefficient in the dcE model. The value of the diffusion coefficient is, unsurprisingly, related to the value of the viscosity coefficient since they both model the random transport in the same gas. Still, the diffusion coefficient is not viscosity.
In view of the above arguments, the Navier-Stokes equations are the result of different modelling choices. To derive the Navier-Stokes equations, Newtons second law is the starting point. To that end, a mass element is required. This is a modelling choice, and it is an accurate one as long as the mass diffusion is negligible. To close the equations, viscous forces have to be modelled. Viscous forces arise from the random movement of the molecules in the gas (diffusion), which causes a diffusive transport of momentum. However, it is not quite the momentum transport that is modelled but the velocity transport; momentum transport would implicitly imply a diffusive mass transport, which is not allowed in the mass element paradigm. For the same reason, the random transport of internal energy can only be effectuated by temperature diffusion.
My point is that the mass element viewpoint, is a modelling choice and an approximation that does not take all effects into account. (The dcE system arises from other modelling choices and of course, it too does not capture all properties of a gas. Both the NSF and the dcE are models and neither is divinely revealed.) Any time the mass diffusion is negligible, one can expect the compressible Navier-Stokes equations to be a good model. In a gas, this would be the case when the density gradients are not too steep.
It has been argued that since mass diffusions is typically negligible, including mass diffusion in the model can at most be of academic interest. My response to that is that 1) It does not hurt. On the contrary, the dcE model is simpler and runs simulations faster. And yes, in most regions of the flow the density gradients are very small and does not affect the solution to the dcE model much. That is why NSF and dcE give the same solutions. 2) In regions where the density gradients are strong, there is a difference (and the NSF is known to be too little diffusive in e.g. shock layers). This difference is a key for the well-posedness of the dcE model. 3) Near vacuum, the mass diffusion is very significant and it prevents negative thermodynamic variables, the prime reason for catastrophic failures in NSF codes, to occur. This property is also very important for the well-posedness of the dcE model. 4) The purpose of any model is to reliably produce solutions that model reality. A model that may or may not have a solution can never tick that box.
In a liquid, the situation is different as discussed above. To close the Navier-Stokes system, viscous forces must be modelled, just as for a gas, but the viscous forces are caused by conduction (collisions without positional changes) rather than diffusion (molecules changing their relative positions). Hence, the mass diffusion in a liquid is indeed negligible (with respect to the macroscopic dynamics) and the mass element approximation is very accurate. This leads to the incompressible Navier-Stokes equations.
One may note, that by setting the density to a constant in the dcE model, it reduces to the incompressible Navier-Stokes equations. However, one should not draw too far-reaching conclusions from this observation. The incompressible limit of a set of compressible equations is a subtle one. It means that when the Mach number (essentially the velocity) is very, very small the gas behaves as if governed by the incompressible equations. It does not mean that the gas has suddenly become a liquid.
Ideal gas. The dcE model is derived for an ideal gas. That is, the mean free path is large in comparison with the molecules; the molecules collide elastically; there are no intermolecular forces; the molecules do not store energy within themselves (rotations and vibrations).
Remark: I have already mentioned that in liquids the mean free path is short. Often there are also strong intermolecular forces and if the molecules are complex, collisions might not be elastic and there are lots of internal modes that may store energy.
The ideal gas approximation is very good for noble gases. It is also commonly used for gases with molecules consisting of very few atoms, like nitrogen and oxygen (air). However, in extreme conditions nitrogen and oxygen may begin to store energy and thus deviate from the ideal gas.
In aerodynamics (with the NSF), it is common to assume that air behaves as an ideal gas and the same assumption can be made for the dcE model. Indeed, the two models produce almost exactly the same results as long as the mass diffusion is negligible.
Real gas. A real gas is any gas that is not ideal. For the NSF model there are a multitude of different models that try to capture some of the effects that occur beyond the ideal regime of a gas. The most obvious is when the ideal gas law no longer applies. Then it might be replaced by van der Waal’s or some other gas law. Most of these models should be possible to use directly in the dcE model. However, I have not yet made any effort to validate such extensions of the dcE model to real gases.
The diffusive compressible Euler equations model an (ideal) gas. As such it is an alternative the the compressible Navier-Stokes equations (for gases). The dcE model does not model liquids. (Attempts to debunk the model by claiming that it does not correctly model some property of a liquid, are thus irrelevant.)
Objectivity, or material frame indifference, is the concept that material properties are independent of the movements of the body. In the molecular perspective, the dynamics of a gas is governed by Newtonian mechanics. In that framework, there is only Galilean invariance. If the reference frame is accelerating, the governing equations will no take the same form. However, objectivity refers to macroscopic constitutive relations that are used in a model. (A constitutive relation models some property of a material. It is not a «law of nature».) Hence, objectivity can only ever be approximately true. It requires that the internal forces inside the material are very strong such that they are hardly affected by the acceleration of the moving frame. A prime example is the stress tensor in a solid.
For gases, objectivity is highly disputed. Nevertheless, the Newtonian stress tensor in the NSF also turns out to be objective. That is, to close the NSF system, a constitutive law for the viscous stresses is needed. For Newtonian fluids, that constitutive law is the stress tensor that models viscous forces by viscosity parameter and the velocity gradients. The result is an objective stress tensor.
In the dcE model, the constitutive laws provide explicit expressions for the diffusive fluxes. That is, they state that the diffusive flux is proportional to the diffusion coefficient times the gradient of the principal variable. (And a temperature diffusion times the temperature gradient.) Diffusive fluxes on this form are objective.
Note that: There are no viscous stresses in the dcE model. There is no viscous stress tensor that requires a constitutive law for its closure. Objectivity refers to constitutive laws that are actually used in a model. Attempts to debunk the model by trying to reinterpret diffusion the diffusion terms as a viscous stress tensor and then claim that it is not objective are thus irrelevant. (It would be an equally moot point to try to debunk the Navier-Stokes equations for not being objective by trying to reinterpret the parabolic terms in the Navier-Stokes equations as diffusion of the conservative variables, which should be objective, and then conclude that the resulting constructs are not objective.)
Kinetic theory. Another question that I have got several times, is whether or not it is possible to derive the dcE model from kinetic theory. The root of this question is of course that the NSF model does follow from the Boltzmann equation. That is, and as far as I know, not any NSF model, only the one for a monatomic ideal gas.
In the original paper, I demonstrated that by using a modification of the Boltzmann equation in the spirit of Klimontovich, the dcE model follows. However, one problem is that the Boltzmann equation is, just like the NSF, obtained from a mass element perspective. Mass diffusion is by definition not allowed. The particle distribution function is describing the velocity distribution of the particles located at that point. For a distribution to make sense, the number of particles at a point have to be large. Clearly, this is an approximation since there can not be a large number of particles at every point in R^3. The Klimontovich-style term introduces a fuzziness in the particle distribution functions, which causes mass diffusion. This is one way to introduce randomness in particle positions but it is not quite how I have derived the dcE model. Rather, it should be possible to derive the model by computing the fluxes of molecules through a finite Eulerian control volume, much like how diffusion constants are calculated in kinetic theory.