Physical foundation of the Navier-Stokes equations
The question, I have asked myself ever since my PhD studies is: Why is there a zero in the continuity equation? It has always puzzled me why mass would not diffuse when diffusion gives rise to momentum (or rather velocity) and temperature diffusion. Many years ago now, I became aware of Brenner’s modification of the Navier-Stokes equations, that introduces mass diffusion and removes the zero in the continuity equation, and that there is a proof that weak solutions exists for this system. (Feireisl&Vasseur, 2010) However, this system was debunked on (in my opinion, erroneous) physical grounds. The Brenner modification is based on the standard Navier-Stokes equations and by introducing separate mass and volume velocities, mass diffusion is introduced into the system. Although, the two velocities can be motivated from physics, it complicates the reasoning in the derivation considerably. What Brenner’s system clearly shows, is that mass diffusion solves many of the mathematical problems of the original system. However, instead of modifying the existing Navier-Stokes system, I went back to the derivation of the Navier-Stokes to see what assumption leads to the zero in the first place.
In textbooks, Navier-Stokes equations are often derived by first deriving the Euler equations. This is, not surprisingly given the name, done in an Eulerian frame. An imaginary box is immersed in the flow, and the flow in and out of it leads to a balance law. To obtain the Navier-Stokes equations, the viscous/diffusive effects must be added to the Euler model. Now, textbooks shift to a Lagrangian frame. That is, the imaginary box now represents a small «mass element» that flows downstream. Being a mass element, it is subject to forces in the flow, which deforms it. This is how, the viscous stress tensor enters the equation. It models the viscous forces acting on the mass element. This view is strikingly similar to structural mechanics and indeed, this was how the equations were derived originally by Stokes.
However, Stokes focused on the derivation of the momentum equation and the continuity equation followed from Newton’s 2nd law and the fact that he considered a mass element. He did not derive the energy equation, which is a later addition to the equations. In the energy equation, the viscous term represent the work done by viscosity. Furthermore, there is a Fourier heat diffusive term, that is typically added to the energy equation in the Eulerian frame.
Countless times have I heard the phrase «mass does not diffuse» as an explanation of the zero in the continuity equation. Given Stokes derivation, one should really say «a Lagrangian mass element can not diffuse mass», but that follows by definition of a mass element. What one also should recognise is that deriving the equations from the dynamics of a mass element is a modelling choice among others.
Returning for a moment to structural mechanics, we note that a mass element makes good sense. In a steel rod, we can «mark» the atoms inside a small box. If we bend the rod slightly the atoms will move, but they will remain in the same internal order. If we let go, the rod returns and all atoms return to their original positions. Atoms in a solid are held in place my electromagnetic forces. Building a continuum theory on mass elements and a stress tensor representing the intermolecular forces, makes perfect sense.
This model of a solid, a collection of molecules arranged in a grid, can also be used to explain heat dissipation. The vibration of molecules is temperature dependent and they will bounce into each other and thereby dissipate temperature through the solid. This process is called conduction and it is perfectly compatible with the mass element view, since all molecules remain in their relative places.
However, a major physical difference between a solid and a gas is that in the latter, the atoms do move. A lot. Arguably, the idea of a mass element does not make sense on the molecular level. Hence, a fluid mass element is not a collection of particles but rather it is a piece of the continuum. That is one has to be careful with what is meant by the «continuum». A continuum variable, such as the pressure, is loosely thought of as a certain local mean value of the cloud of atoms. In fact, one has to be very careful with the meaning of continuum when deriving continuum models, but leaving that aside, we return to the analogy with structural mechanics.
In a fluid the viscous stress tensor arises from the random movements of the atoms. Atoms hit each other and transfer momentum to nearby region, and thereby transferring momentum. However, since gas molecules move relative each other this process is not mainly conductive, but diffusive. This is not contradicting a stress tensor per se. Random movement and collisions of atoms, will give rise to what is perceived as viscosity at the continuum level. An initially sharp interface in the fluid, e.g. between layers of different velocity, will grow fuzzy due to the random transport across the interface. However, if the transfer of momentum of individual molecules across a small distance (on average the mean free path) give rise to a fuzziness of the global momentum, then how come the transfer of mass, by the very same molecules, has no effect at the global scale? Furthermore, the same random movements of molecules give rise to diffusive transport of temperature according to Fourier’s law, which is the only diffusive mechanism acting on the internal energy. However, the internal energy also depends on density and random transport of mass should affect the internal energy as well.
The only way to understand the lack of mass diffusion in the Navier-Stokes equations is if either all collisions occur on the boundary of the mass element such that no molecule crosses, or the same number of molecules pass across the interface in each direction at every moment such that the net transfer is zero. This must be the case, even if the density gradient is large. Physically, that seems very unlikely.
Many common fluids, including air, are modelled as Newtonian fluids. In a Newtonian fluid the viscosity gives rise to a force proportional and perpendicular to velocity gradients. That is, if two sheets of fluid slide against each other, there is a frictional force proportional to the velocity difference between them. In addition, it is assumed that compression or decompression of the mass element diffuse energy. These effects result in the standard stress tensor acting on the mass element in the Lagrangian frame. However, when returning to the Eulerian frame it is very difficult to physically explain some of the terms of the stress tensor. In the Eulerian frame, only forces along the x-axis, should effect the x-momentum, which is not the case unless some additional physical explanation, other the ones mentioned above, is offered.
Kinetic theory
Another argument in favour of the Navier-Stokes equations is that they can be derived from the Boltzmann equation. In particular, no mass diffusion appears in the equations.
However, taking a closer look at this derivation it is not surprising that mass diffusion is lacking. In kinetic theory, the gas is viewed as a collection of «billiard balls» (just as I view a gas when deriving my model). By taking a volume that is small enough so as to be almost a «point» on the macroscopic scale but yet large enough such that it contains a very large number of gas molecules, one can model the molecules inside the box with a particle distribution function for the velocities. That is, it gives the number of particles that has a given velocity in that specific little volume.
To obtain a mathematically workable theory, this idea is then extended to a continuum. That is, there is now a particle distribution function (PDF) that depends on both position and velocity. Since the PDF now represents a continuum it means that at every point in space there is an infinitesimal volume containing a large number of molecules so as to give meaning to the statistical distribution of velocities at that point.
Clearly, this leap constitutes a modelling approximation. There can not even be one molecule inside an infinitesimal volume, let alone a large number of them.
Having defined a continuous PDF, the Boltzmann equation can be derived which describes the temporal and spatial evolution of the PDF. The random motion of the molecules is captured by the collision integral. If that is set to zero, the non-diffusive Euler equation can be derived from Boltzmann. With certain modelling assumptions on the collision integral, the Navier-Stokes equations for a monatomic ideal gas can be derived.
However, the leap to a continuous PDF implies that Boltzmann equation evolves a mass element, since it evolves the infinitesimal volume where a velocity distribution is defined. The collision integral represents collisions at that mathematical point. By construction of the model, there can be no mass diffusion.