Why is there a zero in the continuity equation?
It has always puzzled me why mass would not diffuse when diffusion gives rise to viscosity and temperature conduction.
The answer is given by the modelling framework. When Stokes derived the equations his starting point was Newton’s 2nd law. (F=m*a). This can only work if it has a mass to operate on.
Thus, describes the fluid as a continuum of infinitesimal Lagrangian mass elements. Each mass element has an associated velocity (corresponding to the average velocity of the molecules in that element), which together makes the continuum velocity field. Next, for a mass element, Stokes sets up the force balance that ultimately leads to the momentum equations in the Navier-Stokes equations.
By definition the mass of a mass element is immutable and there can be no (net) transfer of material between any two mass elements. Diffusion, of any kind, is ruled out by definition. This is not rooted in the physical reality but a consequence of the modelling.
The often repeated mantra «mass does not diffuse» should be followed by «in the mass element framework».
Since mass elements are convected by the velocity field, the hyperbolic continuity equation follows trivially.
Stokes was of course aware that a gas consists of many, many particles that travel semi-randomly through space. Furthermore, that their mean velocities (at a point in space and time), is what we perceive as the macroscopic velocity of the gas. Since the random velocity will cause particles to travel in directions other than the mean velocity, they will cause momentum to diffuse. In the Navier-Stokes framework this has to be modelled as a force acting on a mass element. Thus the viscous stress tensor is introduced.
Again, one must pay attention to the modelling choices. Since mass elements are used, friction may only operate on the surface of a mass element. That is, the actual random transport is not modelled, because that would imply that the mass element is no longer immutable. Instead, its effect on the velocity of the mass element through a surface force is modelled.
The same applies to the heat conductive term appearing in the energy equation. (This equation was not derived by Stokes but a later addition.) In the physical gas, the random motion of the particles will transport heat to nearby locations which at the macroscopic scale will appear as a diffusive process. In the Navier-Stokes model this can not happen since the mass elements are immutable. Instead, conduction is modelled, which is the transfer of heat through collisions. Specifically, the energy equation models particles that collide on the mass element boundary and thus transfer their kinetic energy. (Conveniently, the mass elements are infinitesimal, so this can happen at all points. Nevertheless, diffusion is not modelled.)
The choice of infinitesimal mass elements as the fundamental entity, also gives rise to a number of other peculiarities with the theory. Particles are not that small. Thus, particles too must be assumed to be infinitesimally small such that a large number of them fit into the infinitesimal mass elements; as we are studying a gas, they must also be sparsely distributed inside the mass element. Since the elements are uncountably many, the gas must contain an infinite number of particles. Then, however, each particle can not have a finite mass to avoid an infinite total mass. The theory becomes rather intricate.
Kinetic theory
Kinetic theory and statistical mechanics have been employed to give credence to the Navier-Stokes model and also to derive numerical values for its coefficients (viscosity and heat condution).
We begin with some observations on the coefficients. Kinetic theory includes the random motions of the particles and with reasonable assumptions particle distribution functions can be defined, and from them the random transport that cause viscosity and heat conduction. Since kinetic theory models random motions, diffusion is allowed. (To be clear, by diffusion I mean both the random transport of a particle from one point to another a finite length away, and the subsequent collision.) Thus, to derive values for the dynamic viscosity coefficient, the random momentum transfer can not be calculated for a flow with a momentum gradient because that would also include the mass diffusive part. Instead, the gas has to be assumed to have a constant density and a velocity gradient. In this setting, the viscosity coefficient can be estimated. The same goes for heat conduction.
Of course, the same derivations for a mass, or momentum. or energy gradient, would give the random transport including mass diffusion, but that is not what appears in the Navier-Stokes equations so that is not what is calculated in kinetic theory.
The Boltzmann equation
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, which is taken as evidence for the non-existence of mass diffusion in an actual gas.
However, taking a closer look at the Boltzmann equation, 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 in the same way infinitesimal mass elements does.
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. (That is, conduction rather than diffusion.) 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 effectively evolves a mass element. As already noted, mass elements do not allow for diffusion by their very definition. In fact, had the Boltzmann equation not led to mass elements, it would not have led to the Navier-Stokes equations (and it would have been a much less celebrated model).