Seeking Reference on Transport of Momentum by Diffusion of Mass

Question

I'm looking at continuum mechanics from the perspective of De Groot and Mazur's "Non-Equilibrium Thermodynamics" - the first reference that I've come across that seems to do a good job of bringing together thermodynamics and continuum mechanics. In the course of reading I've started wondering why I've never seen transport of momentum by mass diffusion discussed. Transport at the bulk-average velocity $\vec{v}$ is included in standard Navier-Stokes, but it seems to me that a continuum element experiencing mass diffusion should experience additional momentum transport

\begin{align*} \int_{\partial \Omega} \sum_\alpha (\vec{v}_\alpha - \vec{v})(-\vec{j}_\alpha \cdot \hat{n})\ \mathrm{d}A \end{align*} because flux by diffusion of each species ($\vec{j}_\alpha$) carries additional momentum due to its relative velocity $(\vec{v}_\alpha - \vec{v})$. This would give the continuum equation

\begin{align*} \frac{\partial}{\partial t}(\rho \vec{v}) + \vec{\nabla}\cdot(\rho \vec{v} \vec{v}) = \rho \frac{\text{D}}{\text{D}t}\vec{v} = \vec{\nabla}\cdot(\sigma\underbrace{-\sum_\alpha\vec{j}_\alpha(\vec{v}_\alpha-\vec{v})}_\text{New})+\rho \vec{g} \end{align*}

With a some manipulation, I was able to convince myself that this is the same as

\begin{align*} \frac{\partial}{\partial t}(\rho \vec{v}) + \vec{\nabla}\cdot\left(\sum_\alpha \rho_\alpha \vec{v}_\alpha \vec{v}_\alpha \right) = \rho \frac{\text{D}}{\text{D}t}\vec{v} = \vec{\nabla}\cdot\sigma+\rho \vec{g} \end{align*}

i.e., that adding this term is equivalent to using the true momentum transfer term (including diffusion) on the left-hand side. This suggests that I've math'd correctly and that my new term does indicate the difference between the true momentum transfer and the momentum transfer by the bulk velocity.

Can anyone suggest a reference that accounts for this effect when formulating continuum equations? I intend to go further with this, particularly into the non-equilibrium thermodynamics side (e.g. deriving force/flux relations), and would like a source to check my math against. I've had a look around myself, but every time I search for the keywords "mass diffusion" and "momentum transport" together, I am completely inundated with introductory-level results discussing the analogy between diffusion and viscosity.

Answer 1

Hydrodynamic variables are associated with the densities of conserved charges. In a single-component non-relativistic fluid these quantities are the mass density $\rho$, the momentum density $\vec{\pi}$, and the energy density ${\cal E}$. We can use $\vec{\pi}=\rho\vec{v}$ to define the fluid velocity, and view $\vec{v}$ rather than $\vec{\pi}$ as a fluid dynamic variable.

In an $N$ component fluid there are $N-1$ extra variables associated with the partial mass densities $\rho_\alpha$. The corresponding velocities are not hydrodynamic variables; they are determined by the diffusive fluxes $\vec{\jmath}_\alpha$ $$ \vec{v}_\alpha-\vec{v} = \vec{\jmath}_\alpha/\rho_\alpha \sim D_{\alpha\alpha'} \vec{\nabla}\rho_{\alpha'}, $$ where $D$ is the matrix of diffusion coefficients.

This means that the term you are considering is of second order in gradients of hydrodynamic variables (second order in diffusive fluxes), and formally small compared to the Navier-Stokes term $\sigma$. Such a term is part of the second order description of a dissipative fluid, sometimes called the Burnett equation.

The Burnett equation of a single component fluid already contains a lot of terms in the stress tensor, objects like $\sigma\sigma$, $\vec{v}\cdot \vec{\nabla}\sigma$, etc., and the number of terms in a multi-component fluid is obviously even larger. There is some work on this, however, which you can find by searching for "Burnett" and "multi-component".