I'm studying fluid mechanics in more depth during my Ph. D. and there is something related with the diffusive term that has been bothering me for a long time. Looking at the convection diffusion equation: $$ \frac{\partial v}{\partial t} + a\cdot\nabla v - \nabla(\nu \nabla v )=f, $$
and thinking in Fick's law, it's not hard for me thinking in diffusion as the process by which, for example, the particles of a solution with a concentration gradient get apart one from each other to "reduce" the energy of the system; being, each one of these particles, more "comfortable" inside the solvent, which is the same as saying "with the bigger free mean path possible".
But now when I look to the Navier Stokes equation (incompresible and viscous):
$$ \rho \left(\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v}\right)=-\nabla p+\mu\nabla ^2\mathbf{v}+\mathbf{f} $$
I can easily see the viscous term as a diffusive one, but there's no way I can relate it with Fick's law. So, somebody can explain me how can I see viscosity as a diffusive process?
If you are interested in why I'm asking this, it's because in FEM there is a stabilization method called artificial viscosity that add some viscosity to increase diffusion and make the model more stable. So I understand a) why artificial viscosity increases the diffusion, b) why diffusion stabilizes the equation; but I don't understand why viscosity is diffusive (besides the fact that $\mu$ is multiplying $\nabla^2v$)