Governing equations in non-isothermal diffusion

• The balance equations for a fluid mixture are reviewed and the equations for the diffusion fluxes are determined.
• In stationary conditions, with a growth of momentum proportional to the velocity difference, Fick's law is recovered.
• By regarding diffusion fluxes as constitutive functions, a whole set of thermodynamic restrictions are determined.
• Hyperbolic models for diffusion and heat fluxes are established which involve the co-rotational derivative.
• The driving term of diffusion turns out to be the gradient of chemical potential rescaled by the temperature.

Generalizations of Fick's law for the diffusion flux are often considered in the literature by analogy with those for the heat flux. The paper reviews the balance equations for a fluid mixture and provides the equations for the diffusion fluxes. As a consequence, the mass densities are shown to satisfy a system of hyperbolic equations. Moreover, for a binary mixture of ideal gases in stationary conditions, Fick's law is recovered. Next, diffusion fluxes are regarded as constitutive functions and a whole set of thermodynamic restrictions are determined which account for diffusion, heat conduction, viscosity and inhomogeneities. Hyperbolic models for diffusion and heat fluxes are established which involve the co-rotational derivative. The driving term of diffusion turns out to be the gradient of chemical potential rescaled by the temperature.