WIAS Preprint No. 2749, (2020)

On the structure of continuum thermodynamical diffusion fluxes -- A novel closure scheme and its relation to the Maxwell--Stefan and the Fick--Onsager approach


  • Bothe, Dieter
  • Druet, Pierre-Étienne
    ORCID: 0000-0001-5303-0500

2010 Mathematics Subject Classification

  • 76R50 76T30 80A20 80A17 35K57 76V05 80A32 92E20


  • Multicomponent diffusion, irreversible thermodynamics, entropy production, Maxwell-Stefan diffusivities, core-diagonal closure, Darken equation, cross-diffusion, sign of diffusivities




This paper revisits the modeling of multicomponent diffusion within the framework of thermodynamics of irreversible processes. We briefly review the two well-known main approaches, leading to the generalized Fick--Onsager multicomponent diffusion fluxes or to the generalized Maxwell--Stefan equations. The latter approach has the advantage that the resulting fluxes are consistent with non-negativity of the partial mass densities for non-singular and non-degenerate Maxwell--Stefan diffusivities. On the other hand, this approach requires computationally expensive matrix inversions since the fluxes are only implicitly given. We propose and discuss a novel and more direct closure which avoids the inversion of the Maxwell--Stefan equations. It is shown that all three closures are actually equivalent under the natural requirement of positivity for the concentrations, thus revealing the general structure of continuum thermodynamical diffusion fluxes.

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