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

Pattern formation in dense populations studied by inference of nonlinear diffusion‐reaction mechanisms

“…However, it has been proven that this phenomenology for diffusive fluxes is thermodynamically inconsistent. 26 In particular, (1) mass conservation is only satisfied if all diffusivities are the same. (2) The second law is satisfied only in very special cases.…”

“…This theory can be phenomenologically extended to the multicomponent case by assuming flux to be given as a linear function of the gradient fields as follows: $$$$ {\boldsymbol{J}}_i=-\sum \limits_{j=1}^n{D}_{ij}\nabla {c}_j $$$$ where the diffusivity tensor, $$$ \boldsymbol{D}\equiv \boldsymbol{D}\left(\left[{c}_1,\cdots {c}_n\right]\right) $$$ consists of non‐zero off‐diagonal terms: $$$$ \boldsymbol{D}=\left[\begin{array}{cccc}{D}_{11}& {D}_{12}& \cdots & {D}_{1n}\\ {}{D}_{21}& {D}_{22}& \cdots & {D}_{2n}\\ {}\vdots & \vdots & \ddots & \vdots \\ {}{D}_{n1}& {D}_{n2}& \cdots & {D}_{nn}\end{array}\right] $$$$ This results in the following RD model: $$$$ {\partial}_t{c}_i=\sum \limits_j\nabla \cdotp {D}_{ij}\nabla {c}_j+{r}_i $$$$ However, it has been proven that this phenomenology for diffusive fluxes is thermodynamically inconsistent 26 . In particular, (1) mass conservation ...…”

Global existence of weak solutions and weak–strong uniqueness for nonisothermal Maxwell–Stefan systems *

“…(a) To analyse (1) and ( 2), we need to invert the linear system in (5) for the diffusional velocities u i . However, the linear system is singular, yielding infinitely many solutions.…”

“…The isothermal Maxwell-Stefan equations can be derived from the multispecies Boltzmann equations in the diffusive approximation [6]. The high-friction limit in Euler (-Korteweg) equations reveals a formal gradient-flow form of the Maxwell-Stefan equations [17], leading to Fick-Onsager diffusion fluxes instead of (5). In fact, it is shown in [5] that the Fick-Onsager and generalised Maxwell-Stefan approaches are equivalent.…”

Multicomponent incompressible fluids—An asymptotic study

“…The coefficients $$l_j$$ are related to thermo‐diffusion (the Soret and the Dufour effect), while $$\mu _1,\ldots ,\mu _N$$ are the (mass‐based) chemical potentials. For details concerning the modern way of deriving appropriate representations of the fluxes, see [4, 15] and the references given there.…”