Incompressible Maxwell-Stefan-Navier-Stokes system: Modelling & Structure-preserving discretisation

In many applications, incompressible multi-phase/multi-component flows are considered. In the case of only one phase or component incompressibility is a well-defined unique concept, while in multi-phase flows there are at least ttwo conquering concepts of incompressibility.
The first concept assumes constant total mass density (the sum of all partial mass densities) and hence a divergence-free velocity, while the second concept generalises this by assuming the sum of all partial densities weighted by their specific volume at constant reference temperature and pressure is constant.
Abstractly this means that a weighted sum of the partial densities is assumed to be constant, which allows a variable density and leads to a non-divergence-free velocity.
Both models can be written in the same framework using the incompressibility concept as an algebraic constraint on the system.
We will discuss the modelling of such system in the spirit of Druet in the Maxwell-Stefan framework, see [1].
Afterwards, we will derive a variational formulation for both models which preserves the thermodynamic structure and allows for direct structure-preserving approximation by conforming finite elements in space.
The proposed time discretisation allows us to consider a structure-preserving fully discrete method for both incompressibility models and allows us to compare both concepts on concrete examples.
[1] Druet, P.-E.: Global–in–time existence for liquid mixtures subject to a generalised incompressibility constraint, Journal of Mathematical Analysis and Applications 499 (2), 125059