  We develop finite element methods for coupling the steady-state
Onsager--Stefan--Maxwell equations to compressible Stokes flow. These equations
describe multicomponent flow at low Reynolds number, where a mixture of
different chemical species within a common thermodynamic phase is transported
by convection and molecular diffusion. Developing a variational formulation for
discretizing these equations is challenging: the formulation must balance
physical relevance of the variables and boundary data, regularity assumptions,
tractability of the analysis, enforcement of thermodynamic constraints, ease of
discretization, and extensibility to the transient, anisothermal, and non-ideal
settings. To resolve these competing goals, we employ two augmentations: the
first enforces the mass-average constraint in the Onsager--Stefan--Maxwell
equations, while its dual modifies the Stokes momentum equation to enforce
symmetry. Remarkably, with these augmentations we achieve a Picard
linearization of symmetric saddle point type, despite the equations not
possessing a Lagrangian structure. Exploiting the structure of linear
irreversible thermodynamics, we prove the inf-sup condition for this
linearization, and identify finite element function spaces that automatically
inherit well-posedness. We verify our error estimates with a numerical example,
and illustrate the application of the method to non-ideal fluids with a
simulation of the microfluidic mixing of hydrocarbons.
