Finite element methods for multicomponent convection-diffusion

TL;DR

This paper introduces finite element methods for accurately simulating multicomponent low-Reynolds-number flows governed by coupled Onsager--Stefan--Maxwell and Stokes equations, ensuring thermodynamic consistency and computational stability.

Contribution

The paper develops a novel variational formulation with augmentations that enable stable, well-posed finite element discretizations for complex multicomponent flow models.

Findings

Proved inf-sup stability of the linearized system.

Validated error estimates through numerical experiments.

Demonstrated application to non-ideal fluid mixing simulations.

Abstract

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.