High-order finite element methods for three-dimensional multicomponent convection-diffusion

TL;DR

This paper introduces high-order finite element methods for simulating three-dimensional multicomponent flows governed by complex SOSM equations, emphasizing ease of implementation and convergence analysis.

Contribution

It develops a new class of finite element methods reformulating SOSM equations in terms of fluxes and potentials, enabling high-order accuracy and straightforward implementation in 3D.

Findings

Methods are proven convergent and produce symmetric linear systems.

Numerical experiments validate theoretical results.

Application example demonstrates microfluidic non-ideal mixing.

Abstract

We derive and analyze a broad class of finite element methods for numerically simulating the stationary, low Reynolds number flow of concentrated mixtures of several distinct chemical species in a common thermodynamic phase. The underlying partial differential equations that we discretize are the Stokes$\unicode{x2013}$Onsager$\unicode{x2013}$Stefan$\unicode{x2013}$Maxwell (SOSM) equations, which model bulk momentum transport and multicomponent diffusion within ideal and non-ideal mixtures. Unlike previous approaches, the methods are straightforward to implement in two and three spatial dimensions, and allow for high-order finite element spaces to be employed. The key idea in deriving the discretization is to suitably reformulate the SOSM equations in terms of the species mass fluxes and chemical potentials, and discretize these unknown fields using stable $H(\textrm{div}) \unicode{x2013} L^2$ finite element pairs. We prove that the methods are convergent and yield a symmetric linear system for a Picard linearization of the SOSM equations, which staggers the updates for concentrations and chemical potentials. We also discuss how the proposed approach can be extended to the Newton linearization of the SOSM equations, which requires the simultaneous solution of mole fractions, chemical potentials, and other variables. Our theoretical results are supported by numerical experiments and we present an example of a physical application involving the microfluidic non-ideal mixing of hydrocarbons.