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.

翻译：本文推导并分析了一类用于数值模拟多种不同化学物质在共同热力学相中稳态、低雷诺数流动的有限元方法。我们所离散化的基础偏微分方程为Stokes–Onsager–Stefan–Maxwell (SOSM) 方程组，该方程组描述了理想与非理想混合物内部的整体动量输运与多组分扩散过程。与以往方法不同，所提出的方法在二维和三维空间中易于实现，并允许采用高阶有限元空间。推导离散格式的关键思想在于：将SOSM方程组适当地用组分质量通量和化学势重新表述，并采用稳定的 $H(\textrm{div}) – L^2$ 有限元对来离散这些未知场。我们证明了该方法是收敛的，并为SOSM方程组的Picard线性化产生一个对称线性系统，该系统交错更新浓度与化学势。我们还讨论了如何将所提方法推广至SOSM方程组的Newton线性化，后者需要同时求解摩尔分数、化学势及其他变量。数值实验支持了我们的理论结果，并给出了一个涉及碳氢化合物微流控非理想混合的物理应用实例。