We propose a model for the coupling of flow and transport equations with porous membrane-type conditions on part of the boundary. The governing equations consist of the incompressible Navier--Stokes equations coupled with an advection-diffusion equation, and we employ a Lagrange multiplier to enforce the coupling between penetration velocity and transport on the membrane, while mixed boundary conditions are considered in the remainder of the boundary. We show existence and uniqueness of the continuous problem using a fixed-point argument. Next, an H(div)-conforming finite element formulation is proposed, and we address its a priori error analysis. The method uses an upwind approach that provides stability in the convection-dominated regime. We showcase a set of numerical examples validating the theory and illustrating the use of the new methods in the simulation of reverse osmosis processes.

翻译：我们提出一个模型，用于描述部分边界上具有多孔膜型条件的流动与输运方程的耦合。控制方程由不可压缩Navier-Stokes方程与对流-扩散方程耦合组成，我们采用拉格朗日乘子来强制实现膜上渗透速度与输运之间的耦合，同时其余边界考虑混合边界条件。通过不动点论证，我们证明了连续问题的存在唯一性。随后，提出一种H(div)-相容有限元公式，并进行了先验误差分析。该方法采用迎风格式，在对流主导区域提供稳定性。我们展示了一组数值算例，验证了理论的有效性，并说明了新方法在反渗透过程模拟中的应用。