Divergence conforming finite element methods for flow-transport coupling with osmotic effects

TL;DR

The paper develops a divergence-conforming finite element approach for simulating flow–transport coupling with osmotic effects in membrane filtration by coupling Navier–Stokes and advection–diffusion through a membrane boundary condition enforced by a Lagrange multiplier. Existence and uniqueness of the continuous problem are established via a fixed-point framework, followed by an $H(\mathrm{div})$-conforming discretisation with upwind stabilization and an a priori error analysis. Lagrange multiplier stabilization and various FE families are investigated, and extensive 2D/3D numerical experiments including osmotic-channel scenarios validate the theory and demonstrate practical performance for reverse osmosis applications. The results show optimal convergence rates and stable behavior in convection-dominated regimes, highlighting the method's potential for efficient, accurate desalination simulations.

Abstract

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.

Figures (5)

• Figure 1: Cross-flow membrane filtration model with $\Gamma_i$ defined such that there is no corner points on $\Gamma_{\mathrm{m}}\cap\Gamma_{\mathrm{in}}$.
• Figure 1: Example \ref{['sec:simulation2d']}. Scenario 1 (first and third panels) and Scenario 2 (second and bottom panels). Scaled representation of the computed velocity component $\boldsymbol{u}_{h,1}$ and concentration profile in a channel with membrane at $\Gamma_{\mathrm{m}}$ (scenario 1) and $\Gamma_{\mathrm{wall}}=\Gamma_{\mathrm{m}}$ (scenario 2).
• Figure 2: Example \ref{['sec:simulation2d']}. Comparison along $\Gamma_{\mathrm{m}}$ between permeate velocities, concentration profiles and pressures between the two channel scenarios.
• Figure 3: Velocity streamlines (top panels) and concentration profiles (bottom panels) around the spacer in a cavity-type configuration and inlet velocities $u_0=5.0\cdot 10^{-2}$m$/$s (left) and $u_0=1.29\cdot 10^{-1}$m$/$s. The bottom numbers indicate distance from inlet (in $\times 10^{-3}$m).
• Figure 4: Example \ref{['test-spacer']}. Comparison along $\Gamma_{\mathrm{m}}$ between permeate velocities, concentration profiles and pressures between the two velocities scenarios in the channel with cavity-type spacer configurations.

Theorems & Definitions (28)

• Lemma 3.1
• Proof 1
• Lemma 3.2
• Proof 2
• Lemma 3.3
• Proof 3
• Lemma 3.4
• Proof 4
• Theorem 3.5
• Proof 5