Mixed FEM for coupled unsteady fluid flow problems with $p$-type Brinkman-Forchheimer framework and its application for reverse-osmosis desalination

TL;DR

This paper develops and analyzes a fully discrete mixed finite element method for solving coupled unsteady fluid flow and transport problems modeled by the Brinkman-Forchheimer equations, with applications to reverse osmosis desalination, providing stability and convergence results.

Contribution

It introduces a novel Banach space framework and a mixed finite element scheme for nonstationary Brinkman-Forchheimer problems with membrane boundary conditions, including stability and optimal error estimates.

Findings

Numerical results confirm theoretical error estimates.

The method is stable and convergent for the coupled problem.

Effective for reverse osmosis desalination modeling.

Abstract

This work analyzes a fully discrete mixed finite element method in a Banach space framework for solving nonstationary coupled fluid flow problems modeled by the Brinkman-Forchheimer equations, with applications to reverse osmosis. The model couples unsteady $p$-type convective Brinkman-Forchheimer and transport equations with nonlinear boundary conditions across a semi-permeable membrane. A mixed formulation is used for the fluid equation (pseudostress-velocity) and for the transport equation (concentration, its gradient, and a Lagrange multiplier from the membrane condition). The continuous problem is reformulated in Banach spaces as a fixed-point problem, enabling a well-posedness analysis via differential-algebraic system theory. Spatial discretization employs lowest-order Raviart-Thomas elements for fluxes and piecewise constants for primal variables, while linear elements are used for the Lagrange multiplier. A fully discrete Galerkin scheme with backward Euler time-stepping is proposed. Its well-posedness and stability are proven using a fixed-point argument, and optimal convergence rates are established. Numerical results confirm the theoretical error estimates and demonstrate the method's effectiveness.