Coupled Stokes Darcy Problem Research Articles

A stabilized finite element method based on two local Gauss integrations for a coupled Stokes–Darcy problem

In this paper, a stabilized mixed finite element method for a coupled steady Stokes–Darcy problem is proposed and investigated. This method is based on two local Gauss integrals for the Stokes equations. Its originality is to use a difference between a consistent mass matrix and an under-integrated mass matrix for the pressure variable of the coupled Stokes–Darcy problem by using the lowest equal-order finite element triples. This new method has several attractive computational features: parameter free, flexible, and altering the difficulties inherited in the original equations. Stability and error estimates of optimal order are obtained by using the lowest equal-order finite element triples (P1−P1−P1) and (Q1−Q1−Q1) for approximations of the velocity, pressure, and hydraulic head. Finally, a series of numerical experiments are given to show that this method has good stability and accuracy for the coupled problem with the Beavers–Joseph–Saffman–Jones and Beavers–Joseph interface conditions.

Equal order approximations enriched with bubbles for coupled Stokes–Darcy problem

As a remedy to the instability of the Galerkin finite element formulation, symmetric stabilization techniques such as the continuous interior penalty, the subgrid and local projection methods were proposed and analyzed by Burman and Hansbo (2006) [10], Badia and Codina (2009) [11], Becker and Braack (2001) [12], and Nafa and Wathen (2009) [13]. In this work we consider a coupled Stokes–Darcy problem, where in one part of the domain the fluid motion is described by Stokes equations and for the other part the fluid is in a porous medium and described by Darcy law and the conservation of mass. Such systems can be discretized by heterogeneous finite elements in the two parts, such as Taylor-Hood or MINI elements for the Stokes domain, and mixed elements of Raviart–Thomas elements type for the Darcy domain. Here, we discretize by standard equal-order finite elements enriched with bubbles functions and use local projection stabilization technique (LPS) to stabilize the method and control the fluctuation of the velocity divergence vector on the Darcy region. We also suggest a way to control the natural H(div) velocity.

Motion of a solid particle in a shear flow along a porous slab

AbstractThe flow field around a solid particle moving in a shear flow along a porous slab is obtained by solving the coupled Stokes–Darcy problem with the Beavers and Joseph slip boundary condition on the slab interfaces. The solution involves the Green’s function of this coupled problem, which is given here. It is shown that the classical boundary integral method using this Green’s function is inappropriate because of supplementary contributions due to the slip on the slab interfaces. An ‘indirect boundary integral method’ is therefore proposed, in which the unknown density on the particle surface is not the actual stress, but yet allows calculation of the force and torque on the particle. Various results are provided for the normalized force and torque, namely friction factors, on the particle. The cases of a sphere and an ellipsoid are considered. It is shown that the relationships between friction coefficients (torque due to rotation and force due to translation) that are classical for a no-slip plane do not apply here. This difference is exhibited. Finally, results for the velocity of a freely moving particle in a linear and a quadratic shear flow are presented, for both a sphere and an ellipsoid.

Perturbation solution of the coupled Stokes-Darcy problem

Microfiltration of particles is modelled by the motion of particles embeddedin a Stokes flow near a porous membrane in which Darcy equations apply.Stokes flow also applies on the other side of the membrane. A pressure gradient isapplied across the membrane. Beavers and Joseph slip boundary condition applies along the membrane surfaces.This coupled Stokes-Darcy problem is solved by a perturbation method, considering that the particle size is muchlarger than the pores of the membrane. The formal asymptotic solution is developed in detail up to 3rd order.The method is applied to the example case of a spherical particle moving normal to a membrane. The solution,limited here to an impermeable slip surface (described from 3rd order expansion), uses as an intermediate step theboundary integral technique for Stokes flow near an impermeable surface with a no-slip boundary condition. Results of the perturbation solution are in good agreement with O'Neill and Bhatt analytical solution for this case.

A Two-Grid Method of a Mixed Stokes–Darcy Model for Coupling Fluid Flow with Porous Media Flow

We study numerical methods for solving a coupled Stokes–Darcy problem in porous media flow applications. A two-grid method is proposed for decoupling the mixed model by a coarse grid approximation to the interface coupling conditions. Error estimates are derived for the proposed method. Both theoretical analysis and numerical experiments show the efficiency and effectiveness of the two-grid approach for solving multimodeling problems. Potential extensions and future directions are discussed.