The Brinkman model is a unified law governing the flow of a viscous fluid in cavity (Stokes equations) and in porous media (Darcy equations). In this work, we explore a novel mixed formulation of the Brinkman problem by introducing the flow’s vorticity as an additional unknown. This formulation allows for a uniformly stable and conforming discretization by standard finite element (Nédélec, Ravi...

We show that stationary statistical properties for uniformly dissipative dynamical systems are upper semi-continuous under regular perturbation and a special type of singular perturbation in time of relaxation type. The results presented are applicable to many physical systems such as the singular limit of infinite Prandtl-Darcy number in the Darcy-Boussinesq system for convection in porous med...

Abstract. Consider an incompressible fluid in a region Ωf flowing both ways across an interface, I, into a porous media domain Ωp saturated with the same fluid. The physical processes in each domain have been well studied and are described by the Stokes equations in the fluid region and the Darcy equations in the porous media region. Taking the interfacial conditions into account produces a sys...

We investigate mortar multiscale numerical methods for coupled Stokes and Darcy flows with the Beavers–Joseph–Saffman interface condition. The domain is decomposed into a series of subdomains (coarse grid) of either Stokes or Darcy type. The subdomains are discretized by appropriate Stokes or Darcy finite elements. The solution is resolved locally (in each coarse element) on a fine scale, allow...

In this chapter a general model for the two-phase fluid flow in porous media is presented, together with its simplified form, known as the Richards equation, which is applicable (under specific assumptions) to describe water flow in the vadose zone. In each case the governing equations are formulated at the Darcy scale, using the capillary pressure–saturation relationship and an empirical exten...

A third-order in time numerical IMEX-type algorithm for the StokesDarcy system for flows in fluid saturated karst aquifers is proposed and analyzed. A novel third-order Adams-Moulton scheme is used for the discretization of the dissipative term whereas a third-order explicit Adams-Bashforth scheme is used for the time discretization of the interface term that couples the Stokes and Darcy compon...

A reduced basis Darcy-Stokes finite element heterogeneous multiscale method (RBDS-FE-HMM) is proposed for the Stokes problem in porous media. The multiscale method is based on the Darcy-Stokes finite element heterogeneous multiscale method (DS-FE-HMM) introduced in [A. Abdulle, O. Budáč, Multiscale Model. Simul. 13 (2015)] that couples a Darcy equation solved on a macroscopic mesh, with missing...

The goal of this short course is to review the analysis and the systematic discretization of saddlepoint systems that arise in the weak formulation of certain fluid flow problems by Galekrin methods. To demonstrate the applicability of the main arguments in various situations, we consider the slow flow of water through a channel surrounded by a porous aquiver. While the free flow of the fluid m...

This article presents a numerical study of the doublediffusive mixed convection in a vertical channel filled with porous medium by using non-equilibrium model. The flow is assumed fully developed, uni-directional and steady state. The controlling parameters are thermal Rayleigh number (RaT ), Darcy number (Da), Forchheimer number (F), buoyancy ratio (N), inter phase heat transfer coefficient (H...