Finite Element Approximations for Stokes–Darcy Flow with Beavers–Joseph Interface Conditions

Cao, Yanzhao; Gunzburger, Max; Hu, Xiaolong; Hua, Fei; Wang, Xiaoming; Zhao, Weidong

doi:10.1137/080731542

Cited by 183 publications

(92 citation statements)

References 13 publications

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“…This condition relates the tangential velocity along the interface with the fluid stresses, that is, (12) αu f · τ + τ · σ f · n = 0 on Γ, where α is a parameter which needs to be experimentally determined and depends on the properties of the porous medium. An alternative to this third interface condition neglects the second term in (12), giving rise to a no-slip interface condition, (13) u f · τ = 0 on Γ.…”

Section: Free-flow Descriptionmentioning

confidence: 99%

Uzawa Smoother in Multigrid for the Coupled Porous Medium and Stokes Flow System

Luo¹,

Rodrigo²,

Gaspar³

et al. 2017

SIAM J. Sci. Comput.

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Abstract. The multigrid solution of coupled porous media and Stokes flow problems is considered. The Darcy equation as the saturated porous medium model is coupled to the Stokes equations by means of appropriate interface conditions. We focus on an efficient multigrid solution technique for the coupled problem, which is discretized by finite volumes on staggered grids, giving rise to a saddle point linear system. Special treatment is required regarding the discretization at the interface. An Uzawa smoother is employed in multigrid, which is a decoupled procedure based on symmetric Gauss-Seidel smoothing for velocity components and a simple Richardson iteration for the pressure field. Since a relaxation parameter is part of a Richardson iteration, local Fourier analysis is applied to determine the optimal parameters. Highly satisfactory multigrid convergence is reported, and, moreover, the algorithm performs very well for small values of the hydraulic conductivity and fluid viscosity, which are relevant for applications.

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Section: Free-flow Descriptionmentioning

confidence: 99%

Uzawa Smoother in Multigrid for the Coupled Porous Medium and Stokes Flow System

Luo¹,

Rodrigo²,

Gaspar³

et al. 2017

SIAM J. Sci. Comput.

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“…The derivation above indicates that the interface boundary conditions (except for the three obtained via conservation of mass consideration) are in fact variational interface boundary conditions. In the one phase case, Onsager's variation principle reduces to Helmholtz's minimal dissipation principle, and these interface boundary conditions reduce to the well-known Beavers-Joseph-Saffman-Jones interface boundary conditions that have been used in groundwater study and blood filtration [11,13,14,15,19,62,64,65,71,72,73,74]. The Beavers-Joseph-Saffman-Jones type interface boundary conditions can be also derived via homogenization consideration under appropriate assumptions [75] in the one phase case.…”

Section: Application Of Onsager's Extremum Principlementioning

confidence: 99%

“…The last velocity interface boundary condition is exactly the Beavers-Joseph-SaffmanJones interface boundary condition [10,12,14,15,60,61,62,63,64,65] with the slip coefficient β equal to the Beavers-Joseph-Saffman-Jones coefficient α BJSJ . The Cahn-Hilliard-Stokes system can be viewed as the low Reynolds number approximation of the better-known Cahn-Hilliard-Navier-Stokes system for two phase flow [28,35,34,36,40,66,67,68,69,70].…”

Section: Application Of Onsager's Extremum Principlementioning

confidence: 99%

Two‐phase flows in karstic geometry

Han

Sun

Wang

2013

Math Methods in App Sciences

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Multiphase flow phenomena are ubiquitous. Common examples include coupled atmosphere and ocean system (air and water), oil reservoir (water, oil and gas), cloud and fog (water vapor, water and air). Multiphase flows also play an important role in many engineering and environmental science applications.In some applications such as flows in unconfined karst aquifers, karst oil reservoir, proton membrane exchange fuel cell, multiphase flows in conduits and in porous media must be considered together. Geometric configurations that contain both conduit (or vug) and porous media are termed karstic geometry. Despite the importance of the subject, little work has been done on multi-phase flows in karstic geometry.In this paper we present a family of phase field (diffusive interface) models for two phase flow in karstic geometry. These models together with the associated interface boundary conditions are derived utilizing Onsager's extremum principle. The models derived enjoy physically important energy laws. Uniquely solvable first and second order in time numerical schemes that preserve the associated energy law are presented as well.

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“…The convergence we have shown here is not associated with any rate. In [5], the convergence rates of finite element approximations to the time-dependent Stokes-Darcy problem are discussed.…”

Section: The Time Dependent Coupled Stokes-darcy Problemmentioning

confidence: 99%

“…However, a convergence rate is not given here. We consider convergence rates for finite element approximation in [5].…”

Section: Introductionmentioning

confidence: 99%

Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition

Cao¹,

Gunzburger²,

Hua³

et al. 2010

Communications in Mathematical Sciences

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We investigate the well-posedness of a coupled Stokes-Darcy model with Beavers-Joseph interface boundary conditions. In the steady-state case, the well-posedness is established under the assumption of small coefficient in the Beavers-Joseph interface boundary condition. In the time-dependent case, the well-posedness is established via appropriate time discretization of the problem and a novel scaling of the system under isotropic media assumption. Such coupled systems are crucial to the study of subsurface flow problems, in particular, flows in karst aquifers.

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Finite Element Approximations for Stokes–Darcy Flow with Beavers–Joseph Interface Conditions

Cited by 183 publications

References 13 publications

Uzawa Smoother in Multigrid for the Coupled Porous Medium and Stokes Flow System

Uzawa Smoother in Multigrid for the Coupled Porous Medium and Stokes Flow System

Two‐phase flows in karstic geometry

Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition

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