In this paper, we develop an a-priori error analysis of a new unified mixed finite element method for the coupling of fluid flow with porous media flow in R<sup><em>N</em></sup>, <em>N<...In this paper, we develop an a-priori error analysis of a new unified mixed finite element method for the coupling of fluid flow with porous media flow in R<sup><em>N</em></sup>, <em>N</em> ∈ {2,3}, on isotropic meshes. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The approach utilizes a modification of the Darcy problem which allows us to apply a variant nonconforming Crouzeix-Raviart finite element to the whole coupled Stokes-Darcy problem. The well-posedness of the finite element scheme and its convergence analysis are derived. Finally, the numerical experiments are presented, which confirm the excellent stability and accuracy of our method.展开更多
In this work we consider coupled-parallel flow through a finite channel bounded below by a porous layer that is either finite or infinite in depth. The porous layer is one in which Darcy’s equation is valid under the...In this work we consider coupled-parallel flow through a finite channel bounded below by a porous layer that is either finite or infinite in depth. The porous layer is one in which Darcy’s equation is valid under the assumption of variable permeability. A suitable permeability stratification function is derived in this work and the resulting variable velocity profile is analyzed. It will be shown that when an infinite porous layer is implemented, Darcy’s equation must be used with a constant permeability.展开更多
In this paper, we present the analytic solutions of several continuum porous media models that describe the interstitial fluid flow in the interosseous membrane. We first compare the results of the Brinkman, Stokes an...In this paper, we present the analytic solutions of several continuum porous media models that describe the interstitial fluid flow in the interosseous membrane. We first compare the results of the Brinkman, Stokes and Darcy systems in describing the isotropic interstitial fluid flows. Our calculations show that the Stokes equations can well approximate the Brinkman equations when the Darcy number Da 〉 0.2, while the Darcy model is an appropriate approximation to the Brinkman model in the interosseous membrane when Da 〈 2 × 10-4. Yet, in most cases, the anisotropy dominates the interstitial fluid. Therefore, we build an anisotropic Darcy model and show that an isotropic model can be used as a suitable approximation when the ratio between the transverse and longitudinal permeabilities is no larger than 20. Lastly, we take the blood flow in capillaries into consideration as well and introduce the coupled Stokes-Darcy system to describe the cases comprising both the capillary and the interstitial domain. Our results reveal that the profile of the interface exchange flow is not exactly in the linear form as was widely adopted in the numerical simulation, instead, the flux near the artery and the vein is more significant, which in turn results in the increase of the maximum horizontal velocity in the interstitial space while the outflow rate remains the same.展开更多
文摘In this paper, we develop an a-priori error analysis of a new unified mixed finite element method for the coupling of fluid flow with porous media flow in R<sup><em>N</em></sup>, <em>N</em> ∈ {2,3}, on isotropic meshes. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The approach utilizes a modification of the Darcy problem which allows us to apply a variant nonconforming Crouzeix-Raviart finite element to the whole coupled Stokes-Darcy problem. The well-posedness of the finite element scheme and its convergence analysis are derived. Finally, the numerical experiments are presented, which confirm the excellent stability and accuracy of our method.
文摘In this work we consider coupled-parallel flow through a finite channel bounded below by a porous layer that is either finite or infinite in depth. The porous layer is one in which Darcy’s equation is valid under the assumption of variable permeability. A suitable permeability stratification function is derived in this work and the resulting variable velocity profile is analyzed. It will be shown that when an infinite porous layer is implemented, Darcy’s equation must be used with a constant permeability.
基金Project supported by the National Natural Science Foundation of China(Grant No.11202053)the Shanghai Science Foundation(Grant No.12ZR1401100)the National Key Basic Research Program of China(973 Program,Grant No.2012CB518502)
文摘In this paper, we present the analytic solutions of several continuum porous media models that describe the interstitial fluid flow in the interosseous membrane. We first compare the results of the Brinkman, Stokes and Darcy systems in describing the isotropic interstitial fluid flows. Our calculations show that the Stokes equations can well approximate the Brinkman equations when the Darcy number Da 〉 0.2, while the Darcy model is an appropriate approximation to the Brinkman model in the interosseous membrane when Da 〈 2 × 10-4. Yet, in most cases, the anisotropy dominates the interstitial fluid. Therefore, we build an anisotropic Darcy model and show that an isotropic model can be used as a suitable approximation when the ratio between the transverse and longitudinal permeabilities is no larger than 20. Lastly, we take the blood flow in capillaries into consideration as well and introduce the coupled Stokes-Darcy system to describe the cases comprising both the capillary and the interstitial domain. Our results reveal that the profile of the interface exchange flow is not exactly in the linear form as was widely adopted in the numerical simulation, instead, the flux near the artery and the vein is more significant, which in turn results in the increase of the maximum horizontal velocity in the interstitial space while the outflow rate remains the same.
