Asymptotic analysis of incompressible and viscous fluid flow through porous media. Brinkman's law via epi-convergence methods

Brillard, Alain

doi:10.5802/afst.639

Cited by 23 publications

(15 citation statements)

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“…We also mention that, to avoid further technicalities, we only treat the case where the centres of the balls in (1.2) are distributed according to a homogeneous Poisson point process. It is easy to check that our result applies both to the case of periodic centres and to any (short-range) correlated point process for which the results contained in Appendix C hold.After Brinkman proposed the equations (1.3) in [3] for the fluid flow in porous media, an extensive literature has been developed to obtain a rigorous derivation of (1.3) from (1.1) in the case of periodic configuration of holes [2,15,20,16]. We take inspiration in particular from [1], where the method used in [5] for the Poisson equations is adapted to treat the case of the Stokes equations in domains with periodic holes of arbitrary and identical shape.…”

mentioning

confidence: 99%

Homogenisation for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes

Giunti¹,

Höfer²

2019

Ann. Inst. H. Poincaré Anal. Non Linéaire

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We prove the homogenization to the Brinkman equations for the incompressible Stokes equations in a bounded domain which is perforated by a random collection of small spherical holes. The fluid satisfies a no-slip boundary condition at the holes. The balls generating the holes have centres distributed according to a Poisson point process and i.i.d. unbounded radii satisfying a suitable moment condition. We stress that our assumption on the distribution of the radii does not exclude that, with overwhelming probability, the holes contain clusters made by many overlapping balls. We show that the formation of these clusters has no effect on the limit Brinkman equations. In contrast with the case of the Poisson equation studied in [A. Giunti, R. Höfer, and J.J.L. Velázquez, Homogenization for the Poisson equation in randomly perforated domains under minimal assumptions on the size of the holes], the incompressibility condition requires a more detailed study of the geometry of the random holes generated by the class of probability measures considered.in a domain D ε , that is obtained by removing from a bounded set D ⊆ R d , d > 2, a random number of small balls having random centres and radii. More precisely, for ε > 0, we define 2)1 similar size. We emphasize, however, that it neither prevents the balls generating H ε from overlapping, nor it implies a uniform upper bound on the number of balls of very different size which combine into a cluster (see Section 6). The main technical effort of this paper goes into developing a strategy to deal with these geometric considerations and succeed in controlling the term in (1.3). We refer to Subsection 2.3 for a more detailed discussion on our strategy. We also mention that, to avoid further technicalities, we only treat the case where the centres of the balls in (1.2) are distributed according to a homogeneous Poisson point process. It is easy to check that our result applies both to the case of periodic centres and to any (short-range) correlated point process for which the results contained in Appendix C hold.After Brinkman proposed the equations (1.3) in [3] for the fluid flow in porous media, an extensive literature has been developed to obtain a rigorous derivation of (1.3) from (1.1) in the case of periodic configuration of holes [2,15,20,16]. We take inspiration in particular from [1], where the method used in [5] for the Poisson equations is adapted to treat the case of the Stokes equations in domains with periodic holes of arbitrary and identical shape. In [1], by a compactness argument, the same techniques used for the Stokes equations also provide the analogous result in the case of the stationary Navier-Stokes equations. The same is true also in our setting (see Remark 2.2 in Section 2). In [6], with methods similar to [1] and [5], the homogenization of stationary Stokes and Navier-Stokes equations has been extended also to the case of spherical holes where different and constant Dirichlet boundary conditions are prescribed at the boundary of each ball. This corre...

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confidence: 99%

Homogenisation for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes

Giunti¹,

Höfer²

2019

Ann. Inst. H. Poincaré Anal. Non Linéaire

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“…The explicit variation with ∇φ can be generally neglected. In the homogeneous porous medium (with a constant porosity and permeability), the original Brinkman equation [23,24] is shown to actually govern the macroscopic flow in a porous medium with high porosity values (about φ ≥ 0.9) by several upscaling procedures: deterministic homogenization with multi-scale asymptotic expansions [2,21,47,48], stochastic homogenization for random porous media [60,61] or volume averaging method [68]. For φ = 1 or µ f = µ and K = 1/ε I → +∞ when ε → 0, it is shown in [4] that equation (2.6) tends to the Stokes equation.…”

Section: Fluid-porous Incompressible Viscous Flowmentioning

confidence: 99%

Well-posed Stokes/Brinkman and Stokes/Darcy coupling revisited with new jump interface conditions

Angot

2018

ESAIM: M2AN

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The global well-posedness in time is proved, with no restriction on the size of the data, for the Stokes/Brinkman and Stokes/Darcy coupled flow problems with new jump interface conditions recently derived by Angot et al. [Phys. Rev. E 95 (2017) 063302-1–063302-16] using asymptotic modelling and shown to be physically relevant. These original conditions include jumps of both stress and tangential velocity vectors at the fluid–porous interface. They can be viewed as generalizations for the multi-dimensional flow of Beavers and Joseph’s jump condition of tangential velocity and Ochoa-Tapia and Whitaker’s jump condition of shear stress. Therefore, they are different from those most commonly used in the literature. The case of Saffman’s approximation is also studied, but with a force balance for the cross-flow including the Darcy drag and inducing a law of pressure jump different from the usual one. The proof of these results follows the general framework briefly introduced by Angot [C. R. Math. Acad. Sci. Paris, Ser. I 348 (2010) 697–702; Appl. Math. Lett. 24 (2011) 803–810.] for the steady flow.

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“…The case g -0, has been studied in [2], [5], [9], [11], _ [12]. Let P~u~ be the canonical extension of uE taking the value 0 on the inclusions (thanks to the no-slip condition on the boundary of the solid inclusions then the asymptotic behaviour of the fluid is described in the following Theorem : THEOREM 1.1.…”

Section: Volmentioning

confidence: 99%

Asymptotic analysis of incompressible and viscous fluid flow through porous media. Brinkman's law via epi-convergence methods

Cited by 23 publications

References 1 publication

Homogenisation for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes

Homogenisation for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes

Well-posed Stokes/Brinkman and Stokes/Darcy coupling revisited with new jump interface conditions

Asymptotic behaviour of a viscoplastic bingham fluid in porous media with periodic structure

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