On the Homogenization of the Stokes Problem in a Perforated Domain

Hillairet, Matthieu

doi:10.1007/s00205-018-1268-7

Cited by 42 publications

(73 citation statements)

References 18 publications

(79 reference statements)

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“…x i (0) = x 0 i , where the kernel F is the interaction force of the particles. The limit model describing the time evolution for the spatial density ρ(t, x) is given by (9)    ∂ t ρ + Kρ · ∇ρ = 0 ,…”

Section: Main Resultmentioning

confidence: 99%

Sedimentation of particles in Stokes flow

Mecherbet¹

2019

Kinetic &Amp; Related Models

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In this paper, we consider N identical spherical particles sedimenting in a uniform gravitational field. Particle rotation is included in the model while inertia is neglected. Using the method of reflections, we extend the investigation of [10] by discussing the optimal particle distance which is conserved in finite time. We also prove that the particles interact with a singular interaction force given by the Oseen tensor and justify the mean field approximation of Vlasov-Stokes equations in the spirit of [7] and [8].

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Section: Main Resultmentioning

confidence: 99%

Sedimentation of particles in Stokes flow

Mecherbet¹

2019

Kinetic &Amp; Related Models

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“…Then, the problem reduces to homogenizing the Stokes problem in a perforated domain with non-zero boundary conditions (mimicking the particle translation). This particular homogenization problem has been the subject of recent publications (see [8,13,15,19]). Therein, the limit Stokes system including the Brinkman term (1.1) is obtained under specific dilution assumption of the particle phase.…”

Section: Introductionmentioning

confidence: 99%

On the Derivation of a Stokes–Brinkman Problem from Stokes Equations Around a Random Array of Moving Spheres

Carrapatoso¹,

Hillairet²

2019

Commun. Math. Phys.

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We consider the Stokes system in R 3 , deprived of N spheres of radius 1/N, completed by constant boundary conditions on the spheres. This problem models the instantaneous response of a viscous fluid to an immersed cloud of moving solid spheres. We assume that the centers of the spheres and the boundary conditions are given randomly and we compute the asymptotic behavior of solutions when the parameter N diverges. Under the assumption that the distribution of spheres/centers is chaotic, we prove convergence in mean to the solution of a Stokes-Brinkman problem.

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“…This corresponds to the quasi-static regime of holes slowly moving in a fluid, and gives rise in (1.3) to an additional source term µj, with j being the limit flux of the holes. In [6], the holes have all the same radius, are not necessarily periodic, but satisfy a uniform minimal distance condition of the same order of ε as in the periodic setting.In [11], this last condition has been weakened but not completely removed. In particular it is still assumed that, asymptotically for ε ↓ 0, the radius of each hole is much smaller than its distance to any other hole.In [12], the quasi-static Stokes equations are considered in perforated domains with holes of different shapes which are both translating and rotating.…”

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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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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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On the Homogenization of the Stokes Problem in a Perforated Domain

Cited by 42 publications

References 18 publications

Sedimentation of particles in Stokes flow

Sedimentation of particles in Stokes flow

On the Derivation of a Stokes–Brinkman Problem from Stokes Equations Around a Random Array of Moving Spheres

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

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