On the effect of polydispersity and rotation on the Brinkman force induced by a cloud of particles on a viscous incompressible flow

Hillairet, Matthieu; Moussa, Ayman; Sueur, Franck

doi:10.3934/krm.2019026

Cited by 15 publications

(16 citation statements)

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“…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. Due to the shapes of the holes, the problem becomes non-isotropic, i.e.…”

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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“…according to assumption (7). Hence, the first term in the right-hand side vanishes according to (9) and (14). Finally, if we assume that r 0 ρ L 1 ∩L ∞ is small enough, we obtain the existence of a positive constant K < 1/2 such that:…”

Section: The Methods Of Reflectionsmentioning

confidence: 90%

“…Lemma 2.2. Under assumptions (7), (8), (9) and the assumption that r 0 ρ 0 L 1 ∩L ∞ is small enough, there exists a positive constant K < 1/2 satisfying for all…”

Section: The Methods Of Reflectionsmentioning

confidence: 99%

“…where we used Lemma A.1 for k = 2 and assumption (9). In what follows we define the constant L > 0 as the constant satisfying:…”

Section: The Methods Of Reflectionsmentioning

confidence: 99%

“…In the analysis of the associated homgenization problem, it has been proved that the interaction between particles leads to the appearance of a Brinkman force in the fluid equation. This Brinkman force depends on the dilution of the cloud but also the geometry of the particles (see [1,3,8,9]). In the dynamic case, the justification of a mesoscopic model using a coupled transport-Stokes equation has been proved in [12] where authors show that the interaction between particles is negligible in the dilute case i.e.…”

Section: Introductionmentioning

confidence: 99%

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A model for suspension of clusters of particle pairs

Mecherbet

2020

ESAIM: M2AN

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In this paper, we consider N clusters of pairs of particles sedimenting in a viscous fluid. The particles are assumed to be rigid spheres and inertia of both particles and fluid are neglected. The distance between each two particles forming the cluster is comparable to their radii 1 N while the minimal distance between the pairs is of order 1 N 1/3 . We show that, at the mesoscopic level, the dynamics are modelled using a transport-Stokes equation describing the time evolution of the position and orientation of the clusters. We also investigate the case where the orientation of the cluster is initially correlated to its position. A local existence and uniqueness result for the limit model is provided.1991 Mathematics Subject Classification. 76T20, 76D07, 35Q83, 35Q70.

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Large Time Behavior of the Vlasov–Navier–Stokes System on the Torus

Han-Kwan

Moussa

Moyano³

2020

Arch Rational Mech Anal

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114

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We study the large time behavior of Fujita-Kato type solutions to the Vlasov-Navier-Stokes system set on T 3 × R 3 . Under the assumption that the initial socalled modulated energy is small enough, we prove that the distribution function converges to a Dirac mass in velocity, with exponential rate. The proof is based on the fine structure of the system and on a bootstrap analysis allowing to get global bounds on moments.

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On the effect of polydispersity and rotation on the Brinkman force induced by a cloud of particles on a viscous incompressible flow

Cited by 15 publications

References 18 publications

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

A model for suspension of clusters of particle pairs

Large Time Behavior of the Vlasov–Navier–Stokes System on the Torus

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