The Method of Reflections, Homogenization and Screening for Poisson and Stokes Equations in Perforated Domains

Höfer, Richard M.; Velázquez, Juan J. L.

doi:10.1007/s00205-017-1182-4

Cited by 36 publications

(60 citation statements)

References 22 publications

(43 reference statements)

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“…A convergence proof based on orthogonal projection operators is introduced by Luke [15] in 1989. We refer also to the method of reflections developped in [11] which is used by Höfer in [10].…”

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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“…The corresponding spread harmonic measure density ω 0 y (s) is given by Eq. (69). Integrating this density over the sphere ∂Ω 0 , one gets…”

Section: Hitting and Splitting Probabilitiesmentioning

confidence: 99%

“…i.e., partial surfaces ∂Ω i are the connected components of the boundary of the exterior domain Ω − . In the literature, domains like Ω − are called "periphractic domains" [67], "perforated domains" [68,69] and "domains with disconnected boundary" [25]. For interior problems, we consider that N formerly introduced non-overlapping balls Ω i are englobed by a larger spherical domain Ω 0 = {x ∈ R 3 : x − x 0 < R 0 } of radius R 0 , centered at x 0 .…”

Section: Introductionmentioning

confidence: 99%

Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries

Grebenkov

Traytak

2019

Journal of Computational Physics

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We apply the generalized method of separation of variables (GMSV) to solve boundary value problems for the Laplace operator in three-dimensional domains with disconnected spherical boundaries (i.e., an arbitrary configuration of non-overlapping partially reactive spherical sinks or obstacles). We consider both exterior and interior problems and all most common boundary conditions: Dirichlet, Neumann, Robin, and conjugate one. Using the translational addition theorems for solid harmonics to switch between the local spherical coordinates, we obtain a semi-analytical expression of the Green function as a linear combination of partial solutions whose coefficients are fixed by boundary conditions. Although the numerical computation of the coefficients involves series truncation and matrix inversion, the use of the solid harmonics as basis functions naturally adapted to the intrinsic symmetries of the problem makes the GMSV particularly efficient, especially for exterior problems. The obtained Green function is the key ingredient to solve boundary value problems and to determine various characteristics of stationary diffusion such as reaction rate, escape probability, harmonic measure, residence time, and mean first passage time, to name but a few. The relevant aspects of the numerical implementation and potential applications in chemical physics, heat transfer, electrostatics, and hydrodynamics are discussed.

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“…A version of this method has been used in [JO04] to study systems with very large screening lengths ξ. We will use the formulation of the method of reflection in the framework of orthogonal projection that has been investigated in [HV16], where Stokes equations with Dirichlet boundary conditions are considered. In that case, the series representation has been proven to converge if the screening length ξ is sufficiently large (i.e., if the capacity density of the particles is sufficiently small).…”

Section: Outline Of the Proofmentioning

confidence: 99%

Sedimentation of Inertialess Particles in Stokes Flows

Höfer

2018

Commun. Math. Phys.

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We investigate the sedimentation of a cloud of rigid, spherical particles of identical radii under gravity in a Stokes fluid. Both inertia and rotation of particles are neglected. We consider the homogenization limit of many small particles in the case of a dilute system in which interactions between particles are still important. In the relevant time scale, we rigorously prove convergence of the dynamics to the solution of a macroscopic equation. This macroscopic equation resembles the Stokes equations for a fluid of variable density subject to gravitation.We consider a sequence of initial particle configurations indexed by ε and we assume N ε → ∞ and R ε → 0 in the limit ε → 0. Moreover, we assume lim ε→0 ξ ε = ξ * ∈ [0, ∞).Furthermore, we consider the mass density of the particleswhere the particle positions depend on ε. The dynamics (1) implies that the particles are transported by the velocity fieldv ε , i.e, ∂ tρε +v ε · ∇ρ ε = 0.

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The Method of Reflections, Homogenization and Screening for Poisson and Stokes Equations in Perforated Domains

Cited by 36 publications

References 22 publications

Sedimentation of particles in Stokes flow

Sedimentation of particles in Stokes flow

Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries

Sedimentation of Inertialess Particles in Stokes Flows

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