Convergence of the method of reflections for particle suspensions in Stokes flows

Höfer, Richard M.

doi:10.48550/arxiv.1912.04388

Cited by 7 publications

(20 citation statements)

References 10 publications

(27 reference statements)

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“…Annealed L p regularity in form of Theorem 5 below constitutes the main technical input in [15] for our analysis of this sedimentation problem. More precisely, in a general non-dilute regime, this allows us to obtain the first rigorous proof of the celebrated predictions by Batchelor [9] and by Caflisch and Luke [11] on the effective sedimentation speed and on individual velocity fluctuations, thus significantly extending the perturbative results of [24] (see also [32]).…”

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confidence: 54%

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Quantitative homogenization theory for random suspensions in steady Stokes flow

Duerinckx¹,

Gloria²

2021

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This work develops a quantitative homogenization theory for random suspensions of rigid particles in a steady Stokes flow, and completes recent qualitative results. More precisely, we establish a large-scale regularity theory for this Stokes problem, and we prove moment bounds for the associated correctors and optimal estimates on the homogenization error; the latter further requires a quantitative ergodicity assumption on the random suspension. Compared to the corresponding quantitative homogenization theory for divergence-form linear elliptic equations, substantial difficulties arise from the analysis of the fluid incompressibility and the particle rigidity constraints. Our analysis further applies to the problem of stiff inclusions in (compressible or incompressible) linear elasticity and in electrostatics; it is also new in those cases, even in the periodic setting.

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confidence: 54%

“…2 (Deterministic L p regularity in dilute regime). In the dilute regime, the recent work of Höfer [32] on the reflection method easily yields the following version of the above; the proof is a direct adaptation of [32] and is omitted. This also constitutes a variant of the dilute Green's function estimates in [24,Lemma 2.7].…”

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confidence: 98%

Quantitative homogenization theory for random suspensions in steady Stokes flow

Duerinckx¹,

Gloria²

2021

Preprint

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“…Such an expansion appears to be very useful as single-particle operators {q n L } n are essentially explicit. However, as shown in [58,39], based on deterministic arguments, convergence is only expected in the dilute regime -more precisely, for a large enough minimal interparticle distance. For this reason, such simplifying tools are systematically avoided in the sequel.…”

Section: (B) Reflection Methodmentioning

confidence: 96%

“…As checked e.g. in [58,39], the weak solution φ L of the above equations (1.2)-(1.5) can equivalently be written as φ…”

Section: (A) Reformulation By Projectionmentioning

confidence: 91%

“…• Annealed regularity: Due to the nonlinearity of hydrodynamic interactions, the analysis critically requires fine regularity results on the steady Stokes equation with a random suspension. Although deterministic regularity results could be used, the latter always require a large enough minimal interparticle distance -which is physically an unsatisfying restriction -, whether they are based on the reflection method [40,38,53,39] or on other perturbative ideas [29], cf. Section 1.5.…”

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confidence: 99%

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Sedimentation of random suspensions and the effect of hyperuniformity

Duerinckx¹,

Gloria²

2020

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This work is concerned with the mathematical analysis of the bulk rheology of random suspensions of rigid particles settling under gravity in viscous fluids. Each particle generates a fluid flow that in turn acts on other particles and hinders their settling. In an equilibrium perspective, for a given ensemble of particle positions, we analyze both the associated mean settling speed and the velocity fluctuations of individual particles. In the 1970s, Batchelor gave a proper definition of the mean settling speed, a 60-year-old open problem in physics, based on the appropriate renormalization of long-range particle contributions. In the 1980s, a celebrated formal calculation by Caflisch and Luke suggested that velocity fluctuations in dimension d = 3 should diverge with the size of the sedimentation tank, contradicting both intuition and experimental observations. The role of long-range self-organization of suspended particles in form of hyperuniformity was put forward later on to explain additional screening of this divergence in steadystate observations. In the present contribution, we develop the first rigorous theory that allows to justify all these formal calculations of the physics literature. The main difficulty is to account for the nonlinear multibody hydrodynamic interactions in a context when stochastic cancellations are crucial, and to show how the improvement provided by hyperuniformity on the linearized problem is not destroyed by the nonlinearity (as it would be in stochastic homogenization of linear elliptic equations, for instance). On the one hand, in order to analyze stochastic cancellations in this nonlinear setting, we introduce a suitable functional-analytic version of hyperuniformity, which is of independent interest. On the other hand, we appeal to a new annealed regularity theory for the steady Stokes equation describing the viscous fluid in presence of a random suspension. While all previous works build on deterministic approaches like the reflection method or other perturbative ideas to analyze interactions in dilute regimes, we rather appeal to a non-perturbative approach that takes advantage of randomness and yields optimal regularity properties upon averaging over the ensemble of particle positions. This is inspired by recent achievements in the quantitative theory of stochastic homogenization of divergence-form linear elliptic equations. As a corollary, we establish a homogenization result for a steady Stokes fluid in a finite tank with a dense suspension of small sedimenting particles, and we define the associated effective viscosity.

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Mild assumptions for the derivation of Einstein's effective viscosity formula

Gérard-Varet¹,

Höfer²

2020

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We provide a rigorous derivation of Einstein's formula for the effective viscosity of dilute suspensions of n rigid balls, n ≫ 1, set in a volume of size 1. So far, most justifications were carried under a strong assumption on the minimal distance between the balls:We relax this assumption into a set of two much weaker conditions: one expresses essentially that the balls do not overlap, while the other one gives a control of the number of balls that are close to one another. In particular, our analysis covers the case of suspensions modelled by standard Poisson processes with almost minimal hardcore condition.

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Convergence of the method of reflections for particle suspensions in Stokes flows

Cited by 7 publications

References 10 publications

Quantitative homogenization theory for random suspensions in steady Stokes flow

Quantitative homogenization theory for random suspensions in steady Stokes flow

Sedimentation of random suspensions and the effect of hyperuniformity

Mild assumptions for the derivation of Einstein's effective viscosity formula

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