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Corrector equations in fluid mechanics: Effective viscosity of colloidal suspensions
Analysis of PDEs · Submitted 20 Sep 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim More precisely, we establish a homogenization result when particles are distributed according to a given stationary and ergodic random point process.
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The abstract, sentence by sentence
Consider a colloidal suspension of rigid particles in a steady Stokes flow. In a celebrated work, Einstein argued that in the regime of dilute particles the system behaves at leading order like a Stokes fluid with some explicit effective viscosity. In the present contribution, we rigorously define a notion of effective viscosity, regardless of the dilute regime assumption. More precisely, we establish a homogenization result when particles are distributed according to a given stationary and ergodic random point process. The main novelty is the introduction and analysis of suitable corrector equations.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Consider a colloidal suspension of rigid particles in a steady Stokes flow. | background 1.00 | 0.08 | 0.02 | unclear 0.53 |
| s1 | In a celebrated work, Einstein argued that in the regime of dilute particles the system behaves at leading order like a Stokes fluid with some explicit effective viscosity. | background 1.00 | 0.03 | 0.04 | asserted 1.00 |
| s2 | In the present contribution, we rigorously define a notion of effective viscosity, regardless of the dilute regime assumption. | contribution 0.99 | 0.92 | 0.21 | asserted 1.00 |
| s3 | More precisely, we establish a homogenization result when particles are distributed according to a given stationary and ergodic random point process. | method 0.42 | 0.90 | 0.29 | asserted 1.00 |
| s4 | The main novelty is the introduction and analysis of suitable corrector equations. | contribution 0.99 | 0.91 | 0.32 | asserted 0.99 |
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The paper
Original abstract
Consider a colloidal suspension of rigid particles in a steady Stokes flow. In a celebrated work, Einstein argued that in the regime of dilute particles the system behaves at leading order like a Stokes fluid with some explicit effective viscosity. In the present contribution, we rigorously define a notion of effective viscosity, regardless of the dilute regime assumption. More precisely, we establish a homogenization result when particles are distributed according to a given stationary and ergodic random point process. The main novelty is the introduction and analysis of suitable corrector equations.
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How the literature uses it
4 arXiv papers cite this in 7 sentences
Papers citing it · 4 sentences without a confident label
Compares with it 1 paper
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Let us stress here that although quantifying convergences and considering more general cases in comparison to this article, all results in [ref] provide closeness results in low $L^p$ spaces $p≤2$.
A local version of Einstein's formula for the effective viscosity of suspensions · Prior and related results · 0.97
Cites it as background 2 papers
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We refer to [ref] for all details.
Mild assumptions for the derivation of Einstein's effective viscosity formula · Introduction · 0.88
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Time evolution of $X_i = X_i(t)$ is neglected (or frozen), the particle distribution is considered as given, and mean properties of the suspension, such as the average settling velocity [ref] or the effective viscosity [ref] , are deduced.
Analysis of a sedimenting suspension near a vertical wall · Introduction · 0.98
What it cites · 8 sentences without a confident label
Cites as background 2 papers
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In the random setting, we refer to the contributions by Beliaev and Kozlov [ref] , by Basson and Gérard-Varet [ref] , and more recently by Giunti and Höfer [ref] .
Homogenization for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes · General overview · 1.00
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In [ref] (see also the refined version [ref] ), Haines and Mazzucato provide bounds on the difference between a heuristic notion of effective viscosity (defined as some integral ratio with the correct dimensionality) and Einstein's formula.
A local version of Einstein's formula for the effective viscosity of suspensions · General overview · 0.86
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