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Correction to Doi type models for suspensions

David Gérard-Varet, Richard M. Höfer

TL;DR

The paper derives a corrected Doi-type model for suspensions of non-spherical particles in Stokes flow by starting from a microscopic $N$-particle system and addressing the breakdown of the naive mean-field limit for orientation dynamics due to a singular $-3$-homogeneous interaction. A deterministic intermediate system is analyzed via the method of reflections and an energy-based regularization of the singular term, yielding a refined angular dynamics that includes a new term $\phi M B\xi f$ with a microstructure-dependent tensor $B$. When particle configurations are generated by a stationary ergodic marked point process, the singular interaction limit becomes almost surely deterministic and expressible through 1- and 2-point correlations; under mixing, this limit has an explicit representation in terms of the correlations and the flow maps $\Phi$, $\Xi$, enabling a concrete form for $B$. The main result, encapsulated in Theorem thm.f-f_phi, shows that the microscopic empirical measure $f_N$ is approximated to first order in $\phi$ by $f_\phi$, with an $o(\phi)$ error in a negative Sobolev topology, thereby providing a practically usable correction to Doi-type models in dilute suspensions. This links the orientation dynamics directly to particle correlations and two-scale homogenization, suggesting that microstructure information is essential for accurately predicting orientational phenomena in suspensions of nonspherical particles.

Abstract

Starting from microscopic $N$ particle systems, we study the derivation of Doi type models for suspensions of non-spherical particles in Stokes flows. While Doi models accurately describe the effective evolution of the spatial particle density to the first order in the particle volume fraction, this accuracy fails regarding the evolution of the particle orientations. We rigorously attribute this failure to the singular interaction of the particles via a $-3$-homogeneous kernel. In the situation that the particles are initially distributed according to a stationary ergodic point process, we identify the limit of this singular interaction term. It consists of two parts. The first corresponds to a classical term in Doi type models. The second new term depends on the (microscopic) $2$-point correlation of the point process. By including this term, we provide a modification of the Doi model that is accurate to first order in the particle volume fraction.

Correction to Doi type models for suspensions

TL;DR

The paper derives a corrected Doi-type model for suspensions of non-spherical particles in Stokes flow by starting from a microscopic -particle system and addressing the breakdown of the naive mean-field limit for orientation dynamics due to a singular -homogeneous interaction. A deterministic intermediate system is analyzed via the method of reflections and an energy-based regularization of the singular term, yielding a refined angular dynamics that includes a new term with a microstructure-dependent tensor . When particle configurations are generated by a stationary ergodic marked point process, the singular interaction limit becomes almost surely deterministic and expressible through 1- and 2-point correlations; under mixing, this limit has an explicit representation in terms of the correlations and the flow maps , , enabling a concrete form for . The main result, encapsulated in Theorem thm.f-f_phi, shows that the microscopic empirical measure is approximated to first order in by , with an error in a negative Sobolev topology, thereby providing a practically usable correction to Doi-type models in dilute suspensions. This links the orientation dynamics directly to particle correlations and two-scale homogenization, suggesting that microstructure information is essential for accurately predicting orientational phenomena in suspensions of nonspherical particles.

Abstract

Starting from microscopic particle systems, we study the derivation of Doi type models for suspensions of non-spherical particles in Stokes flows. While Doi models accurately describe the effective evolution of the spatial particle density to the first order in the particle volume fraction, this accuracy fails regarding the evolution of the particle orientations. We rigorously attribute this failure to the singular interaction of the particles via a -homogeneous kernel. In the situation that the particles are initially distributed according to a stationary ergodic point process, we identify the limit of this singular interaction term. It consists of two parts. The first corresponds to a classical term in Doi type models. The second new term depends on the (microscopic) -point correlation of the point process. By including this term, we provide a modification of the Doi model that is accurate to first order in the particle volume fraction.

Paper Structure

This paper contains 18 sections, 22 theorems, 310 equations.

Table of Contents

  1. Introduction
  2. Previous results
  3. Outline of the paper
  4. The tensors M and S
  5. Main results
  6. Deterministic results
  7. Further results for stationary ergodic initial data
  8. Proof of the deterministic results
  9. Approximation of the fluid velocity through the method of reflections
  10. Proof of Theorem \ref{['th:app']}
  11. Regularization procedure
  12. Proof of the results for stationary ergodic initial data
  13. Random stationary model for the particles distribution
  14. Proof of Theorem \ref{['theo3']}
  15. Proof of Theorem \ref{['cor1']}
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $q>3$, $g \in W^{1,q}({\mathbb R}^3) \cap L^1({\mathbb R}^3)$ and $f^0 \in \mathcal{P}({\mathbb R}^3 \times \mathbb{S}^2)$ satisfying Assume that ass:separation and ass:diluteness hold and that $\mathcal{W}_\infty(f_N^0,f^0) \to 0$. Let $T>0$. Then, there exist $N_0,C> 0$, depending only on $T$, $g$, $f^0$ and $c$ from ass:diluteness such that for all $t \leq T$ and $N \geq N_0$ and dmin.cont

Theorems & Definitions (37)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • ...and 27 more