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Derivation of the viscoelastic stress in Stokes flows induced by non-spherical Brownian rigid particles through homogenization

Richard M. Höfer, Marta Leocata, Amina Mecherbet

TL;DR

The paper provides a rigorous homogenization-derived derivation of the viscoelastic stress in Stokes flows induced by non-spherical Brownian rigid particles. It introduces a simplified microscopic model with axisymmetric particles, Stratonovich torques, and decoupled translations to uncover the origin of the elastic stress and its mean-field limit. By constructing precise operator approximations and employing the method of reflections, the authors prove convergence results in two Deborah-number regimes: De=O(1) and De→0, obtaining, respectively, a Stokes system with a viscoelastic stress and a coupled stationary Fokker–Planck equation governing particle orientations. The work clarifies how Brownian torques, rotational diffusion, and particle anisotropy combine to generate the elastic stress, and it provides a framework for future extensions toward more complete Doi-type models and non-dilute suspensions, with a rigorous treatment of stochastic boundary data in perforated domains.

Abstract

We consider a microscopic model of $n$ identical axis-symmetric rigid Brownian particles suspended in a Stokes flow. We rigorously derive in the homogenization limit of many small particles a classical formula for the viscoelastic stress that appears in so-called Doi models which couple a Fokker-Planck equation to the Stokes equations. We consider both Deborah numbers of order $1$ and very small Deborah numbers. Our microscopic model contains several simplifications, most importantly, we neglect the time evolution of the particle centers as well as hydrodynamic interaction for the evolution of the particle orientations. The microscopic fluid velocity is modeled by the Stokes equations with given torques at the particles in terms of Stratonovitch noise. We give a meaning to this PDE in terms of an infinite dimensional Stratonovitch integral. This requires the analysis of the shape derivatives of the Stokes equations in perforated domains, which we accomplish by the method of reflections.

Derivation of the viscoelastic stress in Stokes flows induced by non-spherical Brownian rigid particles through homogenization

TL;DR

The paper provides a rigorous homogenization-derived derivation of the viscoelastic stress in Stokes flows induced by non-spherical Brownian rigid particles. It introduces a simplified microscopic model with axisymmetric particles, Stratonovich torques, and decoupled translations to uncover the origin of the elastic stress and its mean-field limit. By constructing precise operator approximations and employing the method of reflections, the authors prove convergence results in two Deborah-number regimes: De=O(1) and De→0, obtaining, respectively, a Stokes system with a viscoelastic stress and a coupled stationary Fokker–Planck equation governing particle orientations. The work clarifies how Brownian torques, rotational diffusion, and particle anisotropy combine to generate the elastic stress, and it provides a framework for future extensions toward more complete Doi-type models and non-dilute suspensions, with a rigorous treatment of stochastic boundary data in perforated domains.

Abstract

We consider a microscopic model of identical axis-symmetric rigid Brownian particles suspended in a Stokes flow. We rigorously derive in the homogenization limit of many small particles a classical formula for the viscoelastic stress that appears in so-called Doi models which couple a Fokker-Planck equation to the Stokes equations. We consider both Deborah numbers of order and very small Deborah numbers. Our microscopic model contains several simplifications, most importantly, we neglect the time evolution of the particle centers as well as hydrodynamic interaction for the evolution of the particle orientations. The microscopic fluid velocity is modeled by the Stokes equations with given torques at the particles in terms of Stratonovitch noise. We give a meaning to this PDE in terms of an infinite dimensional Stratonovitch integral. This requires the analysis of the shape derivatives of the Stokes equations in perforated domains, which we accomplish by the method of reflections.
Paper Structure (36 sections, 35 theorems, 254 equations, 1 figure)

This paper contains 36 sections, 35 theorems, 254 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Previous mathematical results
  3. Heuristics
  4. Formal statement of the main results
  5. Organization of the rest of the paper
  6. The microscopic model
  7. The particles
  8. The resistance tensor
  9. The dynamics
  10. Simplification of the model
  11. Nondimensionalization
  12. Main Results
  13. Assumptions
  14. Notion of solution and well-posedness of the microscopic system
  15. Convergence results
  16. ...and 21 more sections

Key Result

Theorem 3.2

Let ass:phi.log.n--ass:well.separated be satisfied. Then, for all $n \in \mathbb{N}$, there exists a unique solution $(\xi_1,\dots,\xi_n)$ to eq:Particles.T_D and eq:Particles.T_u, respectively. Moreover, there exists $N_0 \in \mathbb{N}$ such that for all $n \geqslant N_0$ and all $s > 1/2$, the op

Figures (1)

  • Figure 1: Heuristic explanation of the viscoelastic stress arising from rotational Brownian motion. The blue arrows represent the motion of the fluid whereas the red arrows represent the motion of the rod.

Theorems & Definitions (81)

  • Definition 3.1: Definition of the velocity field $u_n$
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Lemma 3.5
  • Lemma 4.1
  • Remark 4.2
  • proof
  • Proposition 4.3
  • Proposition 4.4
  • ...and 71 more