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Hydrodynamic limit of multiscale viscoelastic models for rigid particle suspensions

Mitia Duerinckx, Lucas Ertzbischoff, Alexandre Girodroux-Lavigne, Richard M. Höfer

TL;DR

The paper addresses the viscoelastic rheology of suspensions of Brownian rigid rods by deriving a rigorous hydrodynamic limit from the micro-macro Doi--Saintillan--Shelley model in the small Weissenberg-number regime. It combines a formal Hilbert expansion with a rigorous remainder analysis to justify a second-order ordered-fluid model, providing explicit coefficients $\eta_0,\eta_1,\mu_0,\gamma_1,\gamma_2$ that link microscopic parameters to macroscopic rheology. A key advance is establishing well-posedness for the kinetic model and for a perturbative, well-posed class of second-order fluid equations, along with a hierarchical perturbation framework that avoids the usual ill-posedness of ordered-fluid models. The results reveal non-Newtonian features such as nontrivial normal-stress differences and shear-thinning/elongational behaviors that align qualitatively with experiments, and they demonstrate the inappropriateness of Oldroyd--B closures in this regime. The methodology clarifies the conditions under which macroscopic rheology can be rigorously derived from kinetic models, offering a principled bridge between microscopic particle dynamics and measurable viscoelastic responses in active and passive suspensions.

Abstract

We study the multiscale viscoelastic Doi model for suspensions of Brownian rigid rod-like particles, as well as its generalization by Saintillan and Shelley for self-propelled particles. We consider the regime of a small Weissenberg number, which corresponds to a fast rotational diffusion compared to the fluid velocity gradient, and we analyze the resulting hydrodynamic approximation. More precisely, we show the asymptotic validity of macroscopic nonlinear viscoelastic models, in form of so-called ordered fluid models, as an expansion in the Weissenberg number. The result holds for zero Reynolds number in 3D and for arbitrary Reynolds number in 2D. Along the way, we establish several new well-posedness and regularity results for nonlinear fluid models, which may be of independent interest.

Hydrodynamic limit of multiscale viscoelastic models for rigid particle suspensions

TL;DR

The paper addresses the viscoelastic rheology of suspensions of Brownian rigid rods by deriving a rigorous hydrodynamic limit from the micro-macro Doi--Saintillan--Shelley model in the small Weissenberg-number regime. It combines a formal Hilbert expansion with a rigorous remainder analysis to justify a second-order ordered-fluid model, providing explicit coefficients that link microscopic parameters to macroscopic rheology. A key advance is establishing well-posedness for the kinetic model and for a perturbative, well-posed class of second-order fluid equations, along with a hierarchical perturbation framework that avoids the usual ill-posedness of ordered-fluid models. The results reveal non-Newtonian features such as nontrivial normal-stress differences and shear-thinning/elongational behaviors that align qualitatively with experiments, and they demonstrate the inappropriateness of Oldroyd--B closures in this regime. The methodology clarifies the conditions under which macroscopic rheology can be rigorously derived from kinetic models, offering a principled bridge between microscopic particle dynamics and measurable viscoelastic responses in active and passive suspensions.

Abstract

We study the multiscale viscoelastic Doi model for suspensions of Brownian rigid rod-like particles, as well as its generalization by Saintillan and Shelley for self-propelled particles. We consider the regime of a small Weissenberg number, which corresponds to a fast rotational diffusion compared to the fluid velocity gradient, and we analyze the resulting hydrodynamic approximation. More precisely, we show the asymptotic validity of macroscopic nonlinear viscoelastic models, in form of so-called ordered fluid models, as an expansion in the Weissenberg number. The result holds for zero Reynolds number in 3D and for arbitrary Reynolds number in 2D. Along the way, we establish several new well-posedness and regularity results for nonlinear fluid models, which may be of independent interest.
Paper Structure (36 sections, 8 theorems, 248 equations)

This paper contains 36 sections, 8 theorems, 248 equations.

Table of Contents

  1. Introduction
  2. General overview
  3. Informal statement of main results
  4. Previous results
  5. Structure of the article
  6. Main results
  7. Well-posedness of the Doi--Saintillan--Shelley model
  8. Formal Hilbert expansion
  9. Well-posedness of ordered fluid models: hierarchical solutions
  10. Hydrodynamic limit
  11. Proof of the main result
  12. Computational tools for spherical calculus
  13. Proof of Proposition \ref{['prop:first-terms']}
  14. Proof of Proposition \ref{['prop:main']}
  15. Proof of Proposition \ref{['lem:Bous']}
  16. ...and 21 more sections

Key Result

Theorem 1.1

Consider either the Stokes case $\operatorname{Re}=0$ with $d\le3$, or the Navier--Stokes case $\operatorname{Re}\ne0$ with $d=2$. Given an initial particle density $f_\varepsilon^\circ \in C^\infty\cap\mathcal{P}(\mathbb T^d\times\mathbb S^{d-1})$ that is well-prepared in a sense that will be clari the fluid velocity $u_\varepsilon$ satisfies and the particle spatial density $\rho_\varepsilon:=\

Theorems & Definitions (11)

  • Theorem 1.1: Informal statement of the main result
  • Remark 1.2
  • Proposition 2.1
  • Remark 2.3
  • Proposition 2.4: Approximate hierarchical solutions
  • Theorem 2.5: Small-$\operatorname{Wi}$ expansion
  • Proposition 3.1: Well-posedness of hierarchy
  • Proposition 3.2: Error estimates for $\varepsilon$-expansion
  • Proposition 3.3: From hierarchy to second-order fluids
  • Remark 4.1: Connection to Oldroyd--B model
  • ...and 1 more