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Sedimentation of particles with very small inertia I: Convergence to the transport-Stokes equation

Richard M. Höfer, Richard Schubert

TL;DR

The paper rigorously derives the inertial-vanishing limit for sedimenting spherical particles in a Stokes flow, proving that the microscopic inertial system converges to the transport-Stokes equation as $N o ty$ and $R o0$ with $ ext{γ}_N=NR o ext{γ}_*$ and $ ext{λ}_N o ty$. It develops a modulated-energy framework to compare inertial and inertialess dynamics, extends Hauray’s mean-field method to $p$-Wasserstein distances, and obtains uniform resistance-matrix bounds together with a robust control of the minimal inter-particle distance via a Stokes-law-type force representation. The main result provides quantitative convergence estimates for the spatial density $ ho_N$ to $ ho_*$, the phase-space density $f_N$ to $ ho_*ig o ext{delta}_{ ext{velocity}}ig$, and the fluid velocity $u_N$ to $u_*$, including explicit rates depending on the initial discrepancy and $1/ ext{λ}_N$. This work justifies neglecting inertia in the microscopic model within the mean-field regime and deepens the link between microscopic particle dynamics and the macroscopic transport-Stokes description, with implications for modeling sedimenting aerosols and related suspensions.

Abstract

We consider the sedimentation of $N$ spherical particles with identical radii $R$ in a Stokes flow in $\mathbb R^3$. The particles satisfy a no-slip boundary condition and are subject to constant gravity. The dynamics of the particles is modeled by Newton's law but with very small particle inertia as $N$ tends to infinity and $R$ to $0$. In a mean-field scaling, we show that the particle evolution is well approximated by the transport-Stokes system which has been derived previously as the mean-field limit of inertialess particles. In particular this justifies to neglect the particle inertia in the microscopic system, which is a typical modelling assumption in this and related contexts. The proof is based on a relative energy argument that exploits the coercivity of the particle forces with respect to the particle velocities in a Stokes flow. We combine this with an adaptation of Hauray's method for mean-field limits to $2$-Wasserstein distances. Moreover, in order to control the minimal distance between particles, we prove a representation of the particle forces. This representation makes the heuristic \enquote{Stokes law} rigorous that the force on each particle is proportional to the difference of the velocity of the individual particle and the mean-field fluid velocity generated by the other particles.

Sedimentation of particles with very small inertia I: Convergence to the transport-Stokes equation

TL;DR

The paper rigorously derives the inertial-vanishing limit for sedimenting spherical particles in a Stokes flow, proving that the microscopic inertial system converges to the transport-Stokes equation as and with and . It develops a modulated-energy framework to compare inertial and inertialess dynamics, extends Hauray’s mean-field method to -Wasserstein distances, and obtains uniform resistance-matrix bounds together with a robust control of the minimal inter-particle distance via a Stokes-law-type force representation. The main result provides quantitative convergence estimates for the spatial density to , the phase-space density to , and the fluid velocity to , including explicit rates depending on the initial discrepancy and . This work justifies neglecting inertia in the microscopic model within the mean-field regime and deepens the link between microscopic particle dynamics and the macroscopic transport-Stokes description, with implications for modeling sedimenting aerosols and related suspensions.

Abstract

We consider the sedimentation of spherical particles with identical radii in a Stokes flow in . The particles satisfy a no-slip boundary condition and are subject to constant gravity. The dynamics of the particles is modeled by Newton's law but with very small particle inertia as tends to infinity and to . In a mean-field scaling, we show that the particle evolution is well approximated by the transport-Stokes system which has been derived previously as the mean-field limit of inertialess particles. In particular this justifies to neglect the particle inertia in the microscopic system, which is a typical modelling assumption in this and related contexts. The proof is based on a relative energy argument that exploits the coercivity of the particle forces with respect to the particle velocities in a Stokes flow. We combine this with an adaptation of Hauray's method for mean-field limits to -Wasserstein distances. Moreover, in order to control the minimal distance between particles, we prove a representation of the particle forces. This representation makes the heuristic \enquote{Stokes law} rigorous that the force on each particle is proportional to the difference of the velocity of the individual particle and the mean-field fluid velocity generated by the other particles.
Paper Structure (16 sections, 24 theorems, 284 equations)

This paper contains 16 sections, 24 theorems, 284 equations.

Table of Contents

  1. Introduction
  2. Previous results
  3. Statement of the main results
  4. Elements of the proof and outline of the the paper
  5. Generalization of Hauray's mean-field limit result to p-Wasserstein distances
  6. Proof of the main theorem
  7. Notation
  8. Modulated energy argument
  9. Uniform estimates on the resistance matrix mR and the minimal distance dmin
  10. Proof of Theorem \ref{['th:diagonal']}
  11. Estimates for the resistance matrix mR
  12. Estimate for the gradient of the resistance matrix
  13. Control of the minimal distance
  14. Estimates of sums of inverse particle distances
  15. Proof of an Hauray type result for second-order binary systems with asymptotically vanishing inertia
  16. ...and 1 more sections

Key Result

Theorem 1.1

For $N \in \mathbb{N}$ assume that $\lambda_N$, $\gamma_N$, $X_i^0, V_i^0$ satisfy ass:gamma--ass:V_infty. Let $(\rho_\ast,u_\ast) \in L^\infty((0,T);L^q(\mathbb{R}^3))\times L^\infty((0,T);\dot H^1(\mathbb{R}^3))$ be the unique weak solution of eq:transport-Stokes associated to the initial value $\ Moreover the following refined estimates hold and, for all $x\in \mathbb{R}^3$,

Theorems & Definitions (29)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • ...and 19 more