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On the effect of polydispersity and rotation on the Brinkman force induced by a cloud of particles on a viscous incompressible flow

Matthieu Hillairet, Ayman Moussa, Franck Sueur

Abstract

In this paper, we are interested in the collective friction of a cloud of particles on the viscous incompressible fluid in which they are moving. The particles velocities are assumed to be given and the fluid is assumed to be driven by the stationary Stokes equations. We consider the limit where the number N of particles goes to infinity with their diameters of order 1/N and their mutual distances of order (1/N)^{1/3}. The rigorous convergence of the fluid velocity to a limit which is solution to a stationary Stokes equation set in the full space but with an extra term, referred to as the Brinkman force, was proven by Desvillettes, Golse and Ricci when the particles are identical spheres in prescribed translations. Our result here is an extension to particles of arbitrary shapes in prescribed translations and rotations. The limit Stokes-Brinkman system involves the particle distribution in position, velocity and shape, through the so-called Stokes' resistance matrices.

On the effect of polydispersity and rotation on the Brinkman force induced by a cloud of particles on a viscous incompressible flow

Abstract

In this paper, we are interested in the collective friction of a cloud of particles on the viscous incompressible fluid in which they are moving. The particles velocities are assumed to be given and the fluid is assumed to be driven by the stationary Stokes equations. We consider the limit where the number N of particles goes to infinity with their diameters of order 1/N and their mutual distances of order (1/N)^{1/3}. The rigorous convergence of the fluid velocity to a limit which is solution to a stationary Stokes equation set in the full space but with an extra term, referred to as the Brinkman force, was proven by Desvillettes, Golse and Ricci when the particles are identical spheres in prescribed translations. Our result here is an extension to particles of arbitrary shapes in prescribed translations and rotations. The limit Stokes-Brinkman system involves the particle distribution in position, velocity and shape, through the so-called Stokes' resistance matrices.

Paper Structure

This paper contains 21 sections, 14 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. The system
  4. Stokes' resistance matrix
  5. Assumptions
  6. Statement
  7. Comments and organization of the paper
  8. Reminders on the Stokes problem
  9. Standard statements
  10. Remarks on the exterior problem, with different shapes
  11. Proof of \ref{['est:boundinfini']} and \ref{['est:bound']}.
  12. Proof of \ref{['est:decayuinfini']} and \ref{['est:decaynablauinfini']}.
  13. Proof of \ref{['est:conv']} and \ref{['est:convforce']}.
  14. Proof of Theorem \ref{['thm_main']}
  15. Uniform bound for $(E_\Omega(u^{N}))_N$
  16. ...and 6 more sections

Key Result

Theorem 1

Assume eq_ass1 and Assumptions ass:1, ass:2. Assume furthermore that in eq_ass5 -- eq_ass6 we have Then, the subsequence of extensions $(E_{\Omega}(u_{N_k}))_k$ converges weakly in $V_0(\Omega)$ to $\bar{u}$ satisfying, for all $w$ in $V_0$:

Theorems & Definitions (23)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Proposition 3
  • proof
  • Corollary 4
  • proof
  • Lemma 5
  • ...and 13 more