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Derivation of the viscoelastic stress in Stokes flows induced by nonspherical Brownian rigid particles through homogenization

Richard M. Höfer, Marta Leocata and Amina Mecherbet

Vol. 5 (2023), No. 2, 409–460
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 Stratonovich noise. We give a meaning to this PDE in terms of an infinite-dimensional Stratonovich integral. This requires the analysis of the shape derivatives of the Stokes equations in perforated domains, which we accomplish by the method of reflections.

Keywords
homogenization, Stokes flows, viscoelasticity, Doi model, Brownian particles
Mathematical Subject Classification
Primary: 76M50, 76D07, 35R60, 35Q70
Milestones
Received: 21 February 2022
Revised: 13 July 2022
Accepted: 26 August 2022
Published: 26 June 2023
Authors
Richard M. Höfer
Institut de Mathématiques de Jussieu - Paris Rive Gauche
Université Paris Cité
Paris
France
Marta Leocata
Luiss University
Roma
Italy
Amina Mecherbet
Institut de Mathématiques de Jussieu - Paris Rive Gauche
Université Paris Cité
Paris
France