Non-existence of mean-ﬁeld models for particle orientations in suspensions Richard M. Höfer1 Amina Mecherbet1 and Richard Schubert2

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Non-existence of mean-ﬁeld models for particle

orientations in suspensions

Richard M. Höfer∗1, Amina Mecherbet†1, and Richard Schubert‡2

1Institut de Mathématiques de Jussieu – Paris Rive Gauche, Université Paris Cité, France

2Institute for Applied Mathematics, University of Bonn, Germany

October 28, 2022

Abstract

We consider a suspension of spherical inertialess particles in a Stokes ﬂow on the torus

. The

particles perturb a linear extensional ﬂow due to their rigidity constraint. Due to the singular

nature of this perturbation, no mean-ﬁeld limit for the behavior of the particle orientation can

be valid. This contrasts with widely used models in the literature such as the FENE and Doi

models and similar models for active suspensions. The proof of this result is based on the study

of the mobility problem of a single particle in a non-cubic torus, which we prove to exhibit a

nontrivial coupling between the angular velocity and a prescribed strain.

1 Introduction

It is well known that inertialess rigid particles suspended in a ﬂuid change the rheological properties

of the ﬂuid ﬂow. For passive non-Brownian particles this accounts to an increased viscous stress.

In more complex models like active (self-propelled) particles or non-spherical Brownian particles,

an additional active or elastic stress arises that renders the ﬂuid viscoelastic. Over the last years,

considerable eﬀort has been invested into the rigorous derivation of eﬀective models for suspensions.

This has been quite successful regarding the derivation of eﬀective ﬂuid equations in models when

only a snapshot in time is studied for a prescribed particle conﬁguration or for certain toy models

that do not take into account the eﬀects of the ﬂuid on the particle evolution (see e.g. [HM12;NS20;

HW20;GH20;GM22;DG20;GH21;DG21;Gir22;HLM22]).

Much less is known regarding the rigorous derivation of fully coupled models between the ﬂuid

and dispersed phase, although a number of such models have been proposed a long time ago and

some of them have been studied extensively in the mathematical literature. These models typically

consist of a transport or Fokker-Planck type equation for the particle density coupled to a ﬂuid

equation incorporating the eﬀective rheological properties.

The rigorous derivation of such models is so far limited to sedimenting spherical particles where

the transport-Stokes system has been established in [Höf18;Mec19] to leading order in the particle

∗hoefer@imj-prg.fr

†mecherbet@imj-prg.fr

‡schubert@iam.uni-bonn.de

arXiv:2210.15382v1 [math.AP] 27 Oct 2022

volume fraction. This system reads







∂tρ+ (u+g)· ∇ρ= 0,

−∆u+∇p=gρ, div u= 0,

where

(

t, x

)is the number density of particles,

g∈R3

is the constant gravitational acceleration.

Here, the gravity is dominating over the change of the rheological properties of the ﬂuid, which only

appears as a correction to the next order in the particle volume fraction

. More precisely, as was

shown in [HS21], a more accurate description is given by the system







∂tρ+ (u+g)· ∇ρ= 0,

−div((2 + 5φρ)Du) + ∇p=gρ, div u= 0,(1.1)

where Du = 1/2(∇u+ (∇u)T)denotes the symmetric gradient.

For non-spherical particles, the increase of viscous stress depends on the particle orientations (see

e.g. [HW20] for a rigorous result in the stationary case). Moreover, elastic stresses are observed for

non-spherical Brownian particles as well as active stresses for self-propelled particles, both depending

on the particle orientation. Therefore, it is necessary to consider models for the evolution of particle

densities

that include the particle orientation. In the simplest case of identical axisymmetric

particles, the particle orientation can be modeled by a single vector

ξ∈S2

. The model corresponding

to (1.1) then reads







∂tf+ (u+g)· ∇f+ divξ1

2curl u∧ξ+BPξ⊥Duξf= 0,

−∆u+∇p−div(φM[f]Du) = gρ, div u= 0.(1.2)

Here,

Pξ⊥

denotes the orthogonal projection in

to the subspace

ξ⊥

and

is the Bretherton

number that depends only on the particle shape (

= 0 for spheres,

= 1 in the limit of very

elongated particles, see e.g. [Gra18, Section 3.8]). Moreover

[

]is a 4-th order tensor depending

on the particle shape and given in terms of moments of f.

A widely used model for Brownian suspensions of rod-like (Bretherton number B= 1) particles

at very small particle volume fraction

is the so called Doi model (see e.g. [DE88;Con05;HO06;

LM07;Con+07;CM08;ZZ08;OT08;CS09;CS10;BT12;BT13;HT17;La19]) that reads (in the

absence of ﬂuid inertia)











∂tf+ div(uf) + divξ(Pξ⊥∇xuξf ) = 1

De∆ξf+λ1

De divx((Id +ξ⊗ξ)∇xf),

−∆u+∇p−div σ=h, div u= 0

σ=σv+σe=φM[f]Du +λ2φ

De ˆS2

(3ξ⊗ξ−Id)fdξ.

(1.3)

Here,

is the Deborah number,

λ1, λ2

are constants that depend on the particle shape and

is some given source term. Neglecting the eﬀect of the ﬂuid on the particles, the elastic stress

σe

in the Doi model has been recently derived in [HLM22]. There are very similar models for active

suspensions (the Doi-Saintillan-Shelley model) and ﬂexible particles, most prominently the FENE

model. Well-posedness and behavior of solutions to such models have been studied for example in

[JLL02;JLL04;Jou+06;LL07;SS08b;SS08a;LL12;CL13;Mas13;Sai18;CDG22;AO22].

The main purpose of this paper is to draw attention to the limitations of such fully coupled

models like

(1.2)

and

(1.3)

regarding the modeling of the particle orientations through the term

divξ

(

Pξ⊥∇xuξf

)(respectively

divξ

((1

curl u∧ξ

BPξ⊥Duξ

)

)for

= 1). This term derives from

the change of orientation for the particles according to the gradient of the ﬂuid velocity. However,

at least partially, this ﬂuid velocity is arising as a perturbation ﬂow due to the presence of the

particles themselves that cause the viscous and elastic stresses

σv

and

σe

. These perturbations are

typically of order

. On the microscopic level, the perturbed ﬂuid velocity is very singular. More

precisely, to leading order, it behaves like the sum of stresslets. At the i-th particle, it is given by

upert

N(Xi)≈X

j6=i

∇Φ(Xi−Xj) : Sj

where the sum runs over all particles

diﬀerent from

,Φis the fundamental solution of the Stokes

equation and

are moments of stress induced at the

-th particle due to the rigidity constraint

(and possibly activeness or ﬂexibility). Consequently, the change of orientation behaves like

ξi=Pξ⊥

iξi· ∇upert

N(Xi)≈Pξ⊥

iξiX

j6=i

∇2Φ(Xi−Xj) : Sj

As Φis homogeneous of degree

−

1, this behavior is too singular to expect the “naive” mean-ﬁeld

limit to be true that would lead to the term

divξ

(

Pξ⊥∇xuξf

)(respectively

divξ

((1

curl u∧ξ

BPξ⊥Duξ

)

)for

= 1) in the models

(1.2)

and

(1.3)

. These models therefore do not seem to

describe correctly the behavior of the particle orientations to ﬁrst order in the particle volume

fraction

. Instead, it seems necessary to include terms that depend on the 2-point correlation

function for an accurate description up to order φ.

This is reminiscent of the second order correction in

of the eﬀective viscous stress

σv

(see [GH20;

GM22;DG21]). For the evolution of the particle orientation, this phenomenon already appears at

the ﬁrst order, since the particle orientations are only sensitive to the gradient of the ﬂuid velocity.

In this paper, we will make these limitations rigorous for a toy model. More precisely, we consider

a model example in which we show the non-existence of any mean ﬁeld model that incorporates the

change of orientations to the leading order in the perturbation ﬁeld of the ﬂuid. This model example

consists of a suspension of spherical particles in a background ﬂow which is a linear extensional ﬂow.

In a bounded domain Ω⊆R3, the problem would read











−∆u+∇p= 0,div u= 0 in Ω\SiBi,

u(x) = φ−1Ax on ∂Ω,

Du = 0 in SiBi,

ˆ∂Bi

σ[u]ndS=ˆ∂Bi

(x−Xi)∧σ[u]ndS= 0 for all 16i6N.

where

(

)denote the spherical particles and

A∈Sym0

(3) is a symmetric tracefree matrix.

The rescaling with the volume fraction

NR3

is introduced in order to normalize the perturbation

ﬂuid velocity ﬁeld uper induced by the particles.

For mathematical convenience, we consider the analogous problem on the torus

. We attach

(arbitrary) orientations

ξi∈S2

to the spheres and show that no mean-ﬁeld limit can exist by proving

that for periodically arranged particles on

, the particles do not rotate at all, while for particles

arranged periodically on (2Z)×Z2, the particles do rotate with a ﬁxed rate.

1.1 Statement of the main results

We will work on the toroidal domains

T= (R/Z)3,TL= (R/2LZ)3,¯

TL= (R/4LZ)×(R/2LZ)2.

Furthermore we set

B=B1(0), BR=BR(0),

and for deﬁniteness

A=





0 1 0

1 0 0

0 0 0







Theorem 1.1.

For 0

< R <

2and Ω =

or Ω =

, let

u∈H1

(Ω) be the unique weak solution

to the problem











−∆u+∇p= 0,div u= 0 in Ω\BR,

Du =Ain BR,

ˆΩ

udx= 0,

ˆ∂BR

σ[u]ndS=ˆ∂BR

x∧σ[u]ndS= 0.

(1.4)

(i) If Ω = T1, then curl u(0) = 0.

(ii) If Ω = ¯

T1. then there exists ¯c > 0such that



curl u(0) −R3

16 ¯ce3

6CR4(1.5)

for a constant Cindependent of R.

Observe that the factor 16 in (1.5) corresponds to the volume of ¯

T1.

As a consequence of Theorem 1.1 we show the negative result stated in Corollary 1.2 below, that

we outlined in the introduction. We will use the following notation. For

N∈N

, which we think of

as the number of particles in a unit cell, we denote by

φ=NR3

the volume fraction of the particles. Both

and

may implicitly depend on

. We will make this

dependence explicit by a subscript

wherever we feel it is necessary for clarity. For

R >

0and

N∈N, let Xi∈T,16i6Nbe such that

dmin = min

i6=j|Xi−Xj|>2R. (1.6)

This ensures that the balls

Bi=BR(Xi)

do not intersect nor touch each other. Moreover, for 1

6i6N

, let

ξ0

i∈S2

. The associated initial

empirical density f0

N∈ P(R3×S2)is given by

N=1

δXi⊗δξ0

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摘要：

Non-existenceofmean-eldmodelsforparticleorientationsinsuspensionsRichardM.Höfer*1,AminaMecherbet1,andRichardSchubert21InstitutdeMathématiquesdeJussieuParisRiveGauche,UniversitéParisCité,France2InstituteforAppliedMathematics,UniversityofBonn,GermanyOctober28,2022AbstractWeconsiderasuspensionofsph...

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Non-existence of mean-ﬁeld models for particle orientations in suspensions Richard M. Höfer1 Amina Mecherbet1 and Richard Schubert2

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