RevistaMexicanade Física (Suplemento)32 No. SI (1986) SI SI.S2~

HYDRODYNAMICINTERACTIONS AND

TRANSPORT -PROPERTI ESOF SUSPENSIONS

*P. Mazur

Instituut Lorentz, Rijksun!versltelt LeidenNieuwsteeg 18, 2311 S8 Leiden, The Netherlands

RESUMEN

5151

Estudiamos en este curso el problema de las interacciones hidrodinámicas entre esferas moviéndose en un fluído viscoso. La solución está basada en un método de fuerzas inducidas y Su expansión en monopolos-irreducibles. El enfoque puede incorporar los efectos de las paredes.Tratamos tambien la auto-difusión y sedimentación como propiedades detransporte de suspensiones en las cuales las interacciones hidrodinami-cas de muchos cuerpos juegas un papel esencial.

ABSTRACT

In these lec tu res we discuss the problem oi hydrodynamic in-teractions between spheres moving in a viscous fluid. Our treatment isbased on a method of induced forces and their expansion in irreduciblemonopoles. The approach may incorporate wall effects which are alsodiscussed. We also present self-diffusion and sedimentation 35 transport-properties of suspensions for which the many-body hydrodynanic interac-tions play an essential role.

*These lectures are also published 1n the Springer series "Lecturenotes 1n Phys1cs", Proceed1ngs of the Escola de Termodynamica deBellaterra, Spain, 1985.

S152

INTRODUCTION

In a variety oi physico-chemical and physlcal problems ane 15

confronted with the compl1cations caused by hydrodynamic interactions

between spheres moving In a viscous fluid. These interactlons which

are oi importance fer the quantitative understanding of suspensions oE

(uncharged) partieles, were traditlonally studied by the so-calledmethods oi reflections, inaugurated by Smoluchowskl fer those situ-

atieos In which the fluid can be described by the quasi-static Stokes

equation, Le. by the l1nearized Navier Stokes equation for incom-

pressible steady £low.Due to the increasing

the two-sphere case waschowski1), Fax~n2). Dah13)

complexity of

analyzed byand Happe!

the problem. essentially on1y

these methods. Thus Smolu-

and Brenner4) calculated for

remained 1arge1y unnoticed. With regards to the many-sphere

by Muthukumarl2). andworkofmadebe

this case the friction tensors to higher and higher order in the

inverse distance between spheres. while e.g. BurgersS), Batchelor6)

and Felderhof7) evaluated the mobility tensors (which are elements of

the inverse of the friction tensor matrix).

For the discussion of the properties of dllute suspensions. one

needs on1y take pair interactions into account6•8). lt was quite

generally hoped and presumed that pair-wlse addltlvlty of hydrodynamlc

lnteractions would hold In concentrated suspenslons as well (see in

this connexion the excellent revlew of Pusey and Tough9». However In

view of the long-range nature of these interactions. it was rather

questionable whether such an assumptlon was justified. Wlth this in

mind we recently developed a systematic seheme to treat the fu1l many-

sphere problem10). Kynchll) had already by a method of reflection

derlved expressions for three and four spheres. His work however seems

to have

problem mention must a180

Yoshlzaki and Yamakawa13).

In these lectures we consider in a first chapter. the many sphere

hydrodynamic interaction problem and its solution, as glven in ref. 10based on a method of induced forces14). and their expansion In irre-

ducible multipoles. This chapter also includes an extension oí the

5153

approach reviewed to incorporate wall effects, ln particular hydro-

dynamic lnteractions with the wall oi a spherlcal contalner15).In chapter 2 we discuss self-diffusion aud sedimentation as

transport-properties oi suspensions for whlch the many-body hydro-dynamic interactions play ao essentlal role. The discusslon ls res-

trlcted to the evaluatlon oi properties on the short-tlme seale, Le.

for a time regime sueh that the relative configuratlon oE suspended

partieles does not change appreciably. Special attention 15 given tol. the essential non-addltivity of hydrodynamic lnteractions, and2. the influence oi very long-ranged hydrodynamlc interactlons aud of

wall-effects on transport-properties.T.••.o special 1ectures, published elsewhere16), on parts of the

material dealt .••.ith in this course were given respective1y in Taronta,Canada (June 1984) and Lausanne, Switzerland (June 1985).

S154

Chapter 1

HYDRODYNAMIC INTERACTIONS

1.1 Equations oi motion; formal 501ut100

We consider N macroscopic spheres with radii al (i-l.2 ...N),+ +which mave with velocities u1 and angular velocities w1 through ao

otherwise unbounded incompressible vlscous fluid.The motion of the fluid oheys the quasi static Stokes equation

+(;)

1~.P • O

fer al1 1; - Ril >81, 1"'1.2 ••• N (1.1)

+ + (n • O~o v

with

(1.2)

++

Here P 15 the pressure tensor, P the hydrostatlc +pressure. v the

label carte-

denotes the+the fluidj Ri

greek 1ndices

velocity field and ~ the viscosity oi

positton of the center of sphere t, while

siao components of tensorial quantities.+ +

The force Kiand tarque Ti exerted by the fluid on sphere 1 are

given by

+ f dS (1.3)Ki PonSi - i

+ f dS (;- + +-

Ti Ri)APoni (1.4)Si

Here Si 18 the surface of sphere i (to be precise: the surface ofa sphere centred at R

iwith radius ai + E in the limit E • O), and

ni a unit vector normal to this surface pointing in the outwarddirection.

5155

In arder to salve the set oi equatlons (1.1) and (1.2) aud subse-quently determine the torces and torques from eqs. (1.3) aud (1.4),

boundary conditions at che surfaces oE che spheres must be specified.

We assume such conditions

(1.5)

The problem posed by equations (1.1) - (1.5) may be reformulated by

introducing a force denslty F .(~) induced on the spheres and extendingJ

che fluid equations inside the spheres. The fluid equations are then

written in the equlvalent fom.

N

\9- p • E rpl

j-1 for a11 • (1.6)r

9- • (;) • Ov

w1th Fj(;) '" O fer \; - Rjl> ajO Insirle the spheres the fluid veloc1ty

field aud pressure field are extended according to

(LB)

As a consequence oi these extensions che induced force dens1ty 15 of

the fom.

(1.9)

The factor a~2 has been introduced here for convenience.Making use of eqs. (1.3), (1.4), (1.6) and (1.9) one can express theforce Ki and torquethe induced surface

•Ti which the fluid exerts on sphere 1, in terms offorce s 11. Wlth Gauss' theorem one has lndeed

(1.10)

5156

(1.11)

To formally salve the equation oE motion oE the fluid we intro-

duce Fourier transforms oE e.g. the velocity field

(1.12)

The Fourier transformed induced force density F(k) 15 defined in a

reference frame In which the center oE sphere i 15 at the or1g10

F (k)1

(1.13)

With these definitions, the equations oE motion (1.6) become in wave

vector representatlon ..~k2;¡(k)

N -ik.R.- ikp(k) + 1: e J FJ(k) (1.14)

J'1

+ T) (1.15)k-v k O •

By applying the operator .!. - kk, where k = k/k 15 the unit vector in

the direction oE k and 1 the unit tensor, to both sides oE eq. (1.14)

ane obtaios with eq. (1.15)

(1.17)

which has I assumlng that the fluid unperturbed by the motion oE the

spheres 15 at rest, the formal 501ut100

~(k) (1.18)

This equation wl11 serve as the starting point for the calculation of

the forces and torques exerted by the fluid on the spheres, and thus

of the hydrodynamic interactions which are set up between the spheres

by their motion through the fluid.

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1.2 Irreducible tensors; induced force multipoles and veloclty surfacemoments

For the purpose oi evaluating hydrodynamic interactions it 15convenient to introduce irreducible induced force multlpoles, de-fined In terros oi the surface forces 11(~i) according to

U'O) • (2.1 )

Here 'tt'is au irreducible tensor oi rank t, l.e. the tensor oE rankt traceless and symmetric in aoy pair oi lts indices, constructed w!ththe vector b. For k = 1,2,3 ane has, see e.g. ref. 17,

Accord1ng to eqs. (1.10), (1.11) and (2.1)

(2.3)and

F(2a)-1

I ~.. --€'T

2a1 - i' (2.4)

The tensorstionsI0)17)

In eq.

el vita

(2.4) !'iza) 1$ the antisymmetric part oi !.(2) and ,£ the Levi-

tensor, ior which ane has the identity ~:~ = - 21 ."'T'ni satisfy the orthogonality and completeness condi-

t! ó(t,t)(2HI)!! Otro _ , (2.5)

-l 1:4n .l=O

(2.6)

In the abo ve equations (21+1)!! 1,3,5 ••. (H-I).(2HI). The dot 0

5158

~ ~denotes a full l-fold contraetlon between the tensor s 01 and ni'""I'with the conventioo that the last index oE n 15 contracted wlth the

~ U~) 1first index oi ni' etc. A' represents ao isotropic tensor oE

rank 21 that projects out the irreducible part oi a tensor oE rank 1:

(2.7)

For 1~O,1,2 ene has

(2.8)ó • ! (6 6 + 6 6 ) - ! 6 6a~y6 2 ay ~6 a6 ~y 3 a~ y6

With relations (2.5) and (2.6) ane shows that the surface lnduced..• .force dens1ty f1(01) has the f0110w10g expansion ln terma oE irre-

ducible force multipoles

(21+1)!! '~~' o F(1+1)1 -1

(2.9)

This expansion, which 15 written ln a coordinate free way, 15 equiva-

lent to ao expansion ln spherical harmonics, to which it can be re-

duced ti polar coordina tes are lntroduced.+ (~For the Fourier transformed induced force density Ft k), the

expanslon (2.9) leads to (ef. eqs. (1.9) and (1.13»1

(2.10)

with jt(x) the spherieal Bessel funetian of arder ,t. In deriving the

expansion (2.10) from (2.9) use has been made of the identlty18)

'"t'o sinkokt --k-- (2.11)

Next to the irreducible induced foree multipales defined above.

we aIso introduce irreducible surface moments of the fluid velocity

5159

fleId. The irreducible surface moment oí arder ID 15 defined as

(2m+l)!! (2m+l)!!

4n82

(2m+l)!!

( 2.) 3

ik-"R~(~) iv k e , ro ) O • (2.12)

The numerical factor 15 (2m+l)!! lntroduced fer convenlence. The velo-city surface moments are the coefficients oí ao expansion oí the fluidvelocity field at the surfaces oí the spheres in irreducible tensora"'ñl'nI • Usiog the boundary condition (l.5) as well as the orthogonality

condition (2.5) ane has

(2.13)

(2m+l)!! 2 O , for ID ) 3 •

In the next section we aha!l relate the induced force multipoles to

the surface maments oí the fluid velocity field through a hlerarchy oíequations. It 15 this hlerarchy, which 1,1111then enable us to obta!n

expressions fer the mobility tensora which relate the torces and

torques on the spheres to their velocities and angular velocities.

1.3 Determination of induced forces; properties of connectors

To determine the induced force s on the surfaces of the spheres weemploy (2.12) in the following way: we substitute the formal solution(lo 18). together with the expansion (2.10) for Fi(i~.) into the lastmember. and use the results (2.13) for the left-hand slde. Qne thenobtains the following set of coupled equations for the irreducible

5160

force multipoles

6nna (~ó + a £.~w6 ).i i nI i - 1 02

E A{n,m)mol -ij

o F(m)-j

(3.1 )

where the coefficients, the so-called connectors whlch are tensors of

rank n+m, are given by

A(n,m)-ij

38 i (20-1 ) ! ! (2m-l ) ! !

4n2 x

(3.2)

Here iij ij- ii ; for i 1 j , Rij = ¡iiJI ) 81+ aj . In principle

one can determine from the set (3.1) al! force multiples in terms of

the velocities ~i and angular velocities ~i and, In particular, derive

expresslons for the mobil1ty tensors whlch relate the forces and the

torques excerted by the fluid on the spheres to these quantities.

Refore establishing th£'~p f'xpressions we shall discuss a number of(n ,m)properties of the connectors ~ij :

l. One verifies by lnspectlon that these quantities satisfy the sym-

metry relation

A(n,m)aj"ij

-A(m,n)ai-ji

(3.3)

Here e is a generalized transpose of a tensor e of arbltrary rank p

2. The integral (3.2) may be rewritten in the more compact form

A(n,m) = 3a Jdk s(n+l)(k)'S(m+l)*(k) ,-ij i -i -J

(3.4)

(3.5)

5161

(3.6)~i(k) = 2~ (2n-I)!!

-1" 0+1where the k dependent tensor ~i of rank 0+1 15 defined as f0110,,",5

ik •Ri r;:--;"11 • • 1in. kn- (l-kk)k-jn_l(ka)

In (3.5) the asterlsk denotes complex conjugation. Qne then has

l:i,j

l: T(n)< 0 A(n,m)0 T(m) •-i -ij -jn,m (3.7)

Jdk \ l:3aiTin)<0 ~in+I)I 2, O ,~n

where the quantities !(n) are arbltrary k lndependent complex tensors

of rank n. We ahall come back to this inequality in the next section.

3. We sha!l now consider In more detall the self-connectors ~i~,m)that are tensors of rank n+ro independent of the index L lf n+m 15

odd. the intergrand In (3.2) 15 ao odd funetian of k so thatA(n,m) :: O in that case. Since furthermore the spherical Bessel func--11tioos have the property

fdx jZn+vCx)jZm+v(x) ::O for n t ID aud v - 0,1 ,O

(3.8)

it f0110,,",5that the self connectors are aIso diagonal in their upper

indices, or in other words, that there is no direct coupling between

different multipoles in the same sphere

A(n,m)• _ B(n,m)Ó (3.9)--i1 nm

The tensors B(n,n) have been calculated19) explicitly in terms of

tensors ~(~,~r-.The first two are

8(1,1)• _ 1 (3.10)

B(2,2)• _ -! ,(2,2)_ l 5 (3.11)a~r6 lO a~r6 2 a~r6

S16Z

where the tensor ~ oi rank 4, which 1s au anti-symmetrization opera-

tor, has elements

s - ! (6 6 6 6 )a~y6 Z a6 ~y - ay ~6 .

4. Next we discuss the behavlour oi the connectors A(~,m)-iJa funetlan oi the interparticle distance Rij. Expressiona1so be written as fol10w5 fer i * j

(3.1Z)

i • j , as

(3.Z) can

A(n,m)-ij

3ai(Zn-l)!!(Zm-l)!!2n

(3.13)

x

Here the integratlon ayer angles has been carried out, after replacing+the tensors formed with the vector k by dlfferentiations with respect

to RijO Expanding the Bessel functions around kzO, ane has

(Zn-l)!!(Zm-l)!!k-(n+m-Z)jn_l(kai)jm_l(kaj)

ar1a;-I[1 - (4n~Z + 4"';'Z)aialZ] + k3J«k) •(3.14)

where O¿ (z) 1s analytic in the complex plane and bounded fer large

l' I by exp(Zaj '1). Upon substitution of eq. (3.14) into eq. (3.13),the contribution Di J( (z) vanishes in view oi the faet thatRij

> 31+ aJo Straightforward evaluatlon oi the remaining integral thenleads to the results

A(n,m) • G(n,m)R-(n+m-l) + H(n,m)R-(n+m+l)-ij -ij ij -ij ij (3.15)

where thevector rij

tensor s G(n.m) and H(o.m) • which only depend on-ij -ij= Rij/Rij and the radll al and ajO are given by

the uni t

5163

G(o,m) _ (_1)0+1 3 o m-1-lj 4313j (3.16)

H(n,m)-lj

(_1)0 l o m-1R0+m+14 313j lj

(3.17)

The arrow'" on alaR in eq. (3.16) indicates a differentiation to the

left.

(3.18)H(o,m)-lj

The express Ion fer ~(n.m) can easily be further simplified by

carrying out the differentlations. and becomes2 2

( l)m 3 o m-l ( al aj )(2 2 1)"'" - ¡3i3j 20+1 +200+1 0+00- ..

For the tensor C(ntm) the differentiations can In principIe be carrled

Qut in a similar formal way. We 11st here the explicit results for thefirst few oi these tensors

(3.19)

l.

(3.20)

G(25,25) 9 2 [3-lj - - 4alaji i •••

(3.21)

In eq. (3.20) C(1,2s) denotes the part oí G(1.2) that 15 (traceless)-lj -ljsymmetric in lts tast tWQ índices. A similar notation 15 adopted for(25 25)£ij 1 in eq. (3.21)¡ the tensor.Q.. 15 traceless and symmetric in its

first and last two lndices and defined by

(3.22)

5164

Further explicit expressions for G(n,m) , with o,m ,¡;; 3 and n+m " 5 may-ij

be found in ref. 10.

1.4 Mobility tensors

In che linear regime considered, che velocities and angular velo-

citles of che spheres are related to che forces and torques exerted onthem by che fluid in a way described by che £0110w10g set oE linear

coupled equation

(4.2)

(4.1)+ TT + TR T.ui - 1: lOJ.j Kj - 1: lOijj j J

+ RT + RR T.wi - 1: lOij K. - 1: !!.ijJ J J J

In ,he above equations, TI RRlOij and !!.ij

mobility tensors respectively. The1ationa1 and rotational motion. The

are transJational and rotationalTR RT

tensors ~ij and ~ij couple trans-

mobility tensors account for che

hydrodynamlc lnteractlons between che spheres through their dependenceon their relatlve pasitioos.

The analysls given in the previous sections enable us to express

che mobilities in terms of connectors and thereby calculate thesequantities as series io powers oi inverse distances between thespheres.

To carry out this program we first rewrite eq. (3.1) in a morecompact formo For ootationa1 convenience we restrict ourselves for thepresent to the case of equal-sized spheres, Le. al :: aj :: a. Wedefine a formal vector sr of which the components are the irreducibleforce multipoles of the N spheres

{'J¡2 • -[ ~jj 2a.£.1' (4.3)

lITe ¡n • F(.n) 37 j -J ,o;> •

S 165

We also define a second vector U with components

{U} i "1

(4.4)

IU)~ - O , n ) 3 •

Furthermore \ole introduce matrices ...J and d3 with elements (cf. eq.

(3.9»)

{8}nm • B(n,n)Ó ÓlJ - nm lJ '

~}nm • A(n,m) + B(n,n)Ó Ó .lJ -IJ - DI' lJ

WIth these notations the set of equations (3.1) becomes

(4.5)

(4.6)

6nnaU • -87 +.17. (4.7)

Next we define projection operators 91, 3>2' aud a - 1 - 91 -}J2

(4.B)

The anti-symmetrizatlon tensor S of rank 4 has been defined In (3.12).1(n,n) 15 che approprlate unit tensor of rsnk 20 which •.• he n fulIy

contracted with a tensor of rank n. projects out the part irreducible

in the first 0-1 ~ndices. Qne verifles that

Note also that the matrix ~ commutes wlth the projectlon operators

5166

'P al. al 'P , v • 1,2 .v v

Now decompose 7 according to

(4.10)

(4.11)

aod mu1tip1y eq. (4.7) from the 1eft by J\, 12 and {) respective1y.This resu1ts, using also the properties (4.10), io the fo110 •••.iog setof equations

6n~aJi U • Jl1(-/3 +.4) Jll + p/,'P2 '{+ 'P1.Ji Q 'J,

6n~a?¡ U • 'P2 ,) ~ 'J + P2 (- al +¡) 'Pz1 + 'P2ip-1.

ó3OJ • (J )pJ" + Q ;~'J+ QiQT.

One may then salve equation (4.14) for

The matrlx ~ -1 has e1ements

(4.12)

(4.13)

(4.14)

(4.15)

for n t 2 (4.16)

-1Uf\) ¡n2 • B(2s,2s) ó ó .iJ - n2 iJ

(4.17)

-1The tensor B(n,o) ,n t 2 ,15 the generalized 10verse of B(n,o)wheo acting 00 tensors of raok o that are irreducible in their fir~£0-1 lndlces. For its constructlon see ref. 19. The tensor B(2s,2s)ls glven by (cf. 3.11)*

*The tensor-1B(2,2) ltse1f can a1so be found from (3.11) and is given by:

10 (2,2) 3 S6 3 -

9

-1B(2s,2s) _ lQ. ~(2,2)

9 -

5167

(4.18)

Upon substitution oi eq. (4.15) into eqs. (4.12) and (4.13) ane ob-

talns, uslng al so (3.10), (3.11), (4.3), (4.4) and (4.8), eqs. of the

forro (4.1) - (4.2) ""ith the mobl1ities expressed in terma oí con-

nectars according to

TI -1!!.lj : (6nna) [Olj + (4.19)

RR ( 3)-1Eij::l< 241tT)a

{J( ,,-1 • _1¡12 -RT" 1 -'" 0,4) Ij:E.' !!.jl (4.20)

(4.21)

ane obtatos in this way the series

The Onsager relatlon which 15 contained in (4.20) 15 a direct conse-

quence oi the symmetry property (3.3) oi the connectors.By expanding the inverse matrices in (4.19) - (4.21) In powers oi

the connector matrix A the mobilities are obtained in the farel oi a

power series expansion in R-1, where R 15 a typical distance between

spheres.TI

For Eij

+ EkH,j

= -1+ E A(I,m) 0 !(m,m) 0 ~(mj,I)¡+ ...]ro") -ik ~

(4.22)

Each terro in this series has, as a funetlan oi a typical lnter-

particle distance R, a given behaviour which is determined by the

5168

upper indices oi the connectors aud their number. Thus according to

eq. (4.13) a term ln eq. (5.23) •.••ith s connectors. s "" 1,2.3 "',

gives contributions proportional to R-P •.••ith P equal to

1,3 fer s ""1,.¡35 - 2 + Zq fer s) 2. q = 0,1,2 ••.•

(4.23)

Thls lmplles that ~~ canoot conta!n terms proportional to R-2 audR-5• We also note that each term In the express ion (4.22) contalning asequence oi s cannectors lnvolves the hydrodynamic interactlon bet •.••eenat mast s + 1 spheres. Therefore the dominant n-sphere contribution,n ) 2, are oi arder R-3n+5••.••here eq. (4.24) has been appl1ed with

s - 0-1 and q - O.Similar considerations lead to the conclusion that in the series

fer ~r~contrlbutions proportional to R-1, R-J and R-6 are excluded,

and that the dominant n-sphere term is of order R-3n+4•RR -1 -2 -4 -5 -)

For ~ij contributions proportional to R • R • R ,R and R are

excluded; the dominant n-sphere contribution is of order R-3n+3 in

this case.

Expl1cit expressions for the various terms in the expansions of

formula. (4.19) - (4.21) can

(3.14) - (3.17) and forming

three-sphe~e contributlons of

in principIe be found, using formulae

the necessary tensor products. Thus the-) TI

order R to ~ij is given by the product

100 -2 -3 R-2 G(l,2s)8f Rik RU ti -ik

G(2s,2s);:U

G(2s,l)-ti

Into this product one then has to lnsert expressions (3.20) ~nd (3.21)

for the corresponding .Q.-tensors. In re£. 10 a11 contributions to theTI TR RR -)

tensora ~ij' ~ij and ~ij up to arder R are listed explicitly.

5[69

1.5 Wall effects; che spherical container

In che preceding sections, \ole assumed Chat che suspended sphe-

rieal partieles \Jere moving in au unhounded fluid, and calculated che

correspondlng mobLlity tensors .•••hich accounted for che hydrodynamic

lnteractions between che spheres. Characteristic oE these interaction

15 thelr very long range which 15 apparent from Che explicltTTexpressions tor che mobility tensors ~ij' i t j.

Te lOlo/estarder ln che expansion In connectors these are given by

TT~lj •

[ ) k ( [ )'- 1 + v ~-- J3 - 4R1ji t j • (5. [)

as f01101ol5 from (4.22) together with (3.14) - (3.16). As a consequenceof chis long range oature I che Lnfluence of boundary walls can be oi

importance even ln cases where che vessel containing che suspension is

very large. We shall therefore discuss in this section an extension of

the scheme developed

includes the effect oC

The 501utlon of

for the evaluation of mobility tensors, that

a spherical wall bounding the suspension15).

the problem of N spheres moving in a viscous

fluid inside a spherical container may be obtalned froID the solut ion

to the problem oí N+l spheres in an unbounded medium studied above. by

observing that the analysis given remains valid 1f one of the sphe-

rlcal boundaries, the container spec1fied by the index 1••0. encloses

the other N spheres (i"'1.2 •.• N) and the viscous fluid. provided the+indueed force FO on the container is ehosen in suen a way that

far (5.2)

where RO is the center of the container and aO its radl'.1s, and that

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the velocity field has, in addition to the extensions (4.7), theextension

o (5.3)

The analysis oE section 3 then leads to the f0110w10g set oi equa-

tions

N1:j'l

1: A(n,m) G F(m)m=l -iD -'-{)'

N1:

j=l(5.4)

with connectors !¡n,m) Ci,j =: O,I,Z ••• N) defined aga!n by the inte-

grals (3.2) with the additional conditions

for i,j l,2 ••• N, i' j (5.5)

for j=1,2 ••• N (5.6)

The particle-particle connectors remato therefore unchanged; the par-

ticle-container connectors are oi a difierent type but can be eval-

uated as well, usiog properties oi integrals ayer Bessel functions.We only give here the form oi the particle container connectors ior

the case that the centers oE the particle and the container coincide.

ROj =: O. Qne has in that case

o • if n ~ m and n ~ m - 2

A(n,m)(R = O)-JO JO

_ (a./ aoln B(n,n)J -

if n m (5.7)

3 a. n4 (n+l)!(2n-l)! !(-2) (1aO

2a.- -1)aO

6(0+1.0+1)- ' if n=m-2

517l

For the general case we refer to reL 15. In eq. (5.7) B(o,o) denotes

the previously defined self-connector, er. (3.9).

Since the velocity Uo and angular velocity Wo of the container

vanish, ane can reduce the sel oí equations (5.4) for the N+l spheres

(particles and container) to a reduced sel of the form (3.1) for the

particles alene, bul now in terros of new connectors ~¡~;:~c., which

incorporate the effects of the spherical container, Le. the hydro-

dyoamical interaction with the container, and are given by

A{n,m) z A{n,m) +-ij;s.c. -ij

¿ A(l,p) 0 A(p,l) • I,J - 1.2••.N .p-l -10 ~J

(5.8)

~ote thal the 'self'-connectors A(n,m) are fiOl diagonal in their-ii;s.c.upper indices: different multipoles in the same sphere couple via the

container wal1.

lt is in terros of the new connectors that the mobi1ity tensors

~ij which are again of the form (4.19) - (4.22) must now be evaluated.

We give the express ion for the translational mobility tensors~ ..1J(omitting from now on the indices TT) for the case that particle i is

concentric with the container, in an expansion to third order in the

parameters a/aO and a/R (a and R are a typica1 partic1e radius and

interparticle distance respectively):

(5.9)

23~+ '2 2 (1 -

aO

- ---------------------------------------

5172

For a single partiele inside the container, 1""j, and the aboye ex-

pression reduces to

• 3(-.!.) ]'0

(5.10)

Explicit expressions fer more general cases can again be found in ref.15.

1.6 The fluid veroeity field

As we shal! see (cE. seco 2.5) ane needs for a proper discus-

slon oE phenomena such as sedimentation an expression fer the velocityfield oE the fluid al a point ; , caused by the motion oE the spheres.Wlthln the linear regime studied ~(;) may be expressed in terms oE the

forces exerted by the fluid on the spheres in the £0110\<710g way

N- l: 5 (;).¡(

j=l -j j(6.1)

We consider here the case of free rotation of the spheres, l.e. the

case that the torques Tj

a11 vanish. The tensors ~j(;) defined aboye

can be derived fram the general expressions for the translational• •mobilities of N+l spheres by putting ~+l "" r and taking the limit

aN+1 ..•.O

lim• .0N+l

1:N+l .(R~Hl"" n , j=1.2 ..• N• J

(6.2)

This formula expresses the fact that the velocity fie1d can be probed. .(.,with the aid oE an infiniteslma11y sma11 sphere located at r ; v rJin (6.1) is the velocity of this test sphere. Since the mobilities areon1ythatthat

defined. forthe tensors

for all pairs, formula (6.2) im?liesare defined for configurations such

j"'l , ••• N

S 173

(6.3)

The paint r 15 [hen indeed situated in che fluid.

To lowest arder in ,he expanslon 01 [he mobil1 t les In connectars

one has for an unbounded suspension2

S .("r) • 1 {(! + ;j;j) -aj - - 1

-J annIR.-;1 IR fW ("J' j- '3 !)) (6.4 )

Jas fo110,,",5 írcm (6.2) and (5.1) for i"N+l (Le. ior i denoting the

test particle) aod j=1,2...N. In (6.4) rj denotes the unit vectorpolnting frcm ; to che center of sphere j.

Note that 1i ane puts 11 = ;, i~1,2 •••Ngiven sphere j, to lowest arder (aod for 1 R

j-;l

In (5.1), ane has for a> 8

j+ 81)

(6.5)

The left hand sirle represents, per unlt of force exerted on sphere J.~

the velocity of a sphere i at pos1tian r with respect to che fluid

velocity at Chat peint in che absence of sphere 1. Note Chat che R-1

contribution to chis relative sphere veloc1ty cancels, but that a longranged R-3 contr1bution rema1ns. We shall return to this observationand the possible implications thereof in scccion 2.5.

The analysis of the fluid velocity field by means of a testparticle completes our discussion of hydrodynamic 1nteract10ns. Weshall no","discuss on the basis of the resules obtained a number oftransport properties oi suspens10ns.

5174

Chapter 2

ON TRANSPORT PROPERTIES IN SUSPENSIONS

2.1 Diffusion

To apprehend the influence oE hydrodynamic interactions on

properties oE suspensions, we shall study In this chapter certaintransport phenomena. We consider again suspensions oE hard spheres oE

common radius a and mass ID. which except for their short-ranged hard

sphere interaction and their hydrodynamic coupling do fiot exert aoy

direct long ranged forces (e.g. electro-magnetic) on each other. The

phenomena we sha11 study here are diffusion and sedimentation. Anothertransport phenomenon. viscosity, w111 fiot be dealt with, but has been

studied using the same methods by Beenakker20), to whose work werefer. As a starting point for our discussion we write down the stan-

dard correlat10n funetion formula for the wave number dependent (long-.R..t.time) diffus10n coefficient D (k) •

k T ~Ol.t '(k) • _B_ f k' < j (k,O)j(k,<) > • k d< •

NG(k) O(1 .1)

Here kB is Boltzmann's constant aod T the temperature of the system.The brackets < ••• > denote ao average over ao equilibrium ensemble of

suspensions io a volume V. while G(k) aod j(k,<) are the staticstructure factor and the wave vector dependent microscopic flux oi thesuspended part1cles respectively,

G(k)N

N-1 ¡:i,j=l

iiHit-R.]<e j '> (1 .2)

(l.3)

The function G(k) which, as D(k). depends on the magnitude oí the wavevector only is the spatial Fourier transform oí the partiele numberdensity correlation funetion; j(k.t) 1s the spatial Fourier transform

S175

oE the local particle flux j{;). Notiee a1so that OJ.t'(k) 15 ex-pressed in terros cE the longitudinal campanent oE J(k) only. since

diffusion 15 a purely longitudinal phenomenon. Consequently we could

have substracted from j(;) in (l.1) aoy divergence free flux, such as

in particular the volume-flux in au incompressible system.Instead oE the diffusion coefficient given in eq. (1.1) ane can

al so define a time dependent quantity D(k, t) by extending the time

integratlon fiot to 10f101ty, but to the time t. As argued by Pusey andTough21). this quantity has a plateau value ter times t such thattB « t « te' Here te 15 a structural relaxation time in which the

configuratlon oE the particles changes appreciably, while-[

tB : m (6n~a) 15 the corre1ation time of the fluctuating ve10eities.-2 -8For typical suspensions te ~ 10 s whi1e tB'" 10 s. The p1ateau va1ue

of D(k,t) defines a short time diffusion eoeffieient which can be

written as

D(k) Eij

k • < (1.4)

Here use has been made of the re1ation

tf d,O

(1.5)

which holds on the intermediate time scale, as f0110ws by writing down

the Langevin equations for the dynamics of the partic1es. In (l.5) the

brackets denote an average for given position vectors R:N. As stated

previously we omit in what fo110ws the indices TT for the trans1a-

tional mobilities.

It may be remarked that in a 1ight scattering experiment both the

short and long time sea1es in the diffusion regime are accessible, so

that both Di.t• and D can be be measured.

In our analysis we shall restrict ourselves to transport pheno-

mena on the short time sca1e which represent a more manageable prob-

lem. Indeed according to eq. (1.4), we only need for the evaluation of

D(k) to perform an average over given configurations of the suspended

Sl76

spheres. Far che evaluation oE OLe. ane would, according ta (1.1),

also have to cake into accaunt che change of che configuratían oE che

pateieles. This would require che 501ut100. far a11 times, oE che

Langevin equations, Le. oE a highly non linear set oi coupled differ-

ential equations. rhe long time regime therefare constitutes a much

more formidable problem - even wiehout coosidering hydrodynamic inter-8ct100s22).

Lec U5 finally consider the self diffusion coefficient Ds oE ane

sphere, say sphere i. in che suspension. In chat case ane has to re-

place in (1.1) i, • •N by G by Gs and Jet) by J.(,), with

• -ik.R ('t)3 (,) i

G l, = u1(.) e (1.6)• s

00 che short time scale chis chen leads to che short time self

diffusion coefficient (eL che reduction oi (1.1) to (1.4»

-1 NN l:j=l

(1.7)

In the last member of (1.7) we have made use of the faet that in theequilibrium ensemble a11 particles are equivalent. and also of theisotropy of the system. Tr denotes the trace of a tensor.

Comparison of (1.7) and (1.4) shows that

Dslim D(k) •k~m

(l.8)

The physical meaning of short time self diffusion can easily be under-

stood by observing that Ds characterizes self diffusion on a timesea le such that the root mean square displacement of a particularparticle remains much smaller than the average distance between par-ticles. lt 1s so to say the diffusion of a part1cle in a cage formedby the surrounding part1cles. The hydrodynamic interactions with otherparticles playa role, but nat yet the direct hard-sphere interactian.

5177

2.2 Diffusion and long-reuge hydrodynamic interactions

We have mentioned that to lowest arder In the expansion in con-

nectors the (translational) mobilities !::..ij' for bj, conta!o terms oí

arder R-1 and R-3 (eL eq. (5.1) oí Ch. 1). Such terms may, when eva-

luatlog transport properties, give r!se to compl1cations, such as the

accurence oí divergent integrals. For selfdiffusion where ooly ~ii

needs to be considered, the aboye long-rauge terms do not contribute.

\ole shall show here that these terms never cause any difficulties In

the evaluatlon oí D(k), not 001y in the ltmit as k~, t.e. fer Ds' buta150 fer arbitrary values oi k.

From (1. 4.22) and (1.4) if £0110w5 that to lowest arder in the

expanslon in cannectors ane has

(2.1)

with the Stokes-Einstein diffusion coefficient

In (2.1) g(r) 15 the pair correlation function,

n g(r) • N-1 E <6(p - Ri)6(p + ~ - R

j» ,

o ij

!(l,l)(~r)no '"' N/V the average density oi spheres, and

mono pele connector fieId defined as

_A(l,I)(~)A(l,I)(R'~)- ij ij r

(2.2)

(2.3)

the monopole-

(2.4)

Note that since Ai~,l) 15 only defined tor Rij> 2a, the connector

field ls a1so only defined ln this range. But as gC~) "" O ter r < 2a

aoy choice ter the continuatian oi !(l,l)C::) tar overlapping spheres

leaves the integral in (2.1) unchanged. Wlth this in mind \ole may de-

fine !(l,l)Cr) fer a11 ~ as (cf. eq. 0.2) of chapter 1)

5178

(2.5)

with

(2.6)

Equation (2.1) may then be written in the form

~~G(k~D(k) _ 1 • nok' jd; eik.r{g(r) _ l}~(l.l)(;).k

o

(2.7)

Performing the integrations in the second term yields

G(k)D(k) _ 1no

(2.8)

sioce

(2.9)

in view oE (2.6).

Thus to lo•••.est arder in the expansion in connectors ooly a con-

vergent integral remains (g(r) .• 1 for r -+ "" ,sufficiently fast).

whlch vanishes as k ..•.al. AH contributions from higher arder terms in

the expansion In connectors, or essentially higher arder terms in ao

expanslon in interparticle distances, yield convergent contributions.

This completes the proof that the long range l/R aud l/R3 hydro-dynamic-interaction terms do fiot give rise to aoy divergencies ofD(k). in particular oE D(k"'O). the collective diffusion coefficient oE

5179

the suspenslon.

2.3 Virial expansion oi the selfdiffusion coefficient; non addltivity

oi hydrodynamic interactions

If ane inserts into the express ion (1. 7) for the selfdiffuslon

coefficient the series (1. 4.22), together with the explicit form oi

the connectors (see seco 1.3). ane can in principIe evaluate Ds as a

power series in no - N/V (a sa-called virlal expanslon). This has beendone by C.W.J. Beenakker and the author up to aud includlng terms Di

second arder in the density19). Up to this arder ooly two- aud three-

body hydrodynamic interactions need to be considered, since the proba-sbil1ty that a given sphere has s nelghbours In oi arder no • Further-

more Qne needs to this arder ooly knowledge oi the hard-sphere pair

distribution g(r) function to first order in no and of the three

sphere distribution function g(R12, R13, R23) to lowest order. Thus

one must insert into the relevant integrals

o for r < 2a

g(r) - 1 + ~ {8 - 12r/4. + 4(r/4a)3¡ for 2. ( r ( 4••

1 for r > 4a

(3.1)

1 elsewhere

In eq. (3.1) $ ls the volume fraction of suspended spheres

5180

(3.3)

Usiog the aboye expressions for the distribution functions. it wasfound that

D /D = 1 - 1.73 ~ + 0.88 ~2 +e(~3).s o

(3.4)

Ooly two-body hydrodynamic lnteraction contribute to the well-known

term oi arder <\l, and are therefore the ooly ones to contribute at

sufficiently 10\01 densities. At higher densities however, the many-

sphere hydrodynamic interactions may Dot be neglected: two-sphere con-tributions aIone would have led to a value oi -0.93 <\:12for the term oi

2 2arder <\l • lnstead oE the value oi +0.88 <\l in eq. (3.4). This 111u5-

trates dramatically the non-addltiv1ty oi hydrodynamic lnteractions.

We should mentian here that In evaluatlog the coefficients in (3.4).

we have in the expansion oE the mobili ty in inverse power oE inter-

sphere distances negIected, both for the two- and three-body case,terms of order R-B and higher. It can be shown however that the termsnegIected contribute at most a few percent.

2.5. SeIfdiffusion in a concentrated suspension

It is quite cIear from the resuIts of the preceding section thatin a concentrated suspension one fuIIy has to take into account themany-body hydrodynamic interactions between an arbitrary number ofspheres. A vidal expansion however 15 not appropriate at high den-sities. But it is possible to resum algebraicaIIy contributions due tohydrodynamic interactions between an arbitrary number of spheres19).

We consider again the selfdiffusion coefficient Ds• Insertingioto (1.7) formula (4.19) of Ch. 1 oue has

Dsn-o

(4.1)

S181

We shall first rewrite this equation In a more convenient formo For

this purpose we introduce, as we did in 2 for the mODopale case.

connector fields

!:.(n.m)(.r' _ .r) _ A(n,m)(.R • -;, -!') Hj-lj lj - r, •

and local denslties

(4.2)

(4.3)

We no,"" define the fo11o\ol10g matrix elements Cef. che correspondlng

relevant definitlons in seco 1.4)

{d3-I}~.~ - {tl3-I}~tó(r' -r)r,r'

lO )~'~ ' {b) }~im ó(r' - r) .r.r'

With these deflnitions we may rewrlte eq. (4.1) In che form

(4.4)

(4.5)

(4.6 )

(4.7)

DsD-o

1 --1

no (4.8)

where n, A. Q and rB -1 must no••••be interpreted as operators w1th

matrlxelements (4.4) - (4.7). To write (4.1) In che form (4.8) we havealso made use oE translatlonal lnvarlance Di che average.We shall now renonnal1ze che connectors. For chis purpose we first of

a11 write che local density as

(4.9)

The average of 6n(~) 18 zero by definitlon, whl1e

SI82

(4.10)

with g(r) the pair correlation funet1an defined in (2.3).We also make use of the operator identity, valid for ao arbitrary

operator A

(4.11)

If \ole substitue (4.9) into (4.8) aud make use of (4.11) we obta!n for

Ds the alternatlve expresaion

{nA (I-no

(4.12)

where the renormalized matrlx oi connectors v4n 15 defined aso

(4.13)

The renormalizatlon accounts for the fact that fluctuations in thedenslty of the spheres interact hydrodynamically via the suapension,

rather than the pure fluid: the renormalized connector A n incorpo-o

rates already hydrodynamlc interactiona between ao arbitray number of

uncorrelated sphereso Thls corresponda to the algebraic resummation oflnteractlons mentloned aboye. At tbis point we stress aga!n that bare

connector fields are only defioed for l~'-~I>2a. lo expression (4.8)

the N-sphere distributioo fuoctioo vanishes whenever l~'-tI < 2a,

so that as stated withio a slightly different context in sec.2.2, aoy

contiouatioo of the bare coonector field for overlapping spheres

leaves that expression unchanged. The renormallzed connector.;tn ohowever does depend on the choice of the continuation, even though the

full expression (4.12) is agaln invariant for that choice.

Generalizlng the contlnuation for connector fields chosen in f 2to all multlpoles, one can explicitly evaluate JtÍ If one thenno

5183

expands the express ion between curly brackets in (4.12) in powers of

So. ane obtalos ao expansion oE Ds in correlationa of density fluc-

tuatioos of higher aud higher arder. Already the first twa non vanish-iog orders of this fluctuatian expansion glve numerical values fer Dg

which agree well with experimental results fer volume fractiona$ < 0.3. It should be stressed here that the choice of the

continuatlon oi the connector field for I~'-~I< 2a will stronglyinfluence the convergence of the series obtained In the fluctuatian

expansiono An extension such that connector fleIds vanish fer1 ~ '- r 1 < 2a may be expected to glve the best convergence but gl ves

rige to compl1cations fer the evaluatlon of d-n • For more details ono

diffusion io concentrated suspensions see refereoces 19 en 23.

2.5 Sedimentation

As we have seen (sec. 2.2) the long range hydrodynamic interactionterms of order R-1 and R-3 do not contribute to selfdiffusion and giverise to convergent integrals in the evaluation of collective diffu-sion. If one calculates on the other hand the velocity of sedimen-tation in an unbounded medium, this quantity diverges, a fact which issometimes referred to as the Smoluchowski paradox. Pyun and Fixman24)have shown the way to avoid the difficulty caused by the l/R diver-

volumethe backflowaccount

respect to the meanwithlnby considering sedimentation

this way one indirectlygenceflow. takes intocaused by container w~lls. But even then the R-3 terro still gives riseto a conditonally convergent integral, and poses, as noted by Bur-gers25), the problem of a possible dependence of the sedimentationvelocity onBatchelor26)

the shape of the vessel containing the suspension.was able to assign a definite value to the integral in

question, using an argument based on general considerations of aphysical nature - valid for the unbounded system.

Ultimately the difficulties mentioned should be resolved by adirect and explicit evaluation of the influence of container walls on

S184

the mobi11tles of sedimentlng partlcles. Such a calculatlon has beencarrled out fer two geometries. First for the case of aplane wal127)aud then aIso fer the case of a spherical container. The latter case

has been reviewed in sec. 1.5. For a discussion of sedimentationQne also has to consider the fluid flow caused by the motioo oi

suspended spheres (ef. seco 1.6).Que 15 then, using the results found as outlined in the two

sectlons mentioned, in a positian to evaluate the mean particle velo-+ +city vpaud mean fluid velocity v

fof a homogeneous distribution of

identical spheres, 81 aj B, sedimenting inside a sphericalcontalne(28). These two quantities, calculated at the center of the

container and In the limit that lts radius aO tends to infinity, maybe written as conditional averages

+ 11m < E JOij I RiO • O> • F (5.1 )vp a 0..,,<:0 j

+ 11m <E ~p.RoJIRjO > a for a11 j> • F (5.2)vf.a +- jO

Here < 1 RiO •. O> denotes an average over those configurationsfor which RiO •• O, while < .... [RjO > a for a11 j> denotes anaverage over configurations for which no suspended sphere overlaps thecenter of the container; F is the gravitational force (corrected forbouyancy on each of the particles).

To linear order in the volume fraction ~ of suspended spheres+ +calculation of vp and vf on the basis of eqs. (5.1) and (5.2) and eqs.

(5.8) and (6.2) of Ch. 1 yields

(5.3)

(5.4)

We can now also determine the average volume velocity given by+ + ~v • ~v + (1 - ~)vf . From eqs. (5.3) and (5.4) it follows thatv p

5185

(5.5)

Since, because oE incompressibl1ity, the volume flux through aoyclosed surface IDl,.\st vanlsh, this result, namely that there 15 a 000-

vanishing volume ve loe! ty at the center, implies the exlstence oí a

vortex oí convective flow In the spherlcal container.Flnally one may evaluate the mean partlcle veloclty ••••ah respect

to the average volume velocity. The result 15

~ - ~ - {I - 6.55 ~ + O(~2)) F .p v . (5.6)

This 15 the result found tar this quantlty by Batchelor for ao un-

bounded system. lt 15 a1so what 18 found for sedimentation perpendicu-lar to and towards aplane ••••a11, In whlch case v vanishes, In the

vllroit oE ao infinitely dlstant ••••aI129).We therefore come to the fol10...,10gconcluslons:

The average local veloc1ty oE a sedlmenting particle io a homo-geneous suspension depends on the shape of the container, howeverfar the container walls. But this shape dependence disappears forthe sedlmentatlon veloclty wlth respect to the average volumeveloclty.

These results 111ustrate once more the essentlal role played by hydro-dynamlc many-body 10teractlons. They show that, for sedlmentation, the"three-body" hydrodynamlc interactlon of two-particles and thecontaIner can In fact never be omltted from conslderation, not evenfor sufflciently dllute suspenslons.

The dlscussion oí wall effects and of the non-addltlvlty ofhydrodynamic coupllngs in the previous section thus underscores therelevance and usefulness of the scheme developed and summarized in Ch.1 for the evaluation oí many-sphere hydrodynamic lnteractlons.

5186

References

1. M. Smoluchowski, Bu11. loto Acad. Polonaise SeL Lett. lA (1911)28.

2. H. Faxén, Arkiv. Mat. Astron. Fys. 19A (1925) N 13.3. H. Dahl, appendlx of ref. 2.4. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics

(Noordhoff, Leiden, 1973).5. J .M. Burgers, Prac. Kan. Ned. Acad. Wet. (Amsterdam) 43 (1940)

425, 646; 44 (1941) 1045, un.6. C.K. Batchelor, J. Fluid Mech. 74 (1976) l.7. B.U. Felderhof, Physica 89A (19~) 373.8. s.u. Felderhof, J. Phys--:-AU (1978) 929; R.B. Jones, Physica 97A

(1979) U3. -9. P.N. Pusey and R.J.A. Tough, in: Dynamlc Light-scattering and

Velocimetry, R. Pecara ed. (Plenum, New York).10. P. Mazur and W. van Saarloos, Physica lISA (1982) 21.11. G.J. Kynch, J. Fluid Mech. 5 (1959) 19~12. M. Muthukumar, J. ehem. Phy;;. 77 (1982) 959.13. T. Yoshizaki and H. Yamakawa, 1:- Chem. Phys. 73 (1980) 578.14. P. Mazur and D. Bedeaux, Physica 76 (1974)D5; P. Hazur and G.

van der Zwan, Physica 92A (1978), 41f3.15. C.W.J. Beenakker and P~azur, Physica 131A (1985) 311.16. P. Mazur, Can. Journ. Phys. 63 (1985) 24'; Helvetica Physica Acta,

to be published 1985. -17. S. Hess and W. Kijhler, Formeln zur Tensor-Rechnung (Palm und Enke,

Erlangen, 1980).18. P. Mazur and A.J. Weisenborn. Physica 123A (1984) 209.19. C.W.J. Beenakker and P. Mazur, Physica 120A (1983) 398.20. C.W.J. Beenakker, Physica Aj Thesis. Leiden, 1984.21. P.N. Pusey and R.J.A. Tough, J. Phys. A15 (1982) 1291.22. See e.g. the review article by W.Hess---añdR. Klein, Adv. Phys. ~

(1983) 173.23. C.W.J. Beenakker and p. Mazur, Phys. Lett. 98A (1983) 22; Physica

126A (1984) 349.24. C.W. Pyun and M. Fixman, J. Chem. Phys. 41 (1964) 973.25. J.M. Burgers, Prac. Kon. Ned. Acad. Wet-:-44 (1941) 1045. 1177j 45

(1942) 9, 126. -26. G.K. Batchelor, J. Fluid Mech. 52 (1972) 245.27. C.W.J. Beenakker, W. van Saarloos and P. Mazur, Physica 127A

(1984) 451.28. C.W.J. Beenakker and P. Mazur, Phys. Fluids (1985) to be pub-

lished.29. C.W.J. Beenakker and P. Mazur, Phys. Fluids ~ (1985) 767.