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Particle interaction in diluteslowly sedimenting systems

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LOUGHBOROUGH

UNIVERSITY OF TECHNOLOGY

LIBRARY

AUTHOR

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I

PARTICLE INTERACTION IN

DILUTE SLO'NLY SEDU:ENTING SYSTEMS

by

STEPHEN KENNETH COWLA..\1, M. Sc., D. r. S.

A Doctorcl ~hesis submitted in partial ,fulfilment of the requirements

for the award of

Doctor of Philosophy of the

Loughborough Uni veI'si t;y of Technolog,y

May 1976

Supervisor: B. Scarlett, M.Sc. Department of Chemical Engineering

@ by Stephen Kenneth Cowlam, 1976

.... __ .. -------------

For Liz

. ". ,"

AcknowledGements

The work reported in this thesis was conducted

while the author was in receipt of a S.R.C. research

studentship. 'rhe author wishes to thank those who

assis.ted him during the period over which the work was

conducted:

Prof. D.C. Freshwater, Director of Research

l{.r. B. Scarlett, supervisor, f'or his advice

I,;embers of' the Department of Chemical Engineering, LoughborOll[h University of Technology

Frie:lds and colleagues for lively discussion t:nd constructive criticism

lEy parents for their continued encouragement

Pond finally, Liz, for her patience and typing skill.

iii

2.

.13.

4.

5.

- - -- -- -- ---~--. -,-,-----~--------.

Contents

Nomenclature

Introduction

Literature Survey 2.1 Sedimentetion of e Dilute Suspension 2.2 Single Sphere Anc1yses 2.3 The Motion of Two Spheres 2.4 Viscosity Effects 2.5 Concentration Effects 2.6 Statisticel Hydrodynamics of Two Phase

Dispersions 2.7 Models Involving Formation of Clusters

of Spheres 2.8 Models Involvin; Flow Past Arraj's of Spheres 2.9 Other Mathematical MOdels for Sedimentation 2.10 Interpretation of the Literature with

respect to this Work

Expp.rimental Methods 3.1 Introduction 3.2 Details of Particles and Liquid

3.2.1 Particle Selection Bnd Size Measurement 3.2.2 Selection of Sedimcmting Uedium 3.2.3 Menufe.cture of Marker Particles

3.3 Details of Experimental r-ig 3.4 Photography 'for Particle I.Ieasurements

3.4.1 Two DimensionEl Perticle rr,easuremer,ts 3.4.2 7hree Dimensional Partic le ],;eaGuremer.ts

3.5 Fluid Flow lv!easurement s

Experimental Results 4.1 Introduction 4.2 Series 1 of Experiments 4.3 Seri.es 2 of Experiments 4.4 Series 3 of Experiments 4.5 Series 4 of Experiments

Discussion 5.1 ~ualitative Remerks on the Sedimentation

Curves 5.1.1 Series 1 of Experiments 5.1.2 Series 2 end 3 of Experiments 5.1.3 Series 4 of Experiments _

, 5.1.4 The Statistical Accuracy of Results 5.2 Description of the Sedimentation Phenomenon

'5.3 Inferences for Velocity Measurement in Sedimentation Equipment, 5.3.1 Measurement of Velocities 5.3.2 Theory of AverabeVelocity Calculations 5.3.3 Interpretation of other Workers'

Results 5.3.4 Averege Velocities with Time as

the Measured Variable

iv

vi

,1

4 '4

27 50 74 89

109

118 131 142

158

166 ,166 /173 -173 -175 "176 '178 /179 179, _189 /197

204 204 206 225 242 263

264

264 264 268 273 277 282

294 294 295

301

303

v

6. Conclusions and Suggestions for Further Work 310 6.1 Conclusions 310 6.2 Suggestions for Further W~rk 312

References

Appendix 1 Computer Programs for Solid Data

Appendix 2 Principles of Laser Doppler Anemometry

Appendix 3 Computer Proer8m for Fluid Data

314

320

./.323

333

Parameter

a

A

b

c

'C D

c = d

d

D

D =

E =

f g

h

vi

Nomenclature

Description Dimensions

particle radius L

distance between two sphere surfaces L almost in contact

vectors determining unit cell of L s partic le' array

vector area L2

sphere radius (in two sphere system); L distance from particle centre to L vessel axis in Poiseuille flow

reciprocsl lattice vectors (see 0") fP) a(3)) - '- '-volume concentration of solids

drag coefficient

frictional drag coefficient

mean volume concentration of solids

pressure drag coefficient

coupling tensor

particle diameter

mean diameter of particle

vessel diameter drag force

coupling tensor

rate of strain tensor

frictional force resisting particle motion

hydrodynar.lic force vector

. acceleration due to gravity

half di~tance between two spheres

L

L

L

~

1

L

m

m'

n

n

p

Ap

P(C~x)

force/unit mass vector acting on particle

unit vectors in coordinate system (0123)

idemfactor

constant average number of particles/cell

Hadamard-Rybczins~i correction factor

effective volume concentration of two solid species

resistance tensor

distance between particle centres

torque vector

particle mess

fluid displaced particle mass

number of spheres contained in vessel at volume concentration, c

unit normal vector

cell number, number of particles in volumes, Vend A respectively

pressure field

Poisson distribution of k particles

pressure drop

conditional probability that a confiE,uration of. N particle centres

. are found in the range, dC"

P (t),pl(z)Legendre and first associated n n Legendre polynomials

Q

r

R

Re

volumetric flow rate logarithmic distribution function of velocities

position of sphere centre relative to x

-0

vessel radius

Reynolds number

vii

M

L

L

s

8 p

8, ,8"

8

t

T w

T

u

iJ

tTK&

u' k

closed surface

logarithmic variance

particle surface

fluid-solid drag for mass of two solid species

directed surface erea vector

time

wall zhear stress (boundary layer theory)

torque vector

particle velocity

Johne Bnd Koglin average particle velocity

arithmetic mean perticle velocity

arithmetic mean velocity measured with ti;ne as the vE-riable

arithmetic mean velocity measured with time as the variable where T".t

Brenner and Happel wall Corrected velocity

avera[,e Kaye and Boardman particle veloci ty

average fluid velocity

geometric mean particle velocity

harmonic mean pqrticle velocity

Johne and Koglin average particle velocity

Kaye end Boardllicn Dverabe particle velocity

relative velocity of a cell of particles

Ladenburg's corrected velocity for wall and bottom effect

mean fluid velOCity in a parabolic flow profile

viii

T

ML-~-2

LT-~

LT- 1

LT _1

LT -.

LT- 1

. LT -1

LT -1

-.--~~-- ~~----------,....-----------------------

v

v

x

z

fluid velobity at vessel axis in a parabolic flow profile

mean particle velocity

veloci ty relative to Stokes' fOr median size particle, U

Stokes' velocity Us

velocity

instantaneous velocity vector LT- 1

instantaneous velocity vector of LT- 1 particle with centre at ~o and in set eN mean particle velocity vector LT- 1

Koglin velocities in upper and lower LT-1 test sections of sedimentation vessel

velocity vector field LT- 1

velocity at a point, x, in the set spheres, eN, supposinL particle at to be replaced by fluid

volume flui.d return flow veloci ty

of x

x

volume of cell of k particles

velocity contribution to sphere at by all other spheres in set eN at positions x

-0

probebility that nA particles are found within a volume, A

image systems of influence of all spheres in set eN at positions x on a sphere at x .

-0

particle diameter

diameter of cell containing k particles

vector position of perticle centre

axial directions in (Ox1 X Z X 5 ) coordinate system

position vector

LT

L.3 LT- 1

L3

LT- 1

L

L

L

L

L

ix

'le. r &

Brenner End l-leppel ratio of particle distance to vessel axis to cylinder radius

Euler's constant

rotational velocity of a particle

boundary layer thickness

8~) Direc's delta function of x

~ porosity, or voidage

1

1. Introduction

Sediffientation is a phenomenon well characterized in

nature by the fallout of solids and liquids from the atmos-

phere, and the settling out of sediment in the oceans and

inland waterways. It has been used on an industrial scale

in many applications notably in the separation of solids

and liquids in grevi ty settling tanks and in centrifugal

devices for the rapid dewatering of slurries. In the labor-

atory it. is adapted to techniques for characterizing fine

particles.

Observation of the phenomenon may be complicated by

electrostatic or magnetic fields, thermal gradients, eddy

currents, complicated rheological properties of the sedi-

menting fluid or irregular particle shape and density, and

particle concentration.

The effects of gravity 011 a sedimenting system may be

investigated by eliminating other force fields and operat-

ing at constant temperature in a closed vessel. A Newtonian J-'

fluid yields the simplest fluid mechanics, as does the use

of isotropic spherical particles.

This thesis is, therefore, concerned with just such a

system. Earliest investigations into such a two phase

system were concerned with either very low concentrations

or single particles where mathematical ana.lysis wa.s possible

yielding a solution of the equations of motion typified by

Sto~es' law and its modifications for different conditions,

or high particle concentrations in the hindered settling

region typified by observation of bulk properties and analy-

sis from continuity considerations.

2

There exists a region betw~en these two at concen-

trations roughly betwe;?n 0.05% !:cnd 2% by volume of solids

where it was observed that the average particle velocity

did not decline steadily from the Stokes velocity with

increasing concentration, but was in fact enhanced to a

maximum value at about 1% by volume of solids, thereafter

declining with further increases in concentration until

the hindered settling region was

were originally obtained by Kaye

firmed later by Johne(54,55) and

entered. Such results

and Boardman(57) and con-

K l ' (58,59) Th' og ~n ~s

thesis is concerned vlith gravity sedimentation within this

concentration range.

The next section reviews the literature pertaining to

previous work done on this subJect covering many aspects

such as concentration effects, mOdelling as flow past arreys,

and exact solutions for two spheres. It presents the find-

ings and interpretations due to each particular author and

concludes by outlining the relationship of the literature

to this work. These concluding remarks indicate the reasons

for setting up experimental investigations in the manner

described in the experimental section. Previous workers

have measured vertical velocities of particles by measuring

the time taken to pass between fixed points, but this work

was carried out using time lapse photography enabling large

numbers of results to be obtained from any region in the

experimental vessel, demanding the use of semi-automatic

data handlin6 teChniques. Both vertical and horizontal

particle translations were measured. A laser Doppler

3

anemometer was used to obtain dcta for the interstitial

fluid motion.

The discussion analyses ,the results of the experiments

and presents some new facts relating to the reliability and

interpretation of the results of sedimentation tests both

on laboratory and industrial scales. Also given is a

qualitative description of the phenomenon of sedimentation

at these concentrations obtained from the analysis of

experimental results, and thereafter a more quantitative

description based on a force balance over a small volume

of the suspension .

Finally, the author's conclusions from the work

presented in this thesis are given and suggestions made

for further work on particle interaction in dilute

sedimenting systems.

/ ,.'

I I

4

2. Literature Survey

2.1 Sedimentation o~ a Dilute Suspension

The phenomenon o~ sedimentation concerns the motion

o~ particles in a surrounding medium, in this case the

motion o~ small solid particles in a liquid, settling under

the in~luence o~ the ~orce o~ grayity. The motion o~ a

single particle due to gravity was ~ormalized mathematic-

ally by Newton's second law.

2.1.1

where m and m' are the masses o~ the particle and ~luid

displaced by the particle, respectively, u, the particle's

velocity, g, the acceleration due to gravity and ~, the

~rictional force resisting the particle's motion. The

simplest example o~ the motion o~ a particle is the axi-

symmetrical ~low o~ a solid sphere through an unbounded . ,

fluid otherwise at rest and th~s was ~irst treated by

Stokes and solved by him in terms of the stream function,

yielding an expression for the frictional drag on a sphere

of radius, a, in a fluid of viscosity,r'.

2.1. 2

I~ the particle and fluid have densities r:. . f' respectively eq. 2.1.1 becomes

5

2.1.3

An approximate solution of eq. 2.1.3 is obtained when

du : 0, since during gravity settling 99% of the terminal dt settling velocity is seen to be reached very quickly. This

terminal settling velocity is termed the Stokes velocity.

When the inertia term is set to zero, rearranging eq. 2.1.3

Us ': 2o.z{fs-f)~ et"

This solution to the second law of motion has certain

limitations which were summarized by Smoluchowski(8?).

Strictly speaking eq. 2.1.4 is only applicable as the

Reynolds number approaches zero, althoue,h it may be applied

with only }O% error up to a Reynolds number of 0.2. The

lower limit of applicability is reached when the size of

the particle becomes significant with respect to the mean

free path of the molecules,.}. , since slip may occur

between molecules. The Cunningham .correction to Stokes' /

law takes this into account.

2.1.5

When a sphere falls in a containing vessel Stokes' law

must be modified to take into account the wall and bottom

effects of the vessel. For a cylindrical container

Ladenburg(65) gave the necessary corrections. For wall

effect he gave

6

2.1.6

1 + :2.4(~)

and for bottom effect

Us 2.1.7 1t1.1(~)

where d and D are respectively the particle and vessel

diameters. The constant 2.4 in eq. 2.1.6 was corrected in

1921 by Faxeh to 2.1.

These early attempts at solution of the equations of

fluid mechanics provide essential information concerning

the motion of a single particle both in an infinite fluid

. and a finite container, but practical situations involving

a single sphere rarely occur.

Many studies of the sedimentation of particles have

been made, many of them concerning the motion of small

spherical particles in an incompressible Newtonian fluid

under the influence of gravity since such a system is both

easy to observe experimentally and more amenable to the

application of the equations of fluid mechanics.

The usual experimental techniques under the conditions

stated above correspond to the conditions for solving the

continuity equation for incompressible flow

v.'t = 0 2.1.8

and the creeping motion equations

2.1.9

I

I

. i 1 !

7

However, analytical solutions for these equations are only

. available at present for simple particle shapes, such as

spheres, ellipsoids, discs and rods and only for up to two

particles settling in proximity in the fluid. Computer

techniques are available for solving for the motions of a

number of particles in proximity but these are long and

slow since a large number of boundary conditions for the

dispersion must be satisfied simultaneously.

It is found that Stokes' law is obeyed by particles

in a dispersion only up to very low concentrations.

Beyond this limit deviations from Stokes' law are noticed.

The remainder of this section describes the findings of

other workers concerned with sedimentation at low particle

concentrations. The further sections of this literature

survey consider firstly the motion of sinble particles in

a fluid and then the development of theoretical consider-

ations for two particles in proximity showing how velocities

can deviate from Stokes' law giving rise to a distribution

of particle velocities and showing_how a horizontal compon- t ent of particle velocity arises. The following two sections

deal with attempts to describe the chanbing properties of a

dispersion as modifications of velocity and viscosity as

functions of particle concentrations. .The remaining sec-

tions deal with various mathematical ,models to account for

deviations from stokes' law.

A paper by Batchelor(6) summarizes the points for con-

sideration in two-phase mechanics:

1. The random location of discrete elements of a

dispersed phase is an essential feature and probability

8

methods are required to analyse and describe mechanical

properties of the system. A number of different averaging

procedures may be used and their use must be considered

carefully. For instance, for a suspension of solid part-

icles between two rigid planes in steady relative motion

which is statistically homogeneous, except near the bound-

aries it may be shown that the ensemble - averaged stress at a point in the suspension, the observable average stress

over one of the boundaries and the average of the stress

over a plane surface parallel to the. boundaries which cuts

through fluid and solid alike, are all equal, and are not

equal to theaverae of the stress over the fluid portion

of the suspension. As another example consider gravitat-

ional sedimentation such that the Reynolds number is so

small that momentum changes are negligible. The difference

between the fluid pressure avera6ed over the horizontal

bottom and the free surface pressure is equal to the average

weight of a vertical column of unit cross-section containing

the mixture. Some authors have sUpposed that the average

excess pressure gradient in the liquid is (f,.,.-r)9 where Fm is the density of the mixture. However, integration of ~p

gives the magnitude of the excess pressure as

for a dispersion of low solids concentration.

2. Statistical properties of flowing mixtures, such

as average volume fraction of one component,average relative

volume-flux velocity, average stress, usually vary with ~os

ition. Often, however, the distance over which properties

vary considerably is quite large compared to microscopic

9

considerations. It may be possible to find an intermediate

length between the microscopic and macroscopic over which

averaging may be done.

3. The principle of local relative equilibrium may

be exploited. The two components of a mixture are in

approximate relative equilibrium if the characteristic

time for change of the relative velocity is large compared

with other times characteristic of the local relative motion

(such as time for viscous diffusion of vorticity over a

distance comparable with the microscopic structure length)

and the local average relative velocity of the two compon-

emts may be estimated by equating to zero the local forces

tending to make one component move relative to the other.

An element may be accelerating but the point is that under

certain conditions the equations for motion of one component

relative to accelerating axes moving with the element reduce

to the steady motion form.

The limitations to Stokes' law have already been dis-

cussed. It has often been concluded by many authors that /

providing the concentration of the suspension is kept low

es.ch particle falls as it would in an infinite fluid.. An

upper concentration limit of 2% by volume has been suggested

by Orr and Dallavalle(76) but re~earch, notably by Boardman,

suggests that particle interaction above volume concentrat-

ion 0.05% gives the limit to Stokes' law. As the

inter-particle distance is decreased by increasing the

solids concentration a point is reached where hindered

settling takes place. All particles, regardless of size or

shape, seem to settle at the same rate leaving s" liquid-

10

suspension interface. Hindered settling is observed to

take place at about 5% volume concentration. Smoluchowski

showed theoretical~ that when two equisettling spherical

particles move through a fluid, separated only by a few

diameters, the terminal velocity of the pair exceeds Stokes

velocity for either particle individual~. Hall showed

experimental~ that the settling velocity of two equal

spheres 2.6 diameters apart, equivalent to a volume con-

centration of 3% is 20% higher than the Stokes velocity.

Kaye and Boardman(57) attempted to extend Hall's work to

suspensions by dropping groups of spheres into liquid.

With more than four spheres they noticed that the members

of the cluster failed to maintain their positions with

respect to each other and took on a complex rotational

motion. In the case of four spheres released with their

centres in a horizontal plane and surfaces touching, the

spheres situated themselves diagonally opposite one another

and rolled around the line of centres of the second pair

until they touched, whereupon the .second pair performed a

similar movement around the line of centres of the first

pair. Moreover the velocity of the group was much greater

than that ofa single isolated sphere. This shows that

cluster formation in a suspension can result in settling

velocities much higher than those predicted by Stokes' law.

Kaye and Boardman continued by measuring the settling

velocity, Uc ' of 850fID marker particles in a suspension of

850pm median size glass particles in liquid paraffin.

Fig. 2.1.1 shows their results where Us is the Stokes velocity.

.... . :;2.0r-----------""1

"""J -2,

I t, ">-- .-~

O,ot 0.1

vol",,:!e 1.0 10 100

e.ol,(..tt1h~o..,'O,' (/0)

Fig. 2.1.1 Sedimentation of 850fUIl median size particles.

11

A steep rise occurs at about 0.1% corresponding to a part-

icle spacing of approximately 8 diameters between centres

. Uc is the average velocity timed at each concentration.

At 0.25% the curve reaches a peak where the average settling

velocity is about 50% greater than Us and above 3% a bound-

ary was noticed forming between the suspension and the fluid.

Kaye and Boardman attributed the change in settling velocity

to the formation of clusters. They assumed that velocities

measured as slower than Stokes velocity were due to particles

being caught in a return flow current. Coefficients of var-

iation were found and are shown in Fig. 2.1.2. There is a

steady increase in variation up to 3% followed by a rapid

decrease.

Fig. 2.1.1 shows the changing nature of the settling

process. At concentrations less than 0.05% particles settle

as though in an infinite fluid except for occasional inter-

action due to random movement of one particle which takes

in the flow field of another.particle. Here there is no

significant change from the Stokes velocity.

12

'50 .--_______ ..,-_...,

100

(%)

Fig. 2.l~2 Sedimentation of 850fm'median size particles.

With increasing concentration viscous interaction becomes

much more apparent and the average settling velocity inc-

~eases "toO'. 3% solids' at which approximate point velocity

increase is beginning to be checked by return flow. The

coefficient of variation continues to rise despite a slight

. reduction in settling velocity and reaches a maximum value

at 2-3% indicating the formation of large clusters. At

concentrations above 4% effects of return flow become pre-

dominant and instead of localized return flow it diffuses

through a uniform cloud of particles until at 10% the /'

settling characteristics of dense suspensions are established.

Kaye and Boardman carried out further experiments with

the same 850~m glass marker particles in suspensions with

solids containing solids of median sizes 400fm and lOOfID.

Had the results been similar to those quoted above, then

deviations of the same order of magnitude would have been

expected due to the ,volume of particles in the suspension.

But, if the effect were due to cluster formation the magni-

tude of the interaction would have diminished with increasing

13

size ratio. Tney found the latter to be the case. This is

shown in Fig. 2.1.3. Kaye and Boardman tried to snow that

sedimentation can be divided into four distinct zones:

1. Free settling zone at low concentrations.

2. Region of viscous interaction in which particles

settle faster than their Stokes velocity.

3. An unstable region where' clusters form end return flow is irre~ular and localized.

-""""' 1.5.,---.-.:.... ___ -= __ ---, :lIS ~ I.S,.... _________ --, -f ~ 0.5

o~ ____ ~~ ____ ~ ______ ~

0.0< 0.1 1.0 10

VOlumL. co.,c.",,~rC\~:o .... (%) M 40~--______________ ~ __ -,

o '7"; So -~ 20 c~

.~.~ to I;i, :; 400},,,,, ~> O~--~--__ ~ __ ~

0.01 0.1 1.0 10 Vo 10n'"l(' C.Oo'\CC" .rd he., ('''10)

.:z:-

. ~ I"'OJ-------~ "> aJ 0.1; ..,.

:..:: ~ ..

Fig. 2.1.3 Sedimentation of 850pm marker particles in suspensions of median size 400~m and lOOfm.

1001"""

4. A well-defined region at higher concentrations

where hindered settling occurs.

The work of Kaye and Boardman was continued, notably

by JOhne(54,55) and by Koglin(58,59). Johne and Koglin

14

confirmed that the velocity of sedimentation at low con-

c~ntrations was greater than Us' but whereas Kaye and

Boardman found the maximum velocity was only 60% higher

than U , .Tohne found it to be about twice this figure. s . Errors may have arisen in Keye's and Boardman's work,

according to Koglin(5S). He suggests that the spread of

particle size used was not in good agreement with the

median sizes given, and secondly points out that Kaye and

Boardman used as average velocity Uc ' the sum of all the

distances divided by the sum of all the times, whereas

JOhne(54,55) averaged the sum of all the path-time quotients.

Kaye's and Boardman's velocities were, therefore, slightly

too low. Koglin shows that the lower maximum speed of fall

found by Kaye and Boardman cannot be attributed to differ-

ences in the position of the distribution of sizes of spheres

in suspension with respect to the test sphere since their

particle size distribution differs less from the size of the

test sphere than does Koglin's.

Kaye and Boardman did not consider wall effect as

significant, but its importance was' shown by Woodward(9S).

Work by Koglin shows the remarkable effect of wall effect

at values of d, ratio of particle diameter to column diameter, D

of 0.01, 0.005Si 0.0031. This is shown in Fig. 2.1.4.

Kaye and Boardman took measurements only when their

test spheres were not near the cylinder wall and this could

explain the differences between the values obtained by Johne

and Kaye and Boardman. Koglin asserts his theory of wall

effect by plotting D against Koglin's average velocity U d

15

as IT for vertical plane parallel walls and a cylindrical Us

wall as in Fig. 2.1.5.

-" ..2 .,

2.

> 1.

!J i ~

o 1"'O--~"----10"'--;~:-----1....10---;2-----10-:':-:;1--'=-=~1 . Volume.. conc.e ... fr",h'o" (~/o)

Fig. 2.1.4 Wall effect on settling velocity.

..l:--g 2 ., '>

a) > 1t1 ~

0.01 0.005 (I_OOSiS 2 r-----r----,-----.---. 0

rn

1 100

vut

2.00 o cl

Fig. 2.1.5 Mean rate of fall plotted against D for different vessel geometries. d

16

Kog1in uses, instead of Ladenburg's correction for wall

effect, a corrected velocity UBH due to Brenner and Happel.

1+ f(~) 4 o

2.1.10

where ~ is the ratio of the distance of the particle from

the axis of the cylinder to that.of the radius of the cylin-

der. The function f(f} is an infinite series of integrals

for which there is no exact solution. Brenner and Happel

give approximate solutions:

f{f :: 0.15 1- f'>

for ~- 0

from which they produce graphicall,y' an approximate solution

for the entire range O~f$l. This is shown in Fig. 2.1.6

4,0

O,S 0.1

17

consecutive distances of equal length and found that the

par"ticles increased in velocity on settling. He then

plotted the velocity in the upper test section, U1, against

the velocity in the lower test section, U2 , also showing

the straight line between the origin and the mean value.

This is shown in Fig. 2.1.7. There is a correlation between

the speeds of fall in the two sections, as would be expected.

Pairs of values, "fast-fast" (top'right from mean value) and

"slow-slow" (bottom left from mean value) occur markedly

more often than "fast-slow" and "slow-fast" pairs. He

suggests that the mean cluster size increases during sedi-

mentation, and that wall effect has a greater significance

than given by Kaye and Boardman.

" 5

the number of particles in each cell to follow a Poisson

distribution with k particles in each cell.

~(k) = k

2.1.11

18

Denoting rrF , the average fluid velocity, Uk' the relative velocity of a cell of particles, U~, the velocity of a cell

of particles.

U ." U~ - UF k:" '" 2.1.12

and the average velocity is given by

a =:: f. fi

2

f 1-_-

o 10""" 10.... 10' 3 10z . 10" 1 voio .... e. e'::l'lCet\+r"h'otl (to)

Fig. 2.1.8 Plot of velocity against concentration and average number of particles per cell.

Koglin(61) suggests a logarithmic distribution

function, Q, of velocities U Us

where Ug is the geometric

and S2 is the logarithmic

mean velocity

variance 52 =

2.1.16

19

Koglin's values for Ug and 82 against concentration, c,

are shown in Fig. 2.1.9. 4

Fig. 2.1. 9

/ ./ .-..,..

2

1 -0.5

0,4

5" 0.3

0.2.

V --.

-.') \ 1'0)

Geometric mean velocity and variance concentration.

against

20

With the two definitions of averabe velocity due to Johne,

U3 end to Keye and Boardman, Uk5 , Koglin relates them to

his geometric mean velocity.

2.1.17

2.1.18

Koglin(6l) continued the work of Johne utilizing

eq. 2.1.15 end denoting densities of a cluster, a solid

particle and the suspending fluid by ft, f's 'f respectively.

f'~ -r = 2.1.19 Denoting the settling velocity of a cluster by Uk , Koglin

obtains

/'

and from eq. 2.1.19 it is obvious that

UI:2: :: Us

From Ledenburg's correction

2.1.20

2.1.21

2.1. 22

,where U is the velocity of a single particle. Therefore

21

k.x. - 2.1 k:x.. 2.1. 23 :x.k: D

and from a graph of relative velocity against k for various

values of!?, at .Q __ co :x. :x.

Ut! = lA

r{f - 2.1 h D

2

1.5

1

2.

Ut! Us

3 4.

'. Thus

2.1.24

nUMber of po.,.t;d,s if) dustor, k.

Fig. 2.1.10 Relative velocity vs. k for various values of D x

Batchelor(4) considers a system of particles of radius,

a, in a Newtonian fluid of viscosi~y, fA' such that the mean flux across a stationary plane surface gives rise to a zero

mean velocity of material. The velocity, U, of a partic-

ular particle in the dispersion differs from the Stokes

velocity, gs' due to the hydrodynamic interaction among

the various particles and thus U-Us is a random quantity

with a non-zero mean which depends on the particle concen-

tration. He determines the mean velocity of a particle by

2.1. 25

for e sphere with centre

where P(C,,!/ x)dC N is the

at 3So whose conditional

velocity is U(x CN) - ---0

probability that a

22

configuration of N sphere centres being found in the range

dCN about CN where CN refers to a realization of the set of

position vectors, ~, of the centres of N spheres. Calcula-

tions for one or two spheres are feasible but not for more

so Batchelor requires to consider somehow a group of one or

two spheres by reduction of eq. 2:1.25. Integrating this

equation over just one sphere in the configuration, CN ,

on the grounds that the chance of two spheres being simul-

taneously close enough to ~o to influence the velocity of

the test sphere is of order 0 2 and so negligible is invalid-,

ated by the slowness of the decrease to zero of the influence

of one falling sphere on U(~) with increasing distance from

~. The procedure is to look for a quantity whose mean is

exactly known from an overall condition whose value at x -0

has the same long-range dependence on the presence of a sphere

at ~- as the velocity of the test sphere, and once found

the difference between IT and the mean of this quantity can /

be expressed as an integral like eq. 2.1.25 and can then

legitimately be reduced to an integration over the location

of just one sphere in the configuration CN and evaluated

explicitly.

Owing to the asymptotic form of the dependence of

~(~,CN) on the configuration, CN' as the distance of the

spheres in CN that are nearest to ~o becomes large, the

desired quantity becomes obvious. When the spheres of CN

are well away from ~ the sphere may be regarded as immer'sed

in e fluid which, in the absence of that sphere, would have

23

approximately uniform velocity over a region of linear

dimensions 2a. Then U(Xo,CN) would be approximately equal

to the sum of U and that uniform velocity. Thus Batchelor -0

looked at the relationship between U(~o,CN) and the velocity

distribution that would exist in the dispersion if the test

sphere were replaced by fluid of viscosity,/", without

change in the configuration, C~. 'This velocity at the point

~ is denoted by v(x,CN) and will in general be non-uniform

on the spherical surface centred on x with radius a. The -0

resultant vector of the distribution of forces is ~'iT

24

motion of all other spheres than the test sphere, but it is

incomplete because forces acting at the surface of the test

sphere need to be accompanied by image systems in the spher-

ical boundaries of all other spheres to ensure no-slip

conditions are satisfied at those boundaries. A rigid

sphere of radius, a, at a distance, r, from the test sphere

will require an additional translational velocity which in

turn induces a change in the velocity distribution near the

test sphere. All other spheres in the dispersion will have

a similar effect on the test sphere and this additional

velocity, W, representing the image system gives rise to a

new velocity expression

2.1. 28

The problem arises with the non-convergence of ii = ii'-v" where

v'" ~!f Y(~OJ eN) p(C"4hc)dSN 2.1.29

!I S! a2.[V\~(~ICN)l, p(c",l~o)dcN t>.J 0 'Ix.= ~ - ,

V" " 2.1.30

because Ivl behaves as a at a distance, r, from one falling r

sphere and~Iv'~J behaves as Q', the latter giving a r 3

decrease to zero as r --co which is too slow for convergence a

of the integral. Batchelor, therefore, integrates eq. 2.1.29

and eq. 2.1.30 with the aid of exact mean values involving

all the spheres in the conficuration, eN' eventually yielding

25

2.1. 31

and the final expression for the mean velocity of a sphere

in the dispersion is

2.1.33

where V, and Vn are given by eq. 2.1.31 and eq. 2.1.32 respectively and W is given by

2.1.34

These expressions can be evaluated with a knowledge of the

probability density of the location of one sphere relative

to a second sphere in a statistically homogeneous dispers-

ion and the flow field due to two sphere s falling in an infinite fluid. For a dilute suspension the initial con-dition is that

p(~t.d ~) { n ;f ( ~ 20.1 :0 irr

yielding

V" =

SUbstitution of

~ 1 cUs 2-

from rearrangement of eq. 2.1.33 finally yields the

expression"

2.1.40

26

2.1. 36

2.1.37

2.1.38

2.1.39

This equation is correct only to the order of c and suggests

that the average particle velocity declines with increasing

concentration, unlike the experimental findings of Kaye and

Boardman, Johne, and Koglin, described earlier.

27

2.2 Single Sphere Analyses

Hjelmfelt and Mockros(49) considered the forces on a

sphere undergoing arbitrary rectilinear acceleration in a

viscous fluid using the equation first presented by Basset.

if d3 f cl.u

AdV cl\:

where

28

2.2.3

where Re can be regarded as the Stokes terminal Reynolds

number or the nondimensional Stokes velocity. The solution

for (}. f ~ is f

v( t.):: H (-r) [ Vt. t ~(Vt -VI)) eo(2eyfc (~{'t) - 0( (V!. - '10) l~rF~ {(>'-R )] . " . (c(-(:3) d.-{,>

. . )\ 0(2('1:-1:.) ) ~1.('t.1:') r I (..,. .\)] ... 1(0(j3) H ~C- LL (e ~rFc(0/11:1:i. - f e urc \I-> ~ ,,- t.y

w-p . . . / 2.2.4

/'

where 0/ J:>. = 3 r3:i:~5 - 'Of,; 1 I 2(fr.~j) t

. (' 2.

where Vo is the initial velocity, which must be steady state

so that history integral is zero for t

V(-r.) - Vc v..,-Vo

2.2.5

29

For ~ > ~ Cl and [5 become complex, 0(= IHt.V = Z J f3 :: l..\ -,i." :: z:

'" (z) - ,_z' [1. :w: f e "or]

giving

-= H('t)[ 1 - e",,((Vt~u)~) + ~ IwqV~tL\)'\ft)l T ~ H(t-tJI ~w((VTW.)Vt/t(.)

Vt,-Vo L .

~ IJ&t~u)V 1:- Li )1 : V . / . The remaining condition is when Ps ~ 5 ,when C( = f3 such that

f '3 r ,

'" H(t:)ft-1Jifi t (321:-1)e.lfo~e(rC (41t)1 l 'VTf

+ 1f>1::: U(1.-'Ll)1 ~2(l..'t:

30

size and in the same fluid will have terminal velocities

of the same magnitude if their density ratios (Ps) and p

( ~ ) I satisfy f

2.2.7

1.0

0.'5

V-Vc o.~

VI;,-I/" 0.4

0.2

0 0.01 0.1 1.0 10 1CO

't::: tov Fig. 2.2.1 . . .0

2

Theoretlcal veloclty-tlme curves falling in a viscous fluid.

for spheres

More common analyses of spheres in unsteady motion are

not based upon eq. 2.2.1 but on

./

~3 (t'5tf) ~~ + 3/fd f"Y U :: 0./) Tfl';) 2.2.8

or in non-dimensional form

2.2.9

2.2.10

and neglecting the history and added mass terms the solution

eq. 2.2.10 becomes

"

31

= 1 -Q.xp (-1'n~ ) . (~)

2.2.11

A comparison of solutions showing exact and approximate

solutions is shown for different values of ~ in Fig. 2.2.2 fi

f.O

O.'b

V 0.10

Vt,

0.4

0.2

0 0

hi~o"l 1I

32

where Co is a friction factor. He indicates that from

experience when a solid sphere is greater than lmm in

diameter or Reynolds number is greater than 800-1000, CD

essentially becomes constant. Thus rearranging

U ::: 2.2.13

Another special condition arises for particles smaller than

0.05mm or for Reynolds number less than 3, in which case

CD = 24 or ~and eq. 2.2.13 is written as ~ ~ .

2.2.14

the Stokes equation for free settling. Swanson developed

an equation for any value of Reynolds number which coincided

with Stokes' and Newton's laws of settling at their respec-

tive areas of validity and also held in the intermediate

regi ons. As shown in Fig.. 2. 2.3 the St oke s' regi on and

Newton's law region have velocity-diameter relationships

which are described by equations of the form U = k,d 2 and respectively~

Newtoi')!; Iow region

1-----I-~f------,"-:~-

33

Swanson gives a modified Newton's law expression

neglecting Co

2.2.15

where dll- is given by L- and the substitution of this pURe:1

value into eq. 2.2.15 is valid for the rebion of inter-

section of eq . 2.2.15 with the Stekes region. To make

the equation valid throubhout the term. d - J!.... was pUr.e~l

considered. The Newton's law term in d2 was replaced by

and solution gives

2.2.16

but for Re < 1, U becomes negative and Swanson arbitrarily

changed the sign of ~ pLl" .. ~ 1

from negative to positive. Thus

u = L d )A /4,9c.1 (f5-e) \ c.I .. ~ .~ 3('

. p Uee :1

2.2.17

Swanson found by experimentation that eq . 2.2.17 Vias of the

correct form in that the term in brackets was the ratio of

particle diameter to particle diameter plus another term.

which he regarded as the thickness of a fluid layer. From.

boundary layer theory he obtained

x ;. d coscp (1 + lzaA Iv ) V~

. '-.'.-." -~,

2.2.18

)(

Fig. 2.2.4 Boundary layer round a'translating sphere.

where ~ is the kinematic viscosity and U = Us since the flow in the boundary layer is laminar. Thus

~ k?aP ) dpV 4gd(p':.-p) '3f

2.2.19

Defining the Newton's law velocity as UN =./4;Jd(e.,-e) 'V 3(>

the velocity of the particle is expressed as

U = 2.2.20

34

--As diameter increases the term in brackets approaches unity

and Newton's law is approached. As diameter decreases

Stokes' law is approached

. Abraham(2) determined a functional dependence of drag

coefficient for a sphere on Reynolds number, by means of

dimensional analysis such that the drag force, D, on a

sphere is in the form

2.2.21

35

\

giving the relationship

. where" is an undetermined exponent. If drag is independ-

ent of viscosity then -1 =

~~~~~~~~~~~~------------........ 36

Interpreting D as the drag on a sphere of radius, r, moving

at a speed, U, the drag coefficient for the sphere is

C -.: Co (1 ~ 80 ,)Z (Re)'

where eq. 2.2.24 is used.

2.2.26

The parameters Co and 80 must

be obtained through empirical fitting, and Abraham notes

further that for Re 1, c.- COS~ 'which is the functional Ji?c"

dependence of Stokes law c.; 24 for small Re. ge

Brenner(lO,ll) derived the Stokes resistance of an

arbitrary particle considering the instantaneous translat-

ional velocity, Uo ' and instantaneous angular ve10ci ty, W ,

of the particle by solution of the linearized Navier-Stokes

equations and continuity equations for an arbitrary body in

a Newtonian fluid at very low Reynolds number (Re 1)

2 f \l y = Vp 2.2.27

'V.y " 0 2.2.28 ./

where p refers to the dynamic pressure rather than the total

pressure. The effects of hydrostatic pressure are temporar-

ily neglected

. If the particle of arbitrary shape moves in an unbounded

fluid at rest at infinity, the net effect of stresses set

up at its surface is equivalent to a force, F, and a couple,

L; these are the effects of the fluid on the particle.

Calculations require estimates of the rapidity with which

velocity and stress fields are attenuated at great distances

from the particle. Considering a translational motion,

the-particle experiences a hydrodynamic force

2.2.29

37

on its surface, sp' where integration is carried out over

an area an element of which, dS, _is directed into the fluid.

The stress tensor, ~, is, for-an incompressible Newtonian

fluid,

1T " 2.2.30

In the region of applicability of eq. 2.2.28 the relation

V. 1T -= 0 2.2.31

also applies at each point in the fluid. If eq. 2.2.31 be

multiplied by a volume element of fluid,dV, and integrated

over any fluid volume,V,it may be converted by Gauss'

--divergence theorem into a surface integral over the closed

surface, S, containing the volume, V, in its interior.

J ViJ eN "

= 1 d~.-U-s

: 0 2.2.32

V is chosen here to consist of the fluid bounded internally

by the particle surface and externally by a spherical fluid

surface, cr, of very large radius containing the particle-at its centre. Thus,

38

2.2.33

Comparing eq. 2.2.29, eq. 2.2.32 and eq. 2.2.33, one obtains

2.2.34

The solutions of eq. 2.2.27 and eq. 2.2.28 yield for a

point force

'f. '= [" - ,-2 v(r. V) 1 (bTTfr 241fr r

2.2.35

p ~ _1 ([.V)1 4lT r

2.2.36

where r is measured from the singularity. A similar develop-

ment for the fluid at great distances from a rotating

particle follows from

.2.2.37

yielding /

v -- - 2.2.38

If (y', 11') and ( Y.", 1!') be the velocity and stress fields corresponding to any two motions conforming to the

eq. 2.2.27 and 2.2.28, the Lorentz reciprocal theorem allows

that

~ J d?IJ' VI( 2.2.39 '5

39

in which s is a closed surface boundin6 any fluid volume V.

If the two distinct fluid motions are those induced by the

steady translation of a particle through the unbounded

fluid with velocities, g' and U" respectively, as fluid

adheres to the rigid particle it satisfies the boundary

condition

" I, , v=u v-=l..l' - -.) - On 2.2.40

In consequence of the primed motion.the particle experiences

.a force

t'=Jd~.lr' 2.2.41 sI'

with a similar expression for the double primed motion.

SUbstitution of eq. 2.2.40 into eq. 2.2.39 leads to

r"u" . f '5 11' " .:.. _ _ -t- 0 _. = ~ ~ -IS

t't!, + J d. r~ '!.' C1

2.2.42

As the integrands lJ:yll and "!I'l. 'f.' are at most of O( r- 3 ) . as r_= but the area 0' is only of 0(,.2), the surface

integrals of eq . 2.2.42 vanish. Thus,

~/, Elf 2.2.43

implying that F is a linear vector function of U. This may

be shown by considering the body to move with successive

velocities U, ' Uz and (U1 +U 2 ) and denoting F = f(U)

40

= and'

Adding and rearranging

As the velocities are arbitrary, the term in square brackets

must be zero. Thus

This is the definition of a linear vector function and the

force is related to the velocity by a second rank tensor,

~, and further dimensional arguments require that the force

is directly proportional to the viscosity, and general form

is, therefore,

2.2.44

K is termed the resistance tensor and is an intrinsic = property of the body depending only on its external config-

uration. In particular it is independent of orientation

and speed of the body and of the fluid properties. It is

also sy=etric as is shown below. From eq. 2.2.43

L.I:IC.U" -. =:::. - = U " /C U' - .= -

--If!$ denotes the transpose of ~ U~ 1:::. U" : ~/. (\!". ~) = U".K.U' - -= - =

41

Therefore

~:~.u' "" n~ f U. k:".U -= 2.2.45

The general form for ~ is given as

'" ~'!I 1::1, > ~',"2 k"l > ~,(~ l(,; ~ ~z {, 1::"21 ~ !t ~l I::"Z/ t ~l i3 K"Z3 + !;l"l(3' + ~3i.tK.32 t ~2.~3l(33 2.2.46

where i 1 , i 2 and i3 represent unit vectors in any Cartesian

coordinate system, and we have already shown that by

. symmetry K'2 = K21 , KI.3 = K31 , and Kz; '" K3z. It is a property of second rank tensors that rotation of the axes to give new

unit vectors i;, iiand ia may be done such that

2.2.47

and for a spherical particle of radius, a, being an iso-

tropic case, it represents a debenerate case in which the

principal stresses in all directions are eigenvectors.

From Stokes' law one has Kt :: .K2. = K:, = 6 n a, whence

k: "=' I01fo. -- '" E '::

which correctly indicates that the force on a particle is

parallel to the velocity vector.

Eq. 2.2.44 has an analogue for the torque, T, required - , to maintain the steady rotati on of an arbitrary body in an

unbounded fluid at some angular velocity, w The no-slip

condition for a rotating particle is

42

2.2.48

and the torque is given by

T = J r x ( ~ d~ ) 2.2.49 sr

Arguments similar to those used in developing the trans-

lation tensor may be applied to yield an expression relating

the torque to the angular velocity via a positive-definite,

symmetric tensor, the rotation tensor, J1

T = 2.2.50

These were Brenner's earliest considerations for

particle motion. Now, however, we.consider an arbitrary

particle which exhibits anisotropy, that is, the particle

motions are considered from a point which corresponds to

the centre of hydrodynamic stress. For such a particle,

whereas before the centre of gravity and the centre of

hydrodynamic stress have been considered as coincident,

they are not in the general case coincident and in such a

case, measured from the centre of hydrodynamic stress, the

particle makes a contribution to the force exerted due to

its translational and rotational motions and to the torque

from its translational and rotational motions. Thus, the

force on the particle is a consequence of the translational

and rotational contributions, Ko and Eo respectively.

= f -t F -0 -0. 2.2.51

where

f -; I dS.1fo _0 5r

2.2.52

- ir =

F ." d~.~o -C> 2.2.53

In a similar manner the hydrodynamic torques due to the

translational and rotational motions are given by

2.2.54

where

10 = [ ro x (d~ .1[0 ) Sf

2.2.55

To = 1 :Cox(dS."Uo) Sf'

2.2.56

43

For the translational motion applying the same analysis as

in eq. '2.2.39 to eq. 2.2.50, but applying the boundary

condition

v -= Uo ;- Wxr. !)n SP} Y. -- 0 as r - 00

one arrives at the expressions

~ 0 = -f '5. l:Jo To ~ -f ~ . 'do

/

2.2.57

2.2.58

2.2.59

and for the rotational motion using the, boundary conditions

of eq. 2.2.57

F = -poQ.w -0 I - -1"! f"I -" = -f '::,. ~

2.2.60

2.2.61

and it may be shown that

~

where ~ or are known as the coupling tensor. Thus

2.2.62

2.2.63

2.2.64

44

Thus it can be seen that the force and torque on any body

can be described accurately by eq. 2.2.63 and eq. 2.2.64

and in the particular case of a sphere w = 0 and C is = dependent on 0 and 0 = 0 the problem reduces to that given in eq. 2.2.44.

Returning to practical solutions of problems involving

one sphere, Dennis and Walker(30) obtained a semi-analytical

solution solving equations by numerical methods for the

steady flow past a sphere at low and moderate Reynolds numbers.

They solved the simultaneous equations of stream function

and vortici ty in terms of the coordinates (e, e), where

E = log(r), a is the sphere radius, and (r,e) are polar co-a

ordinates in a plane through the axis of the sphere. They

expanded the equations in series of Legendre functions of

arguinent, z = cos e wi thfunctional coefficients in the variable E. This gave two sets of second-order ordinary

differential equations which were truncated and solved

numerically.

eo -2.s. 'Of 5;" e OG

, 2.2.65

45

The expression for vorticity is

2.2.66

The equations satisfied by t and 5 are

2.2.67

and

;: (ee.)eErVrO~ tVeO~ -V,~ -V,,'l" tde] 2 l Qt: ae v:J

The boundary conditions to be satisfied are

Expansions

i -~. ;:

01 = 0 Qc

at

os

c=O

. -.- 00

for 1- and ~ are given as

E ro {1 e.2 .& (CC:) Pn ~) d I:.

." z /

n~ jn (c) P~ Cz.)

2.2.68

2.2.69

2.2.70

where Pn(t) and P~(z) are, respectively, Legendre and first

associated Legendre polynomials of order, n. Over the range

considered Dennis and Walker used as the drag coefficient

They also calculated the separate con-

friction and pressure drag whose coefficients

they gave respectively as

If .

-.1.f ~ (o,e) sj~Ze de /(e.

(>

2.2.71

46

1T .

er == - -L.-{ p (o,e) SL'YlZG dG pUtt>

. 0 .

2.2.72

They found that their numerical values for drag on a sphere

compared with the results of other workers up to the

Reynolds number 0.1.

Bowen and Masliyah(Sl performed similar work to Dennis

and Walker for a number of different axisy=etric particles,

by truncating a series solution for the stream function

obtaining results accurate to 5%.

Ockendon and Evans(71) made a numerical solution of

the Navier-Stokes equations, without loss of the inertia

terms, and the continuity equation for a sphere moving

uniformly with velocity, U, through an infinite incompress-

ible fluid at low Reynolds number and obtained the value

of the coefficient for O(Re2 ) in the expansion in terms of

Reynolds number for the correction to Stokes drag. They

used the method of matched asymptotic expansions with com-

plex Fourier transforms yielding the value of the coefficient

as 0.1879. / /

2.2.73

V P d 1(94) .. erma, andeyan Trlpatll utlllzed a three-

dimensional integral momentum equation .for bodies of

revolution to determine the boundary layer thickness, 6 , . round a sphere. Their calculated values vary by about 50%

from experimental values. The integral momentum is given

by

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

47

2.2.74

where Tw is the wall shear stress and other variables are

shown in Fig. 2.2.5. The expression for 6 is biven as

cS =

- 0,0000212 5:'" 1,6 + O. '6:; 2.2.75

Fig. 2.2.5 Flow past sphere due to Verma, Pandey & Tripathi.

Vleinburger(96) considered the variational properties

of steady fall in Stokes flow for a class of bodies whose

centres of buoyancy and mass are not coincident. In part-

icular he shows that the quasi-steady falling motion of a

48

particle converges to a steady one. Using expressions for

force and torque given in eq . 2.2.63 and eq. 2.2.64 and the

boundary conditions of eq. 2.2.57, the quasi-steady fall of

the particle obeys the equations of motion

fA (~.~ t ~.~) '" Mj \ fA (~.~ j lo.,.~) = m~xrc 2.2.76

dJ = CuxSj dt.

Solving the first two equations and substituting into the

third of eq. 2.2.76 gives

2.2.77

Taking e particular downward direction ~o,there will be a

steady fall in the direction bo provided that

where r C is the position of the centre of mass. The object

is to find a value of~ to make motion stable. If ~ = 0 where

fe:. = .:....!.. 9 xl fx!' dS t 0(3 1Y'I1~12 - Sf' - .

J dS " TYlj Sr

Thus Weinburger shows that

2.2.79

----------------------------------------49

8 and that g converges exponentially to ~ if ~Ijo!> "f

where 5 and I are constants involved in the rearrangement of eq. 2.2.77 - 2.2.79 noting that 1~\2= l~ol2.. For suf-ficiently large ~ (that is, when the centre of mass is

sufficiently low) the steady motion with = ~o is the limit of falling motions except for the unstable motion

with 13. = -.B.o'

./

50

2.3 The Motion of Two Spheres

The first analytical solution of the Navier-Stokes

equation for two spheres was due to Stimson and Jeffery(91).

They considered two spheres, equal or unequal in size,

moving with small constant velocities parallel to their

line of centres. Their approach was to solve the equations

in cylindrical polar coordinates in terms of the stream

function. They obtained a correction factor, k, to Stokes'

law where k is the ratio of the force necessary to maintain

the motion of either sphere in the presence of the other to

the force which would be necessary to maintain its motion

with the same velocity if the other sphere were at an

infinite distance .

2.3.1

where

The factor, k, is obtained by transforming the coordinates

into bipolar cylindrical coordinates,j ,~ , where

:x. - 0.1 :;,; I) 11 cosh] - ccs II

x.+ '

51

where~ and ~ are particular values of 3. The solution is obtained in the form of a Legendre poly~omial giving rise

to the series solution for k given in eq. 2.3.2.

Table 2.3.1 shows typical values of k

t>( centre distance k d~ameter

.

0.5 1.128 0.663 1.0 1.543 0.702 1.5 2.352 . 0.768 2.0 3.762 0.836 2.5 6.132 0.892 3.0 10.068 0.931 co 00 1.000

Table 2.3.1 Sphere size, distance to diameter ratio, and correction factor for two spheres falling parallel to their line of centres, after Stimson and Jeffery.

Happel and Pfeffer(45) carried out experiments at /

/

Reynolds numbers of 0.25 and 0.5 obtaining results within

2% and 3% of the values predicted by Stimson's and Jeffery's

solution at a Reynolds number of zero.

Several workers have studied either theoretically or

experimentally the motion of two spheres in a viscous medium

at Reynolds numbers within the Stokes rebime. Goldman, Co:;{

and Brenner(41) initially studied the solution of equations

of motion Wavier-Stokes and continuity) for two spheres of

equal size separated by a distance, 2h, between their centres.

perpendicular to the direction of gravity. Each has a

translational velocity, U, and an angular velocity. w.

Lack of translational motion in the z direction is only

valid in the Stokes regime.

.. 21,--~..,

u u

Fig. 2.3.1 Two spheres falling side by side.

52

The boundary conditions appropriate to this problem

are

(on sphere I) 2.3.3

(on sphere II) 2.3.4

Id -- QJ / 2.3.5

As indicated in Brenner's earlier papers(lO,ll) the velocities

are separated into translation contributions due to sphere I,

y~ due to sphere II, Yi and the fluid yl and corresponding r V r r

rotat~onal components. YI' ll' 'f. Thus

2.3.7

53

The equation of motion can then be solved in cylindrical

polar coordinates (p, ~, z) for pressure and velocity.

t av; = 0 01.

where V2 == ?/ t 1 a t 1 Ol 0(''>' f Gp p" 04/-

with the boundary conditions

on sphere I

t U='j vlfp = I"- - ~:5;(l~ v[.p "'- on sphere II Vh -: /

They derive a strict mathematical solution,

2.3.8

2.3.9

2.3.10

2.3.11

involving

changes of variable to ease solution, the use of Legendre

polynomials with a resulting series solution. An approx-

imate solution obtained,as a ser~es by the method of

reflections determining the non-dimensional force and

torque acting on sphere I due to two translating spheres.

The approximate forces and torques are at the most 1.41%

and 9.5% respectively different from the exact solutions.

The approximate solutions are

4~5 (~\4 250 2hJ

54

2.3.12

2.3.13

assuming that the higher order terms are expressible as a

geometric series. Similar solutions are obtained for the

rotational motion of the spheres solving eq. 2.3.8 and

. 2.3.9 wi th the boundary condi ti ons

r VIp

r

VIp VIZ

:: W (z-h) cos rfi 1 = -w(z-h)s;" ~ -= -

55

an arbitrary angle of attack, G, e$ in Fig. 2.3.2 using

the subscripts, 1\ and 1. for parallel and perpendicular to the line of centres velocities are calculated

Un = F~ 01Tfa \ Fiji

U1 -:::. rl 0 1Tf'

56

spheres were, the faster each settled. The drng coefficient

for the upper sphere was smaller than that for the lower

sphere, concluding that the validity of the Stokes approx-

imation is smaller than commonly considered. A plot of the

dimensionless velocity of each sphere, U1 , U~ vs. the dimen-Us Us

sionless distance between centres, 2h is shown in Fig. 2.3.3. a

"-___ J Uy

Fig. 2.3.2 Two spheres settling at an angle, El

1.~

1.5

4.3

1.2.

1.1 21-. Z 3 " 5

57

Davis(29) provides a series solution with computation

of coefficients for two unequal spheres translating and

rotating in a direction perpendicular to their line of

centres. The solution is essentially the same as that of

Goldman, Cox and Brenner(41).

Majumdar and O'Neill(68) made theoretical studies of

the Stokes resistance of two equal spheres falling in con-o

tact in El linear shear field. They define the velocity, U,

as the sum of an approach velocity of the undisturbed flow,

!!o and a function of the velocity gradient, 'i/ u

2.3.25

The presence of the spheres disturbs the fluid velocity

field such that it is now

v == U t V 2.3.26

where v is the velocity field for the particles which

satisfy the Navier-Stokes and continuity equations.

2 Vp = f' 'i/ '!.. 2.3.27 'iI.y -:: 0 2.3.28

with the boundary conditions, v = -U on, the sphere surfaces and )yl-o as r~+z2 __ oo where rand z are cylindrical polar coordinates. In spherical polar coordinates (f' ')(. , (?J) :the

spherical resolutes of V are

2.3.29 CO A

E V;'Yls;n fJ'ICl "': 1

eo 00 '" ,1"1 !

V't< = L- V';, C.OS l'YI e ~ 2: V y, s'y, mEl Yn-::C> ",=, 2.3.30

(0 co , L! '" l V"" . V(J - ve cos ..... 6 ~ ~ 5>0 ",,$ m:e ",=1 2.3.31

CO co AM f' ~ '" E -= Q COS me + Q S;t') 1'Y)e

f 11'1>0 ' ",=I 2.3.32

m m . '" where V , , Q are funct~ons of rand N only. Thus

the force and torque in Cartesian components become

~ ,

Fil = 1ff CI~i [ ~r(V; s;.,:t t V~ CJ)s~ - V;) -Q~;~~};;i\ X-dN [" . t:>

T~z

" 2lfrr r "[

59

centres. For the axisymmetric velocity field, vo

where g is the rate of strain dyadic for undisturbed flow and ~ and p represent the changing orientations of two sets of Cartesian axes, one fixed in the fluid and the other mov-

ing with the spheres. The velocity-pressure pairs (vZ,pZ)

and (y3,p3) give no contribution to the mechanical action

. of fluid on the spheres since terms, sin mO and cos m9 in

eq. 2.3.29 - 2.3.32 only. take values m = 0, m = 1. The only asymmetric field pairs which contribute to the forces

and torques are (y"p1), (v"',p4) and (y'\pl!o). Dividing vi

into polar coordinates (r,e,z)

f 2 2 o,rf = r s Q ~>h'1 0(" 05 f' ccs e 2.3.40 v1 r :: ss;n2 0(CD!of!' ~ rQ to ~ (1. t1)1 cos e 2.3.41 Vi e = f Ssin2o( cos~( Y-- 'f)",;"e 2.3.42 V~= s:"ze CO!o~( zQ - 4 cos e , 2.3.43

where X., 1', 1> and Q are func tions of rand z only sati sfy-ing L;i :: L~+ :: L~Q = L~t.:: 0 where

L~/I." 2l ~.! 0 - rf'? t cl Clr~ I 01 ":f'.i 0,22

Solutions for (yA,pA) and (yB,pB) are found in a similar

way as that for (yi,p1) except the functions of ~ and ~ in

eq. 2.3.40 - 2.3.43 are replaced by Scos2.O(cos~ for (vA,pA)

and ScosD/sin~ for (vB,pS).

60

Batchelor and Green(5) calculated theoretically the

hydrodynamic interaction of two small spheres of different

sizes in a linear flow field. They considered two rigid

spheres of radii, a and b, on which no external force or

couple acts. The flow field in the absence of the two

spheres has velocity, U(x,t), a linear function of position,

which can/be characterized instantaneously by a rate of

strain tensor

= 1 (0 !:lj t Ol:!,) z ()x" 0:>: j 2.3.44

'and a rigid-body rotation with angular velocity

1 'iJxu 2: - 2.3.45

Again the Navier-Stokes and continuity equations are solved

with boundary condition

Ui(:Z) ~ E~. :Lj 1- E,"k (U' xI.. "J J as F-I~ro 2.3.46

and no-slip condition I~-~ol = a and !x-x -r\= - -0- b where x -0 is the inst'antaneous position of the centre of the sphere

with radius, a, and ~otr is that of the sphere of radius, b.

The para.meters to be determined are V, the translational

velocity of the sphere of radius, b, to that of the other

sphere, the angular velocities of the spheres of radii, a

r" and b, denoted by and r respectively and also their resp-ective force dipoles, S! . and S~I .

~J ~J The force dipole, S .. ,

~J

for a rigid body with surface, Ao ' and unit outward normal,

n, is defined as

~---~---------------

61

5

2.3.51 for,cases when the spheres are far apart, close

together, in a simple shearing motion, equal spheres in

62

axisymmetric flow, in the presence of a plane wall, b = 1 and b-o.

a

aWacholder and Sather(95) obtained the hydrodynamic

forces and couples acting on two spheres in slow motion

as functions of their relative configuration. They solved

the equations of motion and plotted the components of relat-

ive velocity against the relative trajectories for different

ratios of the two particle

polar coordinates (r,e,~),

si zes a = a,. Using spherical az

the nondimensional velocity of

'sphere 1 relative .tosphere 2 is given by

1!1Z = Uf'

:::

\!1 -t,!2 Ur

~ U1l (:,0. ,r) ccs6 t ~ V,Z (r,c. ,r) s;n6

where the centre of sphere l.moves in the plane

with velocity UtI.' I is the net density (Po-f).

dimensionless time by /

,/

(F.-f')

2.3.52

of .!.r and le Denoting a

2.3.53

the trajectory of particle 1 relative to particle 2 is

given by the equations

'" - \.A1Z ( T,c.,1) COS e

1 V12 (:,0.)1) S;n e r

} 2.3.54 .

Typical results at a = 0.5 are given in Fig. 2.3.4

0.0

0,4

0.2.

- 0,2.

-0,4

O.S

0,/0

0.4

O,l

-0.2..

-0,4

. ____ ---- I~o

_-------- 1:0 -----_--~~---1 =-_------1.5 __ --------- 2.1011';'3

..

53

Fig. 2.3.4 Relative velocity functions against relative trajectory at a = 0.5, after Wacholder and Sather.

When two spheres are almost in contact solutions of

the equations of motion already quoted become inaccurate'.

Cox(25) developed a lubrication theory for two spheres

almost in contact. This involved solution of the usual

64

equations of motion. The surfaces, Wand W', of the two

sphoeres are shown in Fie;. 2.3.5, where a is the distance

between the two surfaces measured along the x~ axis of a

Cartesian system of axes (x1,x2,x3), with origin 0 on W. ,

Surface W may be written, for small values of r = (r, _X~)2

2.3.55

where Ri and R z are the principal radii of curvature of

the surface Vi at O

.3

0' A

o t

w'

-

~=------2.

1

/ ,/

Fig. 2.3.5 The surfaces N, I'l', after Cox.

If the surface, W, is symmetric, the value of x3 remains

unchanged if x1 and x2. are replaced by -x1 and -x2 respec-

tively. In that case

65

, o (r4) .x.3 = - (X~ ) - ( .x.~ ) + 2.3.56

2~1 21':z.

If a new coordinate system is chosen (x~,xi,x~) with v,

origin 0' on W' and the x3 axis coincident with the x~

axis and x~,x~ are chosen to lie in the directions of the

principal curvatures of surface 'Ii', the surface W' may be 1

wri tten, for small values of I'll- = (x~'z _X~2 )2

x:"" 3 2.3.57

where 81 and 82 are the principal radii of curvature at the

surface ",V, at 0'. Again if surface VI' is symmetric, replac-

ing x~ and xi by -x~ and -x~ respectively

2.3.58

If 4 is the angle betwee~ the x1 and x; axes (in the positive sense in this direction)

T xJ.. s:'o'\ ~ "

:x."* ' -:: .x., (,OS t/>

l 1 It- '- -:;y S;Yl f; + x'z. C()sp 2.3.59 x 2 xlt -:. X; - 0-'3 then surface iV' may be rewritten as

2.3.60

G6

The velocity field v = (V1 ,v2 'v3 ) is then written in terms of the Cartesian system (xi 'Xz ,x.3) and the boundary con-

ditions are

'(= U-+Wxr - - - -[ -= (:t".1) :t".2) - :x.~

I ze1 V -:. U t W"X ('I . , - --

2.3.61

2.3.62

2.3.63

2.3.64

where X2 is ~iven by eq. 2.3.60. The velocity field (~,p)

is expanded in terms of the gal? separation, a, in two regions,

an outer region using the coordinates (x1,XZ,x3) as independ-

ent variables and (v,p) as dependent variables and an inner

region using (Xi,X2,Xa) as independent variables and (v,p) as dependent variables where

2.3.65

2.3.66 /

where k is a constant, as yet undefined. Making use of'

eq. 2.3.65, 2.3.66, the equations of motion may be converted

to inner variables where

- ~ N } V = '1.0 t o.'Yl -t . .. . .. .. ,.., ~ '" 2.3.67 P '" Po + 0- Pi t -- - - -

where

(Vo, ~) satisfies \]2 ~ \f ,.., f Vo -= Po 2.3.68 (Vi Pi) satisfies "J,-v V~ ?- Vi -:: P1 2.3.69

67

Relative to the i=er variables of eq. 2.3.65, eq. 2.3.55

takes the form

2.3.70

for the wall W;'"whilst eq. 2.3.60 for the wall IV' takes

the form'

~

1 + x~ ( CO!;}rp -:'1,,2 12 ) x1:12 (1. -1 ) s;"fcosf x3 ': -:I-251 25,2 5) 52 + Q:~ ( S:,,2J t

2$1 ~) 2Sz r 0 (oJ) 2.3.71

In order to solve eq. 2.3.68 with boundary conditions on

IV and W' it is convenient to change variables to (x, ,x2'X3)' where

2.3.72

so that the wall VI becomes

2.3.73 /

,/

and wall IV' becomes

= 2.3.74

where the function hex, ,xz) is defined as

With these new variables eq.' 2.3.68 for (vo,po) takes the

68

form

I" a'-(vo)1 arc -~, % = 0 ()~ ~ iZat. 1 :-(:l(V~)2 a~ - OPo r - :z.. : 0 a-2 oi]. Rz OX3 x?> 2.3.76

op., :: 0 , ~.i3

e(v~ -t CXv,,)~: O(Vo); 1" i.i 0(11;,), + ~ O(V'c)2 ., 0 5:{1 pi:! ~.i~ ~ OX; ~2 0i 3

The boundary conditions eq. 2.3.61 and eq. 2.3.63 may be

written in terms of the inner variables for ~ -0

(0'0), :: ~-k[ ~~U1 - a~j3~ - ~~4)2((~) + (~J 0 (aZ)) 1 (VO)2 "'- -~rj ~ ((~2)(r) 2))1 a a"U:z - aW3~ - ch''-'f :~, t ~ t 0(\ 2.3.77 (VO)3 -= -k [ 1 ( () U3 + 02. WtX.a -W2~)1

/ for Wand a similar set exists for W'. Since the creeping

motion equations are linear and since (v,p), and hence

(~,po) depend linearly on U, g',~ andw~ it follows that

2.3.78

where the flow fields (Ya , Pa ), (Yb''i\), (Yc' pc) each indi-vidually satisfy eq. 2.3.68 these being the resulting flows

f U U ' d U U U' U' " J d ' rom 3, ;\ an 1, 2 , Wp W l , l' 1.' tup t",2.' an tu:!, W 3

respectively. Thus the boundary conditions to be satisfied

69

are, for (va,Pa )

(V"')f (VO)2 ~ 0, C) -1. 0" W

1 -:.0 VD. ;;.,. a U3

(Vo.) I (Va)z ~ 0 ') r) ~k I I 2.3.79 "" Va :, ::. '" U3 Ot) W

for (Vb,Pb)

(Vb), '" 0-_b~

Ut t Cl(a-ht~) , (V~2 '" 0-"'1 (.\2 + o( a-R'~) (vbh cb~( - _ ) '; 0. Lt.:l j :(.1 - U),2, ~1 on W (Vb) 1 -h 1 , o (0 .".~ 2.3.80 - a. 2 U1 t (Vb)2 -b1U' o (0 -b~ "" a 2. 2 t (Vt3 -hi ( ) = 0 2. w,-x:'z - ~.if on VI'

and for '" ~ (vc,pc)

(VG)f '" -k+\ ) o -t.0z,:.t2 (vJz -k~1( -) -= o w3 .::c.., (Vc) 3 -= 0 on W (Vc) 1 _k+l( '_) 2.3.81 ': a -l.0j::(.2 / (Vt) 2 = o-b1( '-) W5X-l (YC)5 .- 0 on iV'

From these boundary conditions one must take k = 0 to

calculate (va'Pa ), k = i for (vb,Pb) and k = 1 for (vc,pc)' , with an error O(a l ) in the eq. 2.3.73, 2.3.74. Three cases

for solution arise. For direct approach of the two surfaces

the flow field (v ,p ) of the inner expansion results from -a a U3 and Uj, where k = O. The' effect of flow field (vb,Pb)

. 70

results from a tangential and rOlling motion of the sur-

faces IV and W' due to components Ut, Uz , Ul, Ul ,(0.)1 and

Ojl. The flow field (~,pc) results from the rotational motion of the surfaces about their mutual normal, and is

produced by the components LU., and w 3

Each of these three cases involves substitution of

the apprdpriate boundary conditions and corresponding

value of k. Considered here is only the direct approach

of the two surfaces. Thus, taking k = 0, the boundary ,~

conditions on v become -a .I

(Vo.\ = (V")2 -= 0 (V (1) 1 .,. {VoL ~ 0 2.3.82

in the limit a~O. Since (va'p ) satisfies eq. 2.3.76 - a

J "

where A, B, C and D are constants" obtained by integrating

wi th respect to Xs and are arbitrary functions of xi and xl' Thus, from eq. 2.3.82

A ~ - ~ (~~ ) ~ (:i1 ) ~2) .B ~ - ! (aft;) h(::t '1_)

2fJ O.x.2 1 } -J.

C" 0

0= 0

2.3.84

71

SUbstituting and integrating with respect to x~,

2.3.85

Since (va )3 = U.3 on X3 = 0 it follows that E = Us. Also, since CVa)g = U3 on x3 = h(x 1 ,X2.)

2.3.86

Substitution of the values of A and B fromeq. 2.3.84 into

eq. 2.3.86 yields

2.3.87

or alternatively ,/

2.3.88

Now h may be written in the form

1 t 2.3.89

where x is the column vector x = (~:)and iT is its transpose and

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

l( 2.3.90

Defining ~ and Al as the eigenvalues of the matrix, K,

they are the roots of

Thus eq. 2.3.89 transforms into

h 1"" "2 "\,,~ . '" ... .1\1 ~ ... /\2 =c.z

which eventually gives the solution

The forces

(F,,) f " (r .. ) , " (F())~ ~

(G .. ) I " (G"')2'"

(G"')3 =

and torques resulting are

o (-e.., 0. ) / /

c (~t\) _ ;:m~(u~-u~) + 0(6-1 )

d(>-ti Az) (AI )..;')!

2.3.91

2.3.92

2.3.93

2.3.94

2.3.95

72

Similar methods are shown by Cox for the solution for the

other two forms of motion. The resulting expressions

for" all the forces and torques may be expressed in terms

73

of the grand resistance matrix defined by Brenner(lO,11,13).

/

74

2.4 Viscosity Effects

In 1906 and 1911 Einstein(32,33) showed that as

concentration of solid particles approaches zero, the

viscosity of a suspension of spherical particles is des-

cribed by

JI : fo ( it 2.Se) 2.4.1

where 1". is the viscosity of the pure liquid and t' is the viscosity of the suspension at volume concentration, c.

Hess suggested the general form

2.4.2

where 1 is a constant numerically greater than one, assum-

ing a simple arrangement for particles in a fluid flowing

in a capillary. Robinson suggested

hbc. i-Co

2.4.3

where Co is the volume of sediment obtainable from unit

volume of suspension, and b is a constant which he tentatively

described as a frictional coefficient. The term c sets the o

upper limiting value, the condition at which sufficient

numbers of particles are in contact with other particles

for the apparent suspension viscosity to approach infinity.

Various workers have found Einstein's equation to have a

constant variable from 1.5 to 3.5. Orr and Blocker(75) ,

75

therefore, suggested an equation

o.ck:. 2.4.4

where a and k are constants which depend on suspension

properties. Orr and Blocker suggest plotting the function

in logarithmic form and obtaining a and k from experimental

data

2.4.5

Further, after experimentation Orr and Blocker found that

k is closely related to the reciprocal of the geometric

standard deviation of the size distribution of the particles J

in suspension and that is closely related to the reciprocal

of co' the limiting concentration.

Studying sedimenting systems of methyl methacrylate and

glass spheres in various liquids Oliver and Ward(74) obtained

the general equation

1 - 1 Pr

- ~c t ~1

/

where the relative viscosity u ~ t for 0.1

76

The value of Kt seems to be a characteristic of a suspens-

ion" of settling spheres. For concentrations below 10%

Oliver and Ward use the expression

2.4.8

although they found values of K1 .to vary from 1. 2 to 3.6.

The magnitude of K1 is closely corlnected to the degree of

instability of the suspension and Oliver and Ward relate

this to f",-P as in Fig. 2.4.1. Po

0,10

0,03

1::1 0.04

o

fa

Fig. 2.4.1 Relationship between Kt

suspension of spherical

o

and Pr.-f fo

particles.

for unstable

The equation of the line is

)

suspensions but is insignificant in downward settling

systems at concentrations less than 30%.

01iver(73) suggests a relative velocity expression

1-c

f're 2.4.9

77

where U I is the relative viscosity of a suspension, and I re.

compares this to another due to Gurel

2.4.10

where fe is a measured relative viscosity. He quotes two

others, due to Hawksley and steinour respectively.

u - (f-CY~i

78

veloci ty round each particle which is necessary to satisfy

boundary conditions. He assumes that the mean velocity, V, is that of a fluid of viscosity,,..... Thus the force of the

containing walls is either the viscosity multiplied by the

normal velocity gradient calculated from V, or the rate of

dissipation of energy calculated with viscosity, /" and

velocity, V, equals the actual dissipation. For steady

viscous flow of an incompressible fluid, neglecting inertia

terms we have the Navier-Stokes and continuity equations,

fA 1J2.~ -= Vp 2.4.13

V.\J '" 0 2.4.14

Thus V% f " 0 2.4.15

This gives rise to a particular integral of eq. 2.4.13

M = rp where r is the position vector of the point 2,... considered. The total solution is

\! =

where v satisfies

rY..

t.:p + Y. 2fA

::0

= 3 p -t rVp = 3p f- !' oy

Or

/

2.4.16

2.4.17

2.4.18

For a pure liquid with infinite boundaries the term in EP

in eq. 2.4.16 becomes infinite, but for boundaries with a

finite distance the velocity 'at the boundary is given by

I . I

I I

I

79

2.4.19

The solution of eq. 2.4.17 is

2.4.20

where G= J.. 41T~

where R is the distance near P where we con-

sider v -p Thus if Uo is the Liain flow between boundaries

which satisfy the boundary conditions and p is the extra

pressure due to a disturbance, and! is also due to the

disturbance,

~ -= !do t p t Y. 2f

2.4.21

In eq. 2.4.30 the area integral extends over the area of

bodies disturbing the main flow. This is only true to a

first approximation, however. The function, G, is changed

considerably. Using the analogy of electrostatics, it is

the potential due to a unit positi~e charge, P, when all

the boundaries are earthed. These earthed particles have

an induced negative charge and the effect of all these neg-

ative charges is to reduce the field and, therefore, the

potential away from P. Thus the effect on the velocity at

P due to particles some way away from Pwhich are shielded

from it is reduced considerably. The shielding effect

becomes more noticeable as the concentration rises. When

particles touch the shielding of P is complete. The motion

of fluid at P is calculated entirely from the motion of the

80

boundaries nearest to it. At a concentration of 10% the

average gap between nearest particles is little more than

their linear dimension. Thus any calculation which assumes

that the effects of the particles on one another are addit-

ive is incorrect.

In consideration of the velocity near a sphere Kynch

assumes the pressure distribution on the surface of a

particle is

2.4.22

where A is a constant," the angular vorticity, x and z

two of the major Cartesian directions, and a, the particle

radius. Kynch obtains, assuming eq. 2.4.22, that

and

the integral of eq. 2.4.20 /

/

V:x.. ;: - )~3 (1t ~ )(aGo) _ \0.5 A 0 (a2~OJ -101' oz A 3Op. ox. 0>-0 z. A

Vj = - A.o.5 A 0 (2l60) - >..o3Aj (02Go) 30;; 6j 0:.:.0 Z. A (n,l. A ch.d2. A

2.4.23

2.4.24

>..0,3 A:l.f\ CC/GCl

2.4.26

in which case

=

=

2.4.27

whereX'(1-~) -= 'A(1tOlC) and the pressure is.

- ~(3)1 25

2.4.28

The two terms ~ and f depend on the concentration

/ ./

2.4.29

2.4.30

81

where b is the interpartic1e distance between centres, and

k the cut-off radius. Notice that the terms increase with

distance from the sphere. If these expressions are used

up to r .. b, the new terms are of the same order of magnitude

as the main terms. Midway between particles Kynch believes

the effects of at least three particles should be considered

whereas here, only one has been considered.

82

To calculate the effective viscosity Y~nch excludes a

thin layer of thickness, d, on the boundary of radius, ra,

sufficiently large for Go to be practically zero along its

narrow rim. Neglecting the term in V'G which is small of o order, c, the only term that remains is in Vz and is

2.4.31

Assuming that Go in the bulk of the fluid does not depend

on the thin surface layer, the necessary expression for it -I,~ ,

is a term in ~ due to the charge at p and a similar term R

due to the image. When kd 1 the integration yields

2.4.32

Assuming that A = -lO~ there is a velocity gradient

2.4.33

and the total velocity gradient near the boundary is

2). (1 + 2.5

83

suspension. The sphere moves with arbitrary constant trans-

1ationalvelocity, U. in the positive z direction parallel

to' the cylinder axis, and with constant angular velocity

relative to the cylinder wall, while

the fluid flows in a 1aminar flow with a superficial velocity,

lUo' in the positive z direction. The sphere is of radius,

a, cylinder radius, Ro' and the centre of the sphere is at

a distance, b, from the cylinder axis in the i direction.

It is assumed that creeping flow exists; thus

2.4.13

2.4.14

where fo is the viscosity of the homogeneous fluid. Bound-

ary conditions are:

At fluid-solid interfaces no relative motion exists.

At large distances from the disturbing influence of the

sphere, z =co, the velocity distribution becomes parabolic.

The boundary value problem is solved by Greenstein and /

Happe1 by the method of reflections, where the general bound-

ary conditions are that U (i.+1) = -![ (.i). Since the equations of motion and boundary conditions are linear, the frictiona