cambridge texts in applied mathematics j. m. ottino-the kinematics of mixing_ stretching, chaos, and...

7/25/2019 Cambridge Texts in Applied Mathematics J. M. Ottino-The Kinematics of Mixing_ Stretching, Chaos, And Transport -

1/387

The kinematics of mixing:

stretching, chaos, and transport

J . M .

O T T I N O

Dcprrrtrnc~nt01' C l ~ r r , ~ i r , u l11qirlrc~ri11g,

Unir .c~r.s i t j

1'

Mu.s.srrc~hu.sc~tt.s

CAMBRIDGE

UNIVERSITY PRESS


2/387

P U B L I S H E D B Y T H E P R E S S S Y N D I C A T E O F T H E U N I V E R S I T Y O F C A M B R I D G E

The Pitt Building, Trumpington Street, Cambridge CB 2 IRP, United Kingdom

C A M B R I D G E U N I V E R S I T Y P R E S S

The Edinburgh B uilding, Cam bridg e CB 2 2RU, United Kingdom

40 West 20th S treet, N ew York, NY 1001 1-421

I ,

USA

I 0 Stamford R oad. Oakleigh. Melbourne 3 166, Australia

O

Cambridge University Press I989

This book is in copy right. Sub ject to statutory exception

and to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place without

the written permission of Cambridge University Press.

First published 1989

Reprinted 1997

Printed in the United Kingdom at the University Press. Cambridge

A catalogue record for this book i s ava ilable from the British Library

Library of Co ngress Cataloguing in Publication data

Ottino,

J.

M.

The kinematics of mixing : stretching, chao s, and transport 1

J.

M.

Ottino.

p. cm.

Bibliography : p.

Includes index.

ISBN 0 521 36335 7. ISBN 0 52 1 36878 2 (paperback)

I. Mixing. I . Title.

TP156.M5087 1989

660.2 '84292-dc 19

88-30253

ISBN 0 521 36335 7 hardback

ISBN 0 52 I 368 78 2 paperback


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Contents

I

Introduction

I I Physical picture

1.2 Scope an d early works

1.3 Appl icat ions an d geometr ical s t ructure

1.4 Approach

Notes

2 Flow, trajectories, and deformation

7.1 Flow

7.7

Velocity. ;icceler;rtion. Lagrangian and Eulerian viewpoints

2.3 Extension to mul t icom ponent m edia

7.4

('l~issical me ans for visuali/ation of flows

2.4.1

Ptrr/ic,lr prrfll. or hii, or frcijec'for.~

2.4.2 S / r ~ ~ t r ~ l l l i ~ ~ ~ ~ . s

-7.4.3

Srrc~trlilirlc~.~

2 . 5 Steady and periodic flows

7 . 0

1)eformation gradient and velocity gradient

2.7 Kinem atics of deform ation-strain

7.X M ot io n a ro u nd a point

2.9

Kinemat ics o f defo rma t ion: rate of s train

7.10 Rates of cha nge of mater ial integrals

2.1 I'hysical me an ing of

Vv. (V v) ' .

a n d

D

Hibliography

Notes

3

('onserration equations, change of frame, and vorticity

' . I

Principle of conservation of mass

.

2

F'l.inciple o f con serv ation o f linear m o m en tu m

3.-;

7'r:iction

t (n .

x.

1 )

3.4 ('a uchy 's eclu:rtion o f motion


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Contents

Principle of conservalion of angular momentum

Mechanical energy equa tion an d the energy equation

Ch ang e of frame

3.7.1 Objecticity

3.7.2 Velocity

3.7.3 Accelerarion

Vorticity distribution

Vorticity dynamics

Macroscopic balance of vorticity

Vortex line stretching in inviscid fluid

Streamfunction and potential function

Bibliography

Notes

Computation of stretching and efficiency

Efficiency of mixing

4.1.1 Properties of e, and e,

4.1.2

Typical behavior of the eflciencj

4.1 .3 Flow classification

Examples of stretching and efficiency

Flows with a special form of

V v

4.3.1 Flows with D( Vv )/D t=

0

4.3.2 Flows with D( Vv )/ Dt small

Flows with a special form of

F ;

motions with constant stretch history

Efficiency in linear three-dimensional flow

The importance of reorientation; efficiencies in sequences of flows

Possible ways to improve mixing

Bibliography

Notes

Chaos

in

dynamical systems

Introduct ion

Dynamicdl systems

Fixed points and periodic points

Local stability and linearized maps

5.4.1 Dejinilions

5.4.2

Stability of area preserriny two-dimensional maps

5.4.3 Fumi1ie.s

c

periodic points

Poincare sections

Invariant subspaces: stable an d unstable manifolds

Str uc tura l stability

Signatures of chaos: homoclinic and heteroclinic points. 1-iapunov

exponents an d horseshoe maps


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C o n t e n t s

vii

5.8.1

Homoi,linic lend heteroclinic connections und points 1 1 1

5.8.2

Sensiticity t o ir~itiul onditions (end Li up un o~ : xpone nts

116

5 . 8 . 3 florseshoe mups

117

5.9 Su m m ary of definitions

of

chaos 124

5.10 Possibilities in higher dimensions

124

Bibliography 125

Notes 128

6

Ch aos in l lamilton ian systems 130

6.1 Intr od uc tion 130

6.2 Ham ilton's eq ua tion s 131

6.3 Integrability of Ilam ilton ian systems 132

6.4 Ge neral stru ctu re of integrable systems 133

6.5 ph ase spa ce of Harn iltonian systems 135

6.6 Pha se space in periodic Ha m ilton ian flows: Poinc are sections an d tori 136

6.7 1,iapunov expo nents 138

6.8 l lom oclinic an d heteroclinic points in H am iltonian systems 140

6.9

Pertu rbatio ns of Harnil tonian systems: M elnikov's m etho d 141

6.10 Behavior near elliptic points 143

6.10.1 PoincurP--H i r k h o th eo rem

144

6.10.2 Th e Kolmogoroa Arnold-Moser theorem (t he K A M theorem )

146

6.10.3 7'he tw ist theorr>m 148

6.1 1 General quali tat ive p icture of ncar integrable chao tic Ham iltonian

systcms 148

Bibliography 152

Notes 152

7

M ixing and chaos in two-dimensional time-periodic flows

7.1

Int roduct ion

7.2 T he tendril-whorl flow

7.2.1

1,occrl c~nu1y.si.s: octcction und slccbilitj o f perioil-1 and period-2

periodic points

7.2.2

(;loha1 unul~.s i.s ind interccctions be tw ee n munifi)kd.s

7.2.3 Formutior1 of horseshoe mccps in the T W mcip

7.3

The blinking vortex flow (BV)

7.3.1 Poinc,ur; seer ions

7.3.2

S luh i li tj o f period-I periodic poitlts ctnd conjugate) lines

7 . 3 . 3

Hor.se.shoe mrcps iri the BV flow

7.3.4

Liupunot. c~.uponenls, rerage cflic.iencbp, [end 'irrecers ihilit y'

7.3.5 Mclcro.sc~opic 1isper.sion o f tracer purtic~les

7.4 Mixing in a journal bearing flow

7.5

Mixing in cavity flows

Bibliography

Notes


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. .

v l l l

Contents

Mixing and chaos in three-dimensional and open flows

In t roduct ion

Mixing in the part i t ioned-pipe mixer

8.2.1 Appro.uirntrtcl 1 ~ ~ l 0 c i t .vlirld

8.2.2

Poirlcrrrc; .sec,tiort.s trrlil three-dimertsiond struc'turr

8.2.3 E\-it tirile dist rib ut ions

8.2.4

L~ocrrl trercllir~g01' rnirr~rirrl ir1e.s

Mixing in the eccen tr ic hel ical an nu la r mixer

Mixing and dispers ion in a furrowed channel

Mixin g in the Kelvin ca t eyes flow

Flows near walls

Streamlines in an inviscid flow

Bibl iography

Notes

Epilogue: diffusion and reaction in lamellar structures and

microstructures in chaotic flows

Transport at s t r iat ion thickness scales

9.1 I

Pirrirrllrter.~rrrttl rtrrirrb1e.s chtrrtrc~rrrizirly rtrrlsporr ut .srnrrll

9.1.2 Re~jir l lc~.~

Comp l ica t ions and i l lus t ra t ions

9.2.1

Distortiorls of Iirrnrllirr structures trrttl rli.stributiort c1/1;.r.t.s

9.2.2 Illustrtrriorts

Passive an d act ive microst ructures

9.3.1 E.~perirnc~rttrrlt~rr1ir.s

9.3.2

Th ro rr ri c~ rl r~rt1ir.s

Active microst ructures a s proto type s

Bibl iography

Notes

Append ix: C ar tes ian oec tors and tensors

Lis t of' ,f iequently used sy mbo ls

Author index

Color sec t ion bc tn~c~enages 153 and 154


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Preface

1.11~bjective of this book is to present a unified treatment of the mixing

o f f luids from a kinematical viewpoint . T he aim is to provide a conceptual ly

clear basis from which t o launch analysis an d t o facilitate the und erstan ding

c )f th c ume rous mixing problemsenc ountered in nature an d technology.

['rcsently, the study of fluid mixing has very little scientific basis;

processes and phenomena are analyzed on a case-by-case basis without

a n y

at tempt to discover general i ty . For example, the analysis of mixing

and 's t i rr ing ' of co nta m ina nts an d t racers in two-dimensional geophysical

flows such as in oceans; the mixing in shear flows and wakes relevant to

aeronaut ics a n d co m bu st io n; the mixing of fluids under the Stokes 's regime

generally e nc oun tere d in the 'blending' of viscous liquids such a s poly m ers;

and

the mixing of diffusing and reacting fluids encountered in various

lqpes of chemical reactors share l i t t le in common with each other, except

possibly the nearly universal recognit ion among researchers that they are

\cry difficult problems.'

Phcre are, however, real s imilar i t ies among the various problems and

the possible benefits from an overall at tack on the problem of mixing

uhing a general viewpoint are substant ial .

7'110poir~t

1'

c.ie\\. rrdoptetl Ilc>r.r s rl~trr fi.oru o kir~c~rntrticalir\z.point ,fluid

/ ~ r i . \ i r l ~ ~

s tho c:[jic,icv~rsrrc~rc~hingrrltl ,fbl tlir lg of ' r?~utrriu l irlr.s urld .surCfucc~.s.

S~lch problem corresponds to the solut ion of the dynamical system

tlx,'tlr = V ( X ,

t ) ,

l~11cre he right ha nd side is the Eulerinn velocity field ( a so luti on o f the

N al ier Stokes equa tions . for ex am ple) an d the initial condit ion corresp onds

the init ial configuration of the l ine or surface placed or fed into the

tlolv ( S represents the location of the init ial condit ion

x

=

X ) .

Seen in this

l ight. the problem can be formulated by merging the kinematical founda-

tions of fluid mechanics (Chapters 3 a n d

3 )

an d the theory of dynam ical

s \lems (C ha pt er s 5 ; ~ n d

).

The approach adopted here i s to analyze

siml'lc protypical f lows to enha nc e intuit ion a n d to extract conclusions


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of general validity.

Mi.uincl i.s strc~tchinyund ,jOMiny and srrc~tchinycrnd

,j i) ldir~g s r l ~ r f ingc~rprint01' chlros. Relatively simple flows can act as

prototypes of real problems and provide a yardst ick of reasonable

expectations for the completeness of analyses of more complex flows.

Undoub t ed l y , I expect that such a program would facil i tate the analysis

of mixing problem s in chem ical, mec hanical , an d aero nau tical engineering,

physics, geophysics, oc ea no gra ph y, etc.

Th e plan of the book is the following: C ha pt er 1 is a visual sum m ary to

motivate the rest of the presentat ion. In Chapter 2 I have highlighted,

whenever possible, the relat ionship between dynamical systems and

kinematics as well as the usefulness of studying fluids dynamics start ing

with the concepts of mot ion a n d , f low.2 Mixing should be embedded in a

kinematical foundat ion. H owever,

I

have avo ided references to curvil inear

co-ordinates and different ial geometry in Chapters 2 and

4,

even though

it could have m ad e the pres entat ion of som e topics m ore satisfying but the

ent i re presentat ion sl ight ly uneven and considerably more lengthy. The

chapter on f lu id dynamics (Chapter 3 ) i s br ief and convent ional and

stresses conceptual points needed in the rest of the work. The dynamical

systems presentat ion (C ha pte rs 5 a n d

6 )

includes a list of topics which

I

have fou nd useful in mixing studies an d should not be reg arded as a

balanced introduct ion to the subject . In this regard, the reader should

note that most of the references to dissipative systems were avoided in

spite of the rather transparent connection with fluid flows.

A few words of caution are necessary. Mixing is intimately related to

flow visualization a n d the ma terial presented here indicates the price on e

has to pay to understand the inner workings of determinist ic unsteady

(albeit generally periodic in this w o rk ) two-dimen sional flows an d three-

dimensional flows in general. How ever, we should no te tha t the geometrical

theory used in the analysis will not carry over when v itself is chaotic.

Though mixing is st i l l dependent upon the kinematics, the basic theory

for analysis would be considerably different. Also, even th oug h m any of the

examples presented here pertain to what is sometimes called 'Lagrangian

turbulence', the reader might find a disconcerting absence of references

to convent ional (o r Euler ian ) turbulenc e. In this regard

I

have decided to

let the reader establish possible connections rather than present some

feeble ones.

I

give full citation to articles, books, and in a few cases,

i f

an idea is

unpubl ished, conferences. When only a last name and a date is given,

part icular ly in the case of problems or examples, and the name does not

ap pe ar in the bibliography, it serves to indicate the so urc e of the p roblem


9/387

or idea. I t is important to note also that some sections of the book can be

rcgnrded as work in progress and that complete accounts most likely will

(allow, expanding over the short descriptions given here; a few of the

problems, those at the level of small research papers are indicated with

an asterisk

( * ) .

In several passages

I

have pointed out problems that need

work. Ideally, new questions will occur to the reader.

Preface to the Second Printing

In preparing the second printing of this book, several typographica l and

formatting errors have been corrected. The objectives of the book

expressed in the original preface remain unchanged. Owing to space con-

straint limitations the amount of material covered remains approximately

the same. The reader interested in the connection of these ideas with

turbulence will find some leads in the article 'Mixing, chaotic advection,

and turbulence',

Annual Reviews

of

Fluid Mechanics,

22,

207-53 (1990);

a succinct summary of extensions of many of the ideas outlined in this

book is presented in 'Chaos, Symmetry, and Self-Similarity: Exploiting

Order and Disorder in Mixing Processes', Science, 257, 754-60 (1992).

I

should appreciate comments from readers pertaining to related articles in

the area of fluid mixing as well as possible extensions or shortcomings of

the ideas presented in this work.

Notes

I

Even the term inology is com plicated. F or exam ple. in chemical engineering the term s mixing,

agi tat ion. and blending are common (Hyman,

1963;

McCabe and Smith,

1956,

Chap.

2. Section

9 ;

Ulbrecht and Pat terson,

1985).

The terms mixing. advection, and

stirring appear in geophysics; e.g.. Eckart.

1948;

Holloway and Kirstmannson.

1984.

Inevitably. different disciplines have created the ir own terminology (e.g., classical reaction

engineering, combustion, polymer processing, etc.).

2 Kinematics appe ars as an integral part of bo ok s in continuum mechanics but m uch less so

in m odern fluid mechanics. There a re exceptions ofco urse : Ch apte rs

V

and VI of the work

of Tietjens based on the lecture notes by Pra nd tl contain and unusually long description

01 deform ation and motion aroun d a point (Pr an dtl and Tiet jens, 1934).


10/387


11/387

-

Acknowledgments

/

This hook grew out from a probably unintell igible course given in Santa

EL : .

Arg entina, in July 1985, followed by a sh ort c ou rse given in Am he rst,

M assach usetts, also in 1985. M ost of the ma terial w as conden sed in eight

lectl~rcsgiven at the California Insti tute of Technology in June 1986,

\vhcrc the bulk of the material presented here was written.

. rhc connect ion between s tretching an d folding, an d mixing an d ch ao s,

bcc;~rnc transp aren t after a con vers ation with H . Aref, then at Brown

IJni\c rsity, du rin g a visit t o Providence, in Sep tem ber 1982.

1

would

particularly like to thank him for communicating his results regarding

the 'blinking v ortex' prior t o pub lication (see Secton 7.3), an d als o for

m a n y research discussions an d his friendship du rin g these years. I would

like to thank also the many c om m en ts of P. Holmes of Corn el l University,

on a rather imperfect draft of the manuscript , the comments of J . M.

Grcenc of

G.

A. Technologies, who provided valuable ideas regarding

syrnnictries as well as to the many comments and discussions with S.

Wiggins and

A.

Le on ard , bo th a t the Cal ifornia Ins t i tute of Technology,

during my stay at Pasadena. I am also gra teful to H . Brenner of the

Massachusetts Insti tute of Technology, S. Whitaker of the University of

California at Davis , W. R . Schow alter of Princeton University, C . A.

Trucsdel l of Johns Hopkins Universi ty, W.

E .

Stewart of the University

of

Wisconsin,

R .

E .

Rosensweig and Exxon Research and Engineering,

and J . E:.. M ar sd en , of the University of California a t Berkeley, for various

comments and s uppor t . I am also part icularly thankful to those who

Supplicd photographs or who permit ted reproductions from previous

Pllhlications (G. M . Corcos of the University of California at Berkeley,

R . ( 'hevray of Columbia University. P.

E.

Dimotakis and L. G. Leal.

the California Insti tute of Technology,

R .

W. Metcalfe of the

Universi ty of Houston, D. P. McKenzie, of the Universi ty of Texas at

Allstin. A. E . Perry of the University of Melbourne, 1. Sobey of Oxford

I J n i \ c r s i t y ,and P. Wellnnder of the Universi ty of Washington).

1

a m

particularly indebted to all my former and present students, but


12/387

x iv A~~kn o w1cd g mc~n t .s

particularly to R. Chella and D. V . K ha kh ar , for work prior to 1986, and

to J .

G .

Franjione, P. D. Swanson , C . W . Leong, T. J . Danielson, and

F. J. Muzzio, who supplied many of the figures and material used in

Chapte rs

7-9.

1

am a l so indebted to

H.

A. Kusch for help with the

proofs and to H. Rising for many discussions during the early stages

of this w ork . Finally,

I

would l ike to express my grat itude to

D.

Tillwick

who helped me with the endless task of typing and proofing, to D.

Tranah, from Cambridge Universi ty Press , for making this project an

enjoyable one , and to my wife , Al ic ia , for he lp and support in many

oth er ways .

Amherst, Massachusetts


13/387

/

1

Introduction

1 .1 .

Physical picture

I n .;pite of its universality, mixing does not enjoy the reputation of being

:I

very scientific subject an d , generally sp ea kin g, mixing pro blem s in na tu re

;\nd technology are at tacked on a case-by-case basis. From a theoretical

viewpoint the ent i re problem appears to be complex and unwieldy and

thcre is no idealized sta rtin g picture fo r an aly sis; from a n app lied viewpoint

i t is

easy to get lost in the complexities of particular cases without ever

seeing the s tru ctu re of th e en tire subject .

Figure 1 . I

. I ,

which we will use repeatedly throughout this work,

describes the most important physics occurring during mixing. In the

simplest case, during mechanical mixing, an initially designated material

region of fluid stretches and folds throughout the space. This is indeed

thc goal of visualization experiments where a region of fluid marked by a

suitable tracer mo ves w ith the m ea n velocity of the fluid. Th is case is also

closcly approximated by the mixing of two fluids with similar properties

and no interfacial tension (in this case the interfaces are termed passive,

see Aref and Tryggvason (1984)) .

Ob viously, a n exac t description of the m ixing is given by th e location

of thc interfaces as a function of space and time. However, this level of

description is rare because the velocity fields usually found in mixing

Processes are complex. Moreover, relatively simple velocity fields can

Produce very efficient area generation in such a way that the combined

act ion of s t retching and folding produces exponent ial area growth.

Whereas this is a desirab le go al in ach ievin g efficient m ixing it als o implies

that initial er ro rs in the loca tion of the interface a re amplified exp onen tially

fast and numerical tracing becomes hopeless. More significantly. this is

"so

a

signature ofchao t ic f lows an d

i t

is impo rtan t to study the condi t ions

under

which they are produced (more rigorous definit ions of chcios are

glvcn

in Chapters

5

a n d

6).

However. without the action of molecular

diffusion, a n instan tane ous cut of th e fluids reveals a ltrti~c~llcirrrlrctlrrc.

(Figure

1.1.1) .

A measure of the state of mechanical mixing is given by


14/387

Figurc I . I . I . Basic processes oc curr in g dur in g mixing of f lu ids: ( 0 ) o r r e s p o n d s

to th e case of tw o simi lar fluids with negligible interfiicial tension an d negligible

interdiffusion: an in i t ia l ly designated mater ia l region s tre tches and folds by

the ac t ion o f

a

f low: ( h ) c o r r e s p o n d s t o

a

blob diffusing in the fluid: in this

case the boundaries become diffuse and the extent of the mixing is g iven by

level curves of concentra t ion ( a p ro fi le n o rm a l t o t h e s t r i a t i o n s is sh o wn a t

the r igh t ) : in

( c , )

the b lo b breaks d ue t o in te r fac ial tension fo rces , p roduc ing

smal le r f ragments which migh t in tu rn s t re tch and break produc ing smal le r

f r a g me n t s . C a se

( h )

s an exce l len t approx imat ion to

( t i )

if diffusion is small

dur ing the t ime of the s t re tch ing and fo ld ing . In

( t i )

the blob is pu.s.sii.e, in

( c )

the blob is trc,tii.c..

s t r ia t ion

th ickness

s t re tch ing

@

concen t ra t ion p rofi le

initial

condi t ion

d i f fus ion

-

i s tance

b r e a k u p


15/387

Scope trtltl eurljs works

3

~ l ~ i ~ . k n e s s e sf the layers, say s, a n d s,, a n d

:(s,

+

s,,) is called the

s l r . ; t ,~ ; o ,~li ick~ios.ssee O t t ino , Ranz , and M acosko , 1979) . The a m ou nt

of intcrfi~cial rea per unit volum e, interpreted as a struc ture d continu um

p rc )p H ty . s ca lle d t he i r l t c ' r r i ~ ~ ~ t ~ r i ~ ~ lIrett dcnsit~,,I,..T h u s ,

i f

S

designates

t1,c area within

a

volume

V

enclosing the point x at t ime

t ,

S

u, . (x ,

t ) = lim .

v - 0

v

~ ~ l l ~ c

f

the abo ve conc epts require m odif icat ion

i f

the fluids a re miscible

o r

immiscible.

I f

the fluids are immiscible, at some point in the mixing

process the st r iat ions o r blobs d o not remain connected an d break into

snlnllcr fra gm ents (F igu re 1.1.1

( c ) ) .

At these length scales the interfaces

arc not prrs,sir.c~ n d inste ad of being convected (pa ssiv ely ) by t he flow ,

they nlodify the surr ou nd in g flow, m akin g the analysis considerably m ore

complicated (in this case the interfaces are termed uc.tioe, see Aref and

Tryg gvu son, 198 4). If the fluids a re m iscible we can still track ma terial

volumes in term s of a (h yp oth etic al) non-diffusive tracer w hich m oves

with the mean mass velocity of the fluid or any other suitable reference

velocity. Designated surfaces of the trac er remain connec ted a n d diffusing

species traverse them in both direc tions . ' Ho we ver , du rin g the mixing

process, connected iso-concentration surfaces might break an d cuts might

rcvcal i slands ra ther than s t r ia t ions (Figu re I . l . l ( h ) ) . In th is case the

specification of the concentration fields of 11-

species constitutes a

com plete desc ription of mixing b ut it is als o clear tha t this is a n elusive go al.

Thus ,

i t

is apparent that the goal of mixing is reduction of length scales

(thinning of material volumes an d dispersion th ro ug ho ut space, possibly

involving br ea ku p) , a n d in th e cas e of m iscible fluids, uniformity of

con cen tration.' W ith this a s a basis, we discuss a few of the idea s used

to describe mixing and then move to examples before returning to the

problem formulation in Section 1.4.

1.2. Scope and early works

A cursory ex am inat ion of a n eclectic an d fairly arbi t ra ry l ist ing of som e

of the earl iest references in the l i terature gives an idea of the scope of

mixing processes and the ways in which mixing problems have been

attacked in the

pas t . "

I .

Tn>l( ) r ( 1 0 3 4 )The

f o r m a t ~ o n f

c m u l s ~ o n s

n

definable fields of

f l ow .

P r o < , .

HI,\,.

Sot... A146.

501-23.

A .

Brc) t l lm: l l l .


16/387

W . R . Hawtho rne , D. S . Wende l l, and H .

C .

Hot tc l ( 1048 ) M ix ing and combus t i on

in tu rbule nt gas jets. p. 26 6- 88 in ' I 'h~ r t l ymp. on C 'o t~~hu. \ r io t~t~clFltrtnc, uttd E.uplosion

Pltc~t~otnc~t~u.

al t imore: Wil l iams & Wikcns .

C . Eckart (1948) An analys is of the s t i r r ing a n d m ~ x i n g rocesses in incompressible

fluids,

. I .

M u r i t l r

Kc,.\.,

VII,

265-75 .

R .

S . Spencer and

R .

M . Wiley (1951 ) T he mix ing of very v i scous l iqu ids . .I . C o l l .

& i . .

6 ,

133-45.

P.

V .

Dan ckwe r t s (1 952) T he defin it ion and measurement of so me charac te r is t i c s

of mixtures , Appl. S ( , i . Ros., A.3, 279 96.

P.

V . Dan ckwe r t s (1953 ) Co nt i nu ou s f low sys tems-d is t r ibu tion of residence t imes ,

Ch[ ,m. Eng. S ( , i . .

2 , 1 13.

P.

Weland cr (1955 ) Studies on the general development of mo tion in a two-dim ensional ,

ideal fluid,

Tollu.s,

7 . 141-56.

S. Corr sin (195 7) Simple theor y of a n idealized turbu lent mixer. A . 1 . C h . E . J . ,

3.

329-30.

W .

D.

M o h r , R . L. S a x t o n , a n d

C.

H . Jcpson (1957) Mixing in laminar f low systems,

I n d . E n g . C h r m . , 49. 1855-57.

T h .

N.

Zweiter ing (19 59) T h e degree of mixing in c ont inu ous f low systems C h r m .

Eng . Sc i . .

I I , 1-15.

It seems at first strange to start a discussion on mixing with Taylor 's

1934 paper. However, there are several reasons. The first one is that the

problem is of practical importance and was attacked with the best tools

of the t ime, b oth theoret ical ly a nd experimental ly. T he second on e is that

the natural extension of his ideas to mixing remain largely unfulfilled.

Taylor 's concerns a re ob viou s fro m the t i tle of the p ape r. He distil led the

essence of the pro blem , in general a complex o ne, and reduced th e question

to a

local

analysis : the deform ation, s tretching, and bre ak up of a droplet

in tw o prototypica l flows - plan ar hyperbolic flow a n d sim ple shear flow.4

Pres um ably , the long r ang e goal w as to m imic a com plex velocity field in

te rm s of po pu la tio ns of these tw o flows."his is sim ilar, in sp iri t, to the

approach adopted in this work, in two respects : ( i ) analysis of simple

building blocks which give useful powerful insights into the behavior

of

complex problems. and

( i i )

dec om po sition of a problem in local a n d

global

com pone nts ( this idea is reconsidered in Ch ap ter 9 ) .

B ro thman , W ollan, an d Feldman (1945) had m ore prac tical and

pressing needs in mind and tried to attack the problem of mixing in a

general and abstract way. They spoke of fluid deformation and [fluid]

rearrangement, and regarded mixing as a three-dimensional shuminB

process . They worked out probabil is tic argu m ents di rec ted p red or ni na nt l ~

to mixing in closed systems, such as stirred tanks, and obtained

kinetic

expressions for the creation of interfacial a rea . Ecka rt (19 48 ) had in mind

substantially larger length scales and started his analysis with the

con tinu um field eq ua tio ns an d calculated the 'mixing times'

of

thermal and


17/387

Scope untl earl), wo rks 5

s ; , l inc >POIS i n oceans wi thout resor t ing to any mechani s t i c descr ip t ion

o f

process. A conceptua l ly s imi la r p rob lem in t e rms of scales, mix ing

i n atmospheric

f lows , was addressed by W eland er

(1955) .

R e m a r k a b l y ,

he did

so

by

cons ider ing the poss ib i li ty of app ly ing Ha m i l to n ia n mechanics

to idea l flu ids an d s t ressed the need fo r s tudying the s t re t ch ing an d fo ld ing

of

elemen ts in the flow an d devised for mu las to follow th e process .

The gront I1 of a mater ia l l ine by fracta l const ruct ion is a l so expla ined in

h i s work as wel l as a t rea tment of mot ion of po in t vor t i ces f rom a

~ a m i l l o n i a n i ew p oin t. H e a l so p e rf o rm e d e x p e r im e n t s a n d o n e o f his

visual izat ions i s reproduced in Figure

1.2.1

.h

7 ' 1 1 ~r~tc .r ac rions ~ r t w r r nu r b u l r n c e a n d c h em i ca l

reactions

is o fu t mo s t

i m p o r t a l ~ c c n combus t i on . A l t hough t he approach t o t hese p rob l ems i n

thc

.iO's ; I I I C ~

60's wa s largely s ta t i s t ical th is wa s not the case an d

I t I S c o r n t o r t ~ l ~ qo k n o b t hn t In be l l t houqh t ou t expc r lmen t a l pape r s such

as th r or lr

h \

Hn \ $ t ho rn r . i l ' endc l l, an d Ho t t c l (1948) o n e f in ds d c s c r ~ p -

t l o n s c ~ r n p h a s i / ~ l ~ qhc qcomc. tr lca1 asp ects of the pr ob lem 7'0 qu ot e

I , '~gure

.:.I.

Reproduct ion of one of the earl?

mixing

exper iments of Welander

(10551: evolution of

an

i n ~ t i a l

ondition in a rotating flow.

He

used butanol

floated o n water and the

~ n ~ t i a l

ond i t ion

(squ are) was

ma d e of

methq l - r ed ;

unfortunatelq f e w

a d d i t ~ o n a l etai ls regarding

the experiment were giten in

t h e o r ~ g ~ n a l

aper.


18/387

6 Introduction

from their paper,

According t o the physical picture of the turb ulen t f lame. eddies . . . are being

draw n in from the sur round ing a tm osph ere an d being broken up in to part ic les

of various sizes

.

.

.

The total area of f lame envelope is many t imes the area

available in a diffusion flame but nevertheless the final intimate mixing of the

gas an d oxygen m ust o ccu r between eddies a s a result of mo lecu lar diffusion.'

Subseqently, statistical theory took over and the geometrical aspects of

the problem were somewhat lost. Of the many possible offsprings of the

statistical theory of turbulence to mixing (e.g., Batchelor, 1953) we might

mention the short but influential paper by Corrsin (1959) which proposed

a simple model to calculate the rate of decrease of concentration

fluctuations in an ideal, yet subsequently widely used mixer.' Another

important concept based on statistical reasoning is that of the mixing

length theory (Prandtl, 1925; Schlichting, 1955, Chap. XIX), which has

found application in an enormous range of problems ranging from

chemical engineering (Bird, Stewart, and Lightfoot, 1960) to astrophysics

(Chan and Sofia, 1987, Wallerstein, 1988). Even though the statistical

treatment does not lend itself easily to visualization, notable exceptions

exist and many of the early works focused on the stretching of material

lines and surfaces, for example, Batchelor (1952; also Corrsin, 1972)

analyzed the problem theoretically, whereas Corrsin and Karweit presented

experimental results (Corrsin and Karweit, 1969). Other work focused on

local deformation to capture the details of the turbulent motion at small

scales. For example, Townsend (195

)

performed an analysis of deformation

and diffusion of small heat spots in order to interpret experimental Eulerian

data in homogeneous decaying turbulence.

Concurrently, at the other end of the spectrum, the stretching of material

lines and volumes was also a concern in the mixing of liquids in low

Reynolds number flows. Spencer and Wiley (1951 focused on the mixing

of very viscous liquids and stressed the idea of being able to describe the

growth of interfacial area between two fluids and the need to relate the

results to the fluid mechanics. However, even though the mathematical

apparatus, largely developed in continuum mechanics, was already in

place for such a program. i t was not until much later that such develop-

ments took place and most of what followed from Spencer and wiley's

work was confined to deformation in shear flows. Two other points worth

mentioning from their work. which have a clear relationship with

dynamicol systems. are the identification of stretching and cutting or

folding as the primary mixing mechanism, the so-called 'baker's trans-

formation' characteristic of chaotic systems (see Chapter 5). and the idea


19/387

Scope trnd

e u r l y

works

7

of rcp resen t ing mix ing p rocesses in t e rms o f mat r ix t r ans fo rm at ions . wh ich

is

r el at ed t o m ap p i n g s ( see C h ap t e r 5 ) an d t ran si ti o n m a t ri c es

(see

p ,

164 n Reichl , 1980). A c lose ly re lated s tudy by M oh r . Sa x to n , an d

Jepson (195 7) a lso focused on v iscous l iqu ids in t he c on tex t of po lymer

p,.oces~il lghe y considered th e s t re tch ing of a f i lam ent of a f lu id in th e

bulk o f ano the r o ne in a shea r flow us ing s imp le a rg um en t s to acc oun t

for

the

Llscos i tyof the f lu ids bu t wi th out tak in g in to accou nt the in ter fac ial

t ens ion . T h e p rob lem is s imi la r in sp i r i t to the on e t r ea ted by Tay lo r

(193 4) an d m uc h wor k cou ld fo l low a lo ng these l ines . Nevertheless th is

simplif ied t reatm ent form s the basis for mos t of the subsequ ent d evelop men ts

in

the

mixing of viscous fluids."

~ t 1s p rob ab ly fai r to say tha t mo st of th e p rev ious w or ks ha ve th e

g e o m c t n c n l i n t e r p r e t a t ~ o ngiven In Fig ure 1 .1 .1 . Ho we ver , a point of

departu re fro m this p icture of s ign~fic ant onsequen ce in chemical engineer-

Ing took place w ~ t hhe papers by Zwei ter ing (1959) a nd Danckwer ts (1958) .

In th is c ase t h e ap p r o a ch b ecam e m o r e ' l u m p ed ' o r m ac r o sco p i c an d t h e

em p h as i s sh if ted t o c o n t i n u o u s f lo w sy s tem s an d t h e ch a r ac t e r ~ za t i o nof

m ~ x i ng y the t emp ora l d i s t r ibu t ion o f exit t imes .' '

Whereas the ob jec t ive of mos t o f the ab ov e wor ks w as t o re la te the f lu id

m ech an ic s t o t h e m i x ~ n g r so m e k n o w l ed g e of t h e p ro ce ss t o t h e o u t p u t .

Danc kwer t s (19 53) ocused p r imar i ly o n t he charac te r i za t ion of the mixed

state . 1.e .. he devised nu m be rs o r indices t o indic ate t o th e user h ow well

mlxed

a

system is (e .g . , ho w well mixe d is th e system of Fig ure 1 .3 .1?

F igu re 1 .3 .3?F igu re 1 .3 .4? ). Even thou gh we a r e go ing to say l it tl e a bo u t

' the measurement o f mix ing ' . th is i s p robab ly the p lace to s t ress our

opinion o n

a

few points: ( i ) the meas u re shou ld be se lected acco rd ing t o

t he specific ap p l i c a t ~ o n n d

i t

is fut ile to devise a s ingle me as ure t o cover

contingencies. a n d ( i i ) the measu re men t has t o be re la tab le to the f lu id

m e c h a n ~ c s .

Examplc'.s

( 1 )

T h e s t r i a t i o n t h ~ c k n e s s , . Fig ure 1 .1 .1 . is imp or ta nt in processes

in \ ,o lv ing d i ffus ing an d react ing f lu ids and represen ts t he d is tance tha t

t he m o lecu le s m u s t d if fu se in o r d e r t o r e ac t w ~ t h ach o the r . In s imp le

Gases. .s can be calculated exact ly with a knowledge of the veloci ty

field ( s ee C h e ll a a n d O t t i n o ( 1 9 8 5 a ) an d ex am p l e s In C h a p t e r 4 ) .

t i i )

Mo lten po lymers ar e o ften mixed ( a n oper;at ion of ten refer red to a s

b lend ing ) t o p ro d uce mate r ia ls with ~ ~ n ~ q u eropert i e s. F o r example .

In th e mnnuf;acture o f barr ier pol ym er f i lms ~t might be desir able to

Produce s t ruc t l l rcs wi th ef fec tive permeabi l i ty . Th is requ i res tha t


20/387

Introduction

the clusters of the m ore perm eable ma terials a re disconnected and d o

not form a percolating structure. However, the details of effective

diffusion near the percolation point depend on the ramification of

the clusters (Sevick, M on so n, and O ttino , 1988). Even though such

measurements can be extracted from electron micrographs (eeg.,

Figure 1.3.4, see color plates) via digital image analysis (Sax and

O ttino , 1985), to da te there are no models allowing the com putation

of such details from the fluid mechanics of the process.

( i i i ) A model for the structure of the Earth's upper mantle (Allegre and

Tur co tte , 1986) postulates that the oceanic crust becomes entrained

in the convective mantle where it is subsequently stretched into

filaments by buoyancy induced motions. It follows that, on the

average, the 'oldest' layers are the thinnest and that the diffusion

processes concurrent with the stretching become important over

geological time scales when the striation thickness is of the order of

0.1-1 m (the typical diffusion coefficients are of the order 10-14-

1 0 - l6 cm 2/s, which implies diffusion o n time scales of the order of

10'6-1020 S. By com pariso n th e time scale based on the age of the

E arth is 1.4

x 10'' s) . In this case a model describing the entrainment

of material in the convective mantle coupled to a model describing

the evolution of the striation thickness as function of time is capable

of describing the gross characteristics of the process.

(iv) Co nsider Eulerian conc entration measurements in a tu rbulent m ixing

layer. In principle, the fluctuations can be taken as an indication of

the mixing between the streams and an index such as Danckwerts's

intensity of se gregation 2 can b e co m pu ted . In the ideal case of a

non-invasive probe with an infinitely fast response and vanishingly

small resolution volume we obtain an indication of the thickness of

the striations passing by the point as a function of time. However,

even

i f

this were possible, the statistics o f the fluctu ation s would be

very complicated and hard to connect to the fluid mechanics of the

proces s itself. W ha t is worse, how ever, is th e inability of th e measure-

ments to give a correct global picture of large scale structures, the

so-called coherent structures (Roshko, 1976). In this case a flow

visualization study base d, for exam ple, on shadowgraphs is in f in te l~

more revealing with regard to the structure of the flow (Brown

and Ro shk o, 1974) (see also Figure 1.3.5; see color plates ).


21/387

A p p l i c t ~ t i o n s

ncl

yeometr ic .n l s t ruc ture

9

1.3. Applications and geometrical structure

I t Is

clear that even restr ict ing our at tention to f luid f luid systems."

miscible o r ~ m m i s c i b l e ,d i f fus ive or non-di f fus ive . reac t ing or not , the

scope

of mixing p roblem s

IS

en or mo us and it is no t possibl e t o deve lop a

complete and useful theory en com pas sing a l l th e ab o v e s i tua t ion s . It is

nevertheless

ev iden t t ha t . in sp i te o f t he eno rm ou s r ange o f l ength an d

t ime scales . the under lying geometr ical s t ructure associated wi th the

process o f red uc tio n of length scales is th at of F igu re 1.1 . I . In this sect ion

we highlight this aspect by means of a few examples.

Mixing is relevant in processes ranging from geological length scales

( ] O h

m ) and exceeding ly l ow Reyno lds num ber s

(10 -20) .

such as in the

mix ing p rocesses occur r ing in t he Ea r th ' s m an t l e , t o Rey no ld s n um be r o f

o rde r 10" co r r e spond ing to mix ing in oceans an d the a tm osph ere . ' * An

example of a s imula t ion of mixing in the E ar th ' s ma nt le is show n in F igure

1.3.1. in which the f low is model led as a two-dimensional layer heated

f rom be low.13 In these cases , ac tua l mixing exper iments a re of course

impossible . However , laboratory models of large scale c i rculat ion in

oceans can be carr ied ou t w i th l iquids in con ta iners p laced o n a ro ta t ing

turn table somet imes involv ing co mb ina t io ns of sources and s inks . F igure

1 .2 .1 , f rom the ear ly pa per by We lander (1955) . sho w s an exam ple ( it is

wor th not ing tha t the output of s imi la r exper iments , were descr ibed as

' ch ao t~c ' , .g. , Veronis , 1973) .

Undoub ted ly .

c.htlotic

i s an apt descr ip t ion of s t re tch ing in t ru ly

turbulent f lows. Figure 1.3.2 shows the s t re tching of mater ia l l ines in a

turbulen t , near ly i so tropic , f low ( Co rrs in an d Karw ei t . 1969) , where th e

e x ~ e c t a t i o n ' ~s tha t of expo nent ia l gro wt h (Batch e lor , 1952) . N ote ,

howevcr . the inherent l imi ta t ion of exper imental techniques in resolving

the smallest scales.

The deform at ion of mater ial l ines has been s tudied a l so in the case o f

chaot ic Stokes ' s f lows . Exper im ents focus ing o n d efo rm at io n of mater ia l

l ines were carr ied out by Chaiken

et t r l .

(1986) an d Chien , Ris ing , and

Ot t i n o ( 1 9 8 6 ). In t h e c as e o f Ch a l k e n

pt (11.

the f low consis ts in a n eccentr ic

Journal bear ing t ime-per iodic tw o-d ime nsio nal f low which is descr ibed in

det ai l in C ha p t e r 7 . F igure 1 .3.3 sho ws the shap e adop ted by a ma te r i a l

line of a t racer by the per iodic d i scont in uous op era t io n of the inn er and

outer cylinders in

;I

coun ter - ro ta t ing sense . N ot e the absence of ' corners '

"d 'branc hes ' in the folded str uc tur e even tho ug h th e f low is essential ly

d i s c o n t i n l l ~ u ~co m pa re wi th Figure 1 .2 .1

).

Figures 1

. ?. I a n d 1 .3 .3 sho w com plex s t re tched a n d fo lded s t ruc tures .

character is t ic

c,f

mixing in two-dimensional f lows. In both cases the


22/387

10 Introduction

structure formed is lamellar and an indication of the state of mixing is

provided by the striation thickness. How ever, in ot he r cases the structu res

obtained are considerable more complex. If the fluids are immiscible and

sufficiently different, interfacial tension plays a dominant role at small

Flgure 1.3.1. Deformation of a tracer In a numerical experiment of motion In

the Earth mantle The sides of the rectangle are Insulating but the bottom is

subjected to a constant heat flux while the temperature of the top surface is

kept consta nt. Th e motion is produced by buoyancy and internal heating

effects (th e fluid is heate d half from below an d half from w ithin) . Th e Rayleigh

number is 1.4

x

lo h , the time scale of the num erical sim ulation corre spond s

to 155 M year, a nd the thickness of the layer is 700 km . An instantan eous

plcture of the strea ml ~n eseveals five cells. (R eprod uced with pe rm ~s slo n rom

Hoffman and McKenzle (1985).)


23/387

Applic3utions ~r nd eometricul structure 1 1

k.jfurc

I .3 .2 .

Cirowth of

a

'mater ial l ine ' composed of smal l hydrogen bubbles

produced

by

a

plat inum wirc s t re tched across an decaying turbulent f low

behind

a g r ~ d l aced a t the ex tr eme l ef t. Th e R eynolds number based on the

diam cter is

1.360.

(Reproduced with permission f rom Corrs in a nd Karweit

( I 9 6 l l J . )

Figure

1.3.3.

Mixing in a creeping flow. Th e l igure sho ws th e deformat ion of

a material region in a journ al bearin g flow w hen it op era tes in a t ime-periodic

hishion lexpcriment from Ch aiken rt r r l . (1986)), for a complete desc r ip t ion see

Scct ion 7 .4 . (Re prod uce d wi th permission. )


24/387

12 Introduction

scales. Figure 1.3.4 (see color plates) shows an image processed two-

dimensional s t ru ctu re produ ced by mixing an d preserved by quenching

o f w o immiscible molte n polyme rs. In this case there has been a com plex

process of breakup and coalescence. As indicated earl ier, the character-

Figure 1.3.6. C on ce ntr atio n of a turb ulent round jet fluid injected into water

at Reynolds number

2,300,

measured by laser induced fluorescence; the cut

is alo ng a plan e including the axis of a sym metry of the jet. (Re produ ced with

permission from Dimotakis. M iake-Lye, an d Papa ntoniou (1983).)


25/387

Approuch 13

iwtion of this structure depe nds on the intended appl icat ion of the blend

and

a

large nu m be r of mixing measures ar e possible."

interp lay betwecn chemical reactions a nd mixing is now here mo re

ev id en t than in the case of fast reactions (C h a p te r 9) . In m any cases of

i n t e r e s t the flows are turbulent and careful studies have been carried out

to probe the interplay between the f luid mechanics and the t ransport

processes a nd by using prototy pical flows such as tur bu len t sh ea r

flows

an d wakes, perturbed o r not . Th e react ion itself can be used t o m a p

ou t the interface of reaction. F or ex am ple, Koochesfahani a nd Dim otakis

(1986; see also 1985) have used laser induced fluorescence an d high speed

real-time digital im age acq uisition techn iques to visualize the interface of

reaction between two reacting liquids undergoing a diffusion controlled

reaction (see also Ch ap te r 9) . A similar technique c an be used in th e case

of diffusing scalar. Fo r ex am ple, F igure 1.3.5 (see colo r pla tes) was

obtained by measuring the concentration of a fluorescent dye, initially

located in one of the streams of the mixing layer, whereas Figure 1.3.6

shows the mixing of turbu len t jet c on tain ing a fluorescent dy e with a clear

surroun ding f luid. Lam ellar s t ructure s (C ha pte r 9 ) are clearly seen, even

at Kolmogorov length scales (D imo takis , Miake-Lye, an d Pa pan toniou ,

1983).

Ano ther instance of interplay between mixing w ith diffusion a n d

reaction. but at smaller length scales and lower Reynolds numbers

(approximately 200 5 0 0 ) , well kn ow n in polymer engineering, is provided

by the impingement mixing of polymers (Lee

et

al.,

1980)

w h e r e t h e

objective is to mix tw o viscous l iquids (reactive m on om er s) in s hort t ime

scales (o rder 1 0 ~ 2 1 0 - 1) with reaction t ime scales of the o rde r of

10 '~-1o 2 . T he geom etrical picture is similar to the previous cases; in

th is case the mixing requi rement i s to produce s t r ia t ions of the order

of 20-501tm so that the reaction can take place under kinetically

condi t ions. 'h Mixing at even smaller scales might take place

d u e to spontaneous emulsi ficat ion (Fields , Th om as , and Ot t in o , 1987;

W icker t , Macosko , a nd Ranz , 1987).

1.4.

Approach

In spite of its overwhelming diversity, fluid mixing is basically a process

l n ~ o lv i n g reduction of length scales accomplished by st retching an d

folding of material l ines or surfaces. In some cases the material surface

O r line in question is placed in the flow and then subseqently stretched

le.%-

igures 1 2 .1 , 1.3. 3). in others , the surface is continuously fed into

'he flow (cg.. Figures 1.3.5

6) .

Thus. at the most elementary level (i .e. .


26/387

witho ut avera ging but at the c on tinu um level) rnising c.onsists of' stretching

lrr~d fbldii~~gf ,fluid filtrmc~nts,rrld distribution tllrou~ghoutspuce, accom.

panied by breakup

i f

the fluids ar e sufficiently different, an d sim ultan eou s

diffusion of species and energy (Figure 1.1.1

) .

In the most general case,

various chemical species might be reacting. We seek an understanding of

this process in terms of simple problems which can serve as a 'window'

for more complicated si tuat ions. Our approach is to combine the kine-

matical fou nda tions of fluid mec hanics with dynamical systems concepts,

especially chaotic dynamics. The objective throughout is to gain insight

into the working of mixing flows. The goal is not to construct detailed

mode ls of specific prob lem s but rathe r to p rovide pro totyp es f or a broad

class of problems. Nevertheless. we expect that the insight gained by the

analysis will be important in practical applications such as the design of

mixing devices and understanding of mixing experiments.

Mixing is also inherently related to flow visualization. However,

contrary to popular percept ion the 'unprocessed ' Eulerian velocity field

gives very little information about mixing and the typical ways of

visualizing a flow (streamlines. pathlines, an d to a lesser degree. streaklines)

ar e insufficient to com pletely u nde rstan d the process. As we shall see, our

problem begirls ra th er rharl ends with the specificurion of v (x , r). T h e so lut ion

of

dxidr =

V ( X , I )

with x

=

X at t ime

t

=

0.

x

=

@ , ( X ) ,which is called the flohv or motion,"

provides th e start in g point for ou r analysis. In even the simplest cases this

'solution' might be extremely ha rd to ob tai n. Actually. the impossibili ty of

integrating the velocity field in the conventional sense is the subject of

much of Chapters 5 a n d

6,

where the modern notion of irlteyrability is

introduced. The kinematical foundat ions l ie in an understanding of the

point t ransformation x = @,(X) .

We consider the following sub-problem s:

( 1 ) Within the framework of x

=

@ , ( X ) :mixing o f a single fluid o r similar

fluids.

Th e basic objective here is to c om pu te the length (o r ar ea ) corresponding

to a set of initial conditions. As we shall see only in a few cases can this

be done exactly and in most of these the length stretch is mild. The best

achievable mixing c orresp onds to e xpo nential stretching nearly everywhere

and occurs in some regions of chaotic flows. However. under these

con dit ions the (exa ct) calculation of the length a n d loca tion of lines an d

are as is hopelessly com plic ate d. As we shall see in C h a p te r

5

even extremely


27/387

simpliljcd

~ O R S

ight be inherently chaotic a n d a

corripleie

char ;~cter izat ion

is n o t

p o ~ i b l c . o r e x am p l e . f r o m a d y n a m i ca l s y st em s vi ewp oin t Re sh all

, ,,that i f th e system possesses horse shoe s we ha ve infinitely m a n y perio dic

and wi th i t the impl ica t ion tha t we cannot poss ib ly ca lcula te

;dl of t h e m For tuna t e ly . a s f a r

;IS

mixing i s concerned we are

intercstcd in low period events, s ince we want to achieve mixing quickly.

~ ~ ~ ~ ~ t h c l e ~ ~here is a lways the int r ins ic l imi ta t ion of being unable to

ca]culatc p ~ c i s enform at ion (m os t prac t ica l p roblem s involve s t re tch ings

of

10'

o r h ighe r ) such a s l eng th s t r et ch an d loca t ion of ma te r i a l

surf ;lccs, No te th at this is t ru e even th ou gh no ne o f the f lows discussed

in Chapters 7 a n d 8 i s turbulent in an Euler ian sense . Rather , the

previous f indings should be used t o es tabl ish the l imits o f wh at might

const i tute reasonable answers in more complicated f lows ( real turbulent

f lows come

immediately

t o m i n d ) .

T h e

prohlcm /irr.cl is /io\c, t o hcsr

~ ~ / 1 1 1 1 . 1 1 c ~ r c ~ 1 ~ ; : 1 ~/ ] o i ~ i - ~ i i ~ q ,t~o\vi t~qgf:)rP/lut1d /iiit u c o t ~ i p l ~ i ~~ / ~ ~ ~ r ~ ~ ~ ~

i,s ; l ~ l p o , ~ , ~ ; l ~ l ~ ~ .

( 2 )

Within the f ramework of

a

family of flows

x,

=

@ , , ( X , ) ,

s

=

1 ,

. .

. .

,1 ;

each of the mot i ons is assu me d t o be topo logical (see Sect ion

2.3) :

mixing

of similar diffusing a n d reactin g f luids.

This case cor respo nds to th e case of mixing of tw o s t ream s. com posed

ofpossibly several species th at ar e rheological ly ident ical , i .e . , they have th e

same dens i ty . v i scos i t y . e t c . , and have no in t e r f ac i a l t ens ion . Concur -

rently wi th th e mecha nica l mixing th er e i s ma ss d i ffus ion , an d poss ib ly .

chemical r eac t ion . H ow ev er . fo r simpli c it y , w e will a s sum e tha t ne i th e r

t h e d i f f u s i o n n o r t h e r e a c t i o n a f f e c t s t h e f l u i d m o t i o n . ' Th i s c a s e i s

d iscussed in C h ap te r 9 a n d c o r r e s p o n d s t o t h e c a s e of

lumellur structures.

( 3 )

Mixing of d i f fe rent f lu ids ; case in w hich the mot io ns a r e n on -

topologica l. i , e , . there is b re ak up a n d o r fus ion of m ater ia l e lemen ts .

In t h i s ca se . t he mix ing o f two o r more f l u ids l eads t o b reakup and

coalescence of mater ia l regions . This problem is complicated and only a

few special cr lses belonging to this category are discussed in Chapter

9.

by d e c o n ~ p o s i n g he p rob lem in to

I o ~ o l

n d

qlobiil

c o m p o n e n t s .

An out l ine of the orga niza t ion o f the res t o f t h e chapte rs i s the fo l lowing:

2

descr ibes t he k inemati cal fo und a t ion s an d C ha p t e r 3 presents

a

b r i e f n i c r v i e , o f flu id m e c h an ic s. W h e r e ; ~ s h e m a te ri al of Ch a p t e r

2

is

' n d i s ~ e n ~ i i b l e .;~rge p;lrts o f C hi lpt er

1

were ; ~ d d e d o provid e b ;l l;~ncc .

4

focuses on

;l

f e ~x; lmp]es which can be solved in deta i l and

ends

in

;I

riit11cr dcfc; lt is t no te t o pr ov ide a bridg e for th e st ud y o f ch ao s.

5 iwcrcnts ;I pener.11 d iscussion o f d y n i~ m ic ;~ lystems ; ~ n d h i ~ p t ~ r


28/387

16

Introduction

6

focuses on Hamiltonian systems. Similar comments apply in this case.

We use more heavily the 1,laterial of Cha pte r 6 but omission of Ch apter 5

would result in serious imbalance and a misleading rep resen tation of facts.

Chapters

7

and

8,

by far the longest in this work, give examples of chaotic

mixing systems in a n increasing orde r of complexity. C ha pte r 7 discusses

two-dimensional flows, Chapter 8 focuses on three-dimensional flows.

Chapter

9

discusses briefly the case of diffusing and reacting fluids and

active microstructures.

Notes

1 Obviously, on e o f the diffusing species can be tem pera ture, as is the case of mixing

of fluids with different initial tem pera tures o r processes involving exothe rm ic chemical

reactions.

2 A gross, but po pu lar , me asure of the concentration variation is given by th e intensity of

srqreqution, I . If ~ ( x )enotes the concentration at point x and ( . ) denotes a volume

average,

I

is defined as

[(((.-

( c ) ) ~ ) ] ' / '

(Danckwerts, 1952).

3 M any ofthese references inspired addition al wo rk. So me , however, were largely ignored.

4 Subsequently, this problem took a life of its own and much research followed. See for

example Rallison (1984).

5 Th is idea was not widely followed and m ost of the mixing work in the ar ea of d ro p breakup

an d coalescence in complex flow fields resorts to pop ulation balances where break up and

coalescence are taken into account in a probabilistic sense.

6 This paper contains many good ideas, however, it has remained largely ignored

by the mixing community.

7 In the past few years there has been a revival of this idea (e.g., Spaldin g, 1976, 1978b;

Ot tino , 1982).

8 The theory also found use in two-phase mixing. An early reference is Shinnar (1961).

9 The analysis of mixing of viscous fluids has been largely confined to polymers

(Midd leman , 1977; Ta dm or and Gog os, 1979).A follow up pape r, written in the context

of the m ixing of glasses. is C oo pe r (196 6). Even tho ugh C oop er's trea tment of the

kinematics is at the same level as Mohr, Saxton, and Jepson (1957), there is

substantially m or e, since it deals explicitly w ith mass diffusion. This pa pe r, how ever,

has remain ed largely ignore d.

10 This app ro ach wo rks well for pre-mixed reactors with slow reactions, but it is not suited

for diffusion controlled reac ions, such a s in com bus tion. Curiously enough, the

par ticipation of chemical engineers in subjects dealing with non-pre-m ixed reac tors has

been relatively minor a nd in spite of comp lex chem istry , the are a has become largely the

do m ain of mechanical a nd aerospa ce engineering researchers. See the discussion ( pp. 100-

102) following the paper by Dan ckw erts (1958 ). Mu ch w ork followed alo ng these

lines. For a su mm ary, see Nau m an an d Buffham (1983).

I I

W ith the possible exception of the very last exa mple of the last cha pter . we do no t consider

fluid-solid systems.

12 See for example Veronis, 1973: Rhines. 1979; 1983. and the articles by Holland

('Ocean circulation models' . pp. 3 4 5 ) . Veronis ( 'The use of tracer in circulation

studies '. pp. 1 6 91 8 8) , an d Rhines ( 'The dynam ics of unsteady currents ' .

pp. 18%318), in Goldberg er

a l . ,

1977.


29/387


30/387

Flow , trajectories, and deformation

In the first part of this ch ap ter we record the basic kinem atical fou nda tions

of fluid mechanics, starting with the primitive concept of particle and

motion, and the classical ways of visualizing a flow. In the second part

we give the bas ic e quat io ns for th e d eform at ion of inf ini tes imal mater ia l

l ines, planes, and volumes, both with respect to spatial , x , and material ,

X, variables, and present equations for deformation of l ines and surfaces

of finite extent.

2.1. Flow

T h e physical idea of

flow

is represented by the m a p or point t ran sforma tion

(Arno ld, 1985, C ha p.

1

)

x

= @ ,(X) with X = @,=,(X), (2.1.1)

i.e., th e initial con diti on of particle X ( a m ean s of identifying a point in

a continuum, in this case labelled by its init ial position vector) occupies

the posi tion x at t ime t (see F ig ur e 2.1 .I ). W e say tha t X is mapped to x

after a time

t . '

In con tinuu m mechanics (2.1 I ) is called the

motion

and is usually

assumed to be invertible an d differentiable. In th e lang uag e of dyn am ical

systems a map ping

@,(XI

+

x

(2.1.2)

is called a

Ck

diffeomorphism

i f

it is

1-1

a n d o n t o , a n d b o th

a,(.)

n d

its inverse are k-times differentiable. If k

=0

the transformation is called

a homeomorphism. In fluid mechanics, k is usually taken equal to three

(see Truesdell , 1954; Serrin, 1959). Also, the transf orm ation (2.1.1) is

required to satisfy

or alternatively,

J = d e t ( D @ , ( X ) )


31/387

whert D d e n o t e s t h e o p e r a t io n i, ),/ax,,. e . , de r iva tiveswi th r e spec t t o t h e

configurat ion, in this case X. I f the Jacobian

J i s

equal to one

flow is called

isochoric.

~ h cequirement (2.1.3) precludes two par t ic les , X , an d X,, f rom

occup) , ing the sam e posi t ion

x

at a given t ime, or one part icle spl i t t ing

into ~LI 'O :.e. ,

notz-topologicul

mo t ions such as b reak up o r coa lescence a re

not a l lowed (Truesdell an d T ou pin , 1960, p .

510).

I n

the lang ua ge of dynamical systems, the set of diffeomorph isms (2.1.1

for all particles X be lon gin g to th e bo dy Vo is called the flow (i.e.,

a

one-p aram eter se t of di f feomo rphism s) an d is represented by

r 1 - 1

, X I

-

@ , (X I ; = @ , ( X i

where X ) is the set of par t ic les belonging to V (Vo

=

{ X } a n d

V, =

( x ) ) ,

Thus. we say that V,, is m ap pe d in to Vr at tim e t ,

f V 1

- @

rv

1

i ri -

r t

01

or that the materia l l ine L o is mapp ed into L, a t t ime

t ,

( L , ) = @ I L o : .

Figure 2.1.1. Deformation of l ines and volumes by a flow x

= Q , ( X ) .

motion

-


32/387

20

F l o w , trujrc.tories ur~ tl lt$)rmtrtior~

Note that f lows can be composed according to

@,

+, (XI

= @,(@AX)) ,

i.e ., X is taken to pos i t ion @ ,(X) an d then to @ ,+, (X ) .T he f low can also

be reversed,

@ ,-,(X I = @ , ( @ - , ( X ) ) = @ l ( @ l - l ( x ) ) x

i .e. , X is taken to x and then back to X. '

2.2. Velocity, acceleration, Lagrangian and Eulerian viewpoints

The velocity is defined as

a n d it is the velocity of the particle X . T he a cce lera tion , a , is defined as

a

-

c ' . ' @ , ( ~ ) / i ? t ' ) I ~a ( X , t ).

Any fun ction G (scala r, vector, ten so r) ca n therefore be viewed in two

different ways:

G(X,

t ) =

Lagrangian or mater ia l

i.e., follows the motion of a particular fluid particle, or

G(x , t ) = ~ u l e r i a n % r sp atial

i .e. , the pro per ty of the particle X th at hap pen s t o be at the spatial location

x at t ime t .

Thus , v(X, t ) is the Lagrangian velocity and v(x, t14 is the Eulerian

velocity. In most classical problems in fluid mechanics it is enough

to

ob tain the spat ial desc ript ion. Th e material (o r Lag ran gia n) derivative

is

defined as

DGIDt

-

('G/?t)1,

representing the cha ng e of pro perty G with t ime while following the mo tion

of particle X, whereas the standard time derivative is

?G/?t = (?G/?t)lx,

an d represents the chang e at a fixed position x. T h e relationship between

the two is easily obtained from the chain rule

DG/Dt = ? G / i t

+

V - V G ,

wh ere V is defined a s V =

(?/?.u,)e,

(see App endix). Th e expression allows

the comp uta t ion of the accelerat ion a t (x ,

t )

without compu t ing the mot ion

first.

T h e expression for the m aterial t ime derivat ive of the Jacob ian o f the

flow is known as Euler 's formula (Serrin, 1959,

p.

131

;

Chadwick,

1976,

p.65).

DJIDt = J ( V . v ) = J t r (Vv)

and is the basis of a kinematical result known as the trtrnsport theorem.


33/387

Veloc,ity,acceleration, Layrangian and Eulerian

consider the integral

1,ix,

t )

d i

where V, represents a m aterial volum e, tha t is. a vo lum e com pose d alw ays

of the sam e particles X belonging to the b ody Vf. M ak in g reference t o

Figure

2.1

I

:V , ]

= 1 : V,}, i.e. . the flow

0,

ransforms { V , } in to

{ V f }

a t

time t . Th e integral ca n be w rit ten, using the definition of the Ja co bia n, as

r P

G ( x ,

)

d c

=

J G ( X , t ) J d V

where d

V

represents a volum e in the reference conf iguration V,,. T he t ime

derivative.

(which is a material derivative since all the particles in

V,

remain there)

can be written as

and since the dom ain V is not a funct ion of t ime, exp and ing the mater ial

derivative we obtain

: I",

(x.

r )

d r

= jv

&

+

G( V.

v)]

dc .

By

means of the divergence theore m.

where n i s the outward normal to the boundary of

V, ,

deno ted ?v , . In

general this result holds fo r an y ar bit rar y co ntr ol volum es V; movin g with

velocity v , ,

Problem

2 . 2

.J

Show that

i f

A (r ) is invert ible, d(det A)/dr

=

idet A ) t r [ ( d A / d r ) ' A ' l .

Problem 2.2.2

that

i f

I

I ; /

=

e l [

~,]

hen

; i v , \

=

@ ,; iV , , ] , .e ., the b ounda ry is

mapped

into the hound;iry. Us ua]]y, this is taken to be that the surfiice

Of

a ma terial bo dy consists o f the siime p;irticles (v on Mises an d


34/387

22 Flow , trajecto ries and defbrmution

Friedrichs, 1971). Th is result is called 'Lagrang e's theo rem ' by P ra nd tl

and Tietjens

( 1

934, p. 97 ).

2.3. Extension to multicomponent media

In th e case of multicom pone nt media we envision m aterial surfaces m oving

with the m ean ma ss velocity (see C ha pte r 9 ). If th e system h as several

components , s = 1 , . . . , N , we assume the existence of a set of motions

a?),

nd Equation (2.1 I ) is generalized as

X, =@?)(X,) (no su m )

where X, represents a particle of species-s and x, its position at time t.5

Each species is assigned a d ensity p,

=

p,(X,, t ) such that p

=

x ,, where

the sum runs from 1 to

N .

Individual

velocities

are defined as

v, - a@ls)(x,)~at)l ,= v,(x,, t )

and the average mass velocity is defined as

v

-

x

P, /P)~ , .

T h e time derivative of an y function G following the m otion of the species

denoted s, is given by

DG'"/Dt = aG/at + v;VG,

and the relative velocities are defined by

Us= v , - V .

The simplest constitutive equation for

us

s

us = - W, DVO,

(i.e. , dilute solution o r eq uim olecula r counter-diffusion, B ird, Stew art, and

Lightfoot, 1960, p. 502) where

w ,

is the mass fract ion (=p/p,) and D is

the diffusion coefficient.

O th er q uantities, such a s individual deform ation tensors for species-S,

etc., can be defined ana logou sly (B ow en, 1976) but they a re not used in

this work.

2.4. Classical means for visualization of flows

There are several ways of visualizing a flow. In this section we record the

three classical ones.

2 .4 .1 .

Particle path, orbit, or trajectory

Given the Eulerian velocity field v = v(x,

t ) ,

the particle path of X is given

by the solution of

dxldt = v(x, t ) with x =

X

a t t = 0. Physically it


35/387

Cla.ssicu1 meuns ,for visualization of Jows

23

orrcspo nds to a long time expo sure ph oto gra ph of a n il luminated f luid

p a r t i ~ l ~ .

AS

seen in texts of differential equa t ions, the solut ion t o the a bo ve

x = a , @ ) , is unique a nd co nt inuo us with respect to the in itial

data i f v (x ) has a Lipschi tz con stant ,

K

> 0.' Under these condi t ions, i f

we d en ote x l = @ , ( X I ) a n d x 2 = @ , ( X 2 ) ,we have the trajectories evolve

according to

I x -

x21 G IX1- X21 ex p( K t) ,

K > 0.

AS we shall see there ar e man y systems (C ha pte r

5 )

tha t diverge from the

init ial condit ions at an exponential rate, i .e. , the non-strict inequali ty

becomes a n eq uali ty. '

2 .4 .2 . Stveamlines

The st reamlines corresp ond to the solut ion of the system of equat ions

dxlds = v(x,

t )

where the time r i s t reated as constant and s is a parameter (that is, we

take

a

'picture' of the vector field

v

at t ime

t ) .

Physically, we can mimic

the streamlines by labelling a collection of fluid particles and taking two

successive photographs at t imes t and t + At. Joining the displacements

gives

v

in the neigh borho od of the point x. Th e st reamlines are tangential

to the instanta ne ous velocity a t every p oint , except a t po ints where v = 0.

2.4 .3 .

Stveaklines

The picture at t ime t of the streakline passing through the point x ' is the

curved formed by all the particles X which h app ene d to pass by x ' dur in g

the time 0 < t ' < t . Physically, i t corresponds to the curve traced out by

a non-diffusive tracer (i .e., the particles X of the tracer move according

to

x =

@ ,( X )) njected a t the posi tion x ' .

Example 2.4.1

Com pute the pathlines, s t reamlines, an d st reakl ines corresp onding t o the

unsteady Eulerian velocity field

v 2

=

1. 8

1 =

x l l ( l + t ) ,

To co mp ute the pathlines we solve

d.u,/dt = x 1 / ( l

+

t ) ,

dx2 /d t = 1

with the condition x,

=

X , ,

x, =

X,, at t =0.

.Y ,

= X l ( l

+

t ) ,

s, = X, +

t

and, el iminating t , we obtain

.Y1 - x 1 x 2= X 1 ( l - X2).


36/387

24

Flo w, tr~ljectories l n d eformrrtion

i .e. , the p articles mo ve in s traigh t l ines. T h e streamlines ar e given by the

solution of

d x , ~ d s

x , / ( l +

t ) , d x , / d s

= 1

with the condit ion

x,

=

x y ,

x,

=

x ; ,

a t

s

=

0,

while holding

r

cons tan t .

Th us, the s treamline passing by

x,

=

xy, x ,

= x;, is given by

x ,

=

xy

exp[s/( l +

t ) ] ,

0

x, = x, +

s,

and e l imina ting the pa ram ete r s,

0

( 1 +

t )

n ( x ,

x y )

=

x ,

- x,,

which show s tha t the s treamline s are

t ime dependent .

T o get the s treakline

pass ing through

x',, x;

we first invert the particle paths at t ime

t'

X I

= x i / ( l +

t ' ) ,

X ,

=

,u;

-

' ,

which indicates that the particlt

X , , X ,

will be foun d a t the p osition xi , x;

a t t ime t ' . T he place occupied by this par ticle at an y t ime t is found again

from the par t icle path as

x , =

x',( l +

t ) J ( l

+

t ' ) ,

x, = .u; -

'

+

t ,

and is interpreted as: the particle which occupied position

x;, x;

at t ime

t'

will be found in position x,,

x ,

at t ime

t .

Eliminating [ 'we get the locus

of the streakline passing by

x',,

x;:

x , x ,

-

x , ( l +

x ;

+

t )

+ x ' , ( I

+

t )

= 0

which shows that the s treaklines ure ulso ,functions

of'

t ime (plots

cor responding to th is example are given by Truesdell a nd To up in, 1960,

p .

333) .

2.5.

Steady and

periodic

flows

A flow is steady i f i t lacks explicit t ime dependence, i .e. , v = v( x ) . N o te

that the concept of steadiness depends on the frame of reference.

An

unsteady f low in o ne f ram e can be s teady in a no the r f rame (mo ving

f rames are s tudied in C ha pt er

3 ) .

W he n the flow is stead y in a given frame

F,

the stream lines an d pathlines coincide when viewed in the frame F .

Fu rthe rm ore , the s treaklines coincide with bo th s treamlines an d pathlines

provided th at th e position o f the dye-injection ap p ar at us is fixed with

respect to the frame

F.

I t is obvious that these statements are a property

of dynamical systems in general and they are not confined to f luid

mechanical systems.

A point x such tha t v(x)

=

0 for all

t

is called a

,fired or ,siri


37/387

Ste ad y and periodic f l o w s

point in fluid mechanics). A point P is periodic, of period

T,

i f

P = @,.(P)

for ,

= T

but not for any r

t l l

~ i h c r eG is an y sca la r , vec to r . o r t enso r func tion

3.2.

Principle of conservation of linear momentum

/rttcgrul cersion ( o r Euler's u x io m )

This

pr incip le s ta tes tha t fo r

a

m at e r i a l v o l u m e

V ,

l i n ea r m o m en t u m i s

conserved. i .e . .

rate of

c h a n g c

of forces

; ~ c t i n g

n

rnomcnturn

body

jb,

~ v d i .

= jL,

f d i .

+

4

b , ,

t d . s

(3.2.1 )

d t

;~ccclerat~on bod

Corccs

contact

Corcca

\ i llere t (n . x .

t )

is a s yet a n undefined vector cal led the

t r r r c l i o n

which

d ep en d s o n t h e p l acem en t x o n t h e b o u n d a r y ,

CV,,

the in s tan taneous

o r ~ c n t a t i o n .n, and t ime ,

t

(see F igure

3 . 2 . 1 ) .

The vec to r f i s the body

force . which in th is work is assu me d to be indepe nden t of the conf igu ra t ion

o f t l i e b o dy ,

V,.

Using the t r anspo r t theo rcm (Sec t ion

2 . 2 )

w e o b t a i n

k > r a n y e gio n

V, ,

a t a n y t im e ( w h e r e

V,

c a n be a n a r b it r a ry c o n t r o l v o l u m e ) .

Pr.ohlcm 3.2.1

f 'l.o \e tha t t he Pr inc ip le of Co ns er va t io n of L inear M om e nt um implies

the co n t inu um version of 'New ton ' s th i rd law ' : t ( x . n , 1 )

=

t ( x . n . t ) .

3.3 Traction t (n , x , r )

It

can be proved tha t t (n .x . I )

= T'

e n ,

where

T

is a tensor (n o te conv ent io n) .

T h ~ sm p li es t h a t t h c i n f o r m a t i o n ab o u t t r a c t io n s a t t h e p oi n t x a n d an y

' . l l~f i~cc i th no rm al n i s con ta ine d in th e tens or T. Th e proof cons is t s of

t ll rcc s t eps which \ye will briefly repeat here.

( i )

Cuuchy's theorcm

( 'ons ide r a ( smal l ) r eg ion

V,

su r round ing a pa r t i c l e

X .

T h e n , if L r e p r e -

h en ts so m e l en g t h s ca l e

o f V, w e

h a v e :


56/387

14

C'onserlution rquu tions, chun ge of ir u tn e , untl lwrtic.it.r

V o l u m e (

V,)

= c o n s t . ,

L"

Surface c ?

V , )=

const . , L2

T h e m e a n v a lu e t h e o re m s t a te s th a t f o r a n y c o n t i n u o u s f u n c ti o n G ( X ,

I )

defined over

C

a n d i?V, we can f ind X ' and X" (a t any t ime

t )

s u c h t h a t

I-

J,.

d o

=

c o n s t . ,

L-

G ( X f ,

) ,

where X ' be longs to V,

(; d s = con st . , L2 G(X", t ) ,

where X" be l ongs t o ?V,

F~gure .2.1

( t i )

Mate r ia l

region In

present configuration.

V,,

ind~cating o r m a l

n a n d tri tctlon I;

( h )

cons t ruc t ion for balancc of a n g u l a r momentum.


57/387

Truction t(n , x , t)

Applying the theorem to E qu at io n (3 .2 .2) we obta in

D

~r

( v ( X 1 , )

cons t . , L")= pf cons t . , L

Thcn, dividing by

L2

and le t t ing

L

go to zero (preserving geometr ica l

similarity) ,

Th at is, the tract io ns are locally in equil ibr ium (see Serr in , 1959, p. 134) .

( i i ) Traction on an arhitvary plane (C au ch y's tetvahedvon constvuction)

By me ans of this co nst ruc tio n, which we will not repeat here (see Se rrin ,

1959, p . 134, for de ta i ls ) , Cau chy was ab le to prove tha t the com po ne nts

of the traction t( n ), [t([n])] =

( r n l ,

rnZ, n3 ) , re re la ted to the n orm al n ,

I n ] =

( n l ,

,, n,) , by the ma trix mu ltip lica tion : '

I " '

= 7 ' . . ) 1 .

11

I

o r equiv alently [ t([n])]

=

[n][T], [t([n])]

=

[T1][n]

whcrc the brackets [ ] represent the display of the components of the

vector n and the matr ix representat ion of the tensor T.

(iii) The third step is to prove that [TI is indeed the matrix

rc~presentutionof a tensor T (see Appendix)

I n

order to prove tha t the T,,s a r e the com po ne nts of a tensor T , we need

to prove tha t the components Ti, transform as a tensor. Since n is just a

frcc vector, which is objective,

n' = Q - n ,

a n d since the tract ion t t ransforms as

t ' = Q . t ,

then. T transforms as '

T ' = Q . T . Q T .

Problem

3.3.1

( Jsc Cauchy 's cons t ruct ion in conjunct ion wi th the 'mass balance ' t o show

that the mass f lux in the direct ion normal to a plane n is given by j-n.

I+

licrc

j

is the mass f lux vector. This allows the definition of the Eulcrian

\clocity a s v

=

/p which can now be regarded a s the primitive qua nti ty

cambridge texts in applied mathematics j. m. ottino-the kinematics of mixing_ stretching, chaos, and...

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