The stability of parallel flows of fluids with memories

(1)

' I1 ''5 ,. , .t 0- --?pr,...Te-' r.,, '1.11.;:cJ.-?J ....-.4.-- ',.' l'_',,, _;,.

/ "

1.);'''-'.2';EtiL- ' I .t-;'. ' - ,...t. , 5 , , , , . 1 ,,,,k, ., i 1 ; -..., .J\ . A ' ...,,N. ,s,,j, , "...I ..A -, ,le - Y . .' 4 ' 4 _. _1., -- ,1 9' , '

f1), _ I T,,.1,ar_A.k.. `a"s`l ,.,r,. '- , _. , ca, 1, \ . ,.. 't _21 , "4.__.". '' 1 ..t. ' . , .4

'

''',r,...,t. ,

.. A I ,{1----,, , - :. , ,

,

,,' e-,---q, , Nri i.r,-,----, ,, _{- A} .

- . . '' -.) -1, , , ,ti ,,, J ....

'

,,:q.-1 , -.,

'

- ,,' ' ',' ' , ° '..;; _'-'., _-:. 065Q5-3-T . _' _-'ff''".. ' ' .,.. ., k "lc. a - f\-ei ,` ,,,, ,JL r i` . ., V ' 1 I. ) : . ,1,:.,.% 4. ,,..,, ' 1 I , ..4 'r ? ' _, 1 k ' '. ' ' F, '1 I I-'4' L . , ,. . .,-, -,,,I'

a"1.]-\\)r, '

' 1-, - _,.. , pp. -,. _).-,,,, ,,,..,./., " N .1.. . t k Vy. ) -,., N; I 1 r . ' k ' \ ' s L. _'4,, s'i Y`i :',,, '' r ' . ...., -1,-,p,, 0 _- _{e F,'} , ... _{; ie,} _'''.0 11 '-'' '4 \ '--- Y: 1 - '1 : r' _v 1.= ;-'56 ..., - .,..._ H

OFFICE OF RESEARCH ADMINISTRATION

ANN ARBOR

0

, ..-I- .',*" - ', ' ,,.,,,,,,

1 I 4' ..- '-t-' . -,, 1 iaktAlri,.--,ttrAftr ,I.S, 1.." I. \ rp ' , v.., ' '1,±4,_,, 7 p 'II ' 41, yr, 4

' - ., , , r i ,,, j.._, _, ] ,. 1 ',: , .4,.1 Y 7 . .. I t. ' 4

\II ,1 ...., I. I COLLEGE OF ' ENGINEERING '.;';'j 1'1'; t'r '.--. _{''.:7'-- '-'-} _.._, - -

''' ''''''

, _''. _{L'' 4'},,,,

;.).0t,,,,, ,,, ,

-' -' ' t1.4(''' ';:, '4 ' --.;,e, '''' 4 '-' ',..; . 4.''...i1

%'!' 0- ',s.-'-,-., Y '..%'' _{DEPARTMENT OF ,ENGINEERING ,MECHANICS i's'} r_

.S:r(lb 17 ' -.?.: ' - '' - ' '- i ' it' 1. ''-'' _I , w " 1,14 ..1 r,e ,, i . . -, Vf , I a ' ';1' \-r '.,,,' i* ,ks- 4 ..P \' r -, . I 0 10 :' , ) _,I ' , ''

---,,,

''' r -j I. 7 1._`..`t'llc- *, 1 - * ri " 1- ' ,t, , - 1 , -A, v tr-i,,,, 1 ,.,. , ... , . ... 01 , ,' ..- ' -:. 1 .r1 ' - A., i . ( -/ ,,,,,:', i 't e -,' tr -.R. ..,1 111:1:4' J-I,', ,p it _{3 rt' ''.p,s i 1} _'1;`:.N''' , 4 . ,

Technical Report 1",',--Y .-, - , -....

- . 1 4 I - .-1. f,t ,-,I , .4 p ,k f I kr -.' s; V "t!... or ,` ,,... 1,C. o ` _a' -,,, 1 . , ,, , , = '"'N. c '''- 1 r i .,,,_1'-'0 41>:,- " ' - 5 tv --,"' r.: ; .,, -- , ' -I. . '. p . i ., 0 , . .4., oi I .-, t l .1 ,Y, o 1,

' ),. ,, '1 ' s-i4 I,* l',. ---' .,k.i,,-(r.. .--\ . , '

..!.1::"I'#' 441:-Iif,

The Stabiliti'af 'Parallel Flows

' :_ -, .:. .

t ...

I r'l - t 6 ''"1, '1 7

`) . '

'-,' A. _-,k ,,- 4 ''_ 4;4:

.1../ . , t , .., ' , ./ ,

/' '',`s, '''

1:- l',1/4" N 1,1,o f Fluids "4/ i th !. Me mori

es ,t'-',,,

'-''- -,.,,,,, : ' ,-v'''-. . .. ._ .. ,. - ... , ,,.t. ( .., ,, ,,, ... , ,,, , , . _ , ,, ,,,. , ,,, iI s ' , .I.., 41, 4. ',

- ",.

j : _-'. \ ''t. 1a _{1 : jj...'')! .} .'''

"'

''..41 ,i- : '1: a ''' ,', --'4''.,

,,,,, ,, -,,Y ,./ :§-,- ,,,,l_ 2 ,/,,,,-. _ :',..4' 1,P ,.., ',.11''',4-.'' ' ' I', ' ,-Y!/ ,,7k '..,,1,1,1. .7,, :1:, 41'v :A t ,, 1--'' ,.' ,_ ) a ri- ',1ci '`f ,", ' .,,A

' r ',(' ''' . i i . e.t. - ',(::- -.7. --t' , ..- ..e rl _-_ Vit-.71',if, (1,, / 1 .) , ,, li, - , z,,

i ' '? Jrt_ ' '

r " -

--1-til", li {- - k , ' . v 1,I i`-, . ,' t; -,1;1 'f" - ..-N af

'' '71i'. -';:i.s., A_--,,-y1,1'".r ..: 1 A .60_' .)'_'t-it.z.'''4.4. ;,. 7- 9 ,41..'"'71, r,,i',.-t, i . r, ^ :'- ' 4I ' fl ' 6 '' ,,N't, \ , ', q r.Y ,-,' ,/, ,1-' x.t 'i -;.. .4:,,..;.., ' i,-,, . . 4 I . '

, ,' .4, ,,s. - , 2 ' , ,,; " r _-I", -, j.14 . '

T " ,

_.-, 0, ''' C 1. ,i V - y V I i

,

' Iii ' I:N ':".:(., rtlfr. -,,' IA,V,r '',.1 ..q. jif i,'' rti.f._.,. r, -'-,, f '',.reg,-- r . -,,,.,...- , - - .: /- 1-dr, -/-- A-7.,, ,,- i.1`. '.p, : r-'7<,4- ,i , 17 7., 1 1 , I, Ai ','-_,, ',,/t ;_. , ; ,.. ---71 , t .. -y-,.. .

y , i. t , ,1 , I L, ,,,, --7,,, i -, - , 'N.. ,. 4: i. -.4- '13 .' ,2'.0. A I -i. sk,, 7,L_ ''',. ' '1' 7' ' .",) - 'N', 1,; ' : ' ' 'ff ' 1.. ) - ,

' 71:4). " ' '1' A .: `f e' veht I.- - 'r '.,! -. _ . '', '2.-1". j` " ' '. %.' ,.4i,' ;-iii-,s; (V),, -, - . , - i t. . .44';' -7-(4

x

7F-' l 4-'''' t;-: . Y4., .1;:, I/, LA . , ilS

,-' ' f 1 )''' 1 -7 1 'f ...

- " \

? ''',',,N i .t1, DEAN T.' MOOK , '' -- ..I.I., "'A' _{'.'1`1- 1--} ' _4-',-`r1 F-h--.):. , , )4'1=14', 'k., ,t41,....,.. - ,. ?''J,11-7 h. -,:k1+-', <4 / - - ' ' -"'f+ - i,, A - . / 1. ' :I - tr - si''

,-..4.1"Adt_ To - - ' , , . W.. P.' GR AEBEL; PROJECT DIRECTOR

"I''''- ',.' -.5c7. 1.'"), .,f.' ' t- ( , , k I

(.., , , , ' 1 ' 'a I ''" .

,-Y-'; II.,' I9-4 :I ', 4: ) fjs, 7,:: ;-al ' 'iN'' .1" ,,,V.:95 i - ' ' ' ''

,, ..., I. , '' r' It' ( ; . kt '. '' ' . '.' \ ', I. 'yr, ' r. OV., 4, .' i I- ' , ' !I "I r ;" s ''; - A .,, ;,.. - ,;;, 'x ---', - , '' '

kli ;

' '1 'IL r' '/:-.- ''.,t r e;41.1. 4,,,,,,,. v , ...; ,, ., . . , - ,y. ,. ,, , t"` ' a-...1 a p . ' - ." .. , i c (-4\ ,. ' ,',74-41 ,-2,, 4-... 1, . r i',',4' -1 '' A 0 . -,.., { : ' ..i. - ' ' -' , ' , --'' Or '-.- k,v,.:' r4 '1,7'. 7 1 - a raj:, ' T'' '' 1 ' '' ' (' I MY: 1, /_.-( V4 ' ' ' ' - 1/ 4- ' ' -a. i.\- ,.1, , . , .v,,, : .; --), , - 4)- ,, ,. ,,'-,, 1\ k-7:-,-; -,,k1 ,': ,-. , ,,,,4_;,,.l I ,.. ,.. 1,P, A y , 4: ₇ , r I ,- ., C ''VN l'1,-, -. ! ;'' I I ' , ',gri ' t. .4 ').. f , . ' ' P* ' et . ' I 1 .-., '-p ,,,,-,_ ,---.,,,, , x f, _N. ' ' '':') fs,r1")-7 i .' .,,I, 1 ' I, r,,,,, I. . - ,4_,..i. 7-"Yo --' ... 7 , . 1, ', i .t r 4 , r-': '-/- I' f ,, i I " !11-1 \z4,.7 la

-/ Under contract with -, . 7%

II . ... r ar-'.., , 7.- ..*/ ,t- .., , , If C 1 I ,, -J \ ''.::,., ' . . I . , ' L'' ' J I a r I ..;' I I , 'S. , P (. .1 P. V I I, . ..a'i ';',.--,e 4 7 I , Or. ;',. trr., , ' .1 , 10 4 y ,t t . . ' -.1":f;-r - ''r I -' 1 ' '' . : .1 ,

Departmenti of the Navy ,:., c,,:', (_,..,, . ,.,--,, ,r, ,-,-;,

t.

;

,.., . ,

...1,, .- 1. / -',4 - i. L4

.

( '''' * v Office/ of Naval' Research

- 1 1 -. '-' , ' .. .,. , , . ( ' - A 1a a , , ,-, i_

a -* Contract No. Nonr-1224(49), NR 'No. 0 -342

I -' 62 1 ,-.E'.,,,,,, - , r','T - r' ' t. f,,,,'. ' 1 s )3C 1.,/q%, , , , ''' -' 1 I ,, I-, -- / $ Washington, D. C. ) _'', -' k, ." $4)' p ,i--.... t t - _{'-"g- i-t^} p . -'. ; 4 ,.1 -a- _,/ ₄ , a '4.',ag '' _I _' ' 'T 'II'. a) iIr 4 - " , 1 , 4i 1 , I/ S.4- ,--a- , \ - Pr 4. a '''',1 ' Vrki7 . ' ' .. --rl: ''' Administered through::'' '7; ---.>.

),'

-.,,. '1 , September,1967:Afl.),-' . ' t .4 1\ 1 c 1 - 17,4 IL 1 - t 5k - . ' ' I A : 1 ; kr.;

(2)

Wctmi.44-"Pneriei

United States of America

Department of the Navy

OFFICE OF NAVAL RESEARCH, BRANCH OFFICE

Keysign House, 429 Oxford Street

London, W. I, England

ONE 3930/1

- .'

(3)

.--THE UNIVERSITY OF MICHIGAN

COLLEGE OF ENGINEERING

Department of Engineering Mechanics

Technical Report

THE STABILITY OF PARALLEL, FLOWS OF FLUIDS WITH MEMORIES

Dean T. Mook

W. P Graebel

ORA Project, ₀₆₅₀₇

under contract with:

DEPARTMENT OF THE NAVY

OFFICE OF NAVAL RESEARCH

CONTRACT NO, Nonr-1224()i-9), NR NO. 062-342

CHICAGO, ILLINOIS

administered through:

OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR

September 1967

(4)

unlimited-TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS iv

ABSTRACT

I., INTRODUCTION 1

II. GOVERNING EQUATIONS 8

III. ASYMPTOTIC SOLUTIONS 20

A. The Determination of the Characteristic Equation for 6

Nailier-Stokes Liquid 20 0

B. The Determination of the Characteristic Equation for a. 1

Viscoelastic Liquid of Type A' or BL 29

C. Solution of the Characteristics Equation 31

D. the Stability of a, Viscoelastic Liquid of Type C' 36

BIBLIOGRAPHY 42

(5)

iv

LIST OF ILLUSTRATIONS

Table Page

1. Velocity and Stresses for Primary Flow of Materials A', B',

and C' 10

Figure

Neutral stability curves for various values of N.. ₃₇

Critical Reynolds number versus the parameter R1. 38 Neutral stability curves and curves of constant c for various

(6)

The Stability of Parallel Flows of Fluids with Memories

D. T. Mook

Virginia Polytechnic Institute

and

W, P, Graebel

The University of Michigan

Abstract

The equations governing the stability of plane parallel flows

are developed for three models of fluids with memories. Asymptotic

solutions valid for large Reynolds numbers are obtaired and the effect

of the memory are shown to be destabilizing. The approach to the

problem allows evaluation of how fast a memory must fade to allow

evaluation of the stresses in power series in the time interval, An alternate approach to inverting convected derivatives is also presented.

(7)

-I. Introduction

A recent paper by Chan Man Fong and Waiters (1965) considered the stability of parallel flows of two visco-elastic fluids with very

short memories The present work extends their analysis to such

fluids with long but still fading memory and also extends the analysis to a newer model which has been proposed by Goddard and Miller (1966). A discussion of the various convected derivatives is also presented in a manner which allows more ready physical interpreta-tion as well as a quicker way of obtaining forms for convected integrals.

Quasi-linear models of visco-elastic fluids are usually written

in the form

E

LM

+

E." L)

z:

where

is related to the total stress t by

t-

I-p

d is the rate of deformation tensor, and

Lt is a time derivative

operator satisfying the principal of material indifference,

constantsXn and r

are related to the stress relaxation times and

rate of deformation relaxation times, respectively. Several forms

for these operators have been proposed in the past 1See Oldrovd

(1968) for a review); we present here briefly three of these definitions in a somewhat different manner which facilitates their physical

interpretation

If 0 is a convected material reference frame, and -la is a set

of covariant base vectors defined by _{(r a position vector) so that}

aea

(8)

Chapter 3)9 then a second order tensor can be written as

or

=

Tr yP

_4r3

ya being the contravariant base vectors defined by

eV.

The absence of a dot or cross between two vectors indicates the

indefinite, or dyadic, product. Oldroyd (1950) proposed two separate

definitions for tt; denoting time differentiation with material

coordinates held fixed by D/Dt they are

yP

OtATI:i

t

rr

u-

t

for the model he called type A , and

t

ri1=1

for the model he called type B. Lattft teLdices and base vectors here

refer to a space fixed reference frame, It is readily found by the

normal tensor transformation laws that

(9)

and that

where CO is the vorticity tensor. Thus Oldroyd's definition of the convected rate is the material rate of those tensor components which an observer would measure with respect to a coordinate system both rotating and deforming with the material; for type A the components

are measured with respect to the contravariant base vectors, while for type B the covariant base vectors are used, the base vectors in

both cases btdng both stretched and rotated with the material. The present iorm of writing the convected derivative allows

ready inversion, for letting

then since

CAID-r

_{-t-t crini}

_cobri.)

_T.J

t

co%

_,

de:Pi

Drj

(0e:k.

4

of t

t

co'L) T",

Dt

-4-

co

l_1(16

pri

D t

(4' + CA 3

(10)

and since the es are constant in time for a material particle,

w0t2,(191-e)

a o4

610f6 c't

axi'" a le

x`

axi

"j"Fek /Aim (X ;

a

ct

t;(I: \Ail% 1 oti

and aiUar1y for the type B derivative. These integral& were

presonted first by Oldroyd (1950); they were used by Walters (1962)

In models designated as A' and B' by writing

or

a)e-

de

2 r

t)

ôxcl,,(xiiilde

(2)

d

is)d-t)1 (3)

respectively, where 4(t) is a material relaxation function.* When

40) coneiste of a combination of exponential. and Dirac delta functions,

and

(11)

equations (2) and (3) are exactly equivalent to equation (1); otherwise

they are generalizations cf equation (I). (For example, when N M 1,

equation (I) is obtained by taking N(p) _a

t

_xi. (1.))

(1-

--Ln

AL

Jaumann (1911) proposed a different definition of _Lt which we

shall designate as Ltc. Introducing new base vectors ro, by

eNIO

where S satisfies the equation

DS:.!

TD-f-

so,gd,P

and reduces to the identity matrix as an initial condition, then, denoting the inverse tensor with a minus unity superscript,

60111101

=NON

r

ofcRi

t

fD4,

_e

oft

as before it is readily found from the transformation laws that

and that ofaTlec

D

t

Dt

CAD". jTe, 5

48;

T

rp

_Tat('

=

-r" Si s -41p

and

(12)

D r...

WWI

t

Since by Ricci's theorem and the above (II ,iekts.0, raising andt.1

lowering of indices commutes with the operation of Jaurnann

differentia-tion.

The above results Call be put in a simpler appearing form by

introducial a further tensor &, defined by

R

sISS1bof!

.

p

_dXi

then

D

CDc:3 (4) arid

R"4

or

-where R2 Is equal initially to the identity matrix. Thus S is the tensor

rotating the material bass vectors Ni into the material base vectors

roc and R is the tensor rotating the material base vectors

ra

into the fixed base vectors g. As has been shown by Goddard and

Miller (l965), R corresponds to an orthogonal transformation, and

hence its inverse and transpose are equ ivaleato Thus the Jaumann

(13)

observer would measurl.n.with respect to the base 'vector .

..._. a.

...

rotating locally with the same rate as the vorticity,, ioea moving.

with the principal. axes of The length of -these base. Vectors changes:

also, but not directly with the material.. Inversion of the derivative, again follows readily from the definition; if now

then and OEM 4=11111=1 -Fes

t.

Jr.

st

Cs i')di

₎

aegd

aix' am'

Rs;

7

Sit

S

c't 69t'd

e

0 Oil

VII

V

/\

j

tidt'

) X ) an

r

=

(14)

the initial conditions on R.. being imposed at time t' - A material of type C' could now be defined by

,t

=2\

(1)(t--C)R!"'t!Kd

(xi i.)01i3,

(5)

Pc.1' -G4i4 t,

j

'$111

We note that equation (5) is the constitutive equation presented by Goddard and Miller (1966), their integration being presented by other

arguments. No simple relation has so far been 'found relating the

various Oldroyd and Jaumann integrals.

II. Governing equations

The solution for steady flow between stationary parallel plates

is next presented for materials of type A' , B' , and C' . The

stability problem for parallel flows is then formulated for each by superimposing a wavy infinitesimal disturbance on the primary flow and then determining under what conditions this disturbance will grow.

Cartesian coordinates are used, the z-axis and the y-axis being

chosen parallel and perpendicular to the plates, respectively. For the

--steady flow the velocity components are assumed in the form

T.J=

v.

0) -\"1

with W = 0 at y

+h. By inspection the motion is

xi. y,

) cold

primed coordinates denoting the position of a particle at time t' The

only non-zero component of the rate of deformation is

(15)

9

dZ3

1" _DW*.)

where D represents differentiation with respect to y. The non-zero

displacement gradients and rotation components are

LE12.

(t-tlIDTA1)

R.1.1

1

) Rz2. ₌

33 = cos

ay,d _z WO.

R32.

Substitution of these into equations (2), (3) and (5) and the equations of motion yields the non-zero partial stress components and velocity as shown in Table I, where

By,

N(r)

= (6)

and

14:c0rNft)[1+

(7"

Dvi)fOter

, (7)

Hence, fluids A' , B' , and C' all predict different normal stresses.

Normal stress differences are consistent with the sudden expansion or contraction of the stream when some non-Navier-Stokes liquids

suddenly emerge from a tube into the atmosphere (the Merrington

effect, which would occur in A' and C') as well as with the differences

in the shape of the free surface for such different liquids undergoing

)

(16)

TABLE I

Velocity and Stresses for primary flow of materials A',

B', and C',

I .._ XX 0 o o PYY -2B1(DW)2 0 -2J1 (DW)2 1 Pzz 0 2B 1(DW) 2 2J1 (DW)z 1 Pyz BoDW

- .

BoDW J10 DW P -a-E z - 2B (DW)2 3z 1 aP 1 aP 1

z - 2J1

(DW)2 cm z 3z W 2 Wo(1 -h z 2 Satisfies DW

Ji

vrY--h

o

(i

-i ) o Bz -ap iz-BD2W o BD2W o (J1 -2J2DW) DzW o 2 yz

(17)

-Collette flow (the Weissenberg effect, which would also occur in At

and C'). Only C' shows a variable effective viscosity,

In the development of the Orr-Sommerfeld equation (the

stability equation for Navier-Stokes liquids), consideration is limittd

to a disturbance that corresponds to a velocity field which is both

temporally and spatially (in t) periodic, Subject to this limitation,

Squire (4933) has shown that it is sufficient to consider a disturbance that correSponds to a two.dimen.sional velocity field, In the

visco-,,,

elastic case, only such disturbances will be considered also, althougli

no proof of the sufficiency of this exists for these fluids, (In fact,

Listrov (1966) has shown that at least for a Stokesian fluid three dimensional disturbances are less stable than two dimensional

disturbances.) Accordingly, the disturbance velocity components are taken in the form

LA*.

_{0) lf*}

E")

(),4

c,,x*

(0( )E.

where E exp iA(z CO; A is the (real) wave number, and c the

(complex) wave speed.

We first derive the stability equation for materials At and B',

The total displacement is assumed to be the sum of the primary flow displacement and the displacement resulting from the disturbance,

namely, and sOPS MIO J

11=-1+

c(t)-I:

2 -{W(&+k,(1,

11 1) ,

(18)

Taking

14:1

1.)E"

and similar expressions for and k. since

0)

the solutions are readily obtained as

4:

_v/LA

_C))

CN6.+1

(1)1/

-

DC,,)

IA (w-C))

(-h,

1,Dv-

vD1.,}iA(v.C),

--uple,i)14\(w-c),

=

}lc

A (WC.),

t

A

(19)

Expanding all quantities about values at time t it follows that to the lowest order and art(' 1.7 E ( 1

-A

_(\AT-

_C)

}Ad

17

Dw

(t- -t )

c' A ("Cr--.6)

alr

At, Al(w_c.-211E-

F),

(9)

ere F

exp iA(C-W) (t-t').

If consideration is restricted to a short time interval, then

eqtzations (8) and (9) can be approximated by expanding F in a power

series in (t-t') and retaining only the first order terms, Hence,

H-V),

A

Thest are the expressions used by Walters (1962). The more general forrifs given by equations (8) and (9) will, however, be used here,

We note also that equations (8) and (9) are well-behaved at the

critical point where W and C have the same value, Applying the limiting process W4C results in

13

(20)

Making use

{Dler*

°Lr

of the continuity equation,

displacements can thus be written as

(

-

1?(14-1)E

tr-tw.

8.7 (1-

r)

c A (N-T -

c).'

1

the non-zero derivatives of the

and

a

41 ow* (AV

C2:711\'

41.

To the same approximation used for the displacements the non-zero

rate of deformation components at time t' are

d

=ci4z

thiEF

and and -4 C

(i41-1)

1,00

4cW

z

ptf

E

(f-f)

C

_"siiirE

_(t-t12

in agreement with the limit as t approaches t to the order of the

linear terms in t-V.

The nonzero rate of deformation components at time t are

-

811z

E.

(21)

Ct.VN

utw

6:(1.-F1

t

A (14---c.

341 1.7E (1.-F)

- ₎

a`vpvire'q-v)

TN,"

C

-V

(t

.1111\ .011.111

Combining these result

'OW F

t

a)

cow

(Dif

1.71)1d

_t")

₍

v

irstv

nvo

\,,T

emw-c)

(1DWII C

2.0vDVer

-s with the con-stitut (equation (2)) leads to the disturbance stresses

15

E:(3.-F")

_2.-tr(o-wr

C L)

1.

iye equations tor A'

2. I<tif -2 A D-W

-

Ty,-

OW -r

(22)

and

The functions _itn (y), are generalizations of Walters' constants and

are defined by 00

plot

1--(Kt*

K21.101.11-+

zko'

2.(Dwri.p,(w-c)(

KA

+2,tDWVIAT(VeS12.*K2'2)

Z (10-W13

t(\ic33 )

t A DW _K1115,

th

A

_{(r)Rzi. LA(vi-cYcl" dr.}

₍₁₀₎ 0

Upon considering the equations of motion, one obtains, after subtracting the equations satisfied by the velocity components of the

primary flow and eliminating the pressure by cross-differentiation,

, (11)

= 4. A

D p

_,

(15+A)

1.1..

In order to write equation (11) in terms of non-dimensional variables, the following dimensionless functions of y are introduced

K1; /Wo

=

Kti-v0/61,-(1

_r

) 123= K3etwo51(01,,s.

(23)

11

Putting these and the previous expressions for 10

p

and

Y yz

into equation (11) results in

(

0,(1)t

fu-

cef; u

c)(2 DU)

-0(13-c)[31)U

-

2.00W- c\P31t1*-IT

(LT

(DUr

(15

2,04:T1J-VdeDIT-+

_{{ 4pv(00 DUI. r317)4,e(IT-cttj}

43(t'U)101),-[..0(U-c.)'

4-

(

c)

-ttirj

_{2 Ge (Dif F.3} t:

T.LEI

₎

_(tz)

where a is the non-dimensional wave number Ah, c is the

non-dimensional wave speed _{C/Wo ,} U is now the non-dimensional primary

flow velocity, W/W09 and l) represents differentiation with respect

to the non-dimensional y For a Navier-Stokes liquid all the p

are zero except p

and equation (12) would then reduce to the

Orr-Sommerfeld equation.

Walters (1962) has suggested that for some viscoelastic

called "Slightly viscoelastic9" it is reasorialbe to expect the CT) to .

(24)

-where and

i

_{[(LIAV-041 DIUI}

15

41-4cA

(V-0Jr-v--4-

e Dili

/

se

R

(

/2

0 1111

which-is the equation given by Walters.

Two remarks are appropriate. Equation (12) was derived

with-out specifying the form of W(y) and hence is not restricted to the

primary flow here considered. Also, if equation (3) (the constitutive

equation for material B') had been used in.-place of equation (2) (the

constitutive equation for material Al), the resulting equation for v

would have been exactly equation (12) - a perhaps surprising result,

vanish rapidly as T increases,

This is the justification he gave for

using the approximate forms for equations (8) and (9) This is

equiva-lent to replacing the upper limit in the expressions for the pn by a finite limit (say T) which may even be quite small, Assuming

44

rri

everywhere in the flow field and neglecting all

(25)

The stability equation for the C' material is -developed in much

the same way. For the perturbation velocities, the solution to equation

(4) is found to be, to the order of the linearization,

R

k

_{cosk.,(t-ttlp\"714- EH siY\'t4F-(t-d1)141}

"

R

_z

= -

sieri

DV.]

Co5

DWI)

where

(1- F)

(

H

A

V'

27.7f7Ctinkt.m" 1:ANi-c))

't A

D'If

+

(t-

_'al:

tikr

_{A (WC)}

From equation (5), the disturbance stresses are

33.7-

VI--)\el".2,0M-13.4kCA

pzz=

4 if

313,1)1,

+2.0/4

P\(\W-C)I-111

4 DIV

it---(C1,44-,C) (12^0

-

Jon

)

_(t3)

'4--

'14

AL10

kt6r-C) 41.1D-41.414

[0\(w.CarRLI,

)6W

+ (A

(26)

where 2.1)1,4

Li,. Du

Jr

--41

(

1-

L(vo

LAW-C)

LI:=7.c,"rN(1-) t144.AKT-c17.T

(en-c.1)7-rdr,

(LS)

Because of the complexity of these stress terms we do not write out the stability equation here, but consider the appropriate approximation in the next section.

III. Asymptotic Solutions

An approximate solution to the stability equation (12) for the

primary flow U 1 - y2 is next obtained, The determination of the

characteristics equation for a Navier-Stokes liquid is first briefly discussed in Part A, the procedure used being that presented by

Graebel (1966) with only slight modifications. The counterpart

characteristics equation for viscoelastic liquids A' and B' is next

presented in Part B , and the procedure used to actually solve this

characteristic equation and to determine the points on the neutral

stability curve is presented in Part Co

Viscoelastic liquid C' is

dis-cussed in Part Do

A. The Determination of the Characteristic Equation for a

Navier-Stokes Liqu id

It is anticipated that both c and 1 /alct will be small for the

case of interest, which suggests a solution in terms of matched

asymptotic expansions. The flow region is first divided into an inner

and an outer region., The inner region is a strip that includes the

(27)

where U(y) = c

. The outer region extends from the inner region

to the centerline between the plates at y = 0 which, as a result of the symmetry of the geometry ad equations, serves as the other boundary.

It is assumed that the two regions overlap.

The procedure is to obtain a solution valid in the inner region and a solution valid in the outer region and then to merge the two

solutions. The inner solution is made to satisfy the boundary conditions

at y

-1 and the outer solution the conditions at y a0 The merging

then produces a characteristic equation which gives a as a function of

R and c

the plot of a versus R for co = 0 being the neutral stability

curve.

In the inner region v is obtained by introducing the change in variable (coordinate stretching)

and by putting

VI

(.4

?(J/A

=

r'(")(1) -1.-/A2(1'

(+-

-where p. is a function of aR r, expected to be small, but unknown at

this point.

Substitution of the above into equation (11) with all of the

Rn

for n I set equal to zero suggests that the proper choice for

p.

Dz. tiaG

0(1: _(1.2)

2((°1

(aR)-1/3 and equation (12) becomes

+

_[

cefto)

tz:L1:oto) 0( 1.4' °( 17- ok.;((1) TYC3c.

+

21 J P. is -= ,

(28)

andi((1) are the solutions of and Hence:, and where

ry

(o)

N.3

-.M=0 (4)

Lb

DU

220)

cit

v

Cci6G11C,14

( Due

kfr( 00

2..(.v((DU,

3

11 io

3

2_ /7. (21 r

3 xl

(29)

P.

H(1) and H(2) are Hankel functions of order one-third. Since

h2

increases exponentially with large positive sit at a much faster rate

than does the outer solution,, .2(.4(°) cannot be merged with the outer

solution and is therefore discarded.

For(1)

0( we have

9

Graebel (1966) gives the 'solutions for ?((:) and 2e31) as

(121'

2. (

DUc).413 c _{PO 01}

10:d4S

(LDUc\l/3T,s

,s3]

and where where 7

_(kr

M=11 .01, 0=00

rod)

(13

it__

9

63

3

i°

(DUG)

S413

10(DU)

tol

÷ v

_rya)

2.0 Uc

3 . 4 (" 2_

(0,

39 /60 (DUcli

_-

_oi67g9

) For large

t

23 en 627

30 (DUc).4

Oi 390

'9 9

=

(30)

and

where

-

A1

and

In the outer region v is obtained by introducing the expansion

li`;:(1)

A=

1n +3 (v).+11y)-+-4-)Ay,#,. +

Du,

-

A,

(Ank3)(n

Luc

13-

, 13 E`)2 1-cjc. -1:etz k c, )

Cf),c,/A)

v`°)()-A-

i(r)

Substituting this into equation (12) gives

(-u-cl(v---01./)-u(°)

-

(0)

_{u-- o)}

and v(0) v(n)

for all n such that 0((p.))

0(43). The Tollmien

solutions (1936) are

(31)

and

In the inner region Graebel used

where the. C. are arbitrary constants,' For the ;inner and outer solu-_-i

tioni to merge for large positive it theproper choice is 1,fu for

to

1n4 for t

a and

1 for eIn the outer region v is given by

1 2°

n

_zetFi.

li( (1)

CL

2c'

_1)-6,

+AR +OGA

_/

tA,

$.

wheile A is' a constant that Cannot be determined until higher order

terms are considered. fZt.0) decreases exponentially with large

positive r( and hence, tao terms are needed to merge with it.'

The boundary conditions are

ooi

D3v=

If V is an even function of y, or

v 0

25 0.14

t.

(0)

-3)

if v is an odd function. of _{The Orr.Som,merfeld equation}

D

allows .separation into, even and odd parts when U is an even function

of y; since v in the outer region satisfies second order equation.

D

a

(32)

which is even in y, satisfaction of only one boundary condition at

y 0 is sufficient. Using symmetrical disturbances since they are

less stable than asymmetrical disturbances, the boundary conditions

give

DF

r

I °

BUG

t

CI)

C1

Cz

2C(3°)

zCz

to the lowest nontrivial order In pa , where (1+ ydip.

and Z s _{yc a} For the existence of a non.trivia/ solution,

DT.Jc DPI

I/ (30) ( vci) 73. I-

2.1)ri. CIO)

-vco

'4

(Ai 6 tgZ Z /p, 1

which is the characteristic equation obtained byPraebel. The plot of

a vs R. for c.

0 obtained from this, however, does not contain the

characteristic loop of a neutral stability curve, but gives only the lower

branch of the curve. This shortcoming is explained by considering the

magnitudes of the 1) terms in the inner expansion, which in turn

requires some knowledge of a With Lin.'s (1945) results

2a 5, H. oPe, 1/20) as a guide it appears that 2(0) does not

adequately describe the solution near y a -1, for Im(14:(2/))

ft 0.3

as compared with the

1420)

0 a

The real parts of pr I)

and

2

and

c,

1/4-3

(33)

1p1( 1 )' are however small compared to .(0) and '

it'

_{respectively,}

2 ₁ ₂

and

4(1)

3 is small compared to V3°'. Thus by including the It.A1) terms

the imaginary part of #2(2 is altered significantly, this additional term apparently being responsible for generating the loop in the stability

curve, _{It seems then that a better choice for the solution in the inner}

region is

. For merging eqUation (16) with the outer solution, again the

?Toper choice is 114 for 1n

for si

and for ,0 .as before,

Then v(y) in the outer 'region is given,.by

Substituting the appropriate boundary

161

as* CIF*,

(zCtFi

'

Dric.

conditions leads to

Due

Cz(1.

_2c(vti):

to)

ct

ot?c13.1(111

ot7(m(i)

vut

r

otfr(

_&I

q

( el)

Du, t

Oct),

1 1 (

/

v(i

_{c1 (}

(34)

and

I

c:0(Ri z DUc

eUc.

+

R'i (DWI z4

4pi

VIRt t..tRzaDV,[12.co(k2J-u-,

okz R3

(u}*

1)2.

R-liRz-2,11.0

Dq{Le,Dzu,

+ 3 °( RAIL

(DUcri a(zz+

'8 '1)

R.71

I

R4. -

3,0( R,-,21-cr,

jR5:11-C3c

-qcor 0( R6

(11L7c)11

d\

al

±

where D

"tr

a DzIU evaluated at y

yc These are valid in the

neighborhood of the critical point for any distribution function that

vanishes as T becomes large, specifically for N(T ) negligible when

T where liA(W-C)Tli d< 1.

A change to a stretched variable and inner and outer expansions

are introduced ss in part A. When all these are substituted into

equation (12), the stability equation in. the inner region is now modified

(35)

and thus to the characteristics equation

a

(

i4 42(2.(1160-1

IDUL14#

ctru

JDF110-ziAkt7,11)R,(4,)

z10 2

E(t01+11)Ucpq(i'yait)(

dl

29

The plot of a versus R resulting from this does have a loop and the

equation is essentially that used by Lin, although he elected to express

the outer solution as an expansion in powers of a2 . With present

computers the expansion in terms of the coordinate rather than a2

seems to be much simpler and more accurate.

13. The Determination of the Characteristic Equation for a Visccelastic

Liquid of Type A orB'

Fez a viscoelastic liquid which does not depart too drastically

from a Navier.Stokes liquid the solution of equation 00 can be carried

out in a manner analogous to the solution presented in part A. Specifi.=

cally, it is anticipated that in equation (12) both lc I and .E) /a I will

be emall. Hence, the flow region can be divided into an inner and an

outer region as previously done.

In the inner region the P can be expressed as series in z by

expanding the denominators arid then integrating term by term. The

(36)

and thus whore CA:12' 21

641X")

4.) Lic

a 0,

7.A t: 0(

8 DC/c,

otlX.

oul

(Al

LL7c.

d't

1

a °

It

(o)

'er(DUe((

or960

(.:0( DUe.

where ualki, The outer equation is unchanged. Since the term

containing R1 multiplies the fourth derivative, Xi and 242 remain as

in part A. Taking E to be small but still of larger order than H.

is approximated by ( dz2,,(01 .0 t 18)

)

(19) I

(37)

and

where

ei*

r di (i6i

dr(4

D Lic

d

e,r)

DU,

'

?CO)

Then So is the same as in Part A, and 3

(

da'4

fr(

cht.'

114-

).

Thus for the solution in the inner region, Xi and 2(i2 as given in Part A

are used and Oo E(/), is used for 2(3°

Because of the additional linearization introduced by equation

(19), the present results are limited to small E and serve mainly-to

indicate a trend. For larger values of E the perturbation scheme used

to solve equation (18) may not be adequate. In this case, one would have to resort to an exact solution of equation (18) as presented in the

Appendix.

Since the outer solution remains unchanged and (.3 is not

directly involved in the merging, the characteristic equation retains

the same form as equation (17), the only difference being ?(3 is now

given by equation (19).

Ca Solution of the Characteristics Equation

For caleulation pruposes, it is convenient to introduce the change

in variable

31

7

(0

(38)

Then the left hand side of equation (17) becomes

a

( (J°1( 6(1 where The functions

h1 and 1.1,2 are discussed, and tables of h1 and h2 and

their derivatives are given in A=4E611945). Putting x

I S

It folio:NS that (". + 8214

s

(zoo )7"1

vroo (20)lp \ 1

f

Ct _{11 tt i 11)} _A 1 :1

::s

_°g s),..Tx

113(3711(1.c=-11

11 il1

Al

A 19 k4(1: 5i 4 Apic-::---:.ci

s

Z .-t- ylifi 5,

il

AzevItt -S2

zoo)''

"c?)

(zoo

(21)

3

(39)

and where MOO.

(2.00y.,

A0

2!3/11( v3)

)

2.00/4,4,113,(3t4,-1),

.sgf Looya.Vv,41.

are also tabulated

For the integration indicated on the left hand side of equation (20)

the above series were integrated term by term for S

5 For

"S > 5

the integration was continued numerically using Simpsonls rule' and the asymptotic expansion of

hl until there were no noticeable

changes, in the value of the integral in the seventh significant figure. These asymptotic expansions are

_) it)

5)

S

e

_Cos

(4

I 3 r

+

2. ) ) 33

(tV"S

(zoo)4"'

,-.2:6/31"3rA

3,-- an 644.4

/3wi

(30,141)

2(-1V".41

(40)

and

1-11(,

s)

^-

S

5-Y

514-1

1311"

÷

where

₅

=(lri)Vi/ ,,f7F

and _11L

ar

4,

The result, of the integration are

c:4-5

k()

oso43 41

I. (61-N+512.00)Lv4-41

+-1 Z.

(-1)"

S7-31-

(6>m+ 4S)

4,

(-Jr"'

<(..vvx*2-(c0tA442) (2-00)2"-

'3 7

(5

c'

,c1-S

RA. ki(cs)

-

0:29099Z

-,r0,5o+361

( (40M+45-K6W"

0(2-00)"."

I

3 I (6m, ++4'41+,51

(2062'3-(-1)-"a2.0.1

,r4A,A,3 ?

A

(41)

and s =

?73588

9Q0

+r0'

C6w14-7-1 (W0)4"

vvt=

a"

1

(-1.)

13.2e.4i

3 1

((at-1+s)

(2.00)"41-((ow+i)

(2007---,;(11,1

1,1,G-s)=

6.6786i-o

o.

3.5.- s)e -S

f_

C(-1)-2.1c9.,1-3)(zoor-oi

Bzw,+t

I ko kv14-5-161,14+6

(ZO01

A

zim

.11

-t4IG

v14#2..)(2,0 60A.14.-z 35

k(1

1Y1

(42)

For given values of X and c the solution of equation (20) was obtained by plotting the real part against the imaginary part for each

side. The intersections of these two families of curves gave a andli

R. was computed from

Vc.

R

Dxu` (

)3

The results are shown in Figure 1. The graph for X st 0 which corresponds to a Navier.Stokes liquid, is seen to be in close agreement with Lis results, and the preceeding statements regard= lag the anticipated silts of the various parameters are seemingly coa.

iliiitdirst with the final results. The results are qualitatively in

agree-ment with those of Chan Man Fong and Walters, shown also in Figure 1,

(The X in their paper is defined as five times the value of the present

ene4 ) The quantitative disagreement of the two results is not

under-stood; it is noted, however, that Chan Man Fong and Walters results

for Ka 0, departing considerably from the results of Lin, do agree

with the results of Stuart (1954). On this basis, it is believed that the

present regeWts are tilt more accurate, ones,

D. The Stability of a Viscoelastic LicelidaLTIELS!

Introducing stretchsd coordinates again as in part B, rear the critical point the stresses become

ovu

z.D\A,/, p--"

3 +

(4-)

110 1CL414

2,(misTej

L..J

(43)

2.2 2.( 0,7 0:6 0.5 0.4 ,41 t-,12 )3, /14 \

_\

\

37

'NEUTRAL STABILITY CURVES

X=0.005,

X=0

CHAN MAN FONG,,I3 'WALTERS '41965)

000,00. UN (1945)

PRESENT RESULTS

X=,02

i

1 IL II 1 5 II I, 1111 tiL V "I ti II: ili

115 ,16 17 18 119 20 211 22, 23 24 25 '26 27 28, 29 30, 31 32

R 1/3

Figure 1: Neutral stabdlity curves for various, values of X..

1.8 (1 1 7 1.51 1 3 a'2 1.2 09' X 0 02 X =0.015 I 1.9 16 1.4 1.1 1.0 0.8 0.3 I I 2.0

(44)

5500 ,1 5000 4500 Rcr 4000 3500 3000

Figure 2. Critical Reynolds number versus the parameter Ri. ^

2500

0 2 3 4

(45)

1.8 1.7 1.6 1.5 1.4 1.3 IL 2 a2 1.1 _1.0 0.9 0.8 0.7 0.6 _0.5 C.030 C.0.29 C.0.27 Figure 3.

Neutral stability curves and curves of constant c for various values of X..

12 13 14 15 17 16 18 19 R 113 20 21 22 23 24 25 26 27

(46)

_

v. I

DWc OV,

_

ID21,\T",

j43c.

iA

'OVA jiac

-3,1c

+

(tMe-r Cnc.

_11-J3t.))1

4-

a

c; A

VAT,

0( v)

where the subscript appended to the Jrn-,;

Idicates

that they are

m

evaluated at y; the J

's and Km s have been assumed to be all of

c n

the same order of magnitude The parameter is defined by

where

_p

_{D'Wa. ((WW1= r}

w°1-) wce

D\J.c.c

Making the distances and velocities dimensionless as before, and

defining

Ji

lc+

J4

2. 1 C.

333c

(Ilap.,

I_I)

substitution of these stresses into equation (11) along with use of the equation for the primary flow results in

D Cfc 07712.(12"11 Cr

DIU

ttr%

(

DUc

t z

(vu-c

0--jc.

R,13-5

6137)-4-

0( ',4z)

0.

4 533c.(.

17-1

8 (

(47)

40-7

Proceeding further as in cas.e.B, with

61.

cKiA

kit

)62= sy,f Nt)

we have now

tuc)

1:ttpq

11 t..6_z

DU

_{c (II}

43 (.) 3 (22)

as the governing equation in the inner region. It is seen then that if

4,0 and oz are bath small compared to unity, but larger that

1s.

again ?(.(0) and Vt°) remain u.nchanged, but a first approximation to

C:2

23(0)

2(

(.03) 62 iz

Duc)

4,0

+

where

cOo, 4)1 are as given in part B.

Away from the critiaal region, the invisctd equation will again

hold, but the primary velocity profile is of course different from the

parabolic one, (For the model typified by equation (1) with M N 1

and _Lt _LtC for instance, DW satisfies a cubic equation where the

coefficients are linear functions of y.) Comparing a fluid of type A'

(or 10 with type C' if N and piaz are the same for both cases, the

fluid of type C' will have a steeper velocity profile than the parabolic

one, To carry out the details of the solution, it is necessary to specify

N. _{When this is known, W can readily be found by numerical methods,}

and equation (17) then used to determine the neutral stability curve,

(48)

-BIBLIOGRAPHY

The Annals of the Computation Laboratory of Harvard University,

Volume II, Harvard University Press, Cambridge, Mass., 1945,

Chan Man Fong, C_ F., and Walters, K., "The Solution of flow

problems in the case of materials withfnemory, Part 11,"

Journal de Mechanique 4, 1965, 439-453o

3, Goddard, J. D., and Miller, C., "An inverse for the Jaumann

derivative and some applications to the rheology of viscoelastic

fluids," Rheologica Acta 5, 1966, 177-183,

4. Graebel, W. P., "On the determination of the characteristic

equa-tions for the stability of parallel flow," Journal of Fluid Mechanics

24, 1966, 497-508.

Jaumann., G., "Geschlossenes System physikalischer und chemischer Differenzialgesetze," Sitzber, Akado Wise, Wien (Ha) 120,

1911, 385-530,

Lin, Co C., "On the stability of two dimensional parallel flows,"

Quarterly of Applied Mathematics 3, 1945, 277-301.

Listrov, A. T., "Parallel flow stability of non-Newtonian media,"

Soviet Physics, Dok.l, 10, 1966, 912.914,

Oldroy& J. G., "On the formulation of rheological equations of

state," Proceedings of the Royal Society (A) 200, 1950, 523-541,

Oldroyd, J. G., "Non-Newtonian effects in steady motion of some idealizedelastioo-viscousliquids," Proceedings of the Royal

Society (A) 245, 1958, 2.78-297.

Sokolnikoff I, S., Tensor Analysis John Wiley and Sons, Inc.,

New York, 19510

Squire, H. B., "On the stability for three-dimensional disturbances

I

70.,

(49)

Stuart, J. T., Proceedings of the Royal Society (A) 2219 1954,

189-205.

Tollmien, W., "General instability criterion of laminar velocity

distributions," Technical Memorandum 792, NAGA, 1936.

Walters, K., "The solution of flow problems in the case of materials

with memory, Part I," Journal de Mechanique 2, 1962, 479-486.

143

(50)

Writing and with solutions

at

P(I

hZ4

APPENDIX

co

e ,

e

-av

the equatiOn for f is the confluent hype rgeometric form

z

=

3)

An exact solution of equation (1,8) is possible and has in fact,

been given by Chan Man Fong: and Walters (i465). A. modified and 'More

complete version of their results is presented here to ,shOw. its use in the present method.

Equation (18) is of the form.

4X

_2./2

e

(51)

and where , em.y.p

--csz

sjds

e

r(g

r

if-2

-

e

u

:"U+1

2,71. OQ g

As z approaches ioo U(z) approaches z and V(z) approaches

zeg

This suggested that U must be the solution corresponding

to X3 and V to 2(4

To verify that this is indeed the case, replace

v in equation (23) by 1 / -1 s \FT%) Then in the limit as X. approaches

zero with arg z 1T and p 2w/3,

r(c) (-1)

eZ

U(-2)

Doe-P

ex.pLls-idi

+

(

-or upon expressing the integral in. terms of Hankel functions,

x-vt

(-1)'

e:

a/2

-UrU

211 -

( -571(t -`,4 d4,1(1-)}

-2I -4-i

X4Idiz(Y( '44

_4-1)1j

WA 2 (1)

\

3 e

H

[ z

143 3 45 CL

zr41

2 a

(52)

+1)

-0"

olv

?(

and using this (D) in the characteristics equation (17).

3

This is the desired result apart from a multiplicative constant. Thus

the general form of eg(0)

3 for arbitrary X is given by integrating

e

Drf.lz

(53)

DISTRIBUTION LIST

(One copy unless noted otherwise)

Chief of Naval Research Chief, Bureau of Ships

Washington, D.C. 20360

Attn: Code 438 Attn: Code 300

Code 463 Code 305

Code 466 Code 335

Code 341

Commanding Officer Code 342A

Office of Naval Research Code 345

Branch Office Code 420

495 Summer Street Code 421

Boston, Massachusetts 02110 Code 44o

Code 442 Commanding Officer Code 634A

Office of Naval Research

Branch Office Chief, Bureau of Naval Weapons

219 S. Dearborn Street Department of the Navy

Chicago, Illinois 6o604 Washington, D.C. 20360 Attn: Code R

Commanding Officer Code R-12 Office of Naval Research Code RR

Branch Office Code RRRE

207 W. 24th Street Code RU

New York, New York 10011 Code RUTO Commanding Officer (25) Commander

Office of Naval Research Military Sea Transportation Service

Branch Office Department of the Navy

Fleet Post Office Box 39 Washington, D.C. 20360

New York, New York 09510

Special Projects Office

Commanding Officer Department of the Navy Office of Naval Research Washington, D.C. 20360

Branch Office Attn: Code 001

1050 E. Green Street Code 201

Pasadena, California 91101

Chief, Bureau of Yards and Docks Commanding Officer Department of the Navy

Office of Naval Research Washington, D.C. 20360

Branch Office Attn: Code D-202

1076 Mission Street

(54)

Commanding Officer and Director

David Taylor Model Basin

Washington, D.C. 20007 Attn: Code 500 Code 515 Code 520 Code 521 Code 522 Code 580 Code 589 Superintendent U. S. Naval Academy Annapolis, Maryland 21402 Attn: Library

Commanding Officer and Director

U. S. Naval Civil Engineering Laboratory

Port Hueneme, California 93041 Commanding Officer and Director

U. S. Navy Mine Defense Laboratory

Panama City, Florida 32402 Commander

Naval Ordnance Test Station China Lake, California 93557

Code 4052 Code 5014 Code 753 Officer in Charge

Naval Ordnance Test Station Pasadena Annex 3205 E. Foothill Boulevard Pasadena, California 91107 Attn: Mr. J. W. Hoyt Research Division Code P508 Code P804 Code P807 Code P80962 Superintendent Director (6)

Naval Research Laboratory

Attn: Code 2027 Commanding Officer

NROTC and Naval Administrative Unit Massachusetts Institute of

Tech-nology

Cambridge, Massachusetts 02139

Commanding Officer

U. S. Naval Underwater Ordnance

Station

Newport, Rhode Island

Attn: Research Division

Commander

U. S. Naval Weapons Laboratory Dahlgren, Virginia 22448

Attn: Tech. Library Div. (AAL) Commander

Boston Naval Shipyard

Boston, Massachusetts 02129 Commander

Charleston Naval Shipyard U. S. Naval Base

Charleston, South Carolina 29408

Commander

Long Beach Naval Shipyard Long Beach, California 90802

Commander

Mare Island Naval Shipyard Vallejo, California 94592

Defense Documentation Center ₍₅₎ Cameron Station

Alexandria, Virginia 22314

Commander

(55)

Commander Pearl Harbor

Navy No. 128, Fleet Post Office San Francisco, California 96614 Commander

Philadelphia Naval Shipyard Naval Base

Philadelphia, Pennsylvania 19112

Commander

Portsmouth Naval Shipyard

Portsmouth, New Hampshire 03804

Attn: Design Division Commander

Puget Sound Naval Shipyard Bremerton, Washington 98314

Commander

San Francisco Naval Shipyard San Francisco, California 94135 Commander

Air Force Cambridge Research Lab.

L. G. Hanscom Field

Bedford, Massachusetts 01751

Attn: Research Library, CRMXL-R

Air Force Office of Scientific Research Mechanics Division

Washington, D.C. 20333 Director

U. S. Army Engineering Resarch

and Development Laboratories

Fort Belvoir, Virginia

Attn: Technical Documents Center

Commanding Officer

U. S. Army Research Office--Durham Box CM, Duke Station

Durham, North Carolina 27706

Commander

Hq., U. S. Transportation Research and Development Command

Transportation Corps

Fort Eustis, Virginia

Commander

Hq., Ti. S. Transportation Research

and Development Command Transportation Corps

Fort Eustis Virginia

Attn: Marine Transport Division

Commandant

U. S. Coast Guard

1300 E Street, N.W. Washington, D.C.

Superintendent

U. S. Merchant Marine Academy Kings Point, L. I., New York

Attn: Department of Engineering Director of Research

National Aeronautics and Space

Administration Headquarters

600 Independence Avenue, S.W. Washington, D.C. 20546

Attn: Code RR

Director

Langley Research Center

National Aeronautics and Space Administration

Langley Field, Virginia

Coordinator of Research Maritime Administration

441 G Street, N.W. Washington, D.C. 20235

Division of Ship Design Maritime Administration

(56)

Research and Development Maritime Administration

National Academy of Sciences National Research Council 2101 Constitution Avenue, N.W. Washington, D.C. 20418

Dr. G. B. Schubauer

Fluid Mechanics Section

National Bureau of Standards Washington, D.C. 20360

Mr. E. S. Turner

National Research Council Montreal Road

Ottawa 2, Canada

Director

Engineering Science Division

National Science Foundation Washington, D.C.

Mr. C. A. Gongwer

Aerojet General Corporation

6352 N. Irwindale Avenue Azusa, California

Mr. W. R. Wiberg, Chief Marine Performance Staff

The Boeing Company Aero-Space Division

P. O. Box 3707

Seattle 24, Washington

Hydrodynamics Laboratory

California Institute of Technology

Pasadena, California 91109

Dr. J. Laufer

California Institute of Technology

Pasadena, California 91109 Department of Engineering University of California Berkeley, California 94720 Professor P. Lieber University of California Berkeley, California 94720 Professor M. S. Uberoi Department of Aeroanutical Engineering University of Colorado Boulder Colorado Mr. R. H. Oversmith Hydrodynamics Laboratory Convair

San Diego 12, California Professor A. B. Metzner University of Delaware

Newark, Delaware

Engineering Societies Library

29 W. 39th Street

New York 18, New York

Mr. P. Eisenberg, President

Hydronautics, Inc.

Pindell School Road Howard County

Laurel, Maryland

Professor L. Landweber

Iowa Institute of Hydraulic Research State University of Iowa

Iowa City, Iowa

Dr. C. Elata

Hydraulics Laboratory

Israel Institute of Technology Haifa, Israel

Professor S. Corrsin

(57)

Dr. R. H. Kraichnan

Peterborough, New Hampshire

Professor A. T. Ippen

Massachusetts Institute of Technology

Professor C. C. Lin

Massachusetts Institute of Technology

Dr. B. Sternlicht

Mechanical Technology, Inc.

968 Albany-Shaker Road Latham, New York

Acting Director

St. Anthony Falls Hydraulic

Laboratory

University of Minnesota

Minneapolis, Minnesota 55414 Professor J. Ripken

St. Anthony Falls Hydraulic Laboratory

University of Minnesota

Minneapolis, Minnesota 55414

Dr. E. R. Van Driest

Missile Development Division

Noth American Aviation, Inc. Downey, California

Dr. T. R. Goodman Oceanics, Inc.

Plainview, L.I., New York Mr. L. M. White

U. S. Rubber Company

Research and Development Department Wayne, New Jersey

Society of Naval Architects and Marine Engineers

74 Trinity Place New York 6, New York

Dr. J. P. Breslin

Stevens Institute of Technology Davidson Laboratory

Hoboken, New Jersey

Transportation Technical Research

Institute

Investigation Office Ship Research Institute

700 Shinkawa Mituka

Tokyo, Japan

Dr. A. Sacks Vidya, Inc.

2626 Hanover Street

Palo Alto, California Technical Library

Webb Institute of Naval

Architecture

Glen Cove, L.I., New York 11542 Mr. H. Crawford

Westco Research

Division of the Western Company

1171 Empire Central Dallas 7, Texas

Professor David T. Pratt Engineering Department

U. S. Naval Academy

Annapolis, Maryland 21402

Mr. Ralph Little

Naval Research Laboratory

Attn: Code 6170

Dr. C. R. Singleterry Naval Research Laboratory

Washington, D.C. 20390 Attn: Code 6170 Dr. C. Se Wells, Jr. LTV Research Center Ling-Temco-Vought, Inc. P. O. Box 5003 Dallas 22, Texas

(58)

Professor S. R. Keim

College of Engineering University of California Davis, California

Mr. W. E. Ferrin

Commercial Exploration Manager

Archer Daniels Midland Company

10701 Lyndale Avenue

South Minneapolis, Minnesota

Office of Research Administration The University of Michigan

Ann Arbor, Michigan

Attn: File Room

Professor R. B. Couch

Department of Naval Architecture and Marine Engineering

The University of Michigan Ann Arbor, Michigan

Professor W. W. Willmarth

Department of Aerospace Engineering The University of Michigan

Ann Arbor, Michigan Dr. C. S. Yih

Department of Engineering

Mech-anics

The University of Michigan Ann Arbor, Michigan

(59)

. _UNCLASSIFIED Security Classification FORM 1473 I JAN 64.1, _UNCLASSIFIED Security Classification

DOCUMENT CONTROL DATA - R&D

(Security clastlication of tale, body of abstract and indexing annotation must be ntred when the °vomit report c ia retied)

I. ORIGINATING ACTIVITY (Corporate author)

The University of Michigan

Department of Engineering Mechanics Ann Arbor, Michigan

2a. REPORT SECURI1Ty C LASSWICATFOnj

Unclassified

2b GROUP 3. REPORT TITLE

The Stability of Parallel Plows of Fluids with Memories

4. DESCRIPTIVE NOTES (Typo of report and inc(uive, date.)

Technical Resort

5. AUTHOR(S) (Last name, firer name, initial)

Mook, Dean T.

1

Graebel, W. P, Project Director

6. REPORT DATE

September 1967

' A 'TOTAL NO. OF PAGES

51

7h. NO. OF REFS

--11a. CONTRACT OR GRANT NO,

1 Nonr-1224(49)

b. PROJECT NO 1 NR 062-342

d.

9.5r ORIGINATOR'S REPORT NUMBER(S)

06505-3-T

oh. OTHER RgPOR'T NOM (Anyothlucturnbera that May be ssiffnad I

1 thi report)

10. A V A ,IL ABILITY/LIMITATION NOTICES,

Distribution of this document is unlimited.

II

11. SUPPLEMENTARY NOTES 12. 'SPONSORING MILITARY ACTIVITY 1

Department of the Navy i

Office of Naval Research

Washington, D.C.

13 ABSTRACT

The equations governing the stability of plane parallel'flows are 11

1

I developed for three models of' fluids with memories. Asymptotic solution 1 Valid for large- Reynolds numbers; are obtained and the effect of the memory

are shown to be destabilizing. The approach to the problem allows evaluation of how fast a memory must fade to allow evaluation of the stresses in power

series in the time interval. An alternate approach to inverting convected, derivatives is also presented.

i , , , I I , 1 -. --- -D

(60)

UNCLASSIFIED

Security Classification

1. ORIGINATING ACTIVITY: Enter the name and address of the contractor, subcontractor, grantee, Department of De-fense activity or other organization (corporate author) issuing the report.

REPORT SECURITY CLASSIFICATION: Enter the

over-all security classification of the report. Indicate whether

"Restricted Data" is included. Marking is to be in

accord-ance with appropriate security regulations.

GROUP: Automatic downgrading is specified in DoD Di-rective 5200.10 and Armed Forces Industrial Manual. Enter

the group number. Also, when applicable, show that optional markings have been used for Group 3 and Group 4 as

author-ized.

REPORT TITLE: Enter the complete report title in all capital letters. Titles in all cases should be unclassified. If a meaningful title cannot be selected without classifica-tion, show title classification in all capitals in parenthesis

immediately following the title.

DESCRIPTIVE NOTES: If appropriate, enter the type of report, e.g., interim, progress, summary, annual, or final.

Give the inclusive dates when a specific reporting period is

covered.

AUTHOR(S): Enter the name(s) of author(s) as shown on or in the report. Enter last name, first name, middle initial.

If military, show rank and branch of service. The name of

the principal author is an absolute minimum requirement.

REPORT DATE: Enter the date of the report as day,

month, year; or month, year. If more than one date appears on the report, use date of publication.

7a. TOTAL NUMBER OF PAGES: The total page count should follow normal pagination procedures, i.e., enter the number of pages containing information.

76. NUMBER OF REFERENCES Enter the total number of

references cited in the report.

8a. CONTRACT OR GRANT NUMBER: If appropriate, enter the applicable number of the contract or grant under which the report was written.

8b, 8c, & 8d. PROJECT NUMBER: Enter the appropriate military department identification, such as project number,

subproject number, system numbers, task number, etc. ORIGINATOR'S REPORT NUMBER(S): Enter the offi-cial report number by which the document will be identified and controlled by the originating activity. This number must be unique to this report.

9b. OTHER REPORT NUMBER(S): If the report has been

assigned any other report numbers (either by the originator or by the sponsor), also enter this number(s).

AVAILABILITY/LIMITATION NOTICES: Enter any

lim-INSTRUCTIONS

imposed by security classification, using standard statements

such as:

(I) "Qualified requesters may obtain copies of this

report from DDC."

"Foreign announcement and dissemination of this report by DDC is not authorized."

"U. S. Government agencies may obtain copies of this report directly from DDC. Other qualified DDC

users shall request through

"U. S. military agencies may obtain copies of this

report directly from DDC. Other qualified users shall request through

"All distribution of this report is controlled.

Qual-ified DDC users shall request through

If the report has been furnished to the Office of Technical Services, Department of Commerce, for sale to the public,

indi-cate this fact and enter the price, if known.

II. SUPPLEMENTARY NOTES: Use for additional explana-tory notes.

SPONSORING MILITARY ACTIVITY: Enter the name of

the departmental project office or laboratory sponsoring (pay-ing for) the research and development. Include address.

ABSTRACT: Enter an abstract giving a brief and factual summary of the document indicative of the report, even though it may also appear elsewhere in the body of the technical

re-port. If additional space is required, a continuation sheet shall

be attached.

It is highly desirable that the abstract of classified reports be unclassified. Each paragraph of the abstract snail end with an indication of the military security classification of the

in-formation in the paragraph, represented as (TS). (s), (C). or (U)

There is no limitation on the length of the abstract.

How-ever, the suggested length is from 150 to 225 words.

KEY WORDS: Key words are technically meaningful terms

or short phrases that characterize a report and may be used as

index entries for cataloging the report. Key words must be

selected so that no security classification is required.

Identi-fiers, such as equipment model designation, trade name, military project code name, geographic location, may be used as key

words but will be followed by an indication of technical

con-text. The assignment of links, rules, and weights is optional

14 LINK A LINK B LINK C

KEY WORDS