Fluid-mechanical studies on core-annular flow

ON CORE-ANNULAR FLOW

G. OOMS

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FLUID-MECHANICAL STUDIES

ON CORE-ANNULAR FLOW

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL D E L F T , OP GEZAG VAN DE RECTOR MAGNIFICUS IR. H . R . VAN NAUTA LEMKE, HOOG-LERAAR IN DE AFDELING DER ELEKTROTECHNIEK, VOOR EEN COMMISSIE UIT DE SENAAT TE VER-DEDIGEN OP WOENSDAG 22 SEPTEMBER 1971 TE

16 UUR DOOR GIJSBERT OOMS NATUURKUNDIG INGENIEUR geboren te Schiedam

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2.2 £-/

DRUKKERIJ DEMMENIE EN Zn. N. V. LEIDEN 1971 BIBLIOTHEEK TU Delft P 1937 2251 638808

page:

I. INTRODUCTION 1

n . CORE-ANNULAR FLOW FOR AN

INFINITE-VISCOSITY CORE 6 A. General 6 B. Hydrodynamic lubrication theory 7

C. Solution of the Reynolds equation - 12 D. Frictional force on the core in the

axial direction of the tube 17

E. Conclusion 19

HI. CORE-ANNULAR FLOW FOR A

FINITE-VISCOSITY CORE 25 A. General 25 B. Reynolds equation for the annular füm 26

C. Reynolds equation for the core 30 D. Boundary conditions at the interfece

between the core and the annular film 31

E. Solution of the equations 34

F. Conclusion 37

IV. HYDRODYNAMIC STABILITY OF CORE-ANNULAR

FLOW 38 A. General 38 B. Application of the general stability theory

to our problem 39 C. Conclusion 51

APPENDIX 1: Simplification of the equations of motion 53

APPENDIX 2: Frictional force on the core 55 APPENDIX 3 : P a r t i c u l a r solution of the Reynolds

equation for the annular layer 57 APPENDIX 4: Solution of the homogeneous part of

the Re}Tiolds equation for the

annular layer 61 APPENDIX 5: Solution of the Reynolds equations for

the core and the annular layer 63

TABLE 1 : Values of the constants 66

SUMMARY 68 SAMENVATTING 70 ACKNOWLEDGEMENTS 72

LITERATURE 73 U S T OF SYMBOLS AND UNITS 74

Albert Einstein I. INTRODUCTION

Pipeline transport of a highly viscous oil has up to now been effected by heating the oil and insulating the pipeline. However, these o p e r a -tions involve considerable capital and operating expenses.

Another solution could be the simultaneous t r a n s p o r t through the pipe of the very viscous oil (to be r e f e r r e d to a s "high-viscosity" liquid) and an immiscible "low-viscosity" liquid. In recent y e a r s experiments have been c a r r i e d out to examine this possibility. Thus, in 1961 Charles, Govier and Hodgson investigated the horizontal flow of an oil-water mixture through a 1-inch diameter laboratory pipe. A s e r i e s of differ-ent flow p a t t e r n s were observed, depending on the oil and water flow r a t e s . The patterns were found to be largely independent of the oil viscosity. At high oil-water r a t i o s , the oil formed the continuous phase and a w a t e r - d r o p s - i n - o i l regime prevailed. As the oil-water ratio was decreased the flow pattern changed: first to concentricoilinwater (more generally called coreannular flow), then to o i l s l u g s -in-water and eventually to o i l - d r c p s - i n - w a t e r flow (see Fig. 1). The m e a s u r e d p r e s s u r e drops through the tube indicated that the addi-tion of water greatly reduced the p r e s s u r e gradient.

The different degrees of wetting of the tube wall by the two liquids also determine the particular flow pattern occurring in p r a c t i c e . However, we shall r e s t r i c t ourselves h e r e to core-annular flow, in which case no influence of a possible preferential wetting by one of the two liquids was noticed when the inlet devices were such that the flow pattern already existed at the inlet of the tube.

Of all the observed flow p a t t e r n s it was found that the flow of the high-viscosity liquid as a core with the low-viscosity liquid flowing only in the annular space between the core and the tube wall was the most desirable one for simultaneous flow. The experiments showed that in the case of such a core-annular flow the p r e s s u r e drop through the tube could be of the same o r d e r of magnitude a s - or even s m a l l e r than - the p r e s s u r e drop for the flow of the low-viscosity liquid alone at the same mean velocity of the m i x t u r e . The annular film can be

WATER-DROPS-IN-OIL

CONCENTRIC-OIL-IN-WATER (ALSO CALLED CORE-ANNULAR FLOW)

OIL-SLUGS-IN-WATER

<»

© < ^

&

OIL-DROPS-IN-WATER

FIGURE 1

POSSIBLE FLOW PATTERNS FOR THE SIMULTANEOUS FLOW OF OIL AND WATER THROUGH A TUBE

very thin; a s a result, the amount of low-viscosity liquid required i s small and the pumping power n e c e s s a r y to move this low-viscosity liquid i s negligible. Core-annular flow also h a s the advantage of minimizing the contact surface a r e a between the low and the h i ^ -viscosity liquid. The d i v e r s i o n and entrainment of one liquid into the other can be expected to be much smaller than in the case of the other flow p a t t e r n s . Subsequent treatment to m e e t low-viscosity liquid content specifications i s thus eliminated or, at least, minimized. All these advantages suggest the use of coreannular flow for the t r a n s -p o r t of a high-viscosity liquid through a tube whenever the designer

can control this flow p a t t e r n . It thus becomes important to investigate the c h a r a c t e r i s t i c s of this type of flow m o r e in detail.

At Koninklijke/Shell-Laboratorium, Amsterdam, experiments have been performed, for instance in a 50-m long, 2-inch diameter pipe, in o r d e r to investigate the possibility of obtaining a stationary and stable core-annular flow. J u s t as in the case of the Charles, Govier and Hodgson experiments the core consisted of a high-viscosity oil, water being used for the annular film. The difference in density between oil and water was about 100 (kg/m^). The amount of water was varied between 1 and 20%. The e3q)eriments confirmed that a stationary and

stable core-annular flow for a high-viscosity core i s possible. A photograph of a c o r e - a n n u l a r flow made during these experiments, i s given in Fig. 2.

This e3q)erimental work was the reason for us to make a theoretical investigation of some general aspects of coreannular flow in a h o r i -zontal tube. One of the main problems i s to understand how the gravity force caused by a possible difference in density between the two liquids i s counterbalanced. This force tends to change core-annular flow into stratified flow: a flow pattern in which the lighter liquid fills the upper p a r t of the tube and the heavier liquid the lower part (see Fig. 3). Chapters 11 and III a r e devoted to the problem of understanding how t h i s gravity force i s counterbalanced. In Chapter n we simplify the problem by assuming the viscosity of the high-viscosity c o r e to be infinitely l a r g e . So we neglect any flow in the core and any

deformaTUBE WALL -FIGURE 2

OIL-WATER CORE- ANNULAR FLOW

LOW-DENSITY LIQUID

HIGH-DENSITY LIQUID

FIGURE 3 STRATIFIED FLOW

tion of the interface between the two liquids. As was observed during the ejqjeriments (see for instance Fig, 2) this interface i s rippled. These ripples a r e due to the growth of instabilities caused by the surface tension at the interface and by the movement of the core with r e s p e c t to the annular l a y e r . The simplification enables u s to make use of the hydrodynamic lubrication theory, which d e s c r i b e s the flow behaviour of a viscous liquid between two moving bodies. The simpli-fication r e n d e r s it also possible to make a free choice of, for instance, the wavelength of the ripple at the interface and the velocity of this ripple with r e s p e c t to the tube wall, although in reality ( i . e . when the core also h a s a finite viscosity) these quantities a r e determined by the operating conditions. In this simplified case we were able to solve the problem and to find expressions for the hydrodynamic p r e s s u r e and the velocity components in the annular l a y e r . In Chapter III an attempt h a s been made to extend our theory of Chapter H by studjdng c o r e

-annular flow with a finite core viscosity. However, in that case the problem becomes so complicated that only a rough approximation of a possible solution of the relevant equations could be derived.

Another problem investigated i s the hydrodynamic stability of c o r e annular flow. As we mentioned before, instabilities occur at the i n t e r -face due to the sur-face tension and the movement of the core with r e s p e c t to the annular l a y e r . These instabilities tend to break up the core into large and small droplets. In the past much i n t e r e s t has been devoted to t h e instability of a long cylindrical column of liquid under the influence of surface tension and the action of another s u r -rounding liquid. F o r instance, famous investigators such a s

2 3

Lord Rayleigh and Niels Bohr contributed to the solution of this problem. Up to now the investigations were r e s t r i c t e d to the case of a free j e t in an infinitely extended region and it was found that a j e t i s unstable and b r e a k s up into droplets. However, in the case of core-annular flow the surrounding liquid has a finite thickness. Actually this thickness i s very small relative to the radius of the tube. So it i s easy to conceive that in this c a s e the p r e s e n c e of the

tube wall h a s a strong influence on possible instabilities at the i n t e r face between the core and the annular layer and that a stable c o r e -annular flow may be possible, a s i s also found from e x p e r i m e n t s . In Chapter IV, which is devoted to this problem, a f i r s t o r d e r p e r t u r b a

-tion calcula-tion is given, in which the gravity and viscosity t e r m s a r e omitted, in o r d e r to investigate the hydrodynamic stability of core-annular flow. Of course, one cannot ejqject a complete proof of the stability of the flow pattern from such a calculation. This would r e q u i r e a further calculation including h i g h e r - o r d e r t e r m s , while also the gravity and viscosity t e r m s wovild have to be taken into account. Obviously, this would lead to a complicated mathematical problem. However, with the aid of the f i r s t - o r d e r calculation we a r e able to show that the tube wall h a s indeed a very strong reducing effect on the growth r a t e of possible instabilities, so that also from a theoretical point of view it becomes somewhat understandable that core-annular flow can be stable.

n . CORE-ANNULAR FLOW FOR AN INFINITE-VISCOSITY CORE A. General

This chapter deals with the problem of understanding how in the case of core-annular flow through a horizontal tube the net gravity force caused by a possible density difference between the two liquids, can be counterbalanced. To solve the problem we assume, as mentioned in the Introduction, that the core viscosity is infinitely large, neglect-ing any flow in the core and any deformation of the interface between the two liquids. Thus, we consider the core to move a s a rigid body at a certain velocity with respect to the tube wall. This assumption enables us to make a free choice of the shape of the interface and the velocity of this interface with respect to the tube wall. Thus, use can be made of the hydrodynamic lubrication theory, which d e s c r i b e s the flow behaviour of a viscous liquid between two moving bodies.

B. Hydrodynamic lubrication theory

The development of the hydrodynamic lubrication theory began in the 1880's, when Osborne Reynolds in England published a p a p e r in which, on the b a s i s of the hydrodynamic theory, he explained the very wide p r e s s u r e v a r i a t i o n s o b s e r v e d in c e r t a i n b e a r i n g s . At about the s a m e

5

t i m e Petroff , in Russia, was able to calculate the friction in a fluid-lubricated b e a r i n g . Important p a p e r s w e r e also published by

fi 7

Sommerfeld and Duffing .

Although the lubricant film i s thin in comparison with the dimensions of the lubricated bodies, it i s a s s u m e d to be thick c o m p a r e d with the dimensions of a lubricant m o l e c u l e . In consequence, no r e c o u r s e to its m o l e c u l a r s t r u c t u r e i s n e c e s s a r y . In the following we shall apply the calculation m e t h o d s of t h i s lubrication theory to our s i m p l i -fied p r o b l e m a s sketched in F i g . 4: a rigid c o r e with an a r b i t r a r y ripple on it moving through a tube.

FIGURE 4 CORE-ANNULAR FLOW

F o r our reference system we choose a co-ordinate system with the cylindrical coordinates r, d and x (see Fig. 4). The original problem of a moving core can be transformed to one in which the core is supposed to be at r e s t with r e s p e c t to the reference system and where the tube wall h a s a velocity W^ in the x-direction. Here h j and hg, being functions of i? and x, r e p r e s e n t the thicknesses of the core and the annular film, respectively, Rg is the radius of the tube, TJg the viscosity of film and Ap the difference in density between the film and the c o r e . The purpose of the calculation i s to investigate the possibility of a stationary core-annular flow. Somehow, viscous forces must therefore be built up such that in stationary flow they neutralize the gravity force on the core caused by the difference in density. This is impossible if the flow of the liquid in the annular layer i s p a r a l l e l to the wall of the tube. To counterbalance the gravity force, t h e r e must evidently be secondary flows perpendicular to the tube axis.

F o r our calculations of the p r e s s u r e variations and the secondary flows in the annular film caused by the movement of the core with r e s p e c t to the tube wall and by the gravity force we r e q u i r e the viscous force on the core and the gravity force to be in balance. The gravity and viscous forces cannot, of course, be e^qjected to be locally in balance when choosing an a r b i t r a r y form of curvature for the wavy interface between the c o r e and the annular layer a s we have done. In view of our simplifying assumptions (an infinite core viscosity), however, we only need a balance between the forces over the total wavelength of a possible ripple on the i n t e r -face.

We start from the time-independent equations of motion

^^^2 ^ F a j l a , d l ^ \ j \ 2 ^^2l / ^ " 2 , ^ 2 ^ " 2 , ^"2 ^2 \

9r = ^[arjr 37<v)j^7^^^-7a5-J-P2(;'2ör^Ts5-^*2a3r--rj

(2)

^2 ^ Fl 3 / ^*2V 1 ^ S / S l / 5*2 ^2^*2 5*2\

aF = ^2[757^'"-SFJ ^ 7 - ^ * ^ J - P2\"2-ir * T i r -^ *2-5rj •

(3)

in which P, stands for the density of the liquid in the annular film and Un, v„ and w , r e p r e s e n t the velocity components in the annular film in the r, ^ and xdirections, respectively. The dependent v a r i -able cpg is given by

tPg = P2 + P2 ^ cosiJ , (4)

where p_ r e p r e s e n t s the hydrodynamic p r e s s u r e in the film. The distance between core and tube wall i s assiuned to be small in com-parison with their radii of c u r v a t u r e . So

hg « Rg and h^ « i , (5)

where i r e p r e s e n t s the wavelength of a possible ripple on t h e i n t e r -face, The conditions given in Eq. (5) a r e the basic assumptions of the hydrodynamic lubrication theory and, r e s t r i c t i n g ourselves to the l a r g e s t t e r m s , we can simplify the equations of motion to

^ = 0 (6)

The c o r r e c t n e s s of such simplifications was investigated by

Sommer-fi 8 feld (1904), discussed extensively in T i p e i ' s book and is reviewed

in Appendix 1. Integration of Eqs. (7) and (8) yields

-2 = 4 ^ ' * ' " ' " ^)^— ^ - (9)

1 ^^2 2

- 2 = 411^^7'^ . K g l n r + K , , (10)

in which K,-K, are functions of J? and x only. The boundary

condi-tions are

for r = h^: Ug = 0, Vg = 0 and Wg = 0 (11)

and for r = R2: Ug = 0, Vg = 0 and Wg = W . (12)

Applying Eqs. (11) and (12) to Eqs. (9) and (10) we find

".-WM'"^ r,K|-hf, r,E|-hJ, ^ ""

1 ^'•'212 2 ( ^ 2 " ^ l ' ^ " R 2 ? ^ w ^ ' ^ ' R i

ï^2 ^2

According to the equation of continuity we have

or, after integration

J è < ^ 2 ) d r = 0 = - J ^ d r - J r ^ d r .

^ 2 , ?25v„ !^2 a

(16)

h^ h , h ,

2 , 2 4 H ^ ( l n H ^ ) ^ ) ö \ 2 \ 4 (1 " H J ) / ö^^a ^ ( 4 H ^ ( l n H ^ ) ^ 4H^ In Hj 19H^ 092 R g ' ( l - H f ) 1 - H ^ '

l«^2Wwp-"l "? p"l ^

Rg (4(lnH^)^ 2.1n H^) 9x (17)

in which H. is equal to h . / R 2 . Eq, (17) is a second-order p a r t i a l differential equation, also called Reynolds equation, relating tiie p r e s s u r e in the annular film to the distance between and the veloc-ities of the boundaries. We assume, that the following conditions for cpg have to be fulfilled:

(1) The thickness of the core h, i s assumed to be a periodic function of t? and x. We r e q u i r e that cp. (and hence the p r e s s u r e in the annular layer) is also a periodic function of ê and x. Of c o u r s e , this is only valid in case the net p r e s s u r e gradient through the annular layer in the axial direction i s much s m a l l e r than the p r e s s u r e gradients due to the local p r e s s u r e variations.

(2) Since the possibility of stationary core-annular flow i s investi-gated, we r e q u i r e that the viscous and gravity forces exerted on the core a r e in balance over a wavelength £ of a possible ripple on the interface. In Appendix 2 we a r r i v e at the conclusion that this condition yields the following relation:

2rT A 2n i i^\

n^j d^J dx(cp2)^^ c o s ^ + T l 2 R 2 | d^J d x j r | , M U sin^ =

0 0 ^ G O 2 2rT Ji

^fd^fdxh^. (18)

'li

in which the left-hand side r e p r e s e n t s the viscous force and the right-hand side the gravity force. By substitution of Eq. (13) in

Eq. (18) we find , 2n I j ^ 2TT i I 2 In - ^

R2 I dj>r dx(cp5!).. ^cos& +-^i dsf dxll - ,„ ^. „)1-^1 sin «?

u d^r dx(cp2)^^j^cos^+^rd^jc^j. — y ^

0 0 2 0 0 ( 1 - I — I

2n I ^ W

A£g f d i ? ! d x h ^ . (19)

o

C. Solution of the Reynolds equation

We shall now investigate what shapes the core may assume under con-ditions of stationary flow. We choose the general shape a s

• 5 - = ^ - j l + ee(,>)j 1 + Ex(x)j , (20)

in which lu ' r e p r e s e n t s the radius of the core when no ripple is p r e s e n t (e = 0), and in which 6 and x a r e periodic functions of i> and X, respectively:

E

(a cos mj? + b sin mi?) (21) m=l m m ' ^ '

L

, 2nmx , , 2nmx. ,„„,

, (c cos—-.— + d s i n — J — ) . (22) m=l m i m H ' ^ '

The p a r a m e t e r E i s a m e a s u r e of the amplitude of the ripple on the interface between the core and the annular layer. This p a r a m e t e r must be much smaller than 1, because otherwise the ripple amplitude is so l a r g e that the core touches the wall. The exact order of magnitude of e with r e s p e c t to I U / R Q will be determined l a t e r on.

We solve the Reynolds equation (17), in which the thickness of the core i s given by Eq. (20), by first finding a particular solution of the equation and then the solution of the homogeneous p a r t of it.

In A p p e n d i x 3 i t i s p r o v e d by m e a n s of a p e r t u r b a t i o n c a l c u l a t i o n , in w h i c h E i s t h e p e r t u r b a t i o n p a r a m e t e r , t h a t a p a r t i c u l a r s o l u t i o n of t h e R e y n o l d s e q u a t i o n i s p r o v i d e d by

^2.pan = | o ^ " r . (23)

i n w h i c h 9*2 ~ c o n s t a n t (24) (1) ^ ^ w ^ ^ 3 y l°m . 2nmx ''m 2nmx\ /ocx

(2) h'^J''

^2 2 ~ 2 n « 2 S mM ! L V ^ v S f : f n _ ^ ^ ^ „ ^ ^ ^ , . „ ^ ^ , a m x . 2;n«^ I _ ^ i __i / _ 2 2_2„v m m ' " n x, n I ^ ni=l n=l (26)

' ( - ^ - ^

GO QD 0 0 ( 3 ) ^ / y w 6 ^ X; E - " 2 2 2 , ( a cosm.>-^b s i n m ^ ) . 2 ^ m=l n=l q=l / „ ar-'n^Rtc '^ "" *"

r'^—^

[{d (c - c ) - c (d +d )3cos-^7^ + I-' q' n+q n-q' q^ n-q n+q" i ^ fd (d - d ) + c (c +c )} sin—-,— ] + etc. , / 2 7 \ ' a^ n+Q n-a' a n+q n - q " -t -' \^'> q' n+q n-q' q n+q n-q of w h e r e C ^ - C - , w h i c h a r e g i v e n i n T a b l e 1 p a g e 6 6 , a r e f u n c t i o n s h i / R 2 o n l y , c a n d d _ a r e z e r o f o r n - q ^ 0. T h i s p a r t i c u l a r s o l u t i o n i s a p e r i o d i c function of i> a n d x; h e n c e , a c c o r d i n g t o c o n d i t i o n (1) t h e s o l u t i o n of t h e h o m o g e n e o u s e q u a t i o n m u s t b e p e r i o d i c t o o . In A p p e n d i x 4 it i s p r o v e d t h a t t h i s i s only p o s s i b l e w h e n

*2,hom = '=°'»^t^"t- (28)

The solution of the Reynolds equation which satisfies the condition (1) is therefore given by Eqs. (23)-(27).

The l a s t condition to be satisfied is the condition (2). We first calcu-late the left-hand side of Eq. (19), which r e p r e s e n t s the viscous force exerted on the c o r e . In z e r o t h - o r d e r approximation we find

o o \ ' 2 0 0

sint? , (29) r = R ,

in which C i s a s given in Table 1. Substituting Eq. (24) in Eq. (29), we see that Eq. (29) vanishes. In the same way it can be proved that the contributions in first, second and t h i r d - o r d e r approximation vanish a s well. However, in fourth-order approximation t h e r e is a t e r m that contributes, viz.

E^C R 2" ^ /Scp(^)\ E^C R 2" ^ / a (2)\

! - ^ J d . J c i x x l ^ j s i n . + - ^ / d . | d x x 2 ( - 5 - j S i n . .

0 0 '^ "2 0 0 2

(30) in which C„ and CQ a r e given in Table 1. As can be seen from Eq. (30) this t e r m is only p r e s e n t when cp is a function of A According to

Eq, (13) this means that t h e r e a r e secondary flows perpendicular to the tube axis. Substitution of Eqs. (26) and (27) in Eq. (30) yields

^ i " ' ^ 2 W w ^ ^ X ( - C 6 C 8 ^ C 5 C 9 ) A , , (31) in which CO 00 A. = Z ! XI 0 0 0 — r [c {d (c ^ - c ) - c (d + d ^ )3 + 1 n = l q = l / 2n2n2R2c \ " 'ï "•^'1 ""^ï ^ n-Q n+q'"' + d {d (d ^ - d ) + c (c , + c )}] (32) n q^ n+q n-q' q^ n+q n-q'''-' ^ '

i s a form factor dependent on the shape of the core and the radius of the pipe. Substituting C^, C„, C-, C„, CQ and CQ from Table 1 in Eq. (31) and r e s t r i c t i n g ourselves to the largest t e r m s , we find that

Eq. (31) becomes

+ \y (33)

2hf

with a new A. given by

A- = - Z i S ° „ o. [c [d (c ^ - c ) - c (d + d )] +

+ d [d (d ^ - d ) + c (c ^ + c ) ] ] , (34) n q^ n+q n - q ' q^ n+q n-q'''-' ' ^ '

in which h^ ' = R- - hi ' is the thickness of the annular layer when no ripple i s p r e s e n t .

Restricting ourselves to the largest t e r m , the right-hand side of Eq. (19), which r e p r e s e n t s the gravity force, gives

2

n / i p g £ h p . (35)

According to condition (2) the viscous force on the core and the grav-ity force must covinterbalance each other. Hence, Eq. (33) is equal to Eq. (35), yielding the following relation

2 A p g i h ( ° ) \ ( ° ) ^

a, = 1 f , (36) g m i - w e A. R '

'2 w 1 2

This relation determines also the order of magnitude of E with respect to h ( ° V R o . F o r instance, when 2Apgehr i s of the o r d e r 0 (1),

2 2 9"fl2W ajAiR2

A(P)V'^2

we find that e is of the order 0|—=— I

V«2/

After substitution of a. from Eq. (36) in Eq. (23), we finally find that the solution of the Reynolds equation which satisfies the conditions (1) and (2) is given by

'^2 = constant + . 2 ,^^2 2rTnx . 2rTnx c o s — j - — c sin—J— cosi9Z^ Z^ 2rTnx 3 A ^ R 2 n = l q = l / 4TT2n2R2\'

[^'Vq-Vq)-'=qWn-q^<'„^)5-^T^

{d (d - d ) + c (0 ^ + c ) ] sin —,— 1 _{q' n+q n-q' q* n+q n-q' i •'} "STI W R^X - / c ^ 2 ^ ^ d ^ 2 ^ ^ \ ' ,0)3 m = i \ m i m i / vh] 91) (o) 2 m^n 2 3ftni2W^R2

^ Ë f; j f m - n - V n \ ^ , „ j , j m ^ , f

•"'^"-"""•"VQ.

^(^-")4 .

_(o)4 m=ln=l(\ m-n / X \ m-n / Z |

' V )rm^n"^mM . aT(m+n)x rm^n"'^n^m\ 2rT(m-t-n)x^ ^

E E

_{2 2 „ 2 \ * m} l n = l / ^ 4 n V R | \ o V . o e. I J ^ H X 2 1 7 0 X 1

(a cosmt7 + b s i n m t ? - a , cosi7)|a cos—7 c sin

TLW R ^ r-» ^ ^ "*^^ 2^ 1^ Li " „ „ fa cosmï? + b sinnn>-a,cosï?). , ^ n x , jcos—j— + . fd (c - c ) - c (d +d ^ ) } !_'• q n+q n - q ' q^ n-q n+q' + fd (d - d ) + c ( c +c )}sin-?!5^ + ^ q* n+q n - q ' q' n+q n-q ' l J + h i g h e r - o r d e r t e r m s . (37)

There are two extra conditions which must be taken into account for

core-annular flow.

(1) According to Eq. (20) the eccentricity of the core with respect to

the tube axis is given by

ea^h(°) . (38)

Naturally, this eccentricity cannot be larger than the thickness ai '

of the annular layer, because otherwise the core would touch the

wall. Hence

Ea^h(°)<h(°>, (39)

or after substitution of Eq. (36),

2 A p g i h f " h ( ° )

5 ,..3

< 1 . (40)

(2) According to Eq. (20) the amplitude of the ripple on the interface

is given by

Ehf). (41)

As this amplitude cannot be larger than the thickness of the annular

layer, we find

eh(°)

D. Frictional force on the core in the axial direction of the tube

In Appendix 2 it is explained that the force exerted on the core in the

axial direction of the tube per wavelength I is given by

or after substitution of Eq. (14),

r r 1 «2r^ r I '"fe)lM

J,= -T12WJ d^J d x ^ + ^ f d ^ r d x 2 + - ^ j ^ M ) .(44)

m ^ \ R2

Substituting Eqs. (20) and (37) in Eq. (44) and r e s t r i c t i n g ourselves to the l a r g e s t t e r m s , we find

Q 2

2rTTl W ^ R / 32n(Ap)2g2xh(°) hf> A„

J - ^J ' + 4 ^ 6 2 • (45)

^ h(°) 2 7 S 4 ^ 2 W , R | A 2

in which A„ is a form factor dependent on the shape of the core and the radius of the tube:

- n^(c^ + d^)

h - S 1 rT2r2r • (46)

_{n=l / 4 n V R ; \}

Consequently, t h e r e i s an axial s t r e s s on the core, to be henceforth called the " p r e s s u r e drop" through the core, which p e r wavelength X

i s given by g J x ^ 2 W w « 2 ^ 3 2 ( A p ) V i h ( ° ) A2

^Pcore = —-2 = , , . .2 + „„ 4 . „ , „ 6 , 2 • (*') (o)2 j^(o)j^(o)2 27E%2W^R2Af

nh^ "2 "1

The first p a r t of Eq. (47) r e p r e s e n t s the contribution by the main flow in the axial direction of the tube. The second p a r t r e p r e s e n t s the con-tribution by the secondary flow, which i s necessary to counterbalance the gravity force. If t h e r e is no density difference between the two liquids the second p a r t vanishes.

E. Conclusion

The movement of ripples on the interface between the c o r e and the annular film with respect to the tube wall induces p r e s s u r e v a r i a -tions and secondary flows in the annular film, which may exert a viscous force on the core perpendicular to the tube axis. This viscous force can counterbalance the gravity force due to a density difference between the core and the film, so that a stationary c o r e -annular flow is possible. So the ripples on the interface a r e essential; a s the amplitude Eh:°^ of the ripple vanishes the viscous force vanishes too (see Eq. (31)), which m e a n s that the gravity force can no longer be counterbalanced. In such a case the core would r i s e or descend in the tube until it touches the wall.

Eq. (37) yields the hydrodynamic p r e s s u r e distribution in the annular film, whereas Eqs. (13), (14) and (15) describe its flow p a t t e r n . F r o m these equations it can be found that the direction of flow in the annular film i s a s sketched in Fig. 5. By the transition of an ascending side of a ripple to a descending side the direction of the secondary flow changes sign. So in general one can say that a small oscillatory flow perpendicular to the tube axis is superposed on the main flow in the axial direction of the tube.

According to the above calculation the core cross-sections are circles and the ripple is symmetrical.

As can be seen from Eq. (36), the eccentricity of the core, which is given by Ea-jbj ' , is dependent on the density difference, the wavelength of the ripple, the thickness of the annular film, the viscosity of the liquid film, the velocity of the c o r e , the shape of the ripple and the radius of the tube. F o r instance, when the c o r e velocity i n c r e a s e s , the eccentricity d e c r e a s e s . Or, when the difference in density

i n c r e a s e s , the eccentricity i n c r e a s e s , etc. Naturally, the eccentricity of the core with respect to the tube axis and the amplitude of the ripple on the interface may not be l a r g e r than the thickness of the annular film, for in that case the core touches the wall. These con-ditions a r e formulated in Eqs. (40) and (42).

The " p r e s s u r e drop" through the rigid c o r e , which i s due to the frictional force exerted on the core in the axial direction by the liquid in the film, i s r e p r e s e n t e d by Eq, (47). In Fig, 6, 7 and 8 we have for some c a s e s plotted this " p r e s s u r e drop" a s a function of the c o r e velocity W , the viscosity Tl„ of the liquid in the annular

w ^ .

film and the average thickness of this film, h^ , respectively, F r o m these figures it can be noticed that for core velocities high enough, film liquid viscosities high e n o u ^ and film thicknesses small enough the contribution by the secondary flow to the " p r e s -s u r e drop" i -s negligible. So in the-se c a -s e -s the " p r e -s -s u r e drop" through the core i s almost equal to the " p r e s s u r e drop" of c o r e -annular flow without a density difference. However, also in these c a s e s the p r e s e n c e of the secondary flow r e m a i n s essential for counterbalancing the gravity force.

In reality, when the viscosity of the liquid in the core is also finite, t h e r e will be a net p r e s s u r e drop through the tube. However, in case the core viscosity i s much l a r g e r than the viscosity of the liquid in the annular layer, this p r e s s u r e drop i s almost equal to the first p a r t of Eq. (47).

The viscous force i s dependent on the shape of the ripple. For instance, when the ripple h a s the symmetrical shape of Fig. 9 ( i . e . C]^=l; CQ=C2= =0 and d =d,=, ..=0) one can easily see from

- 21 35 r 30 25 -20 •^CORE 15 10

[

\

\

\

-0 CONTRIBUTION FROM MAIN FLOW O CONTRIBUTION FROM SECONDARY FLOWS

g =10 R2 = 0.1 i = 0.03 h'g"' = 0.002 •n^ - 0.01 e = 0.01 A/>= 5 1 c , = C2 = d , = 1 \ C3 = C4 = . . . = dg = . . . = 0 1 1 ] ] ] 1 1 i n i - - ^ - r - ^ - i - j _ i _ L L n CONDITION (40): / W > l . 5 4 ( m / » ) /

/

1

/

1 1 1 1 M 1 IJ 5 -0.01 6 8 0.1 6 81 6 8 K) FIGURE 6

D CONTRIBUTION FROM MAIN FLOW O CONTRIBUTION FROM SECONDARY FLOWS

9 = R? =

\

r-hV'

=

Ww = e = Ap-ci = C3 = to 0.1 0.03 0.002 1 0.01 5 Cg = d, = 1 C4 = . . . = d2 =. . . = 0 J I I I I I 111 1 — •-i~-t-i-\-i.u^ 00001 6 8 0.001 6 e 0.01 6 8 0.1 FIGURE 7

A p

35

3 0

25

20

O CONTRIBUTION FROM MAIN FLOW O CONTRIBUTION FROM SECONDARY FLOWS

15 10 5 -g = 10 Rg = 0 . 1 i = 0 0 3 -q = 0.01 e = 0.01 Ap= 5 I I I I I I I I A I I T I T~i-rq a , -= dp -= . . . -= 0 ' 1 - I 1_L M M 0.0001 6 8 0.001 6 8001 8 0.1 Jo) FIGURE 8

PRESSURE DROP OVER THE CORE AS A FUNCTION OF THE THICKNESS OF THE ANNULAR LAYER

FIGURE 9

AN INTERFACE WITH A SYMMETRICAL RIPPLE

FIGURE 10

eq. (34) that the viscous force vanishes. Coreannular flow i s t h e r e -fore only possible when the ripple does not have a symmetrical shape (see, for instance. Fig, 10). The reason is that although a sym-m e t r i c a l ripple e x e r t s viscous forces on the c o r e , these forces balance each other over a wavelength of the ripple.

i n . CORE-ANNULAR FLOW FOR A FINITE-VISCOSITY CORE A. General

The simplification considered in the foregoing chapter, namely a rigid core instead of a deformable one, has important advantages from a theoretical point of view. F i r s t it i s possible freely to select the shape of the ripple and its velocity relative to the tube wall. Secondly, no internal flow in the core need be considered. However, the shape of the ripple and its velocity a r e in fact determined by the operating conditions; besides, a flow i s p r e s e n t inside the c o r e . In the p r e s e n t chapter we shall consider the effect of the internal c o r e flow by extending our theory to core-annular flow for a finite-viscosity c o r e .

The framework of the p r e s e n t calculation differs in many r e s p e c t s from that of the preceding chapter, not only because the core now also h a s a flow pattern, but also because the interface i s deform-able, which means that for stationary flow to be possible the gravity and viscous forces have to be locally in balance. The aim of the following calculation is thus to find out whether stationary c o r e -annular flow i s also possible in the case of two finite-viscosity liquids differing in density. So we r e s t r i c t ourselves to flow p a t t e r n s in which the shape of the interface between the liquids is

supposed to be at r e s t with respect to a certain system. We choose this system a s our reference system and assume that the tube wall only h a s a velocity component in the axial direction of the tube with respect to this system. Just a s in the foregoing chapter we suppose that the thickness of the annular film i s small relative to the radius of the tube and to the wavelength of a possible ripple

on the interface. In consequence, the calculation methods of the theory of hydrodynamic lubrication a r e again applicable.

In the following subscript 1 r e f e r s to the core and subscript 2 to the annular film. h. and h„, being functions of S and x, r e p r e s e n t the thickness of core and annular film, respectively (see Fig. 11); Rp i s the radius of the tube; 1] and T]- a r e the v i s c o s i t i e s of the two liquids; Ap = p - p^ is the difference in density between the low-viscosity liquid and the high-viscosity one; W is the velocity of the tube wall in the axial direction ( x d i r e c -tion) and U., V. and W. a r e the velocity components of the two liquids at the interface in the r - , ^ and x-direction, respectively.

CORE

LOW-VISCOSITY LIQUID FILM

a

LOW-VISCOSITY LIQUID FILM

FIGURE II CORE-ANNULAR FLOW

B. Reynolds equation for the annular film

In t h i s section we shall derive the Reynolds equation for the annular film. According to the chosen reference system the equations of motion for this film a r e time-independent. To facilitate reading these equations, which were already formulated in the foregoing chapter, they a r e repeated h e r e :

^ = 41'^'"^

X) ^2 + - L ! ^ +

a'u.

öx" 2\ / a u 2 V2ÖU2 a u g V 2 ^ \ 2^*2 9r "^ r ö^ ^ 2 Sx " r y

l ! l 2

r^ 9* (48) 1 ^ 2 r " 5 ^ = 11, 1 ^ ^ ^ 2 2 -,„2 r ÖJ/

2|4J7è(V)|

/ 9V2 V 2 a v 2 a v 2 U2V2\ 2ÖU, r^öj? (49)

"5F

-2 9^2 P2 ^ Tl a /

^^2\

1 / awg vgöwg aw2\ (50)

u„, v„ and Wg r e p r e s e n t the velocity components in the r - , ê- and x-direction in the film, r e ^ e c t i v e l y . The dependent variable tp- i s again given by

CP2 = P2 + p2Sr cost? , (51)

where p„ r e p r e s e n t s the hydrodynamic p r e s s u r e in the film. Making use of the basic assumption that the thickness of the film i s small relative to the radius of the tube and to the wavelength of a possible ripple on the interface between the liquids, a consideration of the o r d e r of magnitude of the various t e r m s similar to the one given in Appendix 1 yields the following set of simplified equations:

1 ^ 2 ^ a j i a , J

7-a^ = \ ^ | 7 a 7 ( ' ^ V j (^^^

^^2 ^2 a I ^^2\

Integration of Eqs. (53) and (54) gives

acp K

1 ^'^2 2

^ 2 = 4 ^ - 9 ^ - - ^ K g - l n r + K ^ , (56)

in which K . - K . a r e functions of t? and x only. The boundary conditions a r e that

for r = h j : U2 = U., V2 = V. and w^ = W. (57)

and for r = R „ : Uo = U = 0, v„ = V = 0 and w„ = W . (58) 2 2 w 2 w 2 w '

By Eq. (57) we admit the possibility that at the interface the two liquids flow with r e s p e c t to the wave pattern of this Interface. By Eq. (58) we a s s u m e that the tube wall has a constant velocity c o m -ponent with respect to the shape of the interface in the x-direction only. Applying Eqs. (57) and (58) to Eqs. (55) and (56) we find

1 S^2j , h^^(r^-R2Vnh^ R^(r2-h^)ln R^^ V I ^ - ' - R I ) ,,,,

'2 ( r(h^-R2) ^(^2 1^ r(hl-R2)

- 2 = 4 T i ^ ^ f - ^ 2 ^ h^^ ^ ^ +W,., (60) _{h, w} In p ^ ' In 5

In cylindrical coordinates the equation of continuity r e a d s -, 9v„ aw

lF(-2) = - ^ - ' ^ - a r ' (^^>

or after integration

ƒ

2 r ' ' " ^ 2 c^ "•"2 | p ( r u 2 ) d r = - h , U . = - J ^ d r - J r ^ dr . hj ^1 ^1 Substitution of Eqs. (59) and (60) in Eq. (62) leads to

2 ( 2 4 H 2 ( l n H , ) ^ a \ 2 \ 4 d - H ^ V ' ^

?2, ?2öv„ ?2 aw,

(62) 1 - H : ;r:—> K + 11 - H

R^r 1

1-H2

^9,^2 I M InH, ^ ^^2

i 4H^(lnH^)2 4H^lnH^jaH^aep2

16Tl„i/ H. \/l-Hf Xiav, 16T1 ( l-H? H? 1 aH

411 (2H?lnH + 1 - Hf/a(W.-WJ

T,2 I n H , ax « 2 ' ^ ^ H. r e p r e s e n t s h . / R g and H , is equal to

hg/Rg-Using the assumption that 'ii„( = l - H , ) « 1, we can simplify Eq. (63) considerably and finally we find the following Reynolds equation:

R2 9,^2 ^«2 ,^2 - 2^2 ^^ M - «2 9x ax '

^\'^i 12Tl2Vi aH2 ^ I2TI2H2 9V. ^2Tl2(Wj-W^ 9H2 2Ay\^^ 9W.

„3 „3 h& ^ 3 9!? „2 ax "" „2 9x '(^^^ «2 «2 ^2 ^2 "2

C. Re3Tiolds equation for the c o r e

The purpose of this section i s to derive the Rejoiolds equation for the c o r e . The equations of motion for the core a r e also time-independent with respect to the chosen reference system. The same boundary layer simplifications appear to be applicable to the flow in the c o r e . So also in this case the equations of motion reduce to

99

-57- = 0 (65)

in which v.. and w. r e p r e s e n t the velocity components in the i>- and x-direction, respectively, of the tube.

9i = P i + P i gr cost?, (68)

where p . r e p r e s e n t s the hydrodynamic p r e s s u r e in the c o r e . Integra-tion of Eqs. (66) and (67) yields

1 ^1 ^ 2

^i = 2 ^ ^ ' ^ ( - ^ " i ' ^ ^ ) " V " - r (6»)

- 1 = 4 ^ ^ ^ ' ^ K 3 - 1 ' ^ ' ^ " K 4 ' (^°)

in which KL.-K, a r e functions of t? and x only. The boundary conditions a r e that

for r = 0: V. = 0 and w. is finite (71)

and for r = h^^: v^^ = V. and w^^ = W. . (72)

Applying Eqs. (71) and (72) to E q s . (69) and (70) we find

1 ^^1 . r ^ i ^

1 411^

The equation of continuity r e a d s

1 ^ 1 2 2 ^ av^ aw^ a 7 ( ™ i ) = - - a ^ - ' ^ ^ - (^^) or after integration

J 97 (^l)dr = hU. =-J ^ d r - j r-53^

<r o o d r . (76)

After substitution of Eqs. (73) and (74) in Eq, (76) and r e s t r i c t i n g ourselves to the l a r g e s t t e r m s we finally find the Reynolds equation for the core

^ a \ a \ ^ 9 H ^ 9 9 aHj^9cpj^ „ 2 , ,2 2 2 \ , 2 9t> " 5 ^ "^ 9x ax R2 at? 9x Rg 8TljU. 4TljV. 9H^ 4T1^ av. 411^ 9W. + ^ ^ ; r + - 9 - - ^ - ( " ) 3 3 9t!> 3 9t> 2 9x Kg « 2 " 2 " 2

D. Boundary conditions at the interface between the c o r e and the annular film

The following boundary conditions hold at the stationary interface between the two liquids:

(1) the velocity components of the two liquids parallel to the i n t e r -face a r e equal;

(2) t h e r e a r e no velocity components normal to the interface; (3) the forces which the two liquids exert on each other a r e equal

and opposite.

The boundary conditions (1) and (2) can be reformulated in the following way- a liquid p a r t i c l e flowing on the interface between the two liquids r e m a i n s on the interface. This statement leads to the following

con-sideration: suppose that the shape of the interface i s given by

F(r, t?, X) = r - h^{^, x) = 0 . (78)

Let us follow a liquid particle, which at time t is p r e s e n t in point (r',«>',x') of the interface, so

F(r',t?', X') = 0. (79)

At time (t + dt) this particle i s present in point

V.

(r' + U.dt, I?' + j - i d t , X' + W.dt), (80)

which according to the above statement also belongs to the interface,

V.

F ( r ' + U.dt, ê' + r ^ d t , x' + W.dt) = 0. (81)

F r o m Eqs. (79) and (81) the following general equation is found, relating the velocity components of the liquids on the interface to the shape of this interface:

V. 9H aH

U i = + H ^ - ^ + W . R 2 ^ . (82)

Boundary condition (3) can be written in the following form

( ^ ^ l i k ) V. = ( " i ^ 2 ik> ,, • (^^> 1 i , u c p^jj 1 ^, IK r=h]^

in which T, ., and T „ ., represent the s t r e s s t e n s o r s of the core and the film, respectively, n is the unit vector normal to the interface. By using the basic assumptions that the thickness of the film i s small relative to the radius of the tube and to the wavelength of a possible ripple on the interface, Eq. (83) leads to the following t h r e e conditions

(P.) = (P2) , +'^(4- + ^ ) (84)

1 r=hi '^ r=hi y i l £ 2 /

(85)

(86)

in which a is the effective interfacial tension and R. and Rg a r e the principal radii of curvature of the interface, to be taken a s negative when the respective centre of curvature falls on the side of the annular layer. Substitution of Eqs. (51) and (68) in Eq, (84) gives

^2 = 9^ + A p g R 2 H ^ c o s t ? - a y - + ^ J . (87)

Substitution of Eqs. (59), (60), (73) and (74) in Eqs. (85) and (86) yields

,2 „ \ ..„ „ 2 , 9cp

"HIT

1 ^2/., '^2 , , M , ^ V 2 ! L ,88^

_{r = ^V^-2—2^''h7l\ . 2 „2, (^^)}

~"^v^r\) 2 '^i • ^

21^11'^R^

H I ' ^ R ;

After substituting h.. = Rh„ in Eqs. (88) and (89) and r e s t r i c t i n g o u r -selves to the l a r g e s t t e r m s we find

H2R2acpi H ^ a 9 2

_ H2Rf9cp, H ^ a c p 2

Wi-Ww--^fi^"aF--zfi^-5r- (^^)

E q s . (82), (87), (90) and (91) a r e the r e p r e s e n t a t i o n s of the boundary conditions on the interface.

E. Solution of the equations

The Re5Tiolds equations for the annular film (64) and for the core (77) together with the boundary conditions at the interface, a s formulated by Eqs. (82), (87), (90) and (91), form a complete description of the system from which in principle all possible flow p a t t e r n s can be found. In o r d e r to simplify this set of equations we substitute the boundary conditions and the relation H.+H2 = 1 in the Reynolds equations. Then we find, after straightforward calculations and after retaining the l a r g e s t t e r m s only (by making use of the fact that Hg "^ 1), the following two equations:

/1-fH a^H b \ \ ) 4T1 w an

c p 2 - ^ P g R 2 ( l - H 2 ) c o s t ? + a ( - ^ + ^ ^ - / . R 2 — ^ j | = + - ^ - ^ (92)

2

and

2 a 2 ^ /i+H 1 a V g ^ H V , 8TI W 9H 92 - A p g R 2 ( l - H 2 ) c o s , ? + a ( - ^ + - — l + R ^ — / j | = - ^ 3 3 ^ , (93) 2

in which t h r e e unknown variables a r e present, viz. 9 , , H„ and W . Owing to the fact that t h e r e a r e only two equations for t h r e e unknown variables, an infinite niunber of solutions a r e , in principle possible. However, the problem i s how to find that solution which r e p r e s e n t s the physically r e a l flow pattern. A possible way out of this difficulty

should be the use of a variation principle. However, to the best of our knowledge the validity of such a principle could never be proved, or at most only in a limited range of p a r a m e t e r values. Another problem is that owing to mathematical difficulties it is not possible to give a general solution of Eqs. (92) and (93). The reason i s that these equations a r e non-linear with respect to H2.

In Appendix 5 it i s demonstrated, that a possible solution of Eqs. (92) and (93) is given by

H„ = constant (94)

9 2 = K^x + ipgRgHjCost^ + Kg, (95)

in which K, and K2 a r e a r b i t r a r y constants. The boundary conditions and the relation H.+H2=l yield

(96)

(97)

(98)

(99)

(100)

The flow pattern belonging to this solution can be found by substitution of this solution in Eqs. (73), (74), (75), (59), (60) and (61). One then finds A p g H HgRg/iii „ \

-1= wr^\^-^J^'''^ (101)

APgH. HgRgT

^1= 2igr^ ^i"»^ (102)

K ( r ^ - h ^

" 1 = 47]^ " ^ i (103)

A p ^ ^ I ^ o s ^ j h ^ h^ h f l n h ^ ' - " 2 , _ . . . 2 ,

u„ = = •(•75 In r - -;r- In h , - T + ^;- • 9 i

M

%

K^ = constant = KjX + a / H j + K2 . U. = 0 A p g H ^ H ^ R ^ 2112 H R K W i ^ J. w 2TL • 2 ^ o ) 2 " ' ^ 2 1 4 4 ^2_^.^ 2 , „ . . , 2 . . 2 . . . . . „ 2 . / h^ R ^ l n r R 2 l n h A \^~~2 ^ ~ ^ ~ 5 ^ ) R2ln (Rg-h R 2 / r 1^1 h j l n r h ^ l n h A ) ApgH^H2RpijCost>/. h^ R 2 l n r R g l n h A (104)

ApgR2H^sintX h j ( r 2 - R | ) l n h^ R|(r2-h^)lnR2^ ^2 " 2fL j ' " ^ ' ^ ^ „ ^ 2 „ 2 , ..,„2 ^2 r(h^-R2) r(R2-hi) ApgH^H2R^^(r2-R2) sint? 2ni2r(h^-R2) ,2_^2„„ i ^ 1 j 2 ^ 2 ( « 2 - ^ ) ^ " R j j H l H 2 ^ 2 ^ 1 ^ ° R i W (105) (106) In p -1^2 ^ 2 l - R „

these expressions r e p r e s e n t the flow pattern of Fig. 12. As can be seen

CORE

FIGURE 12

CORE-ANNULAR FLOW FOR A FINITE-VISCOSITY CORE

from this figure, secondary flows perpendicular to the tube axis occur in the annular film and in the c o r e . As in the preceding chapter, these secondary flows a r e n e c e s s a r y for counterbalancing the gravity force due to the density difference between the two liquids.

However, the solution given in Eqs. (94)- (100) and sketched in Fig. 12 gives r i s e to many c r i t i c a l r e m a r k s ,

(1) The flow pattern differs in many r e s p e c t s from the one found in the foregoing chapter (see Fig. 5). F i r s t of all, t h e r e a r e no ripples on the interface between the liquids, whereas in the preceding chapter these ripples were essential.

(2) There i s no coupling between the main flow in the axial direction of the tube and the secondary flows perpendicular to it. This leads to the strange r e s u l t that we a r e quite free to put the p r e s s u r e drop through the tube (K.) equal to z e r o and yet find a velocity field. This m e a n s that in this limiting case a perpetual motion i s involved.

(3) The radial velocity components u. and u , satisfy only one of their two boundary conditions,

(4) In e^qjressions (101) and (102) for the radial and tangential velocity components of the core the viscosity of the core does not occur,

These difficulties would seem to be due to the fact that we have r e s t r i c t e d ourselves to the l a r g e s t t e r m s in deriving the Reynolds equations and the boundary conditions and in substituting the boundary conditions in the Reynolds equation. If we should continue the calculation to h i g h e r o r d e r t e r m s , we would find that these difficulties d i s -appear. However, owing to mathematical problems this extension of the calculations is very difficult.

F . Conclusion

An approximation to a possible solution of the equations of motion and of continuity h a s been found for the case of core-annular flow with a finite-viscosity c o r e . However, its interpretation gives r i s e to many p r o b l e m s . F i r s t , we do not know whether the solution a g r e e s with the physically real flow p a t t e r n . This is due to the fact that from a theo-retical point of view an infinite number of solutions a r e , in principle, possible and one does not know how to select the r e a l one. Secondly, in o r d e r to c a r r y out the calculations we had to make so many

simpli-fying assumptions that the derived approximate solution can only bear a poor resemblance to the physically r e a l flow pattern. To improve t h i s it would be necessary to extend the calculation to h i g h e r - o r d e r t e r m s , which would lead us, however, to still m o r e complicated c a l -culations, The only tangible conclusion of this discussion i s that in the case of core-annular flow for a finite-viscosity core secondary flows perpendicular to the tube axis occur, not only in the annular film, but also in the c o r e .

However, in case the core viscosity i s much l a r g e r than the viscosity of the liquid in the annular layer, we think that the solution given in the foregoing chapter is a good approximation. In that c a s e the net p r e s s u r e drop is almost equal to the first part of Eq, (47); the flow pattern may be a combination of the patterns given in F i g s . 5 and 12.

IV. HYDRODYNAMIC STABILITY OF CORE-ANNULAR FLOW A. Grcneral

The instability of a long cylindrical column of liquid under the influ-ence of surface tension and the action of another surrounding liquid has been the subject of many theoretical and experimental investiga-tions. At the end of the nineteenth century the founder of the

hydro-2 9

dynamic stability theory, Lord Rayleigh ' , became interested in the p r o p e r t i e s of flames. The complexity of the subject forced him first to examine the behaviour of a very simple type of j e t . Neglecting gravity and the effect of the surrounding fluid and assuming that the fluid of the jet was perfect and incompressible and that the surface of the unperturbed jet was not rippled, he found that the fluid column was unstable and broke up into droplets of a certain s i z e . Many y e a r s later he extended the calculation by taking into account the viscosity of the fluid. In a f t e r - y e a r s m o r e extended calculations appeared, in which the influence of the outside fluid was considered

11 12 a s well (see for instance Weber and Tomotika and Meister and

13

developed to determine the surface tension very p r e c i s e l y . It was 3

Nield Bohr who in 1909 made some major contributions in this r e s p e c t .

F r o m experiments it i s found (see Chapter I) that a stable c o r e -annular flow i s possible. Although, owing to the growth of instabili-t i e s , ripples instabili-then appear on instabili-the ininstabili-terface beinstabili-tween instabili-the instabili-two liquids, the growth velocity of these ripples i s damped out in one way or another. So a ripple with a finite amplitude r e m a i n s and the c o r e does not break up into droplets. Now, one of the main differences between core-annular flow and the above-mentioned j e t s i s that the surrounding liquid h a s a finite thickness. Actually, this thickness i s very small relative to the radius of the tube. So it is easy to conceive that in this case the p r e s e n c e of the tube wall has a strong influence on possible instabilities at the interface and that the tube wall may be the cause of the damping effect on the instabilities. The purpose of the p r e s e n t chapter will be to investigate this influ-ence.

Owing to mathematical difficulties we have to neglect the influence of gravity and of the viscosity of the liquids. As is well-known from the general hydrodynamic stability theory, it can be very risky to neglect viscosity because in a certain category of flows viscosity may cause instability. However, in the above-mentioned calculations it was found that such instabilities do not appear in the c a s e of j e t s ; by ignoring the viscosity the r e s u l t s of the calculation were affected only quantitatively,

B. Application of the general stability theory to our problem

The hydrod3fnamic stability theory can be defined on the b a s i s of the response of a certain flow to a small disturbance. Therefore, we shall first find a stationary unperturbed flow and then t r y to a s s e s s the influence of a disturbance on it. We start from the equation of motion (without gravity and viscosity t e r m s )

Dv^ ^ 2

grad p^ = - P ^ ^ p and g r a d p 2 = - p 2 - 5 t - . (107) in which

2

9u. au.. V. 9u. 9u.. V.

"5r + " l - 9 r + ~ " 5 ^ ^ * l ' 5 F - 7 - (r-component)

Dv.. j 9v. 9v. V. 9v.. a v . u. Vj

^ D r = - § r ^ " l - a r + - - 3 y ^ ^ l ^ ' ' - r - (-^-component) (108)

9wj^ awj^ V 9Wj^ 9Wj^ -rr— + u- -r— + ;-^- + w. -r— (x-Component). 9t 1 9 r r 9i> 1 9x ^ ^ '

The same relations hold for v . .

As in the foregoing chapter, p . and p„ r e p r e s e n t the hydrodynamic p r e s -' ^ -»

s u r e s in core and film, respectively, v . and v„ the velocities of the ••• ^ - >

liquids, u . , v.. and w. the velocity components of v. and u„, v„ and w„ the velocity components of Vp, in the r , t? and x-direction, respectively, and P- and p„ the densities of the liquids. The equations of continuity a r e

div Vj^ = 0 and dlv V2 = 0. (109)

We assume that the interface between the two liquids i s given by

F(r,t>, x;t) = r - h ^ ( i > , x;t) = 0. (110)

The velocity components of the liquids at the interface normal t o this interface must be equal. This boundary condition, which has also been discussed in section HID, can be written a s :

f ^ = 0 , • (111) o r | Z n . + | F ^ ^ 9 F ^èF^^ 9r 1,1 9J> h j 9x 1,1 at ^ ' and

|F ,aFXM.|F^ ^aF^

9r 2,1 9J> h . a x 2,1 at ^ '

in which (U. ., V. ., W^ .) and (U„ ., V„ ., W„ .) a r e the velocity

J.,1 i - , i i - , i —, 1 ^ , 1 ^ f i

components at the interface for core and film, respectively. The forces which the two liquids exert on each other at the interface must be equal and opposite. This condition i s given by

P i - P 2 = ^ ( i ; ; - ' ^ ) f''^ ^ = V (114)

in which a is the effective interfacial tension and JR. and R„ a r e the principal radii of curvature of the interface, to be taken a s negative when the respective centre of curvature falls on the side of the annular l a y e r . The last condition to be satisfied i s that the velocity component normal to the surface of the tube vanishes, so

U2 = 0 for r = R2, (115)

in which R . i s the radius of the tube. The problem i s now completely formulated by E q s . (107), (109), (112), (113), (114) and (115).

The flow pattern in the unperturbed state, which i s indicated by the superscript (o), is assumed to have a smooth core

hj°) = \ = constant. (116)

Besides, it i s assumed that t h e r e a r e constant velocity components in the x - d i r e c t i o n only, so

(O) (O) (O) (O) „ / i i r , .

" 1 2 1 2 (11^)

w(°) = w(°). = constant (118)

w(°) = W^S'l = constant. (119)

Z A, 1

Eqs. (107), (109), (112), (113) and (115) a r e then automatically satisfied, Eq. (114) giving for the constant hydrodynamic p r e s s u r e s the following relation

This unperturbed state, as described by Eqs. (116)-(120), represents

the smooth core-annular flow of Fig. 12, but without gravity and viscous

forces,

An infinitely small disturbance is now siqierimposed upon the interface,

so

h^ = Rj + Ef(t>,x;t), (121)

in which the dimensionless quantity:

e « 1, (122)

To find the response of the flow to this disturbance we make use of a

perturbation calculation, in which e is the perturbation parameter.

There-fore, we ejqjand the hydrodynamic variables in the following way

CO ^ 0 3 > -» V m (m) J -» v m (m) ,, „„. V-, = L, E v^ ' and v„ = 2^ e v^„ ' (123) ••• m=o •*• ^ m=o ^

p , = E e ^ ^ p S ^ ) a n d p 2 = t e ° ^ p f ) . (124)

^ m=o ^ ^ m=o ^

To a first-order approximation Eqs. (107), (109), (112), (113), (114) and

(115) become, with the aid of the unperturbed solution

/a^) a^)\

, (1) r 1 ^ (o) 1 I

g r a d p i ' = - p J - 5 ^ + w ^ ^ - 5 ^ 1

\R^ R^ 9t?2 9 x V

pf) - P^^) = - oi^ + - ^ H "" H ) *°'" "• = 1^1 (^^® ^°*'''^) (129)

u^l) = O for r = Rg. (130).

F r o m Eq. (125) it i s found that

(è ^ - r

è ) - ^

^T' ^ o - ^ (è - - r è ) -^ vf)

-

0.

(131)

Eq. (131) i s satisfied identically if rot vj ' = O and rot v( ) = 0; only in that case have we been able to solve the set of Eqs. (125)-(130) by introducing velocity potentials v( ) = grad \|i. and Vg ) = grad i|/„. Substitution of these potentials in the continuity equations (126) yields

A t j = 0 and Ai|;2= 0 , (132)

These equations have been solved by the method of separation of v a r i a b l e s ; a s i s well-known, the general

bination of the basic solution

a s i s well-known, the general solution for if. i s an a r b i t r a r y linear

com-in..^ ik..x

i|f^ =T^(t){B^Ijj (k^r) + B2Kn^(k^r))e ^ e "^ + constant, (133)

in which Tj^(t) i s an a r b i t r a r y function of time; B, and Bg a r e a r b i t r a r y constants, I_ and K_ modified Bessel functions of the first and second

° 1 ^ 1

kind, n, i s an a r b i t r a r y integer and k, an a r b i t r a r y constant.

Obviously, for t o we find a similar solution:

• 2 = T2(t){B3ln (k2r) + B^K^ (k^r))e^'^^e^^"" + constant. (134)

Since ii^ m u s t remain finite for r = 0, the following condition holds

(136) From Eqs. (133) and (134) the velocity components and the hydrodynamic pressures in the two liquids can be found in the following way:

"l -W' = '^lf^lS(V)^« ^ '^l(t)

4 = 7 ^ = i\fBllni(^l^)/^J« ^ Tj(t)

ai|i in t? ik X

^1 - ^ = *J^iS(^i^)^^ ^ T^(*)

. . . d T . , . in.t? i k . X

pl^ = - P l f d r - < \ ' ^ l T i 3 f B , ^ ( k , r ) 3 e e 1

a n d

,1^ ai|i in„i? ik„x 4 - - a r = kgf B3i;,2(k2r) + B4K;,^(k2r)} e e T2(t)

4 =7^=*°2f(B3^(k2'^)+B4V''2''))/''^^ ^ ^2(^>

^iv ai|i in t? ik„x

^2 = ^ = '^2^^3\^S'^ " B4'^„2*''2'^^^ ' ^ ^2(t)

. . . d T . iUpt? ikpX

P2 = - ^ 2 ^ ^ + i < i k 2 T 2 3 f B3l„2(k2r) + B4K„^(k2r)} e e .

The accent sign means differentiation with r e s p e c t to the argument of the B e s s e l functions. These solutions still have to satisfy the conditions given in E q s . (127), (128), (129) and (130). However, we shall first choose a specific form for the disturbance. As is well-known, every disturbance Ef(t>,x;t) for which

2TT -ko

I di? J d x r I ef I i s finite, (138)

o -«1

can be replaced by a F o u r i e r integral

(137)

o -°3

Ef(!9,x;t) = e I dt? r dxr A(n, k;t)e'"'^e*^^ (139)

where A(n, k;t) gives the value of the amplitude a s a function of time, wave number and direction. Thus the whole disturbance i s r e p r e s e n t e d by a spectrum over n and k. The simplest way to investigate the growth of the disturbance i s to study the behaviour in time of one wave of this spectrum. Therefore it i s assumed that

ef(*, x;t) = eRjT(t) e^'^'^ e*^'', (140)

in which T(t) i s a function of the time only. F o r t = 0 we take T(t) = 1, so that the disturbance initially i s a wave with amplitude E R . . wave

y

p 2 n k

(_S_) + k and wave direction (-^5—, —). Substitution of Rl Y% ^

Eqs. (136), (137) and (140) in (127), (128), (129) and (130) gives a s r e s u l t , that the only contribution to the general solution for t and i|u comes from

n. = Ug = n and k. = k„ = k, (141)

and furthermore that

k{B^I|^(kR^))T^(t) = ikRjW(°)jT(t) + \ ^ ^ (142)

kCBgi;(kR^) + B^K;(kR^)}T2 = ^^'Wz^i 1" ^ ^ 1 5 (l'^^)

dT^ dT„

_ p ^ { _ + ikwi*;). T^3{B^I„(kR,)}+ P g t ^ + ikW^^).T2]{B3l^(kR^) + B4K^(kR,)} =

= _ | _ ( l _ n 2 - k 2 R 2 ) T (144)

B 3 r (kR2) + B^K;(kR2) = 0. a 4 5 )

After substitution of Eqs. (142), (143) and (145) in Eq. (144) we find the following ejqjression for T:

T ^ 2 ' ^ \ E + G ; T d t - ' ^ \ ^ E + G ; " ^ ^ ^ ^ 3 ^ ^ ^ ^ ^ ^ »•

(146) in which

I„(kRi)

and

fv'^«i)-ïSoV^i^i)^

^2 G = — Pi i:(kR2) n i:(kR2)

'^«lfy»^Rl)-r(kO''n('^^l>'-'

> 0 . (148) n ' 2' Qt

From Eq. (146) it is concluded that T must have the form of e , in which Q; i s given by

°1,

2 /wf ).E + W(°»G\

4 ^ " E + G^'^)^

a

( l - n ^ - k ^ R ^ ( W n - W - ' . ) ^ - ^ . (149)

i: + kW)-w(°))'.

P ^ R 3 - (E + G) ^ l . i 2 , i ' • ^ ^ ^ ^ ^ 2

The influence of the tube wall is expressed by G. When the density of the surrounding liquid (pg) vanishes, this influence vanishes too.

The above calculation determines the response of core-annular flow to a small disturbance. Particularly, we asked ourselves whether, if the system is disturbed, the disturbance will gradually decay or grow in amplitude. In the former case the system is called stable with respect to the particular disturbance; in the latter case it is unstable. The flow pattern i s called hydrotfynamically stable if it i s stable with respect to all possible modes of disturbance and it is called hydrodynamically unstable even if there is at least one unstable mode. The tirtie dependence of the modes of disturbance for core-annular flow is given by Eq. (149). If a., or a^ has a positive real part for one of the many possible combinations of the variables n and k, core-annular flow is unstable. In the following we shall investigate Eq. (149) for the different kinds of instability that can occur.

F i r s t , we assume that the velocities of the two fluids a r e zero (w(°). = w(2°)j = 0); then Eq. (149) reduces to

In this case we a r e concerned with instabilities at the interface between two different liquids which a r e only due to the surface tension; the liquids a r e at r e s t with respect to each other. This kind of instability i s of the Rayleigh-Taylor type. When the influence of the surrounding liquid is neglected ( i . e . P = 0), the variable G vanishes and Rayleigh's r e s u l t is found. From Eq. (150) it i s seen that core-annular flow is only unstable with respect to symmetric disturbances (n = 0) with a wavelength X (= 2rT/k) l a r g e r than the circumference 2nR. of the unperturbed c o r e . With respect to all other disturbances the flow pattern is stable. So core-annular flow i s hydrodynamically unstable -with respect to the Rayleigh-Taylor instabilities. In this case a stationary pattern of motion prevails at the onset of instability; the instability sets in as a secondary flow. For this case we have investigated quantitatively the influence of the finiteness of the annular film, with P- ^ 0. In Fig. 13, 14 and 15 we have plotted a a s a function of kR.. for different values of R g / R , . The densities of the liquids a r e assumed to be equal;

the dimensionless quantity a/p R is chosen equal to 50. F r o m Fig. 13 we see that a s the tube wall approaches the core, i . e . R „ / R . approaches 1, the growth velocity of the instability d e c r e a s e s considerably. In the limiting case of z e r o thickness of the annular film this gjrowth velocity even vanishes. So the tube wall has a reducing effect on the instabilities, but does not change the critical wavelength X = 2nR..

Secondly, we assume absence of surface tension (a = 0), so (149) reduces to

[/(O) p .. w(0) r^A irnJ°) _ w(c),

1,2

'^y E + G ) E + G ^ ^ • (1^1)

In this case, where we study the c h a r a c t e r of the equilibrium of the interface between two different liquids flowing parallel to each other, we a r e concerned with an instability of the Kelvin-Helmholtz type. Since E and G a r e positive, we see from Eq. (151) that one of the roots always has a positive real part, when W^ '. is not equal to Wp '.. So one may say that the Kelvin-Helmholtz instability, which a r i s e s from the crinkling of the interface by the shear, occurs even at the smallest dif-ferences in velocities of the two liquids. This is in accordance with the

a(i)

+ 2.0 ± 1 . 0 0 ±1.0 ±2.0 n=0 R2/Rl=3.50 / / R 2 / R , = 2.00 V y / / R2/Ri=1.25 V y 1 / R2/R,=I.02 y / ^ ^ - ' ^ ^ R2/R,=1.00 > 0 5 1.0-^ UNSTABLE R2/R,=1.00

i

kRo 1 R2/R,=1.02 | \ STABLE

l\

W ^ e / R ,

=125 y R2/R, =2.00 ' Rg/R, =3.50 FIGURE 13

TIME DEPENDENCE OF DISTURBANCE AS A FUNCTION OF WAVELENGTH AND THICKNESS OF THE ANNULAR FILM

c(i)

±2

P^ - \

^."t

50 n=1 UNSTABLE ±61 1.5 kRo STABLE R2/R,=2.00 R2/R, =3.50 ±8i FIGURE 14

TIME DEPENDENCE OF DISTURBANCE AS A FUNCTION OF WAVELENGTH AND THICKNESS OF THE ANNULAR FILM

a(i)

± 5 r n = 2 = I ^i"? = 5 0 UNSTABLE R o / R , =1.00 _l i ] L. ±5i • ±101 ±151 +20 i 0.5 1.0 1.5 kRo ••R2/R, =1.02 STABLE R2/R,=1.25 R 2 / R , = 2.00 R 2 / R l = 3 . 5 0 FIGURE 15

TIME DEPENDENCE OF DISTURBANCE AS A FUNCTION OF WAVELENGTH AND THICKNESS OF THE ANNULAR FILM