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Int. J. Muhiphase FlowVo l. 15, No. 6. pp. 877-892, 1989 0301-9322/89 3.00 + 0.00
Printed in Great Britain. All rights reserved Copy right 1989 Pergamon Press/Elsevier
E F F E C T O F L I Q U I D V I S C O S IT Y O N T H E
S T R A T I F I E D S L U G T R A N S I T I O N I N
H O R I Z O N T A L P IP E F L O W
N
ANDP.II'SOS
Chemica l Process Engineer ing Research Inst i tu te , P .O. Box 19517, 540 06 Thessa lonik i , Greece
L
WILLIAMS an d
T J
HANRATTY
D e p a r tm e n t o f Ch e m ic a l En g in e er in g , U n iv e r s i ty o f I ll in o is , U rb a n a , IL 6 1 8 0 1 , U .S .A .
(Received 21 M a r c h 1988; in revised orm 27 December 1988)
Alwtract The
effec t of l iqu id v iscosity on the in i t ia t ion of s lug f low was s tudied in ho r izonta l 2 .52
and
9.53 cm pipe l ines. The resu l ts show the s tab i l iz ing e ffect of v iscosi ty predic ted by Lin & H anra t ty , and
are a t var iance wi th ana lyses which use a long-wave length invisc id approxim at ion . Fo r very v iscous liquids
a s tab i l i ty ana lysis which recognizes tha t s lugs or ig ina te f rom a t ra in of smal l -wave leng th s inusoida i waves
seems consis ten t wi th the measurements .
Key Words: two-phase f low, hor izonta l , gas- l iquid , t ransi t ion , s lug f low, exper imenta l , l iqu id v iscosi ty
I N T R O D U C T I O N
This pa per presents the resu l t s of exper imen ts on the e f fect of l iqu id v i scos ity on the t ran s i t ion f ro m
a s t ra t i f ied pa t te rn to a s lug pa t te rn for gas- l iqu id f low in a hor izonta l p ipe l ine . They sugges t a
new m echan i sm, whe reby t he p r ecu r so r t o s l ug fo rma t ion i s t he appea ran ce o f sma l l -wave leng th
K elv in -H elm hol tz (K H) waves . This me chan ism i s appl icable to l iqu ids wi th v iscos it ies > 20 cP,
and could be impor tan t in la rge d iameter p ipe l ines .
Th e t rans i t ion f ro m a s t ra t if ied pa t te rn to a s lug pa t te rn fo r a hor izonta l gas- l iqu id f low has been
the sub j ec t o f a num ber o f t heo re ti c a l s tud ie s . Ea r ly works by Ko rdyb an Ran ov (1970) and
Wal l i s Do bso n (1973) explored a s tab i li ty me chan ism, wh ereby the s lugs a r i se f rom the grow th
of inf in i tes imal d i s turban ces a t the in te r face . S ince the spac ing be twee n the s lugs i s la rge com pare d
to the p ipe d ia m eter it was na tu ra l to res t r ic t the ana lys i s to w aves which have a la rge wavelength
compared to the he ight of the gas space . I f v i scous e f fec t s a re neglec ted the fo l lowing condi t ion
for ins tab i l i ty i s ob ta ined for a hor izonta l rec tangular channel of he ight B:
k p L (U - - C ) 2
c o t h
k h L -t- k p G ( U - - C ) 2
c o t h
k hG
= g ( P L - - P G ) + t r k 2 , [ 1 ]
wh ere k i s the w aven um ber , U is the gas ve loc i ty , u is the l iqu id ve loc i ty , hG i s the he igh t o f the
gas space , h is the height of the l iquid space, PG is the gas densi ty, PL is the l iquid densi ty and
g i s the acce le ra t ion of gravi ty . Fo r la rge w a v e s
( k h L ~ .
l , khC ,~, l ) and for negl ig ib le sur face tens ion
ef fec t s the fo l lowing ins tab i l i ty condi t ion i s ob ta ined f rom [1] :
US PG -- 1 >
ghG P P
and the wa ve ve loc i ty , C, i s
C = Up 6hL + UpLhG
PLhG + hLPG
I f PGhL/pLhG i s smal l com par ed to uni ty [2] and [3] s impl i fy to
(U -- u ) > [ g( P L -- pGhG2~g . j
a n d
C = u + U ~ .
ghL
- - 1 ; [2]
[ 3
[ ]
[ s ]
877
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878 N ANDRITSOS e t a l
The phys ica l i n t e rpre t a t ion of t he K H ins t ab i li t y represented by [4] is as fo l lows: t he presence
of waves a t t he in t e r face causes a ve loc i ty ma ximu m in the gas a t t he c res t and a ve loc i ty min im um
at the t rough . Acc ord ing to the Bernoul l i equ a t ion th is f low var i a t ion i s assoc ia t ed w i th a pressure
minim um a t t he li qu id surface a t t he c res t and a pressure max imu m a t t he t rough . I f t he
des t ab i li z ing inf luence o f t hese forces i s l a rge enou gh to o verc om e the s t ab i li z ing e f fec t o f g rav i ty ,
ins t ab i l it y occurs .
W al li s D ob so n (1973) perfo rm ed t rans i t ion s tud ies in rec t angu lar channel s and conc luded
tha t [4] overpre d ic t s t he c r it ica l gas ve loc i ty by a fac tor of abou t two . A nu m ber o f au th ors have
argued tha t t h i s d i screpancy can b e expla ined b y cons ider ing n on- l inear e f fec ts . The ana lyses of
Ko rdy ba n (1977), M ish ima Ish ii (1980) and W al li s D ob so n (1973) sugges t a c r i t e r ion of t he
sam e form as [4] wi th factors of 0.74, 0.49 and 0.50 on the r .h.s. T ai tel D uk ler (1976) suggest
tha t i ns t ab i l i t y occurs in a channel when
p ~
J
[61
For gas- l iqu id f low in a p ipe they g ive
[ -g 0 L - - 0 ) X T :
L ~ S j . [7]
Here S , i s t he l ength of t he in t e r face
h L _ i )2 3 ''2 ,
hL is the he ight mea sured f rom the bo t tom of the p ipe , D i s t he d i ameter and Ao is t he a rea occ upied
by the gas .
Recen t ly , L in H anr a t ty (1986), H an ra t ty (1987) and W u
et al.
(1987) have re -examined the
poss ib i l it y of expla in ing the t rans i t ion f rom s t ra ti f ied to in t e rm i t ten t f low by l inear s t ab i l it y theory .
T h ey r e t a i n ed t h e a s su m p t i o n t h a t t h e w av e l en g t h i s l o n g co m p ared t o
h o,
o r h L , b u t ab an d o n ed
the assu mp t ion of i nv i sc id f low by inc luding the e f fec t s o f t he drag of t he gas o n the l i qu id , T i, and
of the res is t ing s t ress o f t he w al l on the l iqu id , ~WL. Eq uat io n
[ 2
s t i l l ho lds provided the ve loc i ty
prof i les in the l iqu id and the gas can be app rox ima ted by p lug f lows. The pr inc ipa l d i f fe rences foun d
by Lin H an ra t ty i s t ha t t he wav e ve loc i ty i s no t g iven by [3] or [5] .
The f i rs t te rm on the l .h .s , o f [1] represent s t he des t ab i l iz ing e f fec t o f l i qu id iner ti a . Fo r a i r -w ater
flows,
PG/PL
= 1.2 X 10 -3 and [5] simpl if ies to C = u for the range o f con di t ion s o ver w hich the
t rans i t ion i s observ ed . Co nseq uent ly , t he inv isc id K H ana lys i s p red ic ts no inf luence of li qu id iner ti a
on neutral s tabi l i ty since i t predicts
C/u -
1) ~ 0. I t repre sents a stat ic instabi l i ty . In co ntras t , the
v i sco u s an a l y s is o f L i n H a n ra t t y g i ve s n o n -ze ro v a lu e s o f
C/u -
1) because of t he inc lus ion of
the effects of ~, and 3w. As a resul t , an important destabi l izing effect of l iquid inert ia i s predicted
by the long-wavelength v i scous K H ana lys i s for l ow v i scos ity l iqu ids . The inc lus ion of v i scous
s t resses , t herefore , g ives the surpr i s ing e f fec t o f caus ing the a i r -w ater sys t em to be mo re u ns t ab le ,
in tha t t he in i t ia t ion o f l ong-wa velength in t e r fac ia l d i s turba nces i s p red ic t ed to oc cur a t muc h low er
gas ve loc i t i es (cons i s t en t wi th exper imenta l observa t ion) tha t i s g iven by an inv i sc id KH ana lys i s .
The theore t i ca l s ign if i cance is tha t a mech ani sm for the in it i a tion of i n t e rmi t ten t f low by the grow th
of in f in it es imal d i s turbanc es can not be ru l ed out .
These theore t i ca l resu lt s o f L in H anr a t ty (1986) a re i l lus t ra t ed in f igure I . The sys t em be ing
cons idere d i s a conc urren t f low of gas and l iqu id in a hor i zon ta l rec t angular channel o f he ight B .
The ord ina te i s t he d imensionless l i qu id he ight ,
hL/B. The
dashed curve i s t he superf i c i a l gas
ve loc i ty , Usc , p red ic t ed b y an inv isc id ana lys i s for the grow th o f l ong-wav elength d i s turbance s . Th e
sol id curves a re the v i scous pred ic t ions for a i r -water f low. In th i s case the in t e r face i s covered by
smal l -wavelength waves when in t e rmi t t en t f low i s i n i t i a t ed . Therefore , t he ana lys i s envi s ions a
l o n g -w av e l en g th d i s t u rb an ce su p e r i m p o sed u p o n t h e ro u g h en ed i n te r f ace . T h e e f fec t o f t h e se w av es
is fe l t by an increase in the in te r fac i a l s t ress . The p ara m ete r f / f ~ i s t he ra t io o f t he f r i c tion fac tor
f PU2
[9]
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L I Q U I D V I SC O S IT Y O N T H E S T R A T I F I E D S L U G T R A N S I T I O N
8 9
to the f r ic t ion fac tor for a sm oo th wal l , f , . A t rans i t ion a t lower gas ve loc i ties predic ted by the
viscous analysis is clearly indicated.
Lin H an ra t ty (1986) a l so present s tab i li ty ca lcu la t ions for l iqu ids o the r than water . The
predic ted e f fec t of a n increase in l iqu id v iscosi ty , when presented in a p lo t su ch as f igure 1 , i s to
t rans la te the so l id curves upwa rd . T he v iscous ana lys is thus p redic t s a s tab i liza t ion wi th increas ing
viscos i ty (when v iewed in th i s type of p lo t ) . However , the reason for th i s i s no t because of the
inc rea sed damping o f d i s tu rbances . S ince compar i sons a r e m ade a t f ixed hL/B increases in viscosi ty
are acco m pan ied by decreases in the l iqu id f low ra te for a g iven U sc . The iner t ia of the l iqu id and ,
there fore, i ts destabli l izing effect decre ases with in creasing l iquid viscosi ty. Fo r su ff icient ly high
l iquid v iscos i ty , des tab i l iz ing e f fec ts of l iqu id iner t ia beco me negligible and the L in Ha nra t ty
ana lys is is represented by the dash ed curve . I t , therefore , g ives the surpr i s ing resu l t tha t the inv isc id
theory becomes more accura te as the l iqu id v iscos i ty increases .
W hen the L in Ha nra t t y ana ly s is i s exh ib it ed i n M and han e coo rd ina t e s (UsL vs Us~), r a the r
than in the coord in a tes used in figure 1, the in te rpre ta t ion b ecom es m ore comp l ica ted . An increase
in l iquid viscosi ty has tw o op pos ing effects: for a f ixed Usc, an increase in the l iquid viscosi ty will
be assoc ia ted wi th the des tab i l iz ing e f fec t of an increase in hL for an y g iven USL. Ho wev er , an
increase in viscosi ty is acco m pa nie d by a sm aller USL for an y hL. This d ecreas es the de stabil izing
effect of l iquid inert ia .
L in H anra t t y (1986) t e st ed the i r ana ly si s by ca r ry ing ou t l abo ra to ry s t ud ie s o f a i r and w a te r
f lowing in a 2.52 cm i .d. pipe with a length of 600 pipe diameters and in a 9.53 cm i .d. pipe with
a length of 260 p ipe d iam eters . The y foun d tha t for UsG > 3 .3 m/s the s lugs were fo rm ed by a
non l inea r m echan i sm invo lv ing t he coa l e scence o f l a rge -ampl i tude i r r egu l a r waves . C onsequen t ly ,
i t could be expec ted tha t a t rans i t ion caused by the growth of inf in i tes imal d is turbances could be
appl icable only for UsG < 3 .3 m/s . A co mp ar ison o f the t rans i t ion fo r a i r -wa ter observed un der th i s
res t r ic t ion agrees qui te wel l wi th predic t ions based on the v iscous , long-wavelength KH ana lys is
i f f / f~ i s t aken equa l t o 2 , a r e a sonab l e a s sumpt ion fo r a i r -wa te r f l ow in t h i s r ange o f hL/D
(And r i t sos H an ra t ty 1987b).
How eve r , t he se t e st s w i th a i r and wa te r a r e a l so i n app rox im a te ag reem en t w i th t he non - l i nea r
ana ly se s d is cussed above . Th i s po in t ed ou t t he need fo r add i t i ona l expe r imen t s and p rom pted t he
in i t ia tion of a labo ra tor y s tud y o f the e f fec ts of change s of the l iqu id v iscosi ty . The resu l ts of th i s
ef for t a re presented in th is paper . Th e s t rong effec t of l iqu id v iscos i ty predic ted by Lin Ha nra t ty ,
when t he t r ans i t i on da t a a r e p lo t t ed a s
hL/D
vs Use , i s observed . However , a c loser compar ison
shows d i ff e rences be tween l abo ra to ry obse rva t ions and t he L in Ha nra t t y ana ly s is . Th i s p rompted
addi t iona l de ta i led s tudies of the mec hanism s for the in i tia t ion o f slugs . On the bas is of these tes t s
1.0
0.9
0,8
l j
m
0.7
0.6
. a 0 5
0 . 4
0 . ~
0 . 2
0 . I -
~ f t/ft 1.0
- f i / f 2 . 0
i l l I t t i l l I l
OOl 0~3~ QO3 005 0.07 0.10 0.20 0.30 3,50
U S G
F i g u r e I . S t a b i l i t y a n a l y s i s f o r s t r a ti f i e d fl o w in a c h a n n e l o f h e i g h t B : - - - i n v i s c id K H a n a l y s i s ; - -
v i s c o u s K H a n a l y s is .
Po PL =
1.12 x 10 -3
VG V L =
16.1.
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8 8 0 N ANDR I TSOS e t a l
it is argued that a KH stability analysis can be used for high viscosity liquids only if the assumption
of long-wavelength disturbances is abandoned.
An important ingredient in developing these arguments is the recent study by Andritsos
Hanrat ty (1987a), over a range of liquid viscosities of 1-80 cP, of the initiation of waves in stratified
gas-liquid flows. For air and water the first waves observed with increasing gas velocity are
generated by a sheltering mechanism involving pressure variations in phase with the wave slope.
At larger gas velocities irregular, large-amplitude waves appear. These are initiated by gas-phase
pressure variations 180 out of phase with the wave height (a KH mechanism). The influence o f
pressure variations in-phase with the wave slope becomes less important with increasing liquid
viscosity. In fact, at sufficiently high liquid viscosities the first waves that appear with increasing
liquid viscosity are the KH waves.
This influence of liquid viscosity has been pointed out some years earlier by Francis (1954, 1956)
and Miles (1959). Both Francis and Andritsos Hanratty observed the same mechanism for the
initiation of waves on a very viscous liquid. The first disturbances observed with increasing gas
velocity are small-amplitude, small-wavelength, rather regular 2-D waves. With a slight increase
in gas velocity, these give way to a few large-amplitude waves with steep fronts and smooth troughs,
and with spacings that can vary from a few centimeters to a meter. Ocassionally, several small 2-D
waves can be seen in front of the large waves.
In this paper evidence will be presented that events leading to the appearance of intermittent flow
at low gas velocities or to the appearance of large-amplitude irregular waves at large gas velocities
are the same. Consequently, an attempt will be made to interpret both phenomena by the same
stability analysis. Similar approaches have been adopted previously by Wallis Dobson (1973)
and by Andreussi Persen (1987). The large-amplitude waves seem to have been given a number
of names: slugs by Wallis Dobson (1973); roll waves, by Lin (1985); large disturbance waves,
by Andreussi Persen (1984); and irregular large-ampli tude waves, by Andri tsos (1986). Since the
initiation of these waves has been shown by Miles (1959) and by Andri tsos Hanrat ty (1987a)
to be due to a Kelvin-Helmholtz instability it seems appropriate to call them Keivin-Helmhol tz
(KH) waves.
DESCRIPTION OF THE EXPERIMENTS
The experimental results, discussed in this paper, for the 10 m horizonta l 2.52 cm pipe are from
the studies of Andritsos Hanrat ty (1987a) for the flow of air and liquids with viscosities of 1,
4.5, 16 and 70 cP. The results for 25 m horizontal 9.53 cm pipeline were obtained recently in a new
facility constructed by L. Williams. Transitions were observed with glycerine-water solutions
having viscosities of 1, 20 and 100 cP. Simple T-junction entries were used for mixing the two
phases at the inlet in both studies. In the 9.53 cm pipe the gas and liquid were introduced at two
different locations.
The thickness of the liquid layer flowing along the bottom of the pipe, hL, was measured in one
test section by two parallel-wire conductance probes which extended vertically across the whole
pipe cross section. For experiments in which small-wavelength waves were present the liquid height
was represented as the time-average of the signal from the conductance probes. A second test
section used pairs of short parallel 0.51 mm chromel wires at 45 c intervals to measure the variation
of the liquid height around the pipe circumference. In the 2.52 cm pipe the two sections were
separated by 10.2 cm, and in the 9.23 cm pipe, by 26.7 cm.
EXPERIMENTAL RESULTS
a) Mandhane p lo t s
A map of the different flow regimes observed in the 2.52 cm pipe is given in figure 2. The solid
line is for a liquid of 4.5 cP; the dashed line for 1.0 cP. Both shown a large region where the
pseudo-slugs, described by Lin Hanrat ty (1987a), occur. These are large-amplitude waves which
touch the top wall. They differ from slugs in that they are not accompanied by large pressure
fluctuations and they can lose their coherency as they move down the pipe.
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L I Q U I D
V I S CO S I T Y
O N T H E S T R A T I F I E D - - S L U G T R A N S I T I O N
881
_1
I ,O t
j . ~ .
S l u / ~
O . I - - / / P l e u do - S lu g
~ t A nnu la r
mlzal ion
o . o o l I I I
1.0 IO.0 IO0,0
OsG m/s )
F i g u r e 2 . F l o w r e g i m e s f o r t h e 2 . 52 c r n h o r i z o n t a l p i p e :
- - , 4 c P l i q ui d ; - - - , I c P l i qu i d . T h e t ra n s i t i o n t o
r e g u l a r s m a l l - a m p l i t u d e w a v e s i s n o t s h o w n .
1 .0
i
Slug
o o i i i
" I ~lon
o . o o l I ' - I
1.0 I0.0 IOOD
USG m /s l
F i g u r e 3 . F l o w r e g i m e s f o r a 2 0 c P ) a n d a I c P - - - )
l i q u id i n t h e 9 .5 3 c m h o r i z o n t a l p i p e . T h e t r a n s i t i o n t o
r e g u l a r s m a l l - a m p l i t u d e w a v e s i s n o t s h o w n .
As d i scussed by An dr i t sos Ha nra t ty (1987a) , t he f i rs t wav es tha t appe ar on a s t ra ti f ied f low
wi th increas ing gas ve loc i ty a re smal l -ampl i tude , regular 2 -D waves assoc ia t ed wi th gas-phase
pressure var i a t ions in phase wi th the wave s lope . Thi s t rans i t i on i s no t shown. At l a rger gas
ve loc i ti es l a rge-am pl i tude i r regular waves appea r tha t a re assoc ia t ed wi th pressure var i a t ions in
p h ase w i th t h e w av e h e i g h t (K H w av es ) .
An dr i t sos H an ra t ty (1987a) g ive resu l ts s imi la r t o those show n in f igure 2 for 16 and 70 cP
l iqu ids . These show a decrease in the l i qu id ve loc i ty requi red to in i t i a t e s lug f low a t l ow gas
ve loc it ies . For the 16 cP l iqu id the regular w aves appea r ov er a very narro w range o f gas ve loc it i es
an d fo r t h e 7 0 cP l i q u id t h ey d o n o t ap p ea r a t a ll.
Ob serve d t rans i t i ons in the 9 .53 cm p ipe a re show n in f igures 3 and 4 for 20 and 100 cP l iqu ids .
I t is no te d th a t t he p seud o-s lug reg ion decrease s in size wi th increas ing p ipe d i ameter and increas ing
l iqu id v i scos i ty . In fac t , i t is no t obse rved for t he 100 cP l iqu id in the 9 .53 cm p ipe .
T h e b eh av i o r a t t h e t r an s i ti o n t o s l u g fl o w can b e s ep a ra t ed i n t o t h ree r eg io n s (w h i ch a re m o s t
ev ident for t he h igh v i scos i ty l i qu ids) . The f i r s t i s for gas ve loc i t i es which a re too low for t he
app eara nce of K H wav es in a s t ra ti f ied f low (UsG < 4 m/s) . The second ex tends to the gas ve loc i ty
a t w h i ch p seu d o - s l u g s f i rs t ap p ea r i n t h e 9 .5 3 cm p i p e (U sG = 4 - 1 2 m / s ) . T h e t h ird is fo r
UsG > 12 m/s.
Fo r m ost o f t he f i r s t reg ime the f low i s no t fu l ly -deve loped and the l i qu id f low i s g rea t ly
inf luenced by the hy draul i c gra d ien t t ha t ex i s ts a long the en t i re p ipe , even thoug h i ts length i s 400
pipe d i am eters fo r t he 2 .52 cm p ipe and 260 p ipe d i ameters fo r t he 9 .53 cm p ipe . W hen such
hydraul i c grad ien t s ex i s t t he l i qu id f lowing for a g iven hL/D s l a rger s ince i t i s be ing moved both
by the gas drag a t t he in t e r face and the hydrau l i c grad ien t . B ecause the in i t ia t ion o f s lugs i s
p r i m ar i l y d ep en d en t o n
hL/D
t he USL va lues corre spo nding to s lug f low t rans i t i ons a t l ow UsG in
f igures 2-4 a re h igher tha n wo uld be ob serve d for a fu l ly -dev e loped f low. Thi s e f fec t i ncreased wi th
increas ing l i qu id v i scos i ty , increas ing p ipe d i am eter an d decreas ing l i qu id ve loc i ty . Thi s no n-devel -
ope d co ndi t ion was obse rved for l ow l iqu id v i scos i ti es a t UsG < 2 m/s in the 2 .52 cm p ipe an d a t
Us~ < 3.5 m/s in the 9 .53 cm p ipe . I t was obse rved for a l iqu id wi th a v i scos i ty of 100 cP a t
UsG < 4.5 m/s in the 9.53 cm pipe. Figu re 5 i l lust rates this effect for ai r-w ate r f lo w in the 9.53 cm
pipe . Here i t i s no ted tha t a t l ow gas ve loc i t i es t he he ight o f t he l i qu id i s i nsens i t i ve to changes
in UGs and dep end s on ly on the loc a t ion in the p ipe .
T h e ap p ea ran ce o f K H w av es i n a s t r a t i f i ed f l o w i s a cco m p an i ed b y a l a rg e i n c rea se i n t h e
in t e r fac ia l s t ress and a dras t i c t h inn ing of t he l i qu id l ayer , as i ll us t ra t ed in f igure 6 . Because of t h is
th inn ing of t he l i qu id mu ch l a rger USL are requi red to in i t ia t e s lugs . Th i s i s re f l ec ted by the sharp
rise in the t ran si t ion cur ve f or UsG = 4.5 -- 12 m/s in f igure 4 for a 100 cP l iquid. Th e t ransi t ion
to s lug f low observed wi th increas ing l i qu id f low in th i s reg ion in the 9 .53 cm p ipe for a l l l i qu ids
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F i g u r e 4 . F l o w r e g i m e s f o r a 1 0 0 c P ) a n d a l c P
-- - ) l i q u i d i n t h e 9 .5 3 c m h o r i z o n t a l p i p e .
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F i g u r e 5 . P l o t s h o w i n g t h a t a t l o w g a s v e l o c i t i e s t h e h e i g h t
o f t h e l i q u i d i s i n s e n s i t i v e t o c h a n g e s i n g a s v e l o c i t y a n d i s
m a i n l y d e p e n d e n t o n t h e l o c a t i o n i n t h e p i p e .
an d i n th e 2 .5 2 cm p i p e f o r 1 6 an d 7 0 cP l i q u i d s is v e r y s h a r p an d acco m p an i ed b y l a r g e p r e s s u r e
p u l s a t i o n s . F o r t h e l o w er v i s co s i ty l i q u i d s in t h e 2 . 5 2 cm p i p e t h e ex i s t en ce o f a l a rg e p s eu d o - s l u g
r eg i o n m ak es t h e d e f i n i t i o n o f t h e t r an s i t i o n m o r e d i f f i cu l t .
T h e t r an s i t i o n t o s l u g f l o w w i t h i n c r ea s i n g l i q u i d v e l o c i t y a t USG > 1 2 m / s is h a r d e r t o d e f i n e
b ecau s e o f t h e p r e s en ce o f p s eu d o - s l u g s . S l u g s m o v e ap p r o x i m a t e l y w i t h t h e g a s v e l o c i t y . A t
t r an s i t io n o n ly one s lug w i l l ex i s t in the p ipe l ine . Af te r i t s pas sag e the l iqu id l eve l in the p ipe i s
g r ea t l y r ed u ced . P s eu d o - s l u g s , t h a t a r e p r e s en t b e t w een t h e ap p ea r an c e o f s lu g s , t r av e l a t v e l o c i ti e s
u p t o 3 0 o f th e g a s v e l o c it y a n d c a r r y l i q u id w i th t h e m . C o n s e q u e n t l y , i t m a y t a k e a s m u c h a s
5 m i n a f t e r th e p a s s a g e o f a s l u g f o r th e l i q u i d to t h i c k e n e n o u g h f o r a n o t h e r s lu g t o a p p e a r . T h e
p r e se n c e o f a s lu g w a s d e t e c t e d f r o m m e a s u r e m e n t s o f p r es s u re f l u c tu a t i o n s , a s d is c u s se d b y L i n
& H an r a t t y ( 1 9 87 b ) . I t is n o t e d f r o m f i g u re s 2 - 5 an d f r o m f i g u r e s 1 3 an d 1 4 i n t h e p ap e r b y
A n d r i t s o s & H an r a t t y ( 1 9 8 7 a) t h a t t h e t r an s i t i o n cu r v e i n a M an d h an e p l o t a t U sG > 1 2 m / s i s
a p p r o x i m a t e l y i n d e p e n d e n t o f li q u i d v i c os i ty a n d o f p i p e d i a m e t e r f o r p i p e si ze s f r o m 2 . 5 4 t o
9.53 cm.
b) Effect of liquid height on transitions
I n o r d e r t o o b t a i n a m e ch an i s t i c u n d e r s t an d i n g o f t h e t r an s i t i o n t o s l u g f l ow , it i s u s e f u l t o p l o t
t h e r a t i o o f t h e l i q u id h e i g h t a n d t h e p i p e d i a m e t e r , hL/D, vs super f i c i a l gas ve loc i ty , Usc. Thi s typ e
o f p l o t m ay a l s o h av e t h e ad v an t a g e t h a t i t i s le ss s en s it i v e t o w h e t h e r t h e f l o w i s u n d e r d ev e l o p ed .
F i g u r e 7 p r e s en t s r e s u l ts o b t a i n ed i n t h e 9 .5 3 cm p i p e . T h e h L w e r e m e as u r e d j u s t p r i o r t o
t r an s i t i o n a t l o ca t i o n s i n t h e p i p e w h e r e s l ug s w e r e fi r st o b s e r v ed . T r an s i t i o n s w e r e o b s e r v ed a t t h e
m i d d l e o f t h e p i p e f o r l i q u i d s w i t h v i s co s it i es < 2 0 cP an d u p s t r e am o f t h e m i d d l e f o r l a r g e l iq u i d
v i sco s it ie s . I n s o m e ca se s t h e g a s v e l o c i t y a t w h i ch t h e l i q u i d h e i g h t w as m e as u r e d w as 2 0 s m a l l e r
t h an t h a t r eq u i r ed t o i n i t i a t e s lu g s . T h e r e f o r e , t h e v a l u e s o f h L m ay b e s o m ew h a t h i g h e r t h an t h e
t rue va lues .
T h e o p en s y m b o l s i n fi g u r e 7 r ep r e s en t co n d i t i o n s w h e r e a t r an s i t i o n t o s l u g f lo w w as o b s e r v ed .
T h e s o l i d s y m b o l s r e p r e s e n t c o n d i t i o n s w h e r e a t r a n s i t i o n t o K H w a v e s w a s o b s e r v e d . C o n s i d e r
the r esu l t s fo r a 100 cP l iqu id . I t is s een tha t fo r UsG < ca . 5 m /s an d h , / D > ca . 0 . 4 t h a t t h e h , / D
r eq u i r ed f o r t h e ap p ea r a n ce o f r o ll w av es v a ri e s s t r o n g l y w i t h U se . F o r t h e s e co n d i t i o n s t h e
i n t e r f ace is s m o o t h a t t h e t r an s i t i o n t o s l u g g i n g . H o w e v e r , f o r UsG > ca . 5 m / s t h e i n t e r f ace o f t h e
1 00 c P l i q u id i s r o u g h e n e d w i t h l a r g e - a m p l i t u d e w a v e s w h e n s lu g s a p p e a r . F o r t h e r a n g e o f g a s
v e l o c it i es o f 5 - 9 m / s t h e h L / D a t t r a n s i t i o n i s a p p r o x i m a t e l y c o n s t a n t a n d e q u a l t o 0 . 4 . A t
UsG > ca. 9 m /s the hL/D ag a i n d ec r ea s e s w i t h i n c r ea s i n g U SG - F o r h , / D < ca . 0 .4 the f ir s t t r an s i t ion
w i t h a 1 00 cP l i q u id t h a t i s o b s e r v ed i s t h e i n i t i a t i o n o f l a r g e - am p l i t u d e w av es . F o r ex a m p l e , f o r
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V I S C O S I T Y O N
TH E
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F i g u r e 6 . E f f e c t o f t h e a p p e a r a n c e o f K H
w a v e s o n t h e h e i g h t o f th e l i q u id a n d t h e
i n t e r f a c i a l d r a g .
8 8 3
F i g u r e 7 . I n i t i a t i o n o f s l u g f l o w o r K H w a v e s in t h e 9 . 5 3 c m
p i pe : - - - , T a i te l D u k l e r t r a n s i t io n s ; - - , a p p r o x i m a t e
c a l c u l a t io n s o f t r a n s i t i o n s t o s l u g s ( o p e n s y m b o l s ) o r K H
w a v e s ( s o l i d s y m b o l s ) .
5 0
hL D
= 0 . 3 l a r g e - am p l i t u d e w av es ap p ea r o n a 1 00 cP l i q u i d a t U s e = ca . 6 .5 m / s an d s l u gs d o n o t
a p p e a r u n t i l U ~ = c a . i0 m / s.
F o r s m a l l hL D h e r e i s a t r an s i t i o n f r o m s t r a t if i ed - w av y f l o w t o an n u l a r f lo w a t l a r g e U s c r a t h e r
t h an t o s l u g f l o w . T h i s i s n o t s h o w n i n f i g u r e 7 .
F i g u r e 8 p r e s en t s t r an s i t i o n r e s u l t s f o r t h e 2 . 5 2 cm p i p e . I n t h i s f i g u re t h e i n i t i a t i o n o f s l u g g i n g
o r o f la r g e - a m p l i t u d e w a v e s a r e t r e a t e d t h e s a m e . N o o b s e r v a t i o n s a r e p r e s e n t e d o f t h e i n it i a t io n
o f s lu g s o n a l i q u i d i n t e r f ace t h a t h ad l a r g e - am p l i t u d e w av es p r e s en t . T h e r e s u l t s s h o w n i n fi g u r e
8 t h u s r ep r e s en t e i t h e r a t r an s i t i o n t o sl u g s o r f o r U ~ > 4 m / s t o a s t r a t if i ed f l o w w i t h K H w av es .
T h e i m p o r t an t r e s u l t t o b e n o t ed i n b o t h f i g u re s 7 an d 8 is t h a t l i q u i d v i sco s i t y h a s a s t ab i l i z in g
e f f ec t i n t h a t l a r g e r v a l u e s o f
hL D
a r e r eq u i r ed f o r l a r g e v i s co s i t i e s . I t i s a l s o n o t ed t h a t an
a s y m p t o t i c b eh a v i o r i s o b s e r v ed f o r v i s co si ti e s > ca . 20 cP w h e r eb y t h e t r an s i t i o n i s i n s en s it i v e t o
ch an g es i n v i s co s i t y .
T h e r e s u l t s s h o w n i n f i g u re s 7 an d 8 co u l d h e l p ex p l a i n t h e o b s e r v ed e f f ec t o f l i q u i d v i s co s it y
o n t h e i n i t i a t i o n o f s l u g f lo w d i s cu s s ed in t h e p r ev i o u s s ec t i o n . A n i n c r ea s e i n li q u i d v i s co s i t y
0 , 9
i i 1 i i
O 2 . 5 2 c m F ' L ( c P )
0 .7 ~ ; , . , ~ . ~ . . . . . .
-
0 . 5 - -
0 . 3 - ~ ~ ~ l k ) ~
o I i
0.2 0 5 1.0 2 0 5.0 I 2 0 5 0
U S G m / s )
F i g u r e 8 . I n i t i a t i o n o f s l u g f l o w o r K H w a v e s in t h e 2 . 5 c m p i p e: - - - , T a i t e l D u k l e r t r a n s i t i o n s ; - -
a p p r o x i m a t e c a l c u l a t i o n s o f t r a n s i t io n s t o s l u g s ( o p e n s y m b o l s ) o r K H w a v e s ( so l id s y m b o l s ) .
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884 N ANDRITSOS t a l
i ncreases the hL/D co r r e s p o n d i n g t o a g i v en U SL an d , b ecau s e o f t h is , t en d s t o c au s e t r an s i t i o n a t
l o w er U SE . H o w ev e r , f o r v i s co s i ty ch an g es f r o m 1 t o 2 0 cP t h i s i s co u n t e r b a l an ced b y t h e s t ab i li z i n g
ef fec t sho wn in f igures 7 and 8 . Th i s exp la ins the sma l l e ff ec t on the USL requ i r ed for t r an s i t ion
cau s ed b y v i s co s i ty ch an g es f r o m I t o 4 .5 cP , s h o w n i n fi g u r e 2 , an d f r o m 1 t o 2 0 cP , s h o w n i n
f ig u r e 3 . H o w ev e r , f i g u r e 4 s h o w s a l a r g e e ff ec t o n t h e U sE a t t r an s i t i o n f o r a v i s co s i t y ch an g e f r o m
1 t o 1 00 cP , b ecau s e t h e d e s t ab i l i z in g e f f ec t o f i n c r ea s i n g h L D o u t b a l an ce s t h e s t ab i li z i n g e ff ec t f r o m
an increase in l iqu id v i scos i ty , shown in f igures 7 and 8 .
c) Visual observations
T h e f ir s t s l ug s f o r l i q u i d v i s co s i ti e s < 2 0 cP w e r e a l w ay s o b s e r v ed i n t h e d o w n s t r ea m h a l f o f
t h e p i p e l in e , d e s p i t e t h e f a c t t h a t h e w as l a r g e r i n t h e u p s t r ea m s ec t io n . F o r t h e 1 00 cP l i q u i d
i n t h e 9 .5 3 cm p i p e t h e f i r st s lu g u s u a l l y w as o b s e r v ed s l ig h t l y d o w n s t r e am o f t h e en t r y ,
b u t f r e q u e n t l y t h e y w e r e o b s e r v e d t o b e c o m i n g f r o m t h e m i x i n g s e c t i o n a t t h e e n t r y t o t h e
p ipe l ine .
T h e m ech a n i s m s f o r t h e t r an s i t i o n f r o m a s t r a t if i ed fl o w t o a s lu g f l o w f o r U s~ < 4 m / s i s m o s t
c l ea r l y seen f o r r u n s w i t h h i g h v i s co s it y li q u i d s b ecau s e t h e i n t e r f ace o f t h e s t r a ti f ied f l o w is s m o o t h .
C o n s e q u e n t l y , sp e c ia l c a r e w a s t a k e n t o a v o i d t h e f o r m a t i o n o f s l ug s a t t h e e n t ry w h e r e
o b s e r v a t i o n s c o u l d n o t b e m a d e . T h i s w a s d o n e b y e l e v a ti n g th e e n t r y f r o m t h e r e st o f t h e p ip e
b y a b o u t o n e - t e n t h o f a p i p e d i a m e t e r . T h i s c a u se d t h e l i q u i d h e i g h t a t t he e n t r y t o b e s m a l le r t h a n
i n t h e r e st o f t h e p ip e . T h e f i r st in t e r f ac i a l d i s t u r b an ces w e r e t h en o b s e r v ed f a r d o w n s t r e am o f t h e
en t r y w h e r e h L h ad a f a i r l y co n s t a n t v a l u e .
F o r r u n s w i t h t h e h i g h es t v i s co s i ty l iq u i d s i n b o t h t h e 2 . 5 2 an d 9 . 52 cm p i p e s, t h e f i rs t
d i s t u r b an ces p r i o r t o t h e ap p ea r an ce o f s l u g s w e r e v e r y s m a l l s i n u s o i d a l d i s t u r b an ces w i t h a
w a v e l e n g t h o f 2 - 3 c m . T h e s e w e r e p r e se n t f o r a f r a c t i o n o f a s e c o n d a n d q u i c k l y g a v e w a y t o a
l a r g e - am p l i t u d e w av e w h i ch g r ew v e r y r ap i d l y t o b r i d g e t h e w h o l e p i p e . W i t h i n c r ea s i n g U s ~ t h e
l i q u id l ay e r i s t h i n n e r a t t h e t r an s i t i o n . E v e n t u a l l y , a t s u f f ic i en t ly l a r g e U s c th e l i q u i d l ay e r is s o
t h in t h a t t h e l a r g e - a m p l i tu d e , l a r g e - w a v e l e n g t h w a v e s t h a t e m e r g e o u t o f th e r e g u l a r d i s t u r b a n c e s
a r e n o t ab l e t o g r o w t o b r i d g e t h e p i p e c r o s s s ec t i o n . I n s t ead t h ey a r e t h e i r r eg u l a r w av es , i d en t if i ed
b y F r an c i s an d b y A n d r i t s o s & H an r a t t y , w h i ch cau s e l a r g e r i n c r ea s e s i n t h e i n t e r f ac i a l s t r e s s an d
a t h i n n i n g o f t h e l i q u i d l ay e r .
F o r i n t e r m ed i a t e l i q u i d v i s co s i ti e s an d f o r g a s v e l o c i t ie s < 3 m / s , t h e f i rs t s l u g s ap p ea r ed c l o se
t o t h e m i d s e c t i o n o f t h e p ip e b y t h e f o l l o w i n g m e c h a n i s m . W h e n t h e l iq u i d w a s t h ic k e n o u g h f o r
t h e ap p e a r an c e o f s l u gs , a s h o r t l en g t h o f t h e l iq u i d w as co v e r ed w i t h sm a l l s i n u s o i d a l d i s t u r b an ces .
T h es e p e r s i s t ed f o r a f ew s eco n d s b e f o r e p r o d u c i n g a l a r g e - am p l i t u d e w av e w h i ch ev o l v ed i n t o a
s lu g a f te r t ra v e l in g a d i st a n c e o f a b o u t 1 m . A t U s c = 3 - 4 .5 m / s th e s m a l l - a m p l i t u d e w a v es , w h ic h
f ir s t ap p ea r ed o n t h e s t r a t i f ied l i q u i d w i t h i n c r ea s i n g l i q u i d f lo w , co v e r ed t h e w h o l e l en g t h o f t h e
p i p e li n e . W i t h a s l i g h t i n c r ea s e in U s e t h e s m a l l - am p l i t u d e w av es ch an g e d t o l a r g e - am p l i t u d e K H
w av es . O n e o f t h e s e q u i ck l y ev o l v ed i n t o a s lu g ; o ca s s i o n a l l y , a s l u g w as p r o d u c ed b y t h e
c o a l e sc e n c e o f tw o w a v e s . A s l ug , o n c e f o r m e d , m o v e d q u i c k l y d o w n t h e p i p e a n d t h i n n e d o u t t h e
l i q u i d l a y e r b e f o r e a n o t h e r c o u l d a p p e a r .
F i g u r e 9 a i s a p h o t o g r a p h o f r e g u l a r 2 -D w a v e s w h i c h c o v e r e d t h e w h o l e l e n g th o f th e p i p e, w h e r e
t h e d i r ec t i o n o f fl o w is f r o m f i g h t t o l e f t. I f th e l i q u i d is t h en en o u g h t h e s e r eg u l a r w av es can ev o l v e
i n t o i r r eg u l a r l a r g e - am p l i t u d e w av es s u ch a s s h o w n i n f i g u r e 9 ( f ) . F i g u r e s 9 ( d , e ) s h o w t h e f r o n t
an d t a il o f a l a r g e - am p l i t u d e w av e w h i ch h a s j u s t g r o w n t o f o r m a s l u g o n a t h i ck l i q u i d l ay e r .
F i g u r e s 9 (b , c ) s h o w t h e f r o n t a n d b a c k o f a s lu g t h a t w a s f o r m e d 8 0 p i p e d i a m e t e r s u p s t r e a m f r o m
w h e r e t h e p h o t o g r a p h w a s t a k e n .
T h e f i r st s lu g s t h a t a r e f o r m ed ac t a s a s w eep b y t ak i n g f l u id i n a t t h e f r o n t to i n c rea s e in
s ize and l eav ing a th in l iqu id l ayer beh ind . As shown in f igures 9 (d , e ) there i s a d ip in the f i lm
t h i ck n es s b eh i n d t h e s l u g . A s s h o w n i n f ig u r e 1 0, th e s l u g v e l o c i t y a t l a r g e g a s v e l o c i t y , m e as u r ed
as t h e t i m e f o r t h e s l u g t o m o v e t h e d i s t an ce b e t w een t h e t w o p a r a l l e l - w i r e co n d u c t an ce p r o b es ,
is ap p r o x i m a t e l y eq u a l t o t h e ac t u a l g a s v e l o c i t y j u s t b eh i n d t h e s l u g . A t l o w g as v e l o c it ie s , f o r
w h i ch t h e r e a r e l a r g e h y d r au l i c g r ad i en t s i n t h e p i p e , t h e s l u g v e l o c i t y ap p e a r s t o b e g r ea t e r
t h a n t h e g a s v e lo c i ty a t t h e d o w n s t r e a m l o c a t i o n a t w h i c h w a v e v e l o c i ty m e a s u r e m e n t s w e r e
m a d e .
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LIQUID VISCOSITY ON T H STRATIFIED SLUG TRANSITION 885
For conditions under which at most one slug existed in the pipe, the exit of a slug from the pipe
resulted in a pressure release, even at low gas velocities. The gas acceleration that resulted from
this pressure release instantaneously triggered the formation of regular sinusoidal surface distur-
bances. If the film thicknesses was high enough, a s econd and, sometimes, a third slug were formed.
These "satellites" moved at lower velocities than the original slug because o f the lower liquid levels
in front and behind the "satellite".
For UsG > 4.5 m/s K H waves are present on the liquid layer prio r to the init iation of slug flow.
Lin & Hanr att y (1987a) poin ted o ut that, for these values of Us~, slugs are formed by the
coalescence of the large-ampli tude waves. This process is very clearly seen in experiments with high
viscosity liquids. At low liquid velocities the K H waves are observed to coalesce and accelerate and
then break up and decelerate. However, at sufficiently large liquid flows (or sufficiently high liquid
viscosities) these waves come together in a group and form a disturbance which grows to a
sufficiently large height to cover the pipe cross section. Slugs formed in this way have a short length
and are highly aerated. The film thickness in front and behind them are seen to be approximately
he same, so they do not increase in size as they move down the pipe.
-.
.... - . . . . . . ,
Figures 9(a-c). (a) Regular waves observed 180 pipe diameters from the entry of the 9.53 cm pipe for
ULS = 0.06 m/s, UGs = 4.3 m/s and /~L = 20 cP. (b) Front of the slug that was formed 100 pipe diameters
from the entry of the 9.53 cm pipe, observed at 180 pipe diameters for Uts '-0.08 m/s, UGs = 4.3 m/s and
gL = 20 cP. (c) Tail of the same slug as in (b).
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g8 6 t~. ANDRIT SOS e t a l
,~
: . . : ,
. : . , : q . - . ) .t :- '. ;: ~ =. O .: . ' ~ . - - - . . ' , ~ ~ . i ~ , . : , : . . " " : ' . . . ? : ' . , ~ ' : , ~ .
. . . . . . . . . . . ~ . . j ~ . ~_ ~ . ~ . . . T ~ . ~ . . ~ . q ~ . . ~ . . z ~ . ~ : -- . .. . . . .
.... : .~.. ~.~ ~.~.;;:~ V.~M
F i g u r e s 9 ( d - f ). ( d ) S l u g t h a t h a s j u s t b e e n f o r m e d 3 0 p i p e d i a m e t e r s f r o m t h e e n t r y o f t h e 9 .5 3 c m p i p e
for L ' t .s = 0 . 0 3 3 m ,.'s , L 'o s = 4 m : s a n d ,u t. = 1 [.1 0 c P . ( e ) T a i l o f t h e s a m e s l u g a s i n ( d ) . ( f ) L a r g e - a m p l i t u d e
w a v e o b s e r v e d 3 0 p i p e d i a m e t e r s f r o m t h e e n t r y o f t h e 9 .5 3 c m p i p e f o r U ~ s = 0 . 0 3 3 m ' s U ~is = 7 m / s a n d
/a = 100 c P .
DISCUSSION
a) Comparison with previous analyses
The non-linear analyses that have been presented in the literature predict that a plot of
hL/D
at the stratified-slug flow transition vs Us~, for a given pipe diameter, should be independent of
liquid viscosity. Figures 7 and 8 clearly are contradic tory to this.
For example, the Taitel Dukler (1976) analysis predicts the appearance of intermit tent flow
to the right of the dashed curve and above the horizontal dashed line. It predicts the appearance
of annular flow beneath the horizontal dashed line. It is noted that the results for water at low
Use, are in good agreement with the analysis. However, the analysis underpredicts the
hL/D
required
for the initiation of slugs in high viscosity liquids.
The results in figures 7 and 8 would seem to support the large-wavelength viscous analysis of
Lin Hanrat ty (1986). See, for example, the calculations for a rectangular channel in figure I.
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LIQUID VISCOSITY ON THE STRATIFIED--SLUG TRANSITION 887
E
t . )
2 0 i i i
/
I 0 - - D = 9 , 5 3 c m /
J
5 - ,
~:
& 7 e
&
(CP)
2 - - I
k / 10(3
i I I I
2 5 I 0 2 0
UG ( m / s )
Figure 10. Slug ve loc i ty as a funct ion of the gas
veloci ty behind the s lug.
0 . 9
0 . 7
(3
~ o , 5
0 . 3
i i i , i
- - D,9.53 cm
O& ' ~ - . .
, , o D ,2 ,5 2 c m
Kelv in- Helmholtz
In l tabi l ty ~ ~
. K ~ l , i , : . , ~ . ~ h , . L n J ~ o , ' , ' ~
I n l . j o o i | i l y
I o l " ' 1
0.5 I.O 2 .0 5 ,0 I0 20 50
US ( m / s )
F i g u r e 1 1. C o m p a r i s o n o f K H s t a b il i ty c a l c u l a ti o n s w i t h
t h e o b s e r v e d i n i t i a t i o n o f s l u g s o r l a r g e - a m p l i t u d e w a v e s o n
a 7 0 c P D = 2 . 5 2 e r a ) a n d a 1 0 0 c P D = 9 5 3 c m ) l i q u i d .
0.1
0.2
H o w ev e r , a c l o s e r ex am i n a t i o n r ev ea l s d i s c r ep an c i e s . T h i s i s p a r t i cu l a r l y t r u e f o r ex p e r i m en t s i n
the 9 .53 cm p ipe w i th USG c lose to 3 m /s an d e xpe r ime nts w i th h igh l iqu id v i scos i ti es .
(b) Expe riments w ith high liquid viscosities
F i g u r e s 7 a n d 8 s h o w t h a t c u r v e s r e p r e s e n t i n g t h e t r a n s i t i o n t o K H w a v e s a r e c o n t i n u o u s w i t h
t h e cu r v es r ep r e s en t i n g t h e t r an s i t i o n t o a s l u g f l o w . T h i s r e s u l t w o u l d s eem t o s u g g es t t h a t t h e
s am e i n s t ab i l i t y m ech an i s m i s r e s p o n s i b l e f o r b o t h t r an s i t i o n s .
F i g u r e 11 is a p l o t o f t h e h i g h v i s co s i t y r u n s s h o w n i n f ig u r e s 7 an d 8 . T h e s t r i k i n g f ea t u r e o f
t h i s p l o t i s t h a t t h e d a t a f o r t h e t w o p i p e d i am e t e r s f a l l o n t h e s am e cu r v e . T h i s i s i n co n t r ad i c t i o n
t o t h e a n a l y s is o f L i n & H a n r a t t y a n d T a i t e l & D u k l e r , w h i c h w o u l d p r e d ic t
hL/D
t o b e a f u n c t i o n
o f t h e d i m e n s i o n l e s s g r o u p [(UsG/(gD) 2] [(PG/PL)I'Z]
T h e v i s u a l o b s e r v a t i o n s p r e s en t ed i n t h e p r ev i o u s s ec t i o n s h o w n t h a t a p r ecu r s o r t o t h e
ap p ea r an ce o f s l u g s o r o f K H w av es o n h i g h v i s co s i t y l i q u i d s i s an i n s t ab i l i t y w h i ch l e ad s t o
s m a l l - w a v e l e n g t h s i n u s o i d a l d i s t u r b a n c e s . T h i s w o u l d s e e m t o s u g g e s t t h a t t h e a s s u m p t i o n m a d e
b y L i n & H an r a t t y t h a t s l u g s a t UsG < 4 m / s a r e t h e r e s u l t o f t h e g r o w t h o f an i n f i n i te s i m a l
l a r g e -w a v e l e n g t h d i s t u r b a n c e c o u l d b e i n c o r r e c t.
T h e r e f o r e , s t ab i l i t y c a l cu l a t i o n s , w h i ch a r e n o t r e s t r i c t ed t o t h e a s s u m p t i o n o f la r g e - w av e l en g t h
w av es , w e r e ca r r i ed o u t . N eu t r a l s t ab i l i t y cu r v es o b t a i n ed b y u s i n g a co m p l e t e l y i n v i s c i d an a l y s i s
( [1] w i th ho ap pro xim ate d as in [10] an d [ I 1 ]) a re g iven in f igures 12-14 . Jus t i f i ca t io ns for us in g
an i n v i s ci d an a l y s i s a r e t h a t A n d r i t s o s & H an r a t t y ( 1 9 87 a ) h av e s h o w n t h a t t h i s a s s u m p t i o n g i v e s
r e s u l t s i n ap p r o x i m a t e ag r eem en t w i t h t h e o b s e r v ed c r i t i c a l g a s v e l o c i t y f o r t h e i n i t i a t i o n o f
l a r g e - am p l i t u d e w av es o n r e l a ti v e l y t h i n l iq u i d l ay e rs an d t h a t L i n & H an r a t t y ( 19 86 ) h av e f o u n d
t h a t t h e s t ab i l i ty o f s t r a t if i ed f lo w s t o l o n g - w a v e l en g t h d i s t u r b an ce s i s i n d ep en d en t o f v i s co s it y f o r
h igh v i scos i t i es .
A l l o f t h e s e n eu t r a l s t ab i l i t y cu r v es s h o w a c r i t i c a l U s w h i ch i s ap p r o x i m a t e l y i n d ep en d en t o f
p i p e d i a m e t e r . T h i s c r i t i c a l c o n d i t i o n c o r r e s p o n d s t o a w a v e l e n g t h i n t h e n e i g h b o r h o o d o f
. 5 - 3 . 0 cm . I t is a l s o n o t ed t h a t t h e v a l u e o f U s n eed ed f o r th e g en e r a t i o n o f la r g e - w av e l en g t h
w av es ( g i v en b y [ 4 ]) i s l a r g e r t h an t h e c r it i c al U sG an d is s t r o n g l y d ep e n d e n t o n p i p e d i am e t e r . F o r
hL/D = 0 .7 , t h e d i f f e ren ce b e t w een t h e c r it i ca l U so an d t h e s t ab i l i ty co n d i t i o n f o r l a r g e w av e l en g t h s
i s q u i t e s m a l l . H o w e v e r , a s hL/D d ec r ea s e s t h e d i f f e r en ce b eco m es s i g n i f ic an t an d can b e q u i t e l a r g e
f o r l a r g e d i am e t e r p i p e s .
Th e gas v e loc i t i es a t w hich s lugs were f i rs t obse rved in the 2 .52 an d 9 .53 cm p ipes for h ig h
v i s co s it y l i q u i d s a r e i n d i ca t ed a t t h e b o t t o m o f g r ap h s . I t i s n o t e d t h a t t h e i n i t i a t i o n o f s l u g s
co r r e s p o n d s t o t h e c r it i c a l U s an d n o t t o t h e v a l u e p r ed i c t ed b y [ 4 ] . F o r t h i n l i q u i d l ay e r s
(h/D = 0 .3 ) , it i s s een t h a t t h e c r i ti c a l U so co r r e s p o n d s t o t h e i n i t i a t i o n o f K H w av es . T h e
a g r e e m e n t o f t h is s t a b il i ty a n a l y s i s w i t h t h e o b s e r v e d a p p e a r a n c e o f s lu g s o r K H w a v e s i s a g a i n
s h o w n i n f i g u r e I I w h e r e t h e cu r v e r ep r e s en t s t h e c a l cu l a t ed c r i t ic a l U sG . T h e d a s h ed cu r v es i n d i ca t e
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N A N D R I T S O S t
a l
l O 0 . O
J
0 . 7 0 h h 3 0
I 0 , 0
I , O - -
K - H
S l u o s S l u o s
Waves
o . t l t t
0 . 0 2 . 0 4 0 6 . 0 8 . 0 I 0 0
U S G m / s )
Figure 12. Neutral stability curves for the flow of air and a very viscous iquid (PL = 1.22 g;cm3,/aL= :c.)
through a 2.52 cm pipe.
c a l c u l a t i o n s b a s e d o n [ 4 ] . T h e s e i n d i c a t e t h a t l o n g - w a v e l e n g t h s t a b i l i t y t h e o r y d i s a g r e e s w i t h
e x p e r i m e n t s a n d p r e d i c ts a s t r o n g e f f ec t o f p i p e d i a m e t e r .
(c) Experiments with low l iquid v iscosi t ies
T h e r e s u l t s o f t h e e x p e r i m e n t s w i t h t h e l o w v i s c o s i t y l i q u i d s a r e d i f f e r e n t f r o m t h o s e w i t h t h e
h i g h v i s c o s i ty l iq u i d s . A s a l r e a d y s h o w n b y L i n H a n r a t t y , th e U sG r e q u i r e d f o r t h e i n i t i a ti o n o f
s lu g s is d e p e n d e n t o n p i p e d i a m e t e r . W a t e r t r a n s i t i o n r e s u l ts a t U SG < 3 m / s f o r b o t h t h e 9 .5 3 a n d
2 .5 2 c m p i p e s c o m e t o g e t h e r o n a s i ng l e c u r v e o f
h L / D
v s
[UsG/(gD)~2][pG/p,)~ -].
T h i s d i f f e r e n t e f f ec t o f p i p e d i a m e t e r f o r l o w a n d h i g h v i s c o s it y li q u id s c a n b e u n d e r s t o o d b y
c o n s i d e r i n g t h e f i g u r e 1 2 . N e u t r a l s t a b i l i t y d i a g r a m s f o r l o w v i s c o s i t y l i q u i d s , t h a t c o n s i d e r o n l y
t h e e ff e c t o f p r e s s u r e v a r i a t i o n s i n p h a s e w i t h t h e w a v e h e i g h t h a v e a f o r m , f o r
h t D >1
0 .4 , s i mi l a r
t o t h a t s h o w n i n f i g u r e 1 2 f o r h i g h v i s c o s i t y l i q u i d s f o r
hL/D >1
0 . 7 . T h e l o n g - w a v e l e n g t h v i s c o u s
s o l u t i o n , w h i c h i s d e p e n d e n t o n p i p e d i a m e t e r , w o u l d i n f l u e n c e t h e c a l c u l a t e d c r i t i c a l U ~
1 0 0 . 0
I I I I
I 0 . 0 - -
h h h
1 .0 - -
K H
Stuos Slugs Waves
o .
t l t t t l
0 . 0 2 , 0 4 , 0 6 , 0 8 , 0 I 0 . 0
U s G m / s )
F i g u r e 1 3 . N e u t r a l s t a bi l i ty c u r v e s f o r t h e f l o w o f ai r a n d a v e r y v i s c o u s l i q u i d P L = 1 . 2 2 g / c m ~, ~ L
= O O )
t h r o u g h a 9 . 5 3 c m p i p e .
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LIQUID VISCOSITY ON THE STRATIFIED SLUG TRANSITION 889
I O O . 0
I 0 . 0 -- ~ ~.I
= i . = I - - I .
1.0 - -
0 . = 1 I I 1
a O 2 . O 4 . 0 6 , O 8 . 0 I 0 . 0
U S G m / s )
F i g u r e 1 4 . N e u t r a l s t a b i l i t y c u r v e s f o r t h e f l o w o f a i r a n d a v e r y v is c o u s l i q u i d P L = 1 .2 2 g / c m 3 , ,uL = oc )
t h r o u g h a 5 0 .8 c m p i p e .
f o r a i r - w a t e r f l o w s a t U s G
1
0.5 [10]
ri g = (0 .5D - HL ) + riG[hL O=0.5
h L / D
< 0.5 [I 1]
I n a d d i t i o n , t h e ~ /' t e r m d e f in e d b y L i n H a n r a t t y t o t a k e a c c o u n t o f t h e i n f lu e n c e o f l iq u i d
R e y n o l d s n u m b e r o n t h e i n t e r f a c i a l f r i c t i o n f a c t o r w a s s e t e q u a l t o z e r o t o a g r e e w i t h t h e r e c e n t
r e s u l t s o f A n d r i t s o s H a n r a t t y ( 1 9 8 7 b ) . T h e li q u i d f l o w s f o r I a n d 4 . 5 c P l iq u i d s i n t h e 2 . 5 2 c m
p i p e a n d f o r 1 a n d 2 0 c P l i q u id s i n t h e 9 . 53 c m p i p e w e r e c o n s i d e r e d t u r b u l e n t . A v a l u e o f t h e
f r i c t i o n f a c t o r r a t i o , f / f s e q u a l t o t w o w a s s e l ec t ed a s a r e a s o n a b l e a v e r a g e r e p r e s e n t a t i o n o f t h e
i n t e r f a c i a l s t r e s s f o r w a t e r .
C a l c u l a t e d t r a n s i t i o n c u r v e s f o r d i f f e r e n t l i q u i d v i s c o s it ie s a r e p l o t t e d a s t h e s o l i d c u r v e s in f i g u r e s
7 a n d 8 . A s h a s a l r e a d y b e e n i n d i c a t e d i n f ig u r e I I , th e 7 0 c P c u r v e f o r D = 2 . 5 2 c m a n d t h e 1 00 c P
c u r v e f o r D = 9 . 5 3 c m a g r e e . T h e c a l c u l a t e d c u r v e s f o r l o w v i sc o s i ti e s s h o w t h e s t r o n g e f f e c t o f p i p e
d i a m e t e r a l r e a d y d i s c u s s e d . T h i s i s i l lu s t r a t e d i n t h e c a l c u l a t i o n s f o r a l i q u i d v i s c o s i ty o f I c P s h o w n
i n f i g u r e 1 5 . A p p r o x i m a t e a g r e e m e n t b e t w e e n t h e c a l c u l a t i o n s a n d e x p e r i m e n t a l r e s u l t s i s s h o w n
i n th e f i g u r e s , e x c e p t t h a t t h e e x p e r i m e n t s s h o w n o d i f f e r e n c e b e t w e e n t h e 1 6 a n d 7 0 c P l i q u i d s in
t h e 2 . 5 2 c m p i p e a n d b e t w e e n t h e 2 0 a n d 1 0 0 c P l i q u i d s i n t h e 9 . 5 3 c m p i p e .
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890 N. A N D R I T S O S t a l
0 . 9 , , , I ,
0 . 5
0 . 3
o i - I I 1
I ~ I
0.2 0 .5 1 .0 2 .0 5 .0 I0 20 50
U s o m / s )
F i g u r e 1 5. C a l c u l a t e d t r a n s i t i o n s t o s l u g f l o w o r K H w a v e s f o r a I c P l i q u id .
T h e ca l cu l a t ed r e s u l t s f o r a f u l l y - d ev e l o p ed fl o w a r e p r e s en t ed i n a M an d h an e p l o t i n fi g u re 1 6 .
T h e r e i s q u a l i t a t i v e ag r eem en t w i t h t h e r e s u l t s p r e s en t ed i n fi g u r e s 2 - 4 an d i n fi g u re s 1 2 - 1 4 o f th e
p ap e r b y A n d r i t s o s H a n r a t t y (1 9 8 7 a ). A n i n c r ea s e o f v i s co s it y f r o m I t o 4 .5 cP f o r D = 2 .5 2 cm
an d f r o m 1 t o 2 0 cP f o r D = 9 .5 3 cm h as a s m a l l e ff ec t b ecau s e o f th e co u n t e r b a l a n c i n g o f t h e
d es t ab i l i z a t i o n a s s o c i a t ed w i t h t h e i n c r ea s e i n h L / D an d t h e v i s co s it y s t ab i l i z a t i o n s h o w n i n f i g u re s
7 an d 8 . F u r t h e r i n c r ea se s i n v i s co s it y a r e s h o w n t o c au s e a s h a r p d ec r ea s e i n t h e USL r eq u i r ed f o r
t r a n s i ti o n t o a n i n t e r m i t t e n t f lo w s u c h a s s h o w n i n fi g u re 4. T h e d e t a i le d q u a n t i t a t i v e a g r e e m e n t
b e t w e e n t h e s e c a l c u l a t i o n s a n d m e a s u r e m e n t s i s n o t s o g o o d b e c a u s e i n t h e e x p e r i m e n t a l s y s t e m
t h e f l o w is n o t w e l l -d ev e l o p ed a n d b ecau s e t h e m o d e l u s ed t o c a l cu l a t ed USL f o r a f i x ed h L / D an d
UsG is not exact .
C O M P A R I S O N W I T H O T H E R M E A S U R E M E N T S
W u et al (1987) r ecen t ly s tud ied g as - -cond ensa te f low in a hor iz on ta l 20 .3 cm p ipe a t 75 b
p r e s s u r e . T h e y r ep o r t ed t r an s i t i o n s t o s l u g f l o w a t s u p e r f i c i a l l i q u i d v e lo c i ti e s i n t h e r an g e o f I m / s
f o r su p e r f ic i a l g a s v e l o c it i es o f 0 . 7 -8 m / s . C o m p a r i s o n s s h o u l d b e m ad e a t t h e s am e v a l u e o f
1 ~
P G - U SG . B ecau s e o f t h e h i g h d e n s i t y o f t h e g a s , t h e i r t e s t s s h o u l d c o r r e s p o n d t o r e s u lt s i n t h i s p ap e r
1 , 0 I
I [
E
v
o . i
0 , 0 1
0 , 0 0 1
D
= 2 . 5 2 c m
/ . ~ L C P )
O = 9 . 5 3 c m
~L C P)
o.0 I ,o Io .o 0 .o I .O Io ,o
UsG m/s) USG m/s )
F i g u r e 1 6. C a l c u l a t e d t r a n s i t i o n s to s l u g f l ow o r K H w a v e s p r e s e n t e d i n a M a n d h a n e p l o t .
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L I Q U I D V I S C O S I T Y O N T H E S T R A T IF I E D S L U G T R A N S I T I O N 891
1 0
,~ 0.1
: : : ) 0 .01
0.001
l i |
0 0 0 0 0 0 0 0
O 0 0 0
0 A
O 6
A
I
0 , 1 1 , 0 I 0 IO0
O s moot h s t r t i f i e d o e l o n g a t e d b u b b le w a v y a n n u l a r
w v y s t r t i f i e d e e l u g ~ a n n u l a r
Figure 17. Flow patterns for a 90 cP glycerine/waterand air m ixture in a horizon tal 3.8 era p ipe (Taitel
Dukler 1987).
i n t h e r eg i o n o f U s c > 5 m / s . I t is q u i t e p r o b ab l e t h a t t h ey w e r e o b s e r v i n g f l o w s f o r w h i ch t h e
m e ch a n i s m f o r t h e g en e r a t i o n o f sl u g s is th e co a l e s cen ce o f K H w av es . I f t h i s is t h e c a se , t h en l i n ea r
s t ab i l i ty p r ed i c t s t h e i n i t i a t i o n o f l a r g e - am p l i t u d e w av es , b u t n o t s l u g s .
F i g u r e 1 7 g i v e s a co m p a r i s o n o f t h e ca l cu l a t i o n s o u t l i n ed i n t h e p r ev i o u s s ec t i o n w i t h th e
m e as u r e m e n t s o f T a i t e l D u k l e r ( 19 87 ), d o n e i n a 3 .8 cm p i p e w i t h a l iq u i d v is co s i ty o f 9 0 cP .
G o o d a g r e e m e n t is n o t e d b o t h w i t h t h e o b se r v e d t r a n s i ti o n s t o K H w a v e s a n d t o i n t e r m i t t e n t f lo w .
C O N C L U D I N G R E M A R K S
A c o m p ar i s o n o f ex p e r i m e n t a l r e s u l t s o n t h e s t ab i l i t y o f a s t r a ti f i ed f l o w i s b e s t d o n e o n a p l o t
OfhL D
v s U s(:; r a t h e r t h a n w i t h t h e M an d h an e co o r d i n a t e s , USL VS U s ~ . T h i s i s so , n o t o n l y b ecau s e
an e v a l u a t i o n o f t h eo r i e s is m o r e ea s i l y d o n e , b u t a l s o b ecau s e e f fec t s a s s o c i a t ed w i t h a
n o n - d e v e l o p e d f l o w m a y b e l e s s i m p o r t a n t .
E x p e r i m en t a l r e s u l ts o n t h e e f f ect o f l i q u i d v i s co si t y o n t h e i n i t i a t i o n o f sl u g s s h o w a s t ab i l i z a t i o n
w i t h in c r ea s i n g l i q u i d v i s co s i ty i n t h i s t y p e o f p l o t , a s p r ed i c t ed b y L i n H a n r a t t y . T h i s w o u l d
s eem t o co n t r ad i c t i n t e r p r e t a t i o n s w h i ch u s e a l o n g - w av e l en g t h i n v i s c i d an a l y s i s .
L i n H a n r a t t y a r g u e d t h a t a t s m a l l USG t h e in i t i a ti o n o f sl ug s o c c u r s t h r o u g h t h e g r o w t h o f
l o n g - w av e l en g t h i n f in i t e s i m a l d i s t u r b an ces . T h e y s h o w ed t h a t l i q u i d i n e r t i a is d e s t ab i l i z in g i f
v i s co u s e f fec t s a r e i n c l u d ed an d t h a t t h i s e ff ec t d ec r ea s e s f o r f i x ed v a l u e s o f h E D a n d ( , ~ . F o r v e ry
l a r g e l i q u i d v i s co si t ie s w h e r e i n e r t i a is n e i t h e r s t ab l il i z in g n o r d e s t ab i l iz i n g t h e L i n H a n r a t t y
an a l y s i s g i v e s t h e s am e r e s u l t s a s a l o n g - w av e l en g t h K H an a l y s i s .
A t f i rs t g l an ce t h e o b s e r v ed e f f ec t o f l i q u i d v i sco s i t y an d t h e g o o d ag r eem en t o f ex p e r i m en t a l
r e s u l t s f o r a i r - w a t e r w i t h t h e p r e d i c t i o n w o u l d s e e m t o s u p p o r t t h e m e c h a n i s m e x p l o r e d b y L i n
H an r a t t y . H o w e v e r , a c l o s e r ex a m i n a t i o n o f t h e r es u l ts w i t h v e r y v i s co u s l i q u i d s s h o w s a d i f f e r en t
e f f ec t o f p i p e d i am e t e r t h an i s p r ed i c t ed .
V i s u a l o b s e r v a t i o n s o f t h e s t ab i l i t y o f a v e r y v i sco u s l iq u i d s h o w t h a t t h e f i rs t w av es t h a t a p p e a r
a r e s i n u s o i d a l an d h av e a s u f f i c i en tl y s h o r t w av e l en g t h t h a t t h e y a r e a f fec t ed b o t h b y g r av i t y an d
s u r f ace t en s i o n . W i t h a s l i g h t ch an g e i n t h e f lo w co n d i t i o n s l a r g e - am p l i t u d e i s o l a t ed d i s t u r b an ces
g r o w o u t o f th e s e r eg u l a r s m a l l - w av e l en g t h w av es . I f t h e l i q u i d l ay e r i s t h i ck en o u g h t h e
d i s t u r b an ces t o u ch t h e t o p w a l l an d f o r m s lu g s . O n t h e t h i n l ay e r s t h a t o ccu r f o r USG > 4 m / s t h e s e
d i s t u r b an ces ev o l v e i n t o l a r g e - am p l i t u d e i r r eg u l a r w av es t h a t c au s e a l a r g e i n c r ea s e i n t h e
i n t e r f ac i a l st re s s an d a l a r g e d ec r ea s e i n t h e t h i ck n es s o f t h e l i q u i d . A t s m a l l l i q u i d h e i g h t s ( o r l i q u i d
f l o w s ) t h e r e i s n o t en d en cy f o r t h e s e w av es t o co a l e s ce . H o w ev e r , a t s o m e c r i t i c a l co n d i t i o n t h ey
g r o u p t o g e t h e r t o f o r m a l a r g e d i s t u r b an ce t h a t g r o w s i n t o a s l u g .
T h e e v o l u t i o n o f t h e l a r g e - w a v e l e n g t h d i s t u r b a n c e s f r o m t h e r e g u l a r w a v e t r a i n o c c u r s b y
n o n - l i n e a r p r o c e ss e s w h i ch a r e n o t y e t u n d e r s t o o d . H o w e v e r , i f t h e a p p e a r a n c e o f t h e s m a l l- w a v e -
l en g t h s i n u s o i d a l w av e t r a i n i s a n ece s s a r y co n d i t i o n t h e n l i n ea r t h eo r y s ti ll c an b e u s ed t o p r ed i c t
s t ab i li t y . S ta b i l it y c a lc u l a t i o n s t h a t a b a n d o n t h e l a r g e- w a v e l e n g th a p p r o x i m a t i o n s u p p o r t t h i s
s u g g es t i o n i n t h a t t h e c r i ti c a l USG f o r t h e i n i t i a t i o n o f t h e s m a l l - w av e l en g t h w av es d u e t o g a s - p h as e
p r e s s u r e v a r i a t i o n s i n p h as e w i t h t h e w a v e h e i g h t i s t h e s am e a s t h e o b s e r v ed USG f o r t h e i n i t i a t i o n
o f s l u g s o r o f l a r g e am p l i t u d e i r r eg u l a r w av es .
MF. i ,'t~--C
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89 N ANDRITSOS
e l a l
Observations of the mechanism for the initiation of slugs for liquids with viscosities close to that
of water is complicated because over a large range of flow conditions the interface of the stratified
flow is covered with waves (generated by gas-phase pressure variations in phase with the wave
slope) prior to the appearance of slugs. However, at very low gas velocities, where the interface
is smooth, Lin (1985) has reported the same phenomena that were observed on very viscous liquids.
This result, plus the agreement of experiments with the approximate calculations presented in this
paper, would tentatively suggest that the type of mechanism observed for very viscous liquids could
be operative for all viscosities.
Calculations with a more accurate representation of viscous effects in the liquid, than is used
in the section on approximate calculations, are needed to clarify this question.
Acknowledgements--This
work has been supported by the Shell Companies Foundation and by the
Department of Energy under DOE DEFG02-86E R 13556.
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