liquid–liquid interface stability in accelerating and constant-velocity tube flows

Nuclear Engineering and Design 210 (2001) 37–51

Liquid–liquid interface stability in accelerating andconstant-velocity tube flows

Michael Epstein a,*, James P. Burelbach a, Hans K. Fauske a,Shigenobu Kubo b, Kazuya Koyama c

a Fauske & Associates, Inc. 16W070 West 83rd Street, Burr Ridge, IL 60521, USAb The Japan Atomic Power Company, Tokyo, Japan

c Ad�anced Reactor Technology Co., Ltd. Tokyo, Japan

Received 3 May 2001; received in revised form 25 June 2001; accepted 9 August 2001

Abstract

This paper presents new experimental data on interface stability when one liquid displaces another in asmall-diameter tube. Interface stability theory suggests that the interface is stable when the displacement is directedfrom a dense and viscous liquid to a less dense and less viscous liquid. This theoretical result has importantimplications in the evaluation of a fast breeder reactor (FBR) core disruptive accident in that it rules outcoolant-boiling-driven compaction of the reactor-core fuel, and has been classified by Fauske (Nucl. Safety, 17 (1976)550) as one of several general behavior principles (GBPs) which may be used to argue against the occurrence ofenergetic events that could threaten the reactor vessel. The experimental data agreed well with the GBP foraccelerated liquid/liquid systems with liquid density ratios and interfacial tensions similar in magnitude to those of theFBR mixed oxide fuel/sodium coolant pair. The data agreed well with the GBP when the displacement occurred inthe laminar regime at constant speed and the viscosity of one of the liquid components was destabilizing. Theinvestigation also uncovered behavior that was not in concert with the GBP, although this behavior appears to belimited to small density-difference, low-interfacial-tension liquid/liquid systems and is probably not relevant to theFBR application. © 2001 Elsevier Science B.V. All rights reserved.

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1. Introduction

It has been pointed out by Taylor (1950) andverified experimentally by Lewis (1950) that whentwo superposed immiscible liquids of differentdensities are accelerated in a direction perpendicu-

lar to their interface, this interface is stable orunstable depending on whether the acceleration isdirected from the more dense to the less denseliquid. Lewis (1950) showed experimentally thatthe unstable accelerating interface develops insuch a way that long fingers of the less denseliquid penetrate into the more dense one. Experi-ments in which one liquid (or gas) drives anotherliquid through a small-diameter tube showed thatsingle fingers of the less dense driving liquid can

* Corresponding author. Tel.: +1-630-887-5210; fax: +1-630-986-5481.

E-mail address: [email protected] (M. Epstein).

0029-5493/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.

PII: S0029-5493(01)00437-X

mailto:[email protected]

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 37–5138

be produced (Ford et al., 1971; Fauske et al.,1970; Grolmes and Lambert, 1980). The heavierliquid is not completely expelled or replaced bythe lighter liquid (gas). A film of the heavier liquidis observed to adhere to the tube wall while atongue or finger of the lighter fluid of reduceddiameter advances through the channel core es-tablished by the portion of the heavier liquid leftbehind (see Fig. 1a). If the velocity of the pene-trating fluid finger is high enough the liquid filmon the tube wall is susceptible to entrainment byand mixing with the fingering fluid via a Kelvin–Helmholtz type instability (see Fig. 1b). Since asingle finger appears to dominate the instabilitythat occurs during the displacement of one liquidby a less-dense liquid in a narrow channel, here-after the phenomenon is referred to as a ‘fingerinstability’.

The finger phenomenon is of interest in theevaluation of a fast breeder reactor (FBR) coredisruptive accident (CDA) scenario. The concernis whether the process illustrated in Fig. 1 canoccur during the ejection of molten mixed-oxidefuel from the active core region into the liquid-sodium-filled upper or lower subassembly struc-tures (channels) of an FBR. More specifically,would a liquid sodium film be left behind whenthe advancing, heavier core material drives outthe liquid sodium? If so, the entrapped sodiumfilm may mix with the fuel and rapidly vaporizeresulting in a local pressurization sufficient toreverse the direction of the fuel and drive it backinto the core. In general, as first pointed out byTaylor for interfaces of infinite extent, fingering

will not occur when a dense liquid accelerates alighter one. A linear stability theory for an inter-face confined to a narrow channel is presented inthis paper which shows that the interface is liableto be stable (no fingering) if the driving liquid ismore dense and more viscous than the drivenliquid. Since molten fuel is more dense and moreviscous than liquid sodium, the fuel–sodium in-terface should be stable to fingering and the twomelts should be hydrodynamically separated dur-ing fuel expulsion from the FBR core. The fingerinstability or its absence is one of a class ofso-called general behavior principles (GBPs) usedby Fauske (1976) to show that energetic fuel–coolant interaction and recriticality events arehighly unlikely during a postulated FBR CDA.

Despite the importance of the hydrodynamicstability of the interface between superposed liq-uids in flow in a narrow channel to the FBRCDA, very few careful and quantitative experi-ments have been performed in which the behaviorof the interface was observed directly. Henry et al.(1976), Peppler and Will (1991) injected, respec-tively, molten UO2 and molten Al2O3 generatedby thermite reactions into rod bundles initiallycontaining sodium. Since sodium-boiling-inducedflow reversal was not observed in these importantproof-of-principle tests it is reasonable to assumethat the liquid sodium was near-completely dis-placed from the surfaces of the rods by the ad-vancing thermite melt and/or thermite-reactiongenerated gas. While X-ray cinematography al-lowed the observation of transient material move-ments (Peppler and Will, 1991), the localevolution and shape of the sodium/driving fluidinterface could not be observed in these tests.Grolmes and Lambert (1980) performed a limitednumber of experiments involving water accelerat-ing mercury and vice versa through a tube. Inter-face displacement measurements obtained with amovie camera clearly indicated fingering when thedisplacing liquid was water. When mercury dis-placed water, no measurable amount of water wasleft behind after the interface had passed.

It is the purpose of this paper to present newexperimental results on interface stability whenone liquid displaces another liquid in a small-di-ameter tube. The results provide new information

Fig. 1. Sequence of film entrapment and mixing events whenliquid 1 is accelerated by a less-dense liquid 2.

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 37–51 39

on the stability of the interface as a function ofthe density difference between the two liquids. Tosimulate the high-density-difference UO2/sodiumsystem, superposed liquid columns of gallium anddecane and columns of gallium and pentane wereaccelerated in a U-tube apparatus. As expected,the finger instability occurred or did not occuraccording as the acceleration was directed fromthe organic liquid to the much more dense galliumor vice versa. Non-isothermal tests were also per-formed to study the interface behavior when a hotdense liquid (gallium) accelerates a volatile andboiling lighter liquid (pentane). It was determinedthat such boiling does not result in the temporaryreversal of the liquid flow. Finally, the stability ofthe interface between superposed liquids in lami-nar, constant-speed flow was examined experi-mentally. The purpose of performing theseexperiments is to check the validity of the stability(fingering) theory in a regime where both viscosityand density differences control the stability of theliquid/liquid interface. Viscous silicone oil andheavier zinc bromide salt solution proved to be anideal liquid pair for the laminar flow study. An-other mechanism of interface instability besidesfingering was inferred from the data on the accel-eration of two liquids with similar densities (viz.water and decane). This mechanism and its impli-cations with regard to the FBR application arediscussed.

2. Mechanisms of liquid entrapment

Shown in Fig. 2 are three different entrapmentmechanisms. These mechanisms are discussed be-low in the order in which they appear in thefigure.

2.1. Entrapment due to surface irregularities

Channel surfaces which are usually regarded asperfectly smooth geometrical planes are in realitymarked with surface irregularities. The valleysand peaks on real surfaces are generally on a scaleconsiderably larger than the molecular scale. It isprobably reasonable to suppose that the displacedliquid continues to occupy the valleys on the

Fig. 2. Mechanisms of liquid entrainment.

surface so that the driving liquid moves over apartly ‘liquid surface’ (see Fig. 2). The thicknessof the entrapped liquid film should be of the orderof the wall roughness. The size of the surfaceroughness is typically so small that pressurizationsresulting from the vaporization of volatile liquidfilms (e.g. sodium) contained within the surfacehollows would not be sufficient to reverse thedirection of the advancing and displacing liquid(e.g. UO2).

2.2. Entrapment �ia a wetting limitation

The subject mechanism is illustrated in Fig. 2.It should be stated at the outset that this mecha-nism has never been observed in a liquid– liquidsystem nor has it been observed in a tube flow. Allthe visual information on the wetting limitationhas been derived from studies on the continuouspenetration of a solid surface into a liquid pooland the associated air entrainment. The reasonwhy this mechanism is discussed here is that itprovides a plausible explanation for the significantentrapment observed in test runs using the lowdensity water–decane pair.


Consider a liquid front (interface) advancingover a solid surface and displacing the liquid (orgas) that initially covered the surface. A ‘contactline’ is formed at the intersection of the interfacewith the solid. The interface intersects the solid ata well-defined angle called the apparent dynamiccontact angle (see Fig. 2a, b). This angle is typi-cally measured through the displacing liquid (liq-uid 2). The behavior of the contact line andcontact angle as a function of liquid speed hasbeen studied by Perry (1967), Burley andKennedy (1976), Blake and Ruschak (1979). Theprocedure involved moving the solid surface inthe form of a continuous tape through a liquidbath. The effect of the plunging tape was todisplace the liquid interface downward. The dy-namic contact angle produced by this displace-ment could be measured from photographs takennormal to the edge of the tape. Photographstaken normal to the plane of the tape revealed thebehavior of the contact (or wetting) line.

At low tape speeds, the dynamic contact angleequals the stationary contact angle and the con-tact line appears to be stable everywhere. As tapespeed is increased, the apparent dynamic contactangle monotonically increases, ultimately reachinga nominal value of 180°. At this point the shapeof the interface (as viewed in the direction normalto the edge of the tape) appears to be semi-circu-lar. However, the previously smooth contact lineadopts a sawtooth shape with short lengths of‘line’ remaining temporarily stuck while inter-spersed portions break loose and move rapidlyover short distances. At sufficiently high speeds,air bubbles are entrained from the vertices of thesawtooth wetting line that are the farthestdownstream.

In liquid coating applications it is generallynecessary to avoid air entrainment and, therefore,based on experiments such as the plunging tapestudies, to avoid apparent dynamic contact anglesof 180°. Consequently the factors that determinethe value of this angle are of interest. A model forpredicting the onset of air entrainment or, equiva-lently, the dynamic contact angle is not available,although theoretical approaches to the problemhave been attempted (Kistler and Scriven, 1982;Hansen and Toong, 1971; Blake and Haynes,1969).

2.3. The Taylor- or Finger-mechanism ofentrapment

In this mechanism of entrapment the contact(or wetting) line remains attached to the channelwall. However, the liquid/liquid interface leavesthe wetting line behind as the displacing liquid 2enters the displaced liquid 1 due to an interfacethat exhibits a global instability (see Fig. 2c).

The following simple argument may be made inorder to understand the basic mechanism of theinstability. Consider upward-vertical-liquid mo-tion in a narrow channel of hydraulic diameterDh. The flow involves the displacement of a liquidof viscosity �1, and density �1, by a second liquidof viscosity �2 and density �2. We limit ourselvesto the forces of gravity, acceleration and frictionalresistance. Under suitable continuum and quasi-steady assumptions, the flow on each side of theinterface may be taken to satisfy Newton’s law,which for a one-dimensional vertical flow in they-direction may be written for liquid 1 as

(�P)1

�y= �1 (g + a) +

12� 4

Dh

�f1 �1 �2 (1)

and for liquid 2 as

(�P)2

�y= �2 (g + a) +

12� 4

Dh

�f2 �2 �2.

(2)

In the above equations ‘a ’ and � are, respec-tively, the instantaneous flow acceleration andvelocity, g is the gravitational constant and f isthe channel friction factor evaluated for the phys-ical properties of either liquid 1 or 2. The differen-tial quantities (�P1)/�y and (�P2)/�y are the local,instantaneous pressure gradients in liquid 1 andliquid 2, respectively. Note that an incrementalpressure decrease with increasing vertical incre-mental distance �y is taken to be positive. Nowsuppose the interface separating the two liquidsexhibits an initial tendency to allow liquid 2 topenetrate a distance �y into liquid 1, as illustratedin Fig. 3. If the pressure gradient (�P)1/�y inliquid 1 just outside the protruding element ofliquid 2 is larger than the pressure gradient (�P)2/�y within the protrusion, then the protrudingelement of liquid 2 will accelerate more rapidly


than liquid 1 and the small disturbance will am-plify. Thus the condition for interface instability is

(�P1)�y

�(�P2)

�y(3)

or from Eq. (1) and Eq. (2)

(g + a) (�1 − �2) +2�2

Dh

( f1 �1 − f2 �2)

� 0. (4)

Eq. (4) is a rather general condition for thefinger instability. If the inequality signs in Eq. (3)and Eq. (4) are reversed and the flow is highlyturbulent so that f1= f2 we obtain the followingcondition for interfacial stability which may bedesignated as an FBR general behavior principle(GBP; see Fauske, 1976):

�1��2 (5)

Eq. (5) states that stability always results when aless dense liquid 1 is accelerated by a more denseliquid 2. For the specific case of flow in a parallel-plate channel Dh=4b, where b is the half-widthof the channel, and Eq. (4) becomes

(g + a) (�1 − �2) +�2

2b( f1 �1 − f2 �2)

� 0. (6)

For laminar flow through a parallel plate channel

f =6�

b � �. (7)

Substituting this result into Eq. (8) gives

(g + a) (�1 − �2) +3�

b2 (�1 − �2) � 0.

(8)

for the instability condition. We see from Eq. (8)that a combination of unfavorable density and/orviscosity ratios and flow direction can render theinterface unstable. For example, for upward verti-cal motion (g+a�0 and ��0) involving thedisplacement of a dense, viscous liquid by alighter, less viscous one we have �1��2 and�1��2 which always results in finger formation.On the other hand, if the displacing liquid is moredense but less viscous than the displaced liquid(�1��2) while (�1��2), gravity and accelerationare stabilizing forces and viscosity is destabilizing,leading to a critical velocity �c above which thereis instability:

�c =(g + a) (�2 − �1) b2

3 (�1 − �2). (9)

The experiments reported herein were per-formed in a tube rather than in a parallel-platechannel. The friction factor for laminar flow in atube of radius R is

f =8�

R � �(10)

Hence from Eq. (4) the desired instability condi-tion is

(g + a) (�1 − �2) +8�

R2 (�1 − �2) � 0

(11)

and for the laminar flow case with �2��1 and�1��2, the critical velocity �c for a tube flowfollows as

�c =(g + a) (�2 − �1) R2

8 (�1 − �2). (12)

The experimental program described below wasessentially a flow visualization study aimed atverifying quantitatively the stability conditionsgiven by Eq. (5) and Eq. (12).Fig. 3. Schematic of liquid– liquid interface stability model.


Fig. 4. Schematic illustration of initial fluid configuration for atypical U-tube experiment.

pressure leg). The U-tube configuration allows forsimultaneous study of two different situations (i.e.heavier-into-lighter and lighter-into-heavier) dur-ing a single experiment. The U-tube measuredabout 45 cm in height and 7 cm in width from thecenter of one leg to the center of the other leg.The internal diameter of tube was 1.0 cm.

The imposed pressure �P across the liquidcolumn in the U-tube was achieved using differenttechniques depending on the desired magnitude ofthe pressure drop. At first liquid head in the ‘highpressure’ leg was used to impose a pressure differ-ential and thereby accelerate the fluids. In thosecases liquid 1 was not present in the high-pressureleg. Instead, the high pressure leg was filled withliquid 2 to a height of about 1.5 m using anattached elevated hose. Flow was initiated byopening a downstream ball valve, allowing theliquid 2 water column to push the relatively shortliquid 1 leading slug up the low pressure leg andinto a receiver chamber (liquid catch vessel). Sub-sequent tests achieved higher accelerations by at-taching a vacuum chamber to the low pressureleg. The high pressure leg was open to the atmo-sphere. Flow was again initiated by opening adownstream ball valve, ‘sucking’ the contents ofthe U-tube into the receiver chamber.

The accelerations which could be achieved us-ing the vacuum technique were somewhat limited,so the apparatus was reconfigured to use an over-pressure source to drive the liquid slugs throughthe U-tube. A pressurized gas chamber was at-tached to the high pressure leg and the flow wasinitiated by opening a supply side ball valve. Theline leading from the low pressure side of theU-tube contained a liquid catch vessel which wasopen to the atmosphere. Supply pressures up to193 kPa (28 psig) were used to generate liquidcolumn velocities approaching 11 m s−1.

A common industrial-type video camera wasinitially used to record the experiments. The mov-ing liquid/liquid interface was blurred whenrecorded at the normal 1/60 s shutter speed, butfortunately the camera was equipped with a 1/1000 s electronic high-speed shutter which pro-vided clear images of the accelerating interface.Video recording was still limited by the 1/30 sframe speed, so a high-speed digital camera was

3. Experimental techniques

The experiments may be classified into threecategories. In the first category are isothermalturbulent-flow tests which studied the accelerationof a heavier (higher density) fluid into a lighter(lower density) fluid, as well as the oppositelighter-into-heavier cases. The second category issimilar, except that conditions are non-isothermal(the more dense liquid is ‘hot’ relative to theless-dense liquid). In the third category are(isothermal) laminar, constant-speed flow testswhich examine the stability of the liquid/liquidinterface between a more-dense, low viscosityfluid and a less-dense, high viscosity fluid.

The heart of the experimental apparatus is theU-tube configuration illustrated in Fig. 4. Thisconfiguration was first employed by Grolmes andLambert (1980) in their experimental study of thefingering phenomenon. Note that liquid 1 is thelower-density fluid and liquid 2 is the higher-den-sity fluid. In an experiment the higher-densityfluid is accelerated into the lower density fluid atthe interface between the liquid 1 ‘leading slug’and liquid 2 (in the low-pressure leg). Similarly,the lower-density fluid is accelerated into thehigher density fluid at the interface between theliquid 1 ‘trailing slug’ and liquid 2 (in the high-


used to resolve interface behavior at high veloc-ities. The camera system with a black and whitecamera head allowed for frame speeds up to1/8000 s. It was found that a frame speed of 1/250s provided a good compromise between framespeed, image size, and recording time window.Image contrast was enhanced where possible byusing an aqueous dye (food coloring). For low-speed tests (laminar flow) a macro lens videofeature proved useful, although it did limit thefield of view. Uniform lighting of the glasscolumns and the adjacent length scale proveddifficult at first and was adjusted on a case-by-case basis. Back-lighting of the U-tube columnswith simultaneous low-wattage front-lighting ofthe length scale appeared to be optimal.

3.1. Isothermal acceleration experiments

Isothermal acceleration experiments were per-formed using decane (�=730 kg m−3) as theless-dense fluid (liquid 1) and either water (�=103

kg m−3) or gallium (�=5900 kg m−3) as themore-dense fluid (liquid 2).

Gallium/decane experiments were first at-tempted by melting the gallium (m.p.�30 °C)and carefully placing it in the U-tube, with decaneto be placed on top. It turned out that galliumreadily ‘wets’ and sticks to the dry glass surface,precluding initial setup of the configurationshown in Fig. 4. Also, gallium seemed to quicklyform an oxide film when exposed to air. It wasnoted that when decane contacted the glass tubefirst, subsequent contact by molten gallium didnot result in significant ‘sticking’ of gallium to thetube walls. Thus the U-tube tests were successfullyperformed by placing the decane (liquid 1) in theU-tube first, and then adding gallium (liquid 2)below the decane surface. (Also, the molten gal-lium was stored with a decane ‘supernate’ toprevent oxide formation.) This was accomplishedwith a syringe and flexible tubing that could be‘snaked’ into the U-tube. ‘Excess’ decane in theU-tube could be similarly removed with a syringeto achieve the desired decane slug lengths. Inorder to ensure that the gallium remained molten,the U-tube was partially submerged in a waterbath on a hot plate.

3.2. Non-isothermal acceleration experiments

Non-isothermal acceleration tests were per-formed using pentane (�=630 kg m−3, normalboiling point=36 °C) as the less-dense fluid (liq-uid 1) and gallium (�=5900 kg m−3) as themore-dense fluid (liquid 2). For these tests thegallium was heated above the pentane boilingpoint prior to initiation of flow. Pentane wasplaced in the tube first, as suggested by previousexperience with the gallium/decane system. Care-ful addition of preheated gallium (via a syringe)created the desired liquid slug configuration, ex-cept that pentane boiling began immediately atthe pentane/gallium interface. Other techniques,such as adding the bulk of the pentane (cold) last,and heating the gallium in situ using electricheating wire resulted in essentially the same initialconditions. (Future experiments of this nature willlikely require an isolation valve to initially sepa-rate the hot and cold liquids.) The gallium tem-perature was monitored using a bare type Kthermocouple fixed to the outer tube wall. Thismeasurement gave somewhat lower temperaturereadings (by up to 20 °C at 60 °C) than a similarsheathed thermocouple which was temporarilyplaced inside the U-tube.

3.3. Isothermal constant-�elocity experiments

Isothermal constant-velocity experiments wereperformed using silicone oil (�=1050 kg m−3,��1.0 kg m−1 s−1) as the less-dense fluid (liquid1) and zinc bromide solutions (�=1800 and 2200kg m−3, �=0.01 kg m−1 s−1) as the more-densefluid (liquid 2). Here a 4 l vessel in the line leadingfrom the low pressure leg of the U-tube alternatedas a pressure/vacuum chamber. A needle valvejust upstream of the chamber was used to regulatethe U-tube flow, thereby maintaining low veloc-ities (laminar flow). Once the needle valve was setit was left alone; flow in the U-tube was initiatedby opening the adjacent ball valve. The directionof flow was determined by whether the 4 l vesselwas pressurized or evacuated. Thus, several testscould be performed on the same fluid slugs by‘cycling’ the pressure/vacuum chamber, takingcare not to discharge the fluids from the U-tube.


Note that a small amount of aqueous dye (foodcoloring) was added to the ZnBr2 solutions toprovide visual contrast with the silicone oil. Thecoloring eventually coagulated into clumps orclouds which were readily dispersed by agitation.This actually provided a visual aid in observingthe flow streams near the liquid/liquid interface,but did not appear to affect the stability behaviorof the interface.

4. Column acceleration experiments: results andinterpretation of data

Twelve acceleration experiments were per-formed in the U-tube apparatus and in each ex-periment attention was focused on the interfacedisplacements of the leading column (liquid 1),although some observations of the behavior of thetrailing column were also obtained. The experi-mental conditions are summarized in Table 1.Initially, the experiments were conducted withwater and decane. While this low-density liquidpair is nonprototypical with respect to the FBRapplication, it is relatively easy to use compared

with, say, the gallium/decane pair (see Section 3.1)and made it possible to carry out extensive ‘repro-ducibility’ and apparatus performance checks.

4.1. Isothermal water/decane experiments

Fig. 5 shows the data from Test 2 when plottedas displacement �y versus time squared t2. Astraight line in the �y-t2 coordinate system corre-sponds to constant liquid-column acceleration.The slopes of the curves in Fig. 5 gradually de-crease with time indicating that frictional resis-tance has an important effect on the interfacedisplacement histories. The initial accelerationsreported in Table 1 were obtained by estimatingthe slope of the �y vs. t2 curves at the origin. Thedisplacements in Test 1 were recorded with thehigh-speed digital camera. The open points repre-sent the displacement history of the upper decane/air interface while the dark points represent thedisplacement history of the lower water/decaneinterface. Both displacements are measured fromthe initial positions of these interfaces so that attime zero the displacements are zero. It is obviousfrom Fig. 5 that the lower water/decane interface

Table 1Column acceleration experiments

Test type Liquid 1Test ID c Liquid 2 �P (kPa) Initial acceleration Camera type(m s−2)

57 VideoWater (�=103 kgDecane (�=730 kgIsothermal1m−3) m−3)

Isothermal Digital2 57WaterDecaneIsothermal Decane Water 34 11 Video3

VideoDecane4 WaterIsothermal 34 105 Decane Gallium (�=5900Isothermal 34 10 Digital

kg m−3)6 Video434GalliumDecaneIsothermal

34GalliumDecane 4Isothermal7 VideoIsothermal Decane Gallium 193 768 DigitalIsothermal Decane Gallium9 117 44 DigitalNon-Isothermal Pentane (�=630 kg Video10 1069Gallium (60 °C) a

m−3 bp 36 °C)Video51411 Gallium (45 °C) aPentaneNon-Isothermal

Non-Isothermal12 Pentane Gallium (40 °C) a 14 5 Video

a At outside surface of tube at location of hot gallium; actual gallium temperature may be 10–20 °C hotter.


Fig. 5. Interface displacement versus time squared for leading decane slug (Test 2).

moves faster than the upper decane/air interface.These measurements clearly indicate that decaneis being entrained by the displacing water column.

The behavior conveyed in Fig. 5 is somewhatsurprising and is not in concert with the theory ofthe finger instability described in Section 2.3.Since the density of the displacing water columnexceeds that of the displaced decane column, astable interface and no liquid entrapment areanticipated. That is, from the instability theoryone would have expected the dark and openpoints in Fig. 5 to occupy the same positions. Thehigh-speed camera images, however, did not re-veal the telltale sign of the finger instability, as atongue of water (liquid 2) did not appear topenetrate the overlying decane column (liquid 1),although the decane/water interface did deforminto a hemispherical shape. Apparently the fingerinstability is not responsible for decane entrap-ment by water but, instead, the process is causedby the wetting limitation discussed in Section 2.2.It is speculated that the sequence of events leadingup to the entrainment of decane by water has thegeneral characteristics shown in Fig. 6, with thelocal disappearance of the contact line being ulti-mately responsible for the ‘leakage’ of decane intothe water column, either in the form of droplets

(as illustrated in Fig. 6) or a film. Recall that thephenomenon depicted in the figure was observedwhen a continuous solid surface enters an air/liq-uid interface (Burley and Kennedy, 1976). If theabove interpretation is correct, to the best of theauthors’ knowledge the data in Fig. 5 representthe first observations of entrainment caused bybreakdown of the wetting line in a liquid/liquidsystem.

Further support for the idea that the decaneentrainment process is not a finger-type instability

Fig. 6. Postulated nature of decane liquid 1 entrainment bywater 2 as inferred from tape plunging studies.


Fig. 7. Interface displacements versus time squared for leading decane slug (Test 8).

comes from observation of the trailing decanecolumn on the opposite side of the U- tube. Herelighter decane is displacing heavier water andindeed the instability appears to begin with thepenetration of the decane/water interface into theunderlying water column. This is followed almostimmediately by vigorous mixing between the wa-ter and decane which made it virtually impossibleto visually keep track of the decane/water inter-face. The other decane/water experiments (Tests1, 3, and 4) were recorded with the normal-speedvideo camera. The behavior of the leading decanecolumns were observed to be identical to thebehavior recorded in Test 2 with the high-speedcamera and discussed above.

4.2. Isothermal gallium/decane experiments

Five experiments were performed, and the con-ditions of the experiments (Tests 5–9) are pre-sented in Table 1. The initial column accelerationsranged from 4 to 76 m s−2. Displacement vs. timesquared data obtained from Test 8 are shown inFig. 7. As can be seen from the figure, the dis-placement histories of the lower and upper inter-faces of the displaced decane column are, within

the accuracy of the measurements, identical. Thecoincident displacement curves are typical of allthe gallium/decane tests and indicate that only aninsignificant quantity (if any) of decane was leftbehind (entrapped) after the gallium/decane inter-face past. This behavior is in perfect agreementwith finger-instability theory. The entrainmentprocess observed during the acceleration of de-cane/water columns, which we believe is linked tothe breakdown of the wetting (contact) line, is notactive in an accelerating decane/gallium column.Apparently, the contact line is more stable ininterfacial regions that separate two liquids with alarge density disparity and with a high interfacialtension. In this regard the decane/gallium systemis similar to the sodium/UO2 system. Thus wetentatively conclude that entrainment by contactline separation from the channel wall does notoccur in the FBR application and the potentialfor sodium entrapment is insignificant, in accordwith finger-instability theory. Further support forthis statement comes from the experimental find-ing that liquid mercury completely displaces water(Grolmes and Lambert, 1980). Of course, a chal-lenging and interesting question pertains to theconditions that control the onset of contact lineseparation in liquid/liquid systems.


In one of the tests the camera recorded theearly motion of the trailing decane column. Thedecane was observed to penetrate the gallium,thereby giving additional evidence for the fingerinstability when the displacing liquid is the lessdense of the two liquids.

4.3. Non-isothermal gallium/pentane experiments

In these tests gallium was heated to a tempera-ture above the boiling point of pentane (36.2 °C)before the pentane was accelerated upward by theheavier gallium. If the pentane/gallium interface ismade unstable by the acceleration a film of pen-tane will be left behind (entrapped) and will beheated by the penetrating core of hot gallium, boilvigorously, and perhaps cause a significant pres-sure spike that reverses the flow direction. Ofcourse for heavy gallium accelerating much-less-dense pentane we anticipate from stability theorya stable gallium/pentane interface, no pentaneentrapment and no flow reversal. This anticipatedresult was borne out by the (video taped)experiments.

The experimental conditions are summarized inTable 1. We turn our attention first to Fig. 8which shows the interface motions for the dis-

placed pentane slug during Test 12. In this test thegallium superheat temperature (relative to thepentane boiling point) was low and pentane boil-ing off the gallium/pentane interface was minimal.Thus both the gallium/pentane (lower) interfaceand the pentane/air (upper) interface were clearlyvisible on the video tape. The fact that the mea-sured displacement histories for the upper andlower interfaces coincide indicates no entrapmentof pentane by gallium, much like what was ob-served when gallium accelerated decane.

Fig. 9 shows interface displacement histories forthe relatively high initial gallium superheat in Test11. As the heated pentane was displaced upwardby the accelerating gallium, the pentane columnexhibited a divided two-phase morphology. Thelower segment of the column appeared as anexpanding vapor-continuous zone while the up-per-segment retained its liquid appearance. Inother words a growing vapor slug or ‘bubble’separated the liquid gallium from the liquid pen-tane. The expansion of the pentane vapor zonewas undoubtedly due to the heated pentanecolumn experiencing a decreasing pressure as itmoved upward toward the exit of the tube. Fig. 9for Test 11 shows the vertical dimensions of thepentane bubble as well as the displacements of the

Fig. 8. Interface displacements versus time for leading pentane slug (Test 12).


Fig. 9. Interface displacements versus time for leading pentane slug (Test 11).

gallium surface (presumably separated from thepentane vapor bubble by a thin evaporating filmof liquid pentane) and the liquid pentane/air sur-face, all as a function of time. The displacementhistories observed during Test 10 were limited tothe gallium/pentane (vapor) and the liquid pen-tane/air interfaces. The motion and the rate ofpentane vapor growth in Test 10 were too fast tobe captured by the video camera.

The important points to be made about thenon-isothermal test series are: (i) at low galliumsuperheats there was negligible change in pentanecolumn length (i.e. negligible pentane entrapment)and; (ii) at high gallium superheats pentane vaporgeneration did not inhibit or reverse the galliumflow. Points (i) and (ii) are self-consistent and arethe expected results based on the stability theory(GBP).

5. Isothermal constant-velocity experiments

The purpose of this set of experiments was toexamine the validity of Eq. (11) for interfacestability during constant-speed laminar flow (ac-celeration suppressed; i.e. a=0). This is perhaps amore interesting situation than accelerating turbu-

lent flow because both liquid density and viscositydetermine the onset of the instability and, as such,these experiments represent a more stringent testof the instability theory.

The results of the constant-speed experimentsare given in Table 2. In these experiments siliconeoil represented the less dense and more viscousliquid while zinc bromide (ZnBr2) salt solutionsserved as the more dense but less viscous liquid.In some of the U-tube apparatus experimentsboth the downward and upward displacements ofthe silicone oil/ZnBr2 interfaces were recordedwith the video camera. The last column of Table2 gives the observed behavior of the interface forthe downward vertical displacement of denseZnBr2 (liquid 2) by lighter, more viscous siliconeoil (liquid 1). Referring to Eq. (11), ��0 (upwardvelocity is positive), (�1−�2)�0, and (�1−�2)�0. Thus the left-hand side of Eq. (11) is alwaysless than zero and the interface is always stable.This theoretical result is perfectly consistent withthe observations of a stable interface during allthe downward vertical displacements.

The next-to-the-last column in Table 2 conveysthe results for the vertical upward displacement ofa light, more viscous silicone oil (liquid 1) by aheavy, less viscous ZnBr2 salt solution (liquid 2).


Table 2Isothermal constant-velocity experiments

ZnBr2 density kgTest ID c Column velocity Initial liquid length (cm) Significant fluidcm s−1 penetration?m−3

Leading Silicone Oil Trailing Silicone Oil 2�1 1�2Slug Slug

3013 5.02200 4.5 Yes Bulged1.7 – 4.5 – No14 22002.5 4.5 –2200 No15 –

220016 3.3 4.5 – No –5.0 4.5 – No –17 22001.9 6.0 1.41800 No18 No

180019 3.5 6.0 1.4 Yes No180020 5.6 6.0 1.4 Yes No

Liquid 1: Silicone oil (�=1050 kg m−3, ��1.0 kg m−1 s−1); Liquid 2: Zinc Bromide (�=1800 and 2200 kg m−3, ��0.01 kg m−1

s−1).

Again, referring to Eq. (11), ��0, (�1−�2)�0and (�1−�2)�0. Thus gravity is a stabilizingforce, while viscosity is destabilizing, leading to acritical velocity �c above which there is instability.The critical velocity is given by Eq. (12).

The results of the experiments on vertical up-ward constant-speed flow are shown in Fig. 10 inwhich the measured values of the column velocityare plotted as ordinates and the right-hand side ofEq. (12) as abscissa. Eq. (12) is seen to do areasonable job of dividing the unstable interfaceobservations from the stable ones. It should benoted that the agreement between theory andexperiment is somewhat sensitive to one’s choiceof the silicone oil temperature, owing to the sensi-tive relationship between the silicone oil viscosityand temperature. The liquid temperature was notmeasured during the tests and the placement ofthe data in Fig. 10 is based on the value 20 °C(�1=0.89 kg m−1 s−1). Had we chosen, say,23 °C the data points in the figure would beshifted significantly to the right and Eq. (12)would appear as an even more accurate neutralstability boundary. While the comparison betweentheory and experiment is encouraging, it is clearthat more data are required to better define thestability boundary in Fig. 10.

In closing we remark that the nature of theinstability observed in the constant-speed experi-

ments was unambiguous; it developed into a longround-ended finger of ZnBr2 penetrating lessdense but more viscous silicone oil, as illustratedin Fig. 1a. In one of the experiments the fingerwas observed to penetrate steadily through thesilicone oil column until the upper surface wasreached causing ‘break-through’ of the silicone oilcolumn.

Fig. 10. Comparison of Eq. (12) with interface stability obser-vations for the displacement of viscous silicone oil by denserZnBr2 salt solution.


6. Conclusions

When a liquid initially occupying a verticalhigh-aspect ratio channel is driven forward by thepressure of another liquid column, the interfacebetween them is predicted to be stable if thedriving liquid is the more dense and the moreviscous of the two. In the converse situation theinterface is predicted to be unstable and the insta-bility develops into a single round-ended finger ofliquid penetrating into the more dense and moreviscous one, resulting in the entrainment (entrap-ment) of the displaced liquid by the displacingone. This predicted behavior provides an impor-tant argument for an energetically benign coredisruptive accident scenario in a fast breeder reac-tor (FBR), and in this context is sometimes re-ferred to as a general behavior principal (GBP).To examine the validity of this GBP, an experi-mental study of the interface between two super-posed liquid columns in vertical, accelerated orconstant-speed motion in a narrow tube has beenconducted. The results are summarized as follows:1. The single finger, global interfacial instability

was not observed when water accelerates de-cane. This result by itself would appear to bein agreement with the GBP since decane is lessdense than water. However, significant en-trainment of decane by water was inferredfrom the liquid-decane displacement measure-ments. Contact (wetting) line separation fromthe channel wall is probably responsible forthe decane entrainment, much like has beenobserved during previous studies of air en-trainment into liquid by a moving continuoussolid surface (see, e.g. Burley and Kennedy,1976). This channel-wall wetting limitation isprobably not important in the high-interfacialtension, high-density-difference UO2/sodiumsystem. However, further work is probablyneeded in this area.

2. The single finger instability readily occurredwhen decane accelerated water. Since decane isless dense than water this result is in agree-ment with the GBP.

3. The finger instability was not observed whengallium accelerated decane or pentane. Sincegallium is much more dense than the organic

liquids these results are in agreement with theGBP. Interface displacement measurementsclearly indicated that no measurable amountof decane or pentane was entrained by thegallium.

4. The single finger instability readily occurredwhen decane accelerated gallium in agreementwith the GBP.

5. When hot gallium accelerated volatile liquidpentane, pentane vapor production wasconfined to the pentane column and did notreverse or inhibit the upward flow of gallium.This is an important result in terms of theFBR application and is the expected resultbased on the GBP.

6. A critical displacement velocity for instabilityis apparent when viscous silicone oil is drivenforward at constant-speed by heavier but lessviscous ZnBr2 salt solution. This result is inperfect accord with the GBP, but additionalexperimental data using the silicone oil/ZnBr2

pair is required to better evaluate the accuracyof the GBP for this most interesting case ofthe viscosity-controlled finger instability.

Acknowledgements

This study has been performed as a part of theresearch and development program for FastBreeder Reactors under sponsorship of the nineJapanese electric power companies, Electric PowerDevelopment Co., Ltd. and the Japan AtomicPower Company.

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liquid–liquid interface stability in accelerating and constant-velocity tube flows

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