Dissertation

DISS. ETH NO. 15460

Continuous Drop Formation at a Capillary Tip andDrop Deformation in a Flow Channel

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH

for the degree of

DOCTOR OF SCIENCES

presented by

Carsten Cramer

Dipl.-Ing. Universitat Karlsruhe (TH)

born May 21, 1974

citizen of Germany

accepted on the recommendation of

Prof. Dr.-Ing. Erich J. Windhab, examiner

Dr. Jo Janssen, co-examiner

Dr. Peter Fischer, co-examiner

2004

Copyright c 2004 Carsten CramerAll rights reserved.

Continuous Drop Formation at a Capillary Tip andDrop Deformation in a Flow Channel

ISBN: 3-905609-22-3

Published and distributed by:Laboratory of Food Process EngineeringSwiss Federal Institute of Technology (ETH) ZurichETH Zentrum, LFO8092 Zurich, Switzerlandhttp://www.vt.ilw.agrl.ethz.ch

Printed in Switzerland by:bokos druck GmbHBadenerstrasse 123a8004 Zurich, Schweiz

ii

Danksagung

An dieser Stelle mochte ich die Gelegenheit nutzen, mich bei allen zu bedan-ken, die einerseits zum Gelingen dieser Arbeit, aber auch allgemein zu einersehr schonen Zeit in Zurich ihren Beitrag geleistet haben.

Prof. Dr.-Ing. Erich J. Windhab gab mir jeglichen wissenschaftlichen Frei-raum, bereicherte die Arbeit mit vielen Ideen und motivierte mich stets furneue Taten. Fur die Begeisterung und das grosse Vertrauen mochte ich michherzlich bedanken.

Dr. rer. nat. Peter Fischer betreute die Arbeit und hatte immer ein offenesOhr auch fur die kleineren Problemchen. Sein experimentelles Geschick undFingerspitzengefuhl, aber auch sein wissenschaftliches Verstandnis waren mireine sehr grosse Hilfe.

Dr. Jo Janssen danke ich fur die Ubernahme des Korreferats und fur dievielen Anregungen.

Weiterhin danke ich den Partnern vom Europaischen Projekt fur die guteZusammenarbeit sowie die interessanten und schmackhaften Projekttreffen.

Wahrend meiner Arbeit wurden zahlreiche Stromungszellen gefertigt, wasohne den technischen Fahigkeiten und den konstruktiven Anregungen unsererWerkstatt, Daniel Kiechl und Jan Corsano, nicht moglich gewesen ware.

Datenerfassung und Datenauswertung stellt bei einer experimentellen Arbeiteinen sehr wichtigen Bestandteil dar. Uli Glunk und Bruno Pfister waren hierimmer hilfreiche und kompetente Ansprechpartner.

Ohne die tatkraftige Unterstutzung von Studenten ware die Sammlung vonderart umfangreichen experimentellen Datenmengen in dieser Zeit undenkbargewesen. Die Betreuung dieser Arbeiten hat mir nicht nur wissenschaftlichsehr geholfen und viel Laborarbeit abgenommen, sondern hatte mir immersehr viel Spass bereitet. Vielen Dank an Barbara Beruter, Philipp Erni, Matt-hieu Stettler, Lida Brich, Jurg Gujan, Armin Tiemeyer und Simon Studer!

Fur die Durchfuhrung der zahlreichen Grenzflachenspannungsmessungen dan-

v

Danksagung

ke ich Dr. sc. nat. Rok Gunde.

Bei allen Mitarbeitern des Labors fur Lebensmittelverfahrensechnik mochteich mich fur die super Stimmung innerhalb der Gruppe und die stete Hilfs-bereitschaft bedanken. Bei vielen Bergtouren, zahlreichen Abenden auf derDachterasse oder im bqm ging der Kontakt weit uber das geschaftliche hin-aus. Danke!

Desweiteren bedanke ich mich beim Bundesamt fur Bildung und Wissen-schaft fur dir finanzielle Unterstutzung des Projektes (QLK1-CT-2000-01543).

Zurich, Marz 2004

Carsten Cramer

vi

Contents

List of figures xvi

List of tables xvii

Notation xixLatin Letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixGreek Letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxIndices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiDimensionless Numbers . . . . . . . . . . . . . . . . . . . . . . . . xxi

Abstract xxiii

Zusammenfassung xxv

1 Introduction 1

2 Background 52.1 Fluid mechanics in multiphase flow . . . . . . . . . . . . . . . 6

2.1.1 The equation of continuity . . . . . . . . . . . . . . . . 62.1.2 The equation of motion . . . . . . . . . . . . . . . . . . 62.1.3 Dimensionless groups . . . . . . . . . . . . . . . . . . . 8

2.2 Drop formation at a capillary tip . . . . . . . . . . . . . . . . 92.2.1 Transition from dripping to jetting . . . . . . . . . . . 102.2.2 Dripping . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.3 Jetting . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Deformation and breakup of single droplets in laminar flow . . 222.3.1 Parameters describing drop deformation and flow stresses 232.3.2 Drop breakup of single droplets in 2-dimensional shear

and elongational flow . . . . . . . . . . . . . . . . . . . 242.3.3 Drop deformation in 2-dimensional flow . . . . . . . . . 27

2.4 Interfacial tension . . . . . . . . . . . . . . . . . . . . . . . . . 322.4.1 Interfacial tension of pure liquids . . . . . . . . . . . . 33

vii

Contents

2.4.2 Dynamic interfacial tension . . . . . . . . . . . . . . . 33

3 Materials and Methods 413.1 Analytical methods for fluid characterization . . . . . . . . . . 41

3.1.1 Viscosity measurements . . . . . . . . . . . . . . . . . 413.1.2 Density measurements . . . . . . . . . . . . . . . . . . 423.1.3 Surface and interfacial tension measurements . . . . . . 42

3.2 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2.1 Hydrophilic phases . . . . . . . . . . . . . . . . . . . . 433.2.2 Hydrophobic phases . . . . . . . . . . . . . . . . . . . 463.2.3 Surfactants . . . . . . . . . . . . . . . . . . . . . . . . 463.2.4 Steady interfacial tension of material systems . . . . . 48

3.3 Experimental setup and methods of data acquisition . . . . . . 493.3.1 Drop formation experiments . . . . . . . . . . . . . . . 503.3.2 Drop deformation experiments . . . . . . . . . . . . . . 59

4 Results and Discussion 654.1 Drop formation . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.1.1 Transition from dripping to jetting . . . . . . . . . . . 664.1.2 Drop breakup at the capillary tip dripping . . . . . . 744.1.3 Drop breakup from an extended filament jetting . . . 96

4.2 Drop deformation in laminar channel flow . . . . . . . . . . . 1164.2.1 Transient drop deformation under pure elongational

flow conditions - contraction flow . . . . . . . . . . . . 1174.2.2 Transient drop deformation under shear flow conditions 1184.2.3 Residence times of the droplets in the flow channel . . 129

5 Conclusions and Outlook 1355.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Bibliography 141

viii

List of Figures

1.1 Continuous production of tailor-made droplets in size and shapeand the inherent process - microstructure - rheology relationship. 4

2.1 Drop formation mechanisms at a capillary tip: a) dripping andb) jetting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Droplet breakup from an extended filament. . . . . . . . . . . 17

2.3 Parameters describing drop deformation: a) initial dropletshape and b) deformed droplet. . . . . . . . . . . . . . . . . . 24

2.4 Critical Capillary number as a function of the viscosity ra-tio. The data representing simple shear flow conditions weretaken from Grace (1982), whereas the data specifying pureextensional flow originate from Bentley and Leal (1986). Thedotted line identifies the viscosity ratio = 3.6. . . . . . . . . 26

2.5 Schematic of an oil/water interface. The surfactant is primar-ily oil-soluble. ~v is the mass-average bulk velocity, ci is theconcentration of species i in the bulk, ~ji represents the molardiffusion flux, ci(0, t) is the subsurface concentration, ~v

s speci-fies the mass average interface velocity, si is the concentrationof species i at the interface and ~jsi is the surface-excess speciesflux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.6 Interfacial tension and surface-excess concentration of speciesi as a function of the bulk surfactant concentration accordingto Eq. 2.38 and 2.39 (Anbarci and Armbruster, 1987). . . . . 37

2.7 Qualitative illustration of the dynamic interfacial tension andthe surface-excess concentration as a function of the surfaceage (Anbarci and Armbruster, 1987). . . . . . . . . . . . . . . 39

3.1 Viscosities of three different PEG/H2O/C2H5OH solutions asa function of the shear rate (solution 1: PEG/H2O/C2H5OH(27/33/40 wt.-%); solution 2: PEG/H2O/C2H5OH (16/46/38wt.-%); solution 3: PEG/H2O/C2H5OH (11/55/34 wt.-%)). . . 44

ix

List of Figures

3.2 Lecithin molecule adsorbing at an interface. . . . . . . . . . . 48

3.3 Measured interfacial tension as a function of the drop forma-tion time (DFT) for two fluid systems where either de-ionizedwater or an aqueous solution of 1.5 wt.-%-Carrageenan wasused as disperse phase. . . . . . . . . . . . . . . . . . . . . . . 49

3.4 Schematic of the setup for the drop formation experiments(flow channel I and flow channel II). . . . . . . . . . . . . . . . 51

3.5 Schematic of the setup for the drop formation experiments(flow channel III). . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.6 Sequence of the continuous output signal from the photore-ceiver displayed on the computer at fixed flow conditions (flowchannel II, dcap = 0.02 mm). . . . . . . . . . . . . . . . . . . 55

3.7 Determination of the jet length (flow channel III). . . . . . . 56

3.8 Dimensionless drop diameter as a function of the Reynoldsnumber of the continuous phase; Ohdisp 0.42, Redisp = 0.15(water/-Carrageenan 0.38 % in AK 35, AK 50, AK 100 andAK 250; flow channel II). . . . . . . . . . . . . . . . . . . . . . 58

3.9 a) Schematic and b) photograph of the double capillary injec-tion tool. In the schematic the injection tool is attached tothe flow channel. . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.10 Schematic of the setup for the drop deformation experiments. 61

3.11 Image analysis of a) almost spherical droplet and b) deformeddroplet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1 Drop formation in a co-flowing environment for a) drippingand b) jetting flow conditions. . . . . . . . . . . . . . . . . . . 66

4.2 Determination of the transition point between jetting and drip-ping conditions by decreasing the velocity of the continuousphase (water in SFO; flow channel II). . . . . . . . . . . . . . 68

4.3 Droplet diameter as a function of the velocity of the continuousphase; Qdisp = 0.25 ml/min (water/-Carrageenan 0.68 % inSFO; flow channel I). . . . . . . . . . . . . . . . . . . . . . . . 68

4.4 Critical jetting velocity of the continuous phase as a functionof the disperse flow rate (water/-Carrageenan 0.68 % in SFO;flow channel I). . . . . . . . . . . . . . . . . . . . . . . . . . . 69

x

List of Figures

4.5 Images of the drop detachment for systems with different vis-cosities of the continuous phase. The velocity of the continu-ous phase was vcont = 0.125 m/s, the flow rate of the dispersephase Qdisp = 6.25 l/min. The flow conditions were adjustedmarginal below the critical jetting velocity for the system withAK 250 as continuous phase (water/-Carrageenan 0.38 % insilicone oil; flow channel II). . . . . . . . . . . . . . . . . . . . 70

4.6 Critical jetting velocity of the continuous phase as a functionof the viscosity of the continuous phase; Qdisp = 6.25 l/min(water/-Carrageenan 0.68 % in silicone oil; flow channel II). . 71

4.7 Critical jetting velocity as a function of the viscosity ratio atdifferent flow rates of the disperse phase (water/-Carrageenanin SFO; flow channel I). . . . . . . . . . . . . . . . . . . . . . 72

4.8 Critical jetting velocity of the continuous phase as a functionof the disperse flow rate for two material systems with iden-tical viscosity ratio but different interfacial tension (water/-Carrageenan 0.68 % and water/PEG 12.5 % in SFO; flow chan-nel I). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.9 Critical jetting velocity of the continuous phase as a functionof the weight fraction of surfactant. The disperse flow rate waskept constant at Qdisp = 6.25 l/min (water/-Carrageenan0.68 % in SFO; flow channel II). . . . . . . . . . . . . . . . . . 74

4.10 Sketch of the drop breakup at the capillary tip and the gener-ation of monodisperse droplets. . . . . . . . . . . . . . . . . . 76

4.11 Droplet patterns at different velocities of the continuous phase:a) vcont = 0.02 m/s, b) vcont = 0.025 m/s, c) vcont = 0.04 m/s,d) vcont = 0.05 m/s, e) vcont = 0.075 m/s, f) vcont = 0.125 m/sand g) vcont = 0.25 m/s; Qdisp = 6.25 l/min; din,cap = 0.02mm (water/-Carrageenan 0.5 % in SFO; flow channel II). . . 77

4.12 Droplet time intervals at different velocities of the continuousphase as a function of the drop number; Qdisp = 6.25 l/min;din,cap = 0.02 mm (water/-Carrageenan 0.5 % in SFO; flowchannel II). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.13 Time return maps for several velocities of the continuous phase;Qdisp = 6.25 l/min; din,cap = 0.02 mm (water/-Carrageenan0.5 % in SFO; flow channel II). . . . . . . . . . . . . . . . . . 79

4.14 Drop diameters and coefficient of variation as a function of thevelocity of the continuous phase; Qdisp = 6.25 l/min; din,cap= 0.02 mm (water/-Carrageenan 0.5 % in SFO; flow channelII). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

xi

List of Figures

4.15 Drop formation time as a function of the velocity of the contin-uous phase; Qdisp = 6.25 l/min (water/-Carrageenan 0.38% in AK 50; flow channel II). . . . . . . . . . . . . . . . . . . 80

4.16 Drop diameter as a function of the velocity of the continuousphase; Qdisp = 6.25 l/min (water/-Carrageenan 0.38 % inAK 50; flow channel II). . . . . . . . . . . . . . . . . . . . . . 81

4.17 Drop formation at different velocities of the continuous phase,Qdisp = 0.05 ml/min (water/-Carrageenan 0.68 % in SFO;flow channel I); a) stretching and constriction of the neck,b) detachment of the primary drop and c) generation of satel-lite drops through the burst of the thread. The contact linebetween disperse fluid and capillary is pinned to the outerdiameter of the capillary. . . . . . . . . . . . . . . . . . . . . . 82

4.18 Drop formation at different velocities of the continuous phase(0.03 m/s, 0.06 m/s and 0.125 m/s from left to right) anddifferent viscosities of the continuous phase (AK 50, AK 100,AK 250 from top to bottom); Qdisp = 6.25 l/min (water/-Carrageenan 0.38 % in SFO; flow channel II). . . . . . . . . . 84

4.19 Drop diameter as a function of the viscosity of the continuousphase; Qdisp = 6.25 l/min (water/-Carrageenan 0.38 % inSFO; flow channel II). . . . . . . . . . . . . . . . . . . . . . . 85

4.20 Drop breakup time as a function of the disperse flow rate fordifferent velocities of the continuous phase (water/-Carrageenan0.75 % in SFO; flow channel I). . . . . . . . . . . . . . . . . . 86

4.21 Drop diameter as a function of the disperse flow rate for differ-ent velocities of the continuous phase (water/-Carrageenan0.75 % in SFO; flow channel I). . . . . . . . . . . . . . . . . . 86

4.22 Drop formation for different viscosities of the disperse phasebut identical flow conditions (water/-Carrageenan in SFO);vcont = 0.15 m/s and Qdisp = 0.025 ml/min; a) stretching andconstriction of the neck and b) generation of satellite drops. . 87

4.23 Droplet diameter as a function of the viscosity of the dispersephase at different velocities of the continuous phase; Qdisp =0.025 ml/min (water/-Carrageenan in SFO; flow channel I). . 88

4.24 Droplet diameter as a function of the velocity of the continuousphase for fluid systems with different interfacial tension. The-Carrageenan content in the aqueous solutions was 0.38 %.Qdisp = 6.25 l/min (flow channel II). . . . . . . . . . . . . . . 89

xii

List of Figures

4.25 Droplet diameter as a function of the velocity of the contin-uous phase for fluid systems containing different concentra-tions of Lecithin in the continuous phase; Qdisp = 6.25 l/min(water/-Carrageenan 0.38 % in SFO; flow channel II). . . . . 91

4.26 Droplet diameter as a function of the weight fraction Lecithinat different velocities of the continuous phase; Qdisp = 6.25l/min (water/-Carrageenan 0.38 % in SFO; flow channel II). 91

4.27 Droplet diameter as a function of the velocity of the continuousphase for fluid systems containing different concentrations ofImbentin-AG/100/30 in the continuous phase; Qdisp = 6.25l/min (water/-Carrageenan 0.38 % in SFO; flow channel II). 93

4.28 Images of the drop detachment and generation of satellitedroplet generation of fluid systems containing different sur-factants. The surfactant concentration was 1 wt.-% in thecontinuous phase; Qdisp = 6.25 l/min and vcont = 0.09 m/s(water/-Carrageenan 0.38 % in SFO; flow channel II). . . . . 94

4.29 Droplet diameter as a function of the velocity of the continuousphase for fluid systems containing different surfactants in thecontinuous phase; Qdisp = 6.25 l/min (water/-Carrageenan0.38 % in SFO; flow channel II). . . . . . . . . . . . . . . . . . 95

4.30 Droplet diameter as a function of the velocity of the continuousphase. The disperse phase was injected via capillaries of differ-ent inner diameter; Qdisp = 6.25 l/min (water/-Carrageenan0.38 % in SFO; flow channel II). . . . . . . . . . . . . . . . . . 95

4.31 Breakup of a liquid jet: Propagation of capillary waves andsubsequent generation of primary and satellite drops (0.68 %-Carrageenan in SFO; flow channel I). . . . . . . . . . . . . . 97

4.32 Images of jet breakup at the pinch-off point at high velocitiesof the continuous phase (0.68 % -Carrageenan in SFO; flowchannel I). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.33 Images of jet breakup at different flow rates of the dispersephase; vcont = 0.004 m/s (AK 50 in PEG 500; flow channel III). 99

4.34 Jet diameter as a function of the flow rate of the dispersephase. The theoretical data are calculated according to thecontinuity equation (see Eq. 4.2); vcont = 0.01 m/s (AK 50 inPEG 500; flow channel III). . . . . . . . . . . . . . . . . . . . 100

4.35 Images of a jet at different times during one experiment (AK50 in PEG 500; flow channel III). . . . . . . . . . . . . . . . . 101

4.36 Jet length as a function of the experimental time (AK 50 inPEG 500; flow channel III). . . . . . . . . . . . . . . . . . . . 102

xiii

List of Figures

4.37 Jet length as a function of the velocity of the continuous phasefor different jet diameters (AK 50 in PEG 100; flow channelIII). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.38 Jet length as a function of the velocity of the continuous phasefor different viscosities of the continuous phase; dF = 1 mm(AK 50 either in PEG 50, PEG 100 or PEG 500; flow channelIII). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.39 Jet length as a function of the velocity of the continuous phasefor different viscosities of the disperse phase; dF = 1 mm (ei-ther AK 50, AK 100, AK 250 or AK 500 in PEG 50, flowchannel III). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.40 Growth rate of a viscous jet as a function of the viscosityratio (AK 50, AK 100, AK 250 or AK 500 in PEG 50 andAK 50 in PEG 50, PEG 100 or PEG 500; flow channel III). . . 106

4.41 Drop breakup at different flow rates of the disperse phase: a)Qdisp = 2.61 ml/min and b) Qdisp = 3.33 ml/min; vcont = 0.02m/s (AK 50 in PEG 500; flow channel III). . . . . . . . . . . . 107

4.42 Distances of successive drops; vcont = 0.02 m/s (AK 50 in PEG500; flow channel III). . . . . . . . . . . . . . . . . . . . . . . 108

4.43 Distances of successive drops; vcont = 0.02 m/s (AK 50 in PEG500; flow channel III). . . . . . . . . . . . . . . . . . . . . . . 108

4.44 Ratio of drop diameter to jet diameter as a function of thedisperse flow rate (AK 50 in PEG 500; flow channel III). . . . 109

4.45 Images of droplets generated by jet breakup at different veloci-ties of the continuous phase. The droplet patterns for each flowcondition are illustrated by two images; Qdisp = 2.5 ml/min(0.68 % -Carrageenan in SFO; flow channel I). . . . . . . . . 112

4.46 Number average drop diameter and the theoretical predictionaccording to Tomotika (1935) as a function of the velocity ofthe continuous phase. Further the coefficient of variation ofthe drop diameters is displayed for each flow condition (0.68% -Carrageenan in SFO; flow channel I). . . . . . . . . . . . 113

4.47 Drop diameter as a function of the velocity of the continuousphase for different flow rates of the disperse phase (0.68 %-Carrageenan in SFO; flow channel I). . . . . . . . . . . . . . 113

4.48 Drop diameter as a function of the velocity of the continuousphase for different injection capillaries (0.68 % -Carrageenanin SFO; flow channel I). . . . . . . . . . . . . . . . . . . . . . 114

4.49 Drop diameter as a function of the velocity of the continuousphase for different channel widths (0.68 % -Carrageenan inSFO; flow channel I). . . . . . . . . . . . . . . . . . . . . . . . 115

xiv

List of Figures

4.50 Dimensions of the channel geometry used for the drop defor-mation experiments. All specifications are given in millimeter. 116

4.51 Impact of pure elongational stress on droplet deformation withinthe flow channel at position x = x3. The droplets were injectedon the centerline of the channel (water in SFO). . . . . . . . . 118

4.52 Enlargement of a cut of the channel geometry used for the dropdeformation experiments (for length scales see also Fig. 4.50). 119

4.53 Experimentally determined relative radial droplet position andthe calculated velocity on the centerline as a function of thex-position along the channel at Qcont,def = 1400 ml/min; ddrop= 0.27 mm (water in SFO). . . . . . . . . . . . . . . . . . . . 120

4.54 Droplet images (ddrop = 0.27 mm) at different axial channelpositions at two different flow rates of the continuous defor-mation phase; a) Qcont,def = 800 ml/min and b) Qcont,def =1400 ml/min (water in SFO). . . . . . . . . . . . . . . . . . . 122

4.55 Droplet deformation as a function of the axial channel posi-tion of a droplet at two different flow rates of the continuousdeformation phase Qcont,def . The drop diameter was ddrop =0.31 mm (water in SFO). . . . . . . . . . . . . . . . . . . . . . 122

4.56 Droplet deformation (ddrop = 0.27 mm) along the channel asa function of the Capillary number at two different flow ratesof the continuous deformation phase, Qcont,def (water in SFO). 123

4.57 Images of droplets of different size at x = x3 at different flowrates of the continuous deformation phase, Qcont,def (water/-Carrageenan 1.75 % in SFO). . . . . . . . . . . . . . . . . . 125

4.58 Drop deformation as a function of the Capillary number fordifferent droplet sizes at a fixed channel position x = 150 mm(water/ -Carrageenan 1.75 % in SFO). . . . . . . . . . . . . . 126

4.59 Angle of the drop major axis, L, versus the flow axis as afunction of the Capillary number at a fixed channel positionx = 150 mm (water/ -Carrageenan 1.55 % in SFO). . . . . . 126

4.60 Drop deformation as a function of the Capillary number at afixed channel position x = 150 mm. The viscosity ratio wasvaried by using disperse phases of different viscosities. Thedrop diameter was between ddrop = 0. 22 mm and ddrop = 0.27 mm (water/ -Carrageenan 0 %,0.5 %, 0.75, 1.5 % in SFO). 127

4.61 Drop deformation at different flow rates of the continuous de-formation phase for a system with 0.2 % Lecithin as emulsifierin the continuous phase. The images were taken at x = 150mm (water/ -Carrageenan 0.5 % in SFO). . . . . . . . . . . . 129

xv

List of Figures

4.62 Drop deformation at different flow rates of the continuous de-formation phase for a system with 1 % Lecithin as emulsifierin the continuous phase. The images were taken at x = 45mm (water/ -Carrageenan 0.5 % in SFO). . . . . . . . . . . . 130

4.63 Distance of a droplet (ddrop = 0.27 mm) from the centerline ata flow rate of the continuous deformation phase of Qcont,def =800 ml/min. The experimental data were fitted using linearapproximations in the different regions x x3 and x > x3. . . 132

4.64 Residence time of droplets within different areas of the flowchannel as a function of the flow rate of the continuous defor-mation phase, Qcont,def . . . . . . . . . . . . . . . . . . . . . . . 133

5.1 Interfacial tension as a function of the drop formation time.The data points were obtained from the measured drop diam-eters according to Eq. 5.1 and 5.2. The solid lines indicate theequilibrium interfacial tension values measured with a dropvolume tensiometer. The disperse phase was an aqueous solu-tion of -Carrageenan (0.38 wt.-%). Different concentrationsof Imbentin-AG/100 were dissolved in the continuous phase,AK 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

xvi

List of Tables

3.1 Zero shear viscosities, 0, and densities, , of various polyethy-lene/water/ethanol solutions at T = 25 C. . . . . . . . . . . . 44

3.2 Zero shear viscosity, 0, and density, , of polyethylene glycolat T = 25 C. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 Zero shear viscosities, 0, and densities, , of various aqueous-Carrageenan solutions at T = 25 C. . . . . . . . . . . . . . 45

3.4 Zero shear viscosity, 0, and density, , of sunflower oil at T= 25 C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.5 Zero shear viscosity, 0, and density, , of silicone oils at T =25 C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.6 Zero shear viscosity, 0, and density, , of 1-Octanol at T =25 C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.7 Steady interfacial tension values and the corresponding dropformation times (DFT) of the pure material systems. Mea-surements were performed using the drop volume method atT = 25 C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1 Effect of the process and material parameters on the transitionpoint between dripping and jetting. . . . . . . . . . . . . . . . 75

4.2 Effect of the process and material parameters on the size ofthe primary drops in the dripping mode. . . . . . . . . . . . . 96

4.3 Residence time of a droplet within the different areas of theflow channel for Qcont,def = 800 ml/min and Qcont,def = 1400ml/min. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

xvii

Notation

Latin Letters

Symbol SI-Unit MeaningA m2 areaaL mol m

3 Langmuir parameterB m minor axis of ellipseCV % coefficient of variationci mol m

3 molar concentration of species ici,CMC mol m

3 critical micelle concentration of species iD rate-of-strain tensorD deformation parameterd m diameterG s1 rate of deformationg m s2 gravitational accelerationh m channel depthI unity tensorI0, I1 Bessel function~ji molar diffusion flux vector of species iKi interface partition coefficient of species i~k external body force vectork m1 wavenumberkad m s

1 rate constant of adsorptionkdes mol m

2 s1 rate constant of desorptionL m major axis of ellipsel m lengthM kg massp Pa pressureQ m3 s1 flow rateR J K1 mol1 universal gas constantr m radius

xix

Notation

Symbol SI-Unit Meaning(cont.) (cont.) (cont.)ri molar production rate vector of species irparticle m particle trackT stress tensorT K temperatureTn s drip intervalt s timeV m3 volume~v velocity vectorv m/s mean velocityvi m/s velocity componentvcont m/s maximum velocity of continuous phase on

the centerline of the channelvcrit,jetting m/s critical jetting velocityW J workX m characteristic lengthx m general coordinate of placey m general coordinate of place

Greek Letters

Symbol SI-Unit Meaningsi mol m

2 surface-excess molar densityi, mol m3 surface concentration limit of species i s1 shear rate N m1 interfacial tension m2 s1 diffusion coefficient s1 elongational rate Pa s viscosity0 Pa s zero shear viscosity m wavelength, drop distance s1 growth rate kg m3 density N m1 surface tension shear stress tensorresp s drop response time rotation angle surface perturbation

xx

Notation

Indices

Symbol Meaningsubscripts0 initial valuec capillarycap capillary, tubecont continuous phasecont, size continuous phase dragging the drops from the capillarycont, def continuous phase responsible for drop deformationdisp disperse phaseeq equilibriumF filamentin innerint interfacial tensionj replacement charactermax maximalm, n control variableT temperatureo outersuperscriptss surface (related to a 2-dimensional area)T transpose

Dimensionless Numbers

symbol MeaningBo Bond numberCa Capillary numberOh Ohnesorge numberRe Reynolds number ratio of inner and outer diameter of capillary ratio of inner diameter of capillary and hydraulic channel di-

ameter viscosity ratio

xxi

Abstract

Emulsions are meta-stable systems of liquids dispersed in another immisciblematrix fluid. They are encountered in a large variety of application areasincluding food, cosmetics, pharmaceutics and polymers. The mean size andthe size distribution of the droplets represent important characteristics ofan emulsion which affect the emulsion quality significantly. These physicalproperties are adjusted by a proper choice of the dispersing apparatus and theprocess conditions. Examples of well-established dispersing devices includerotor-stator systems or homogenizers. In these operation units droplets aresubject to shear and elongational stresses and fragment into smaller dropletsprovided that the flow-generated stresses are supercritical. Generally, thedispersed droplets underlie a certain size distribution. Recently, the demandfor almost monodisperse emulsions has been rising due to new advances inthe production of microcapsules or specially structured multiphase systems.

In this study an emulsification process was developed aiming to gener-ate monodisperse emulsions. The disperse phase was injected via a capillaryinto a co-flowing matrix fluid. The capillary was positioned on the centerlineof a flow channel where the velocity of the continuous phase is at a maxi-mum. Two different drop formation mechanisms were distinguished: Eitherthe drops break up close to the capillary - dripping - or they break up froman extended liquid jet - jetting. The effect of the various process parameterson the droplet size depends on the breakup mechanism and was investigatedfor each flow domain separately. Consequently, the transition point betweenthe flow domains represents an important operating point which was deter-mined experimentally by varying the material and process parameters. Inthe dripping mode the interplay of the counteracting forces, drag force ofthe continuous phase and interfacial tension force at the capillary, governsthe drop breakup. Through the application of a well-defined flow field of thecontinuous phase the droplet size could be controlled externally. The effectof the process and material parameters on the mean droplet size and the sizedistribution in the dripping regime was studied. It could be shown that dropformation at a capillary tip represents a promising technique for the produc-

xxiii

Abstract

tion of monodisperse droplets. In the jetting domain the drop breakup occursdue to the propagation of interfacial waves. Besides the droplet size and theirsize distribution, the jet length and jet diameters represented further param-eters to be investigated. It was found that the droplet size distribution isnot necessarily as narrow as in the dripping mode. Nevertheless, flow condi-tions were found where almost monodisperse droplets were generated in thejetting mode, too. The experimental results were compared with theoreticalvalues obtained from stability theories. It is shown that under specific flowconditions the validity of the theory is limited.

This work was embedded in the project Structure engineering of emul-sions by micro-machined elongational flow processing. The focus of thisproject was on generating, deforming and fixating droplets in order to ma-nipulate the microstructure of a dispersion. Therefore, in addition to inves-tigations on drop formation, this work involves studies on drop deformationin laminar channel flow. Droplets were injected eccentrically into a nar-rowing flow channel via a specially designed injection tool where monodis-perse droplets were generated. Since the droplets left the injection tool on awell-defined streamline, they experienced same stresses and adopted identicalshapes along the flow channel. The drop deformation could be correlated tothe process and material parameters. Further, the possibility of imprintinglarge deformations on droplets in laminar channel flow is demonstrated.

In summary, a flow device was constructed where droplets were bothgenerated and deformed. This study provides new insight into the behaviorof multiphase systems in flow. A comprehensive study on the drop formationin a co-flowing liquid-liquid system was still lacking although it is of relevancein a large variety of dispersing processes. This work may be regarded as afirst experimental step in pointing out the effect of the various parameterson the drop breakup at a capillary in a co-flowing environment.

xxiv

Zusammenfassung

Emulsionen sind metastabile Fluidsysteme, in welchen eine Flussigkeit ineiner kontinuierlichen Matrixflussigkeit dispergiert vorliegt. Sie spielen einewichtige Rolle unter anderem bei der Herstellung und Verarbeitung von Le-bensmitteln, Pharmazeutika, Kosmetika oder Polymeren. Die mittlere Grosseund die Grossenverteilung der dispergierten Tropfen sind Charakteristika,welche die Eigenschaften und Qualitat einer Emulsion massgeblich beeinflus-sen. Diese physikalischen Grossen werden durch geeignete Wahl des Disper-gierverfahrens und der Prozessbedingungen kontrolliert. Oft werden Rotor-Stator-Systeme oder Hochdruckhomogenisatoren zur Herstellung von Emul-sionen verwendet. In diesen Prozessen sind Tropfen einem Scher- und Dehn-stromungsfeld ausgesetzt. Dabei werden die Tropfen deformiert und bei hin-reichend grossen Spannungen im Stromungsfeld zerteilt. Normalerweise un-terliegen die entstehenden Tropfen einer gewissen Grossenverteilung. Auf-grund wissenschaftlicher Fortschritte in der Produktion von Mikrokapselnoder speziell strukturierten Mehrphasensystemen ist die Nachfrage nach Emul-sionen mit einer besonders engen Tropfengrossenverteilung in den letztenJahren stark gestiegen.

In dieser Arbeit wurde ein Emulgierverfahren zur Herstellung monodis-perser Emulsionen entwickelt. Hierbei wurde die disperse Phase durch eineKapillare in eine parallel stromende kontinuierliche Phase eingespritzt. DieKapillare wurde in der Mitte eines Stromungskanals platziert, wo das para-bolische Geschwindigkeitsprofil der Matrixflussigkeit sein Geschwindigkeits-maximum aufweist. Es wird zwischen zwei Tropfenaufbruchmechanismen un-terschieden: Entweder bilden sich die Tropfen unmittelbar an der Kapillar-spitze oder sie brechen am Ende eines Flussigkeitsstrahls ab. Der Einflussder Material- und Prozessparameter auf die Tropfengrosse und die Trop-fengrossenverteilung hangt stark vom Aufbruchmechanismus ab und wur-de deshalb fur die beiden Tropfenbildungstypen gesondert untersucht. Dem-zufolge stellt der Grenzwert zwischen den Aufbrucharten einen wichtigenBetriebspunkt dar, welcher in dieser Arbeit durch Variation der relevantenParameter experimentell ermittelt wurde. Die Tropfenabscheidung direkt an

xxv

Zusammenfassung

der Kapillare wird durch das Zusammenspiel der entgegengerichteten Krafte,Zugkraft der kontinuierlichen Phase und Grenzflachenspannungskraft an derKapillare, gesteuert. Durch die Erzeugung eines Stromungsfeldes der ausserenPhase konnte die Tropfengrosse durch geeignete Wahl der Prozessparameterextern eingestellt und kontrolliert werden. Im Rahmen dieser Arbeit wurdeder Einfluss der Prozessparameter und Materialeigenschaften auf die Trop-fengrosse bei der Tropfenabscheidung an der Kapillare quantifiziert. Deswei-teren wurde herausgestellt, dass die Tropfenbildung an der Kapillare eine viel-versprechende Technik zur Herstellung monodisperser Emulsionen darstellt.Unter gewissen Stromungsbedingungen bilden sich die Tropfen am Ende ei-nes Flussigkeitsstrahls. Hier fuhrt die Ausbreitung von Kapillarwellen an derGrenzflache zur Tropfenabscheidung. Neben der Tropfengrosse und Tropfen-grossenverteilung bildeten die Lange und der Durchmesser des Flussigkeits-strahls weitere Grossen, die in dieser Arbeit untersucht wurden. Es wurdegezeigt, dass die Grossenverteilung der Tropfen, welche durch den Aufbrucheines Flussigkeitsstrahls enstehen, nicht notwendigerweise genauso eng ist wiebei der Tropfenabscheidung unmittelbar an der Kapillare. Dennoch wurdenStromungsbedingungen herausgearbeitet, unter welchen sich fast monodi-sperse Tropfen am Ende des Flussigkeitsstrahls bildeten. Desweiteren konn-ten die experimentellen Ergebnisse mit Resultaten einer Stabilitatstheoriefur Flussigkeitsstrahlen verglichen werden. Dabei wurden Gultigkeitsgrenzender Theorie klar aufgezeigt.

Diese Arbeit war Bestandteil eines Projektes, in welchem Tropfen erzeugt,deformiert und in verformten Zustand fixiert werden sollten, mit dem Ziel,die Mikrostruktur einer Dispersion gewunscht zu beeinflussen. Aus diesemGrund stellte die Tropfendeformation in einem laminaren Stromungsfeld einweiteres Forschungsgebiet dar, welches im Rahmen dieser Arbeit untersuchtwurde. Hierfur wurden monodisperse Tropfen durch eine speziell konstruierteEinspritzvorrichtung exzentrisch in einen sich verengenden Stromungskanaleingespritzt. Da die Tropfen die Einspritzvorrichtung auf einer genau defi-nierten Stromungslinie verliessen, erfuhren alle Tropfen dieselben Spannun-gen und Deformationszustande entlang des Stromungskanals. Korrelationenzwischen Tropfendeformation und Material- und Prozessparametern konntenherausgearbeitet werden. Desweiteren wurde demonstriert, dass es moglichist, den Tropfen in einer Kanalstromung sehr grosse Deformationen aufzu-pragen. Somit ist es gelungen, eine Stromungszelle zu konstruieren, in welchermonodisperse Tropfen kontinuierlich produziert und anschliessend deformiertwurden.

Im Rahmen dieser Arbeit wurden neue Einblicke in das Verhalten vonMehrphasensystemen in Stromung gewonnen. Eine umfassende experimen-telle Untersuchung der Tropfenabscheidung an der Kapillare in einem Stro-

xxvi

Zusammenfassung

mungsfeld der kontinuierlichen Phase fehlte bis zur Fertigstellung dieser Ar-beit ganzlich, obwohl fundiertes Wissen auf diesem Gebiet essentiell fur vieleDispergierprozesse ist. Die vorliegende Arbeit kann als ein erster experimen-teller Schritt angesehen werden, welcher den Einfluss der zahlreichen Para-meter auf die Tropfenabscheidung an einer Kapillare in einem Stromungsfeldbeschreibt.

xxvii

Chapter 1

Introduction

Emulsions are meta-stable systems of liquids dispersed in another immis-cible liquid. Besides the weight proportions of the different fluids and thechemical compounds, the mean droplet size and droplet size distributionare the most important factors influencing the physical properties and thequality of emulsions. For example, monodisperse droplets or droplets witha narrow size distribution lead to higher quality products and often sim-plify the further processing. In particular new advances in the productionof microcapsules and specially structured multiphase systems require specificphysical properties of the drops. The droplet size and the size distribution iscontrolled by the type of dispersing device and by a proper choice of the pro-cess parameters and process conditions. The well-established emulsificationprocesses (e.g. rotor-stator mixers or homogenizers) often make use of theburst of droplets in emulsions caused by shear and elongational stresses gen-erated by a flow field in the dispersing device. To achieve narrow droplet sizedistributions enormous effort has to be undertaken (e.g. Bibette, 1991). Re-cently, new dispersing techniques have been developed in order to producealmost monodisperse emulsions. In these operation units the fluid phasesflow separately into the dispersing device where a new interface betweenthe immiscible liquids is created. When applying these techniques, the finalproduct properties are not dependent on the properties of a pre-emulsion.Further, the flow field and consequently the stresses acting on the dropletsare well-defined. Membrane emulsification represents an example of a directproduction technique of emulsions where the polydispersity lies in the rangeof 10 % of the average droplet size (Yuyama et al., 2000). In the research fieldof microchannel emulsification, polydispersities of droplets below 5 % wererecently achieved, with the droplet size depending primarily on the capillarysize and channel geometry (Sugiura et al., 2002).

Another dispersing technique is realized by injecting the disperse phase

1

Chapter 1 Introduction

via a capillary into the continuous phase (e.g. Basaran, 2002; Cramer et al.,2004). It is distinguished between two different drop generation mechanisms:Either the drops break up at the capillary tip - dripping - or they are gener-ated from an extended fluid jet - jetting. The fundamentals of dripping andjetting have been investigated extensively but these drop formation mecha-nisms have rarely been considered as a promising dispersing tool with rele-vance for technical applications. Most developed applications deal with theinjection of a liquid into surrounding air (e.g. ink-jet printing (Le, 1998)).

In the present work, the disperse phase is injected via a needle into aflowing ambient continuous phase as shown in Fig. 1.1 (drop generation).New interface between the fluids is created at the capillary tip. The dropletbreakup from the capillary tip is accelerated by the drag force of the contin-uous phase in comparison to the injection into a quiescent surrounding fluid.Thus, the material properties, such as interfacial tension, viscosities and den-sity of the fluids are not the only governing parameters but the droplet sizeis rather controlled externally by the flow velocity of the continuous phase asprocess parameter. Under certain flow conditions, a jet of the disperse phasestreams from the capillary and the droplet breakup is caused by capillarywaves at the interface. The filament is excited by naturally occurring pertur-bations in the flow field. As soon as the wave amplitude equals the jet radius,a droplet is separated. The velocity of propagation of the interfacial wavesdepends on the dimensions of the jet and the material properties (Tomotika,1935). Since the dynamics of drop formation in the dripping mode differssignificantly from the dynamics of drop generation in the jetting domain, thetransition point between the flow regimes represents an important operat-ing point. Consequently, the jetting domain has to be demarcated from thedripping mode by studying the effect of relevant process and material pa-rameters on the transition point. Goal of this study is the development of adispersing device for the generation of monodisperse droplets. Consequently,the effect of the various process and material parameters on the mean size ofthe droplets and their size distribution is studied both in the dripping modeand in the jetting regime.

The flow behavior of dispersed systems is not only influenced by the sizeof the dispersed particles but also by their shape. For example, it is knownthat the rheology of suspensions changes drastically depending whether thedisperse phase consists of spheres or fibres (Metzner, 1985). The dispersedparticles build a specific microstructure according to the flow conditions andtheir shape. Fibres have the ability to orientate in flow assuming that thehydrodynamic forces exceed the structural forces. The orientation kinetics ofparticles are influenced primarily by their size and their shape. Even a smallamount of complex shaped particles added to a dispersed system is assumed

2


to have a large impact on the rheology of the entire multiphase system.In contrast to solid particles, droplets are deformable in flow when they

are subject to stresses. The generated stresses depend on the geometry ofthe flow device, the drops trajectory in the flow and the strength of the flow.In the present study the droplets are injected eccentrically into a narrowingflow channel. The flow stresses are controlled by the position of the injectionpoint of the droplets into the channel and by the flow rates of the phases. InFig. 1.1 (drop deformation) the droplet is deformed in shear flow. The dropadopts a vermicular shape which axial expansion depends on the shear stress.When considering the dispersing technique used in this work the dropletsexit the capillary on a well-defined streamline which is determined by theinjection point. As a consequence, all droplets follow the same path andexperience same stresses. Therefore, the deformation history of all dropletsis identical under steady flow conditions along the flow channel. A well-defined, predictable deformation of the droplets is desired to investigate theinfluence of the particle shape on the product properties.

Because of the interfacial tension between the immiscible liquids thedroplets lose their imprinted deformation as soon as they enter an area ofzero stress in the flow channel or for example during storage of the dispersion.Moreover, they quickly relax back to their preferred spherical shape. Whensuperimposing a drop fixation step to the drop deformation process, dropshapes are conserved. Droplet fixation may be induced by physical or chem-ical reactions within the disperse fluid. For example, gelation takes placewhen a heated emulsion of a gel forming cold-set biopolymer is cooled belowa specific temperature and a suspension is formed. In this case a network isbuilt within the disperse phase inducing the solidification process. Previousinvestigations (Walther et al., 2002) in a four-roller apparatus have shownthat complex drop shapes could be fixated in a shear and/or elongationalflow field (see Fig. 1.1 (drop fixation)).

Considering the entire process as illustrated in Fig. 1.1, the resultingdroplet size and droplet shape are complex functions of the material andprocess parameters. In addition, the generated microstructure affects signifi-cantly the product properties such as product stability, rheology and sensoryanalysis. Consequently, dispersions with desired properties are produced bya proper choice of the process conditions (see lower part in Fig. 1.1).

The goal of the project is the continuous production of tailor-made dropletsin size and shape in order to manipulate the microstructure of the dispersion(see also Walther et al. (2004)). Therefor the flow kinetics, fixation kineticsand kinetics at the interface (e.g. surfactant adsorption) have to be coupled.A prerequisite for a successful approach is a comprehensive knowledge of eachof the process steps illustrated in Fig. 1.1. Because of the complexity and

3


drop generation drop deformation drop fixation

gellingencapsulation

shear flowelongational flow

Process Microstructure

RheologySensory AnalysisProduct Stability

Figure 1.1: Continuous production of tailor-made droplets in size and shapeand the inherent process - microstructure - rheology relationship.

the extent of the entire process the work was distributed to five participantshaving expertise in various areas of research1. The present part of the projectfocuses on continuous drop generation and drop deformation.

1This study was embedded in the European Project Structure engineering of emul-sions by micro-machined elongational flow-processing(QLK1-CT-2000-01543). Furtherparticipants were the Swedish Institute for Food and Biotechnology, Institute of Environ-mental Chemistry of the University of Essen, Unilever Research Vlaardingen and TetraPak Processing Systems AB.

4

Chapter 2

Background

Both drop formation at a capillary tip and deformation of a drop suspendedin a continuous matrix phase represent typical examples of multiphase flowproblems. The theoretical framework describing fluid flow is built around theequations of continuity and the equations of motion. Due to the complexity ofthe full transient equations of motion, several approaches to the problem havebeen made either by simplifying the equations in truncating certain termsor by setting up simplified boundary conditions. After stating the relevantequations, this chapter provides an insight into the different approaches fordetermining the unknown variables. A literature review is given which listssome of the most important of the innumerable investigations dealing withdrop formation on the one hand and drop deformation on the other hand.It is attempted to highlight possible simplifications and essential boundaryconditions depending on the different flow problems. Whereas the pioneeringstudies in this area of research starting with investigations of Rayleigh (1879)focused on analytical limiting solutions, large progress has been achieved dueto the access to fast computers and as a consequence thereof the developmentof extensive numerical algorithms especially during the last 25 years.

An essential boundary condition for solving the equations of motion offlow problems in multiphase systems is provided through a normal stressjump at the interface due to the interfacial tension. When surfactants arepresent in one or more fluid phases interfacial species transport processesplay an additional role and the analysis becomes very complex. Therefore,section 2.4 is devoted to interfacial tension effects with special regard todynamic adsorption kinetics of surfactant molecules.

Although the present study is based primarily on experiments, the the-oretical background discussed in this chapter is mandatory to find possibleexplanations of observed phenomena.

5

Chapter 2 Background

2.1 Fluid mechanics in multiphase flow

In rigid body mechanics forces acting on a surface are resolved into a singleforce vector affecting the motion of the center of mass. In contrast, fluidshave to be considered as continuum where the velocity, ~v, the pressure, p, thedensity, , and the temperature, T , are continuous functions of position andtime. Discontinuities may only occur at phase boundaries. For determiningthe mentioned variables in a flow field, equations for conservation of mass,momentum and energy have to be solved. Additionally, constitutive equa-tions are required which describe the material properties. In the following abasis will be provided for understanding the fundamentals of multiphase flowby listing the basic equations and pointing out possible simplifications (seesection 2.1.1 and 2.1.2). Detailed reviews on fluid dynamics are provided forexample by Middleman (1998) and Bird et al. (2002). A common way to char-acterize flow properties and to reduce the large parameter space representsthe description of variables in dimensionless form. Relevant dimensionlessgroups are specified in section 2.1.3.

2.1.1 The equation of continuity

The continuity equation states that the density in the neighborhood of apoint in a continuous medium may change only through unbalanced flows inthat region:

t+ ( ~v) = 0 (2.1)

where represents the fluid density, t the time and ~v the velocity vector.Eq. 2.1 describes the time rate of change of the fluid density at a fixedposition in space. The first term specifies the rate of increase of mass perunit volume, the other term represents the net flux of mass per unit volumeby convection. Considering incompressible fluids in isothermal flow ( = 0and /t = 0) the continuity equation adopts the simple form:

~v = 0. (2.2)

2.1.2 The equation of motion

The equations describing the transport of momentum in a fluid are obtainedwhen balancing the incoming and outgoing momentum over a volume ele-ment:

6

2.1 Fluid mechanics in multiphase flow

(~v

t+ (~v )~v

)= ~k + T . (2.3)

In Eq. 2.3 ~k specifies an external body force vector. Most frequently it is givenby the gravitational force, but also for example by a magnetic force or anelectrostatic force. The linear vector function, T , represents the stress tensor.The left hand side of Eq. 2.3 specifies the rate of increase of momentum andthe convective contribution, ~k is an external body force acting on the fluidand T specifies the rate of momentum addition by molecular transport.

The stress tensor, T , may be subdivided into an isotropic component, thepressure, p, and an extra stress tensor, , which contains the flow inducedstresses:

T = p I. (2.4)where I is the unity tensor. For determining the single components of theextra stress tensor constitutive equations are required. Considering a New-tonian fluid, the shear stress tensor, , adopts the following form:

= 2 D (2.5)where D is the rate-of-strain tensor which consists of the symmetrical partof the velocity gradient tensor:

D =1

2 (~v + (~v)T ). (2.6)

In Eq. 2.5 is the fluid viscosity. When inserting Eq. 2.4, 2.5 and 2.6 into themomentum balance (see Eq. 2.3), the Navier-Stokes equations are derived forincompressible, Newtonian fluids:

(~v

t+ (~v )~v

)= ~k p+ ~v. (2.7)

At very small Reynolds numbers when inertial effects are negligible (see sec-tion 2.1.3) the convective derivative in Eq. 2.7 may be neglected and theStokes equations are obtained:

0 = p+ ~v. (2.8)In literature Eq. 2.8 is often denoted as creeping flow approximation. Fromthe conservation of mass and the conservation of momentum four equationsare provided to determine the unknown variables ~v, , p and . For the caseof compressible flow an additional equation has to be taken into account,

7


the energy balance. In this work we investigated isothermal, incompressibleflow which is described adequately in terms of Eq. 2.1 and 2.3. Analyticalsolutions of flow problems exist only for simple geometries such as tubesand for materials which exhibit non-complex flow behavior. Most frequently,numerical procedures have to be used to compute the velocity field and flowstresses by solving Eq. 2.1 and 2.3. Common numerical approaches includefinite element method (FEM), finite difference method (FDM), finite volumemethod (FVM) and boundary element method (BEM).

Regarding flow problems where two or more fluids are in motion, Eq. 2.1and 2.3 have to be solved for all fluids separately (e.g. Richards et al., 1995).The equations specifying the flows of the different fluids are connected by es-sential boundary conditions describing the interfacial properties. A commonboundary equation states a normal stress jump at a liquid/liquid interfacewhich is balanced by the interfacial tension stress (see also section 2.4) andthe density contrast of the fluids. Additionally, the velocity at the interfaceis assumed to be continuous. In most cases the liquid/liquid interface is mov-ing, too. Examples include deforming liquid droplets dispersed in anotherimmiscible flowing liquid, growing droplets at a needle surrounded either byair or another immiscible liquid. The computation process becomes morecomplicated for this free boundary problem and bulky numerical procedureshave to be involved (e.g. Renardy et al., 2001).

2.1.3 Dimensionless groups

The equations of motion and mass conservation (Eq. 2.1 and 2.3) involvean extensive set of variables. Therefore, dimensionless groups are derivedwhich reduce the large parameter space and allow a complete investigationof problems incorporating fluid motion. In this section only dimensionlessparameters are listed which are relevant in the following literature review.Common dimensionless groups describing the fluid mechanics of multiphaseflow are:

Reynolds number Rej =j vj X

j, (2.9)

Capillary number Caj =j vj

,

Bond number Bo = r2in,cap g

and

Ohnesorge number Oh =disp

disp rin,cap .

8

2.2 Drop formation at a capillary tip

j is the fluid density, vj is the average velocity of the liquid, X specifies acharacteristic length scale, j is the shear viscosity, is the interfacial tension, specifies the density difference of the fluids, rin,cap is the inner radius of thecapillary tube, g the gravitational acceleration and is the surface tension.These dimensionless groups must be formulated both for the disperse and thecontinuous phase. The index j refers either to the disperse phase (j = disp)or to the continuous phase (j = cont). In our experiments, the characteristiclength, X, is specified by the inner radius of the injection tube, rin, for thedisperse phase and is given by the hydraulic diameter of the channel, dh, forthe flow of the continuous phase.

The Reynolds number, Re, measures the relative importance of inertialforces to viscous forces, the Capillary number, Ca, represents the ratio ofviscous forces to interfacial tension forces and the Bond number, Bo, mea-sures the importance of gravitational forces to interfacial tension forces. TheOhnesorge number, Oh, is used most frequently in problems dealing withspraying or atomization and specifies the ratio of viscous forces to the squareroot of surface tension and inertial forces. An additional important materialparameter constitutes the ratio of the viscosity of the disperse phase, disp,and the viscosity of the continuous phase, cont:

=dispcont

. (2.10)

The dimensionless groups are important for estimating the importance of thedifferent parameters. As a consequence, the transient equations of motion(Eq. 2.1 and 2.3) may be simplified depending which effects are negligible.


Drop formation at a capillary tip has been studied extensively over more thana century since it finds applications in several engineering processes. Ex-amples include ink-jet printing (Le, 1998), separation (Heideger and Wright,1986), spraying and atomization or liquid/liquid dispersing devices (Basaran,2002; Cramer et al., 2004). In the following two different drop formationmechanisms at a capillary tip are distinguished - dripping and jetting. Insection 2.2.1, both flow domains are demarcated and relevant investigationsare listed. Subsequently, a brief literature review is given for both flow do-mains separately in section 2.2.2 and 2.2.3 highlighting the scientific progresswhich has been made since the pioneering studies of Rayleigh (1879). Thetheoretical fundament describing the flow in both flow domains is given bythe equations of mass and momentum conservation (Eq. 2.1 and 2.3). The

9


focus of this review is to point out differences in the analytical approach andthe description of the boundary conditions. Furthermore, it is attempted toshow potential analogies between the drop formation in air on the one handand the more complex problem of drop formation into another immiscibleliquid (flowing or at rest) on the other hand.

2.2.1 Transition from dripping to jetting

When a disperse phase is injected via a capillary tube at low Reynolds num-bers into another immiscible liquid two different drop formation mechanismsare observed (see Fig. 2.1): Either drops are formed directly at the needletip (dripping) or they break up from an extended filament due to capillarywaves (jetting). The effect of the process and material parameters on the dy-namics of drop formation depends on the breakup mechanism and has to beinvestigated separately for each of the mentioned flow domains. Importantparameters affecting the drop breakup mechanism include the viscosity anddensity of the disperse phase, disp and disp, the mean velocity of the dispersephase at the capillary tip, vdisp, the viscosity and density of the continuousphase, cont and cont, the velocity of the continuous phase, vcont, and the cap-illary diameter. The transition point between the flow domains is of greatimportance both for theoreticians, experimentalists and operators since thedynamics of drop formation change significantly. The derivation of a suitablemathematical expression which describes the transition point between drip-ping and jetting as a function of all relevant parameters represents a verycomplex problem. When dealing with the problem theoretically a criticaltransition condition has to be set up. A common approach is to define thetransition point when the initial rise velocity of a drop is sufficient low thatthe drop will rise less than one drop diameter during the time of formation ofthe next drop. In this case the breakup length of a drop, defined as the dis-tance from the capillary tip to the furthest point of a pendant drop from thecapillary, exceeds the maximum reachable distance by the drop (Clanet andLasheras, 1999; Scheele and Meister, 1968). The generated drops will thenmerge to jet. Clanet and Lasheras (1999) set up an equation of motion for apendant drop in air based on a force balance which states that the momen-tum of the drop is a function of the action of gravity, surface tension and thejet momentum. These calculations are restricted to the inviscid limit whereall viscous effects are negligible. The agreement of their calculated transitionpoints with experimental data was fairly satisfactory. For a viscous liquidinjected into another immiscible viscous liquid, both fluid viscosities, dispand cont, and the velocity of the external flow field, if any, have to be takeninto account. The model of Scheele and Meister (1968) describing the tran-

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Qdisp

dispdisp

contcont

vcont

vdisp

vcont

dispdisp

contcont

Qdispa) b)

vdisp

Figure 2.1: Drop formation mechanisms at a capillary tip: a) dripping andb) jetting.

sition point for a system of two immiscible liquids is based on a simple forcebalance and has to be considered as an approximation. A determination ofthe transition point according to the approach of Clanet and Lasheras (1999)is not available when injecting a liquid into another immiscible liquid. In thiscase, the critical jetting condition is to be inserted into the full equations ofmomentum conservation (see Eq. 2.3). A comprehensive description of thetransition point of a liquid injected into another immiscible liquid includingviscous effects is still lacking. Moreover, the transition point between drip-ping and jetting of a liquid injected into a flowing ambient fluid has not beenconsidered at all.

2.2.2 Dripping

In the following, formation of a single droplet at a capillary tip and thegeneration of a series of droplets is distinguished. In latter case interactionsbetween the detaching droplet and the successive growing droplet are allowedwhereas in the single drop approach only the flow dynamics are assumed toaffect the drop generation.

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Single drop detachment at a capillary tip

At low flow rates of the disperse phase the drops form at the capillary tip,grow and eventually break off. Qualitatively the drop formation may be di-vided into two stages: 1) the nearly static growth at the capillary and 2)the necking and detaching. A fundamental study of dripping was performedby Harkins and Brown (1919) who investigated dripping into air. They cal-culated the weight of a falling drop, Mdrop, based on a macroscopic forcebalance by including an empirical correction function f :

Mdrop =2pircap

g f(rcap/V 1/3drop). (2.11)

In Eq. 2.11 rcap is the radius at the capillary tip, where the contact linebetween fluid and capillary is pinned, is the surface tension, g is the gravi-tational acceleration and Vdrop specifies the volume of a detached drop. Thecorrection function f accounts for the non-spherical shape of a pendant dropand the residual fluid which remains at the capillary after drop detachment.As soon as the gravitational force exceeds the surface tension force, the equi-librium of forces is lost and the drop starts to neck and it breaks off. Thehigh accuracy of their measurement led to the drop-weightmethod to de-termine the surface tension of a fluid. This set of experiments was furtherdeveloped for the application to liquid/liquid systems and to a wide rangeof capillary diameters, fluid viscosities and surface and interfacial tensions(Hayworth and Treybal, 1950; Null and Johnson, 1958; Wilkinson, 1972).These approaches showed good results in determining the drop sizes at longdrop formation times when the disperse fluid is injected at very low flowrates. Only in this case is the momentum of the disperse phase negligibleand the quasi-steady assumption in the force balance satisfied. Eq. 2.11 wasextended by Scheele and Meister (1968) who added two additional termsarising from the kinetic force associated with the flowing disperse phase anda drag force exerted by the quiescent continuous phase:

Fbuoyancy + Fkinetic = Fdrag + Fint. (2.12)

In Eq. 2.12 Fbuoyancy specifies the buoyancy force, Fkinetic is the kinetic forceof the disperse phase, Fdrag represents the drag force of the continuous phaseand Fint is the interfacial tension force. Fkinetic is derived based on theassumption that all fluid energy is transmitted in vertical direction. The dragforce of the quiescent continuous phase, Fdrag, decelerates the growing dropand is evaluated when considering the forming droplet as a solid sphere. Anadditional term is presented taking into account that a considerable amountof fluid flows into a drop during the necking process when equilibrium of forces

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is lost. The result shows an improvement to previous theories in predictingdrop sizes at varying flow rates of the disperse phase, but still large deviationsof theory and experiment were observed at high disperse flow rates where theassumption of static conditions fails.

It is evident that a static approach as mentioned above can not providesufficient insight into the dripping dynamics. The evolution in time of thedrop shape, the velocity and pressure field inside the forming drop, the gener-ation of a liquid thread between the detaching drop and the capillary and theformation of satellite drops is not considered. A comprehensive simulation ofthe dripping dynamics involves the solution of the equations of motion andmass conservation (see Eq. 2.1 and 2.3) with specified boundary conditions.Because of the considerable computation times, investigations on the drip-ping dynamics have been emerged only with the access to fast computersduring the last decade. They are summarized in the following.

Eggers and Dupont (1994) have derived and solved one-dimensional equa-tions of mass and axial momentum conservation using a finite differencescheme to simulate the evolution of the shape of a drop in surrounding airin the dripping regime. Their model includes viscous body forces as wellas viscous boundary conditions. Although neglecting radial velocity com-ponents and variations in the axial component of velocity and pressure, thegood agreement between some calculated drop profiles with specific experi-mental observations justifies the approximations provided that the disperseflow rate is vanishingly small. An experimental study of dripping dynamicsof a liquid in air is presented by Zhang and Basaran (1995). They showedthat the viscosity of the disperse phase plays an important role in stabiliz-ing a growing drop by damping interfacial oscillations, but has virtually noeffect on the size of the primary drop. Consequently, the length of the liquidthread that forms during necking and breakup (second stage) rises consider-ably with increasing fluid viscosity, disp, but also with increasing flow rateof the disperse phase, Qdisp, and the capillary radius, rcap.

Zhang (1999b) solved the full transient two-dimensional equations of mo-tion for an incompressible fluid injected into surrounding air (see Eq. 2.2and 2.7) by using a finite difference formulation with the assumption of con-stant surface tension along the free surfaces. Drop profiles were calculated byvarying the Reynolds number, the Capillary number and the Bond numberto investigate the effect of inertial, viscous, gravitational and surface ten-sion forces on the droplet breakup. The algorithm is able to calculate thegeneration of the liquid thread between primary drop and capillary duringthe second stage of drop breakup. Well-defined flow conditions were deter-mined where the liquid thread rolls up and coalesces with the liquid residualat the capillary without secondary breakup and satellite drop formation.

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Good agreement was found between calculated drop profiles and few, se-lected experimental data. Whereas Zhang (1999b) based the finite differencealgorithm on an Eulerian mesh which is fixed in space, Wilkes et al. (1999)computed the transient Navier-Stokes system using a finite element methodincorporating a mesh which conforms to and evolves with the changing dropshape. These authors focused on calculating the drop contour at the pinchoff point. The calculations were very accurate and it was shown that the in-terface of a drop of finite viscosity can overturn before the drop breaks, whichwas previously observed by several authors for inviscid fluids (e.g. Schulkes,1994; Day et al., 1998).

When considering the injection of a liquid at a capillary tip into anotherimmiscible liquid, additional effects arise from the viscous properties of thesurrounding fluid. Zhang and Stone (1997) solved the governing equations ofmotion in the low Reynolds number flow limit (see Eq. 2.8). The numericalstudies were based on a boundary integral method for Stokes flows. Theinfluence of the viscosity ratio, , Bond number, Bo, and Capillary number,Ca, on the breakup length and the primary drop volume was investigatedby varying one dimensionless group while keeping the other two parametersfixed. Analogous to investigations of dripping into air (Zhang and Basaran,1995) it was shown that the viscosity ratio has virtually no effect on theprimary drop volume but influences significantly the necking and breakupbehavior. At low viscosity ratios droplets detach directly at the capillary tip,whereas at higher viscosity ratios a thread between primary drop and cap-illary is formed. After the breakup of the primary drop, secondary breakupof the thread due to the unbalanced capillary force causes the generation ofsatellite drops. The primary drop volume decreases approximately linearlywith rising Bond number and increases with rising Capillary number of thedisperse fluid.

Motivated by observations from Oguz and Prosperetti (1993), who showedthe possibility of reducing the size of bubbles formed at a capillary tip byapplying a flow field to the continuous phase, Zhang and Stone (1997) cal-culated the influence of an external viscous flow on the droplet size. In thepresence of a flow field of the continuous phase the generated droplets arestretched in the direction of the flow and the breakup length rises. Theprimary drop volume decreases with increasing velocity of the outer phasedue to the higher drag force. Zhang (1999a) solved the full Navier-Stokesequations without limitations to low Reynolds numbers by applying a finitedifference algorithm. The computed evolution of drop profiles was verified bycomparison with specific experimentally determined drop shapes. Analogousto his previous work (Zhang, 1999b), flow conditions were found where satel-lite drop formation is inhibited. He showed that the volume of satellite drops

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is less than 1 % of the primary drop volume when injecting into quiescentambient fluid but it reaches almost 10 % in the presence of an external flowfield.

The drop generation at a capillary tip into a co-flowing surrounding liquidwas investigated experimentally by Umbanhowar et al. (2000). The effect ofthe velocity of the external flow and the interfacial tension on the primarydroplet size was studied by dissolving surfactants in the continuous phase.An integral force balance was set up accounting for the drag force of theflowing ambient fluid. The drag force was calculated according to a modifiedversion of the Stokes formula for a solid sphere. Providing that the flow rateof the disperse phase, Qdisp, is sufficient low, the force balance is reduced tothe following equation:

3 pi cont vcont (ddrop dcap) drag force

= pi dcap interfacial tension force

. (2.13)

dcap specifies the diameter where the contact line between capillary and dis-perse phase is pinned and vcont is the velocity of the continuous phase. Eventhough this model has to be considered as an approximation, they foundgood agreement between theoretical predictions and experimental data byintroducing a fitting parameter. Moreover, they pointed out the possibil-ity of generating almost monodisperse emulsions by this technique. But thespace of relevant parameters covered by this study was rather small and theauthors failed to point out limits of this technique (transition from drippingto jetting). Further, dynamic interfacial tension effects are not taken intoconsideration.

Periodic dripping dynamics

All studies discussed above describe the incident of detachment of a sin-gle drop at a capillary tip which is governed by the competition betweenall of the forces acting on a drop. In this approach interactions betweensuccessive droplets are not considered. Several investigations deal with theformation of several drops per sequence into air often entitled as drippingof a leaky faucet (e.g. Martien et al., 1985; Katsuyama and Nagata, 1999;Ambravaneswaran et al., 2000; Renna, 2001). In contrast to the studies dis-cussed above, the dynamics of single drop formation is neglected and thesystem is regarded as a black box generating a stream of droplets. Whena drop breaks up from a needle, a residual fluid is left at the needle whichrebounds and oscillates (dInnocenzo and Renna, 1996). The mechanical vi-brations are transfered to the new growing drop and affect its growth and

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detachment. Parameters proven to influence the oscillations are the fluidtemperature and the disperse flow rate (Katsuyama and Nagata, 1999). Ex-perimentally, the oscillations are detected by measuring the drip intervalbetween successive droplets. Observed phenomena vary from regular con-stant drip intervals to chaotic dripping where no evidences of regularities inthe drip intervals are discovered. One-dimensional mass-spring simulationspropose that the frequency of the vibrations decreases with increasing dropmass. At the incident of drop breakup, the subsequent drop is set into sim-ilar oscillations (Martien et al., 1985; Kiyono and Fuchikami, 1999; Tufaileet al., 1999; Renna, 2001). These theoretical approaches are able to calculatesome of the experimentally obtained dripping patterns, but they are inad-equate in describing complex drop formation dynamics. Ambravaneswaranet al. (2000) solved the one-dimensional axisymmetric Navier-Stokes equa-tions based on a slender-jet approximation using a finite element method tosimulate the generation of hundreds of drops in a sequence. They computedvarying drop breakup lengths and different sizes of the generated droplets.The agreement of their calculated data for the breakup distance of the dropfrom the nozzle with experimental data was excellent. They calculated theformation of water/glycerin droplets in air at flow conditions where periodicdripping occurs. The more complex dripping types as chaotic dripping wereobserved experimentally (Ambravaneswaran et al., 2000; Katsuyama and Na-gata, 1999). Calculations based on fluid dynamics (Ambravaneswaran et al.,2000) supply interesting information, but the computation times are stillconsiderable although simplifications have already been implemented intothe theory. Therefore, the application of these methods to the even morecomplex problem of the formation of drop sequences in a liquid-liquid sys-tem is still a future task.

2.2.3 Jetting

Disintegration of a liquid jet has been provided a huge platform for researchsince more than a century. In the following, jetting of a liquid both intoair and into another immiscible liquid is described in detail because of thesimilar theoretical treatment of both problems. In the jetting flow domaina liquid column streams from a capillary which eventually breaks up intodroplets at a certain distance from the nozzle (see Fig. 2.2). The jet decay isprovoked by Rayleigh instabilities in terms of interfacial waves. The drivingforce for the axisymmetric amplification of interfacial waves is the fact thatthe interfacial area per unit length (and thus the interfacial energy) decreaseswith rising amplitude of the oscillations, provided that the wavelength, , ofthe distortion is larger than the circumference of the cylinder:

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L

d Fxr

Figure 2.2: Droplet breakup from an extended filament.

> pi dF (2.14)where dF represents the average filament diameter (see also Fig. 2.2). Assoon as the amplitude of the oscillations reaches the dimensions of the jetradius, the filament necks and a droplet is separated. Oscillations may beinduced by any sort of background noise, perturbations of the flow field orpresent vibrations.

Quantitative description of interfacial wave propagation has been initi-ated by Lord Rayleigh (Rayleigh, 1879, 1892) for the injection of a liquidinto air. The jet surface is expressed in the following form:

rF = r0 (1 + ) (2.15)where is the surface perturbation and r0 represents the radius of the undis-turbed jet. Based on experimental observations Rayleigh (1879) states thata perturbation of any wavelength fulfilling Eq. 2.14 grows exponentially withtime. Following the assumption that the disturbance to the jet radius isperiodic along the flow axis, x, and it grows monotonically in time, t, thesurface perturbation of an infinite jet is given as:

= 0 exp(t+ ikx) (2.16)where 0 specifies the amplitude of the initial perturbation, k =

2pi

is thewavenumber and represents the growth rate of the perturbation. Accord-ing to Rayleigh (1879) the incident of jet breakup is determined by the fastestgrowing wave. is calculated by inserting Eq. 2.16 into the modified equa-tions of motion (see Eq. 2.2 and 2.7) and neglecting all viscous terms andbody forces (Middleman, 1995):

2 =

r3F(krF (1 k2r2F )

I1(krF )

I0(krF )

). (2.17)

In Eq. 2.17 is the surface tension and I0, I1 are Bessel functions. A jetis unstable for 0 < krF < 1 where has a real, positive solution. The

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maximum instability correlating with the fastest growing wave is obtainedfrom the maximum of Eq. 2.17 at krF 0.7. Consequently, the wavelengthof the instability governing the disintegration of an inviscid jet in air resultsfrom

=2pi

k= 4.508 2rF = 4.508 dF . (2.18)

Weber (1931) generalized the instability analysis of Rayleigh (1879) by in-cluding the effect of the jet viscosity into the analysis. It is pointed out thatthe viscosity decreases the magnitude of the growth rate of the instabilities,, and the wave growth is dampened completely for an infinite jet viscosity.These linear theories predict the drop generation induced from a single har-monic waveform. They are able to calculate the jet breakup time reasonablywell whereas nonlinear phenomena such as the generation of satellite dropsand the jet shape evolution near the breakup point are not covered.

The linear theory was extended by Yuen (1968) who included nonlinearterms into the description of the surface perturbation (see Eq. 2.15):

=

m=1

m0 m(x, t). (2.19)

The perturbations m(x, t) are obtained by solving the equations of motionfor an inviscid fluid in each order. Yuen (1968) considered the first threeterms of Eq. 2.19. The nonlinear effects were found to cause a non-sinusoidalsurface deformation and as a consequence, the jet breaks up into dropletswith ligaments in between. The size of the ligaments is negligible at krF= 1 and rises with decreasing wavenumber. The detachment of a ligamentis accompanied by its following burst into satellite drops. Experiments ofRutland and Jameson (1970) showed qualitative agreement with the non-linear theory of Yuen (1968), but in contrast to the theory satellite drops wereobserved at all wavenumbers. Further higher-order perturbation formulationsof the problem of an infinite unstable cylinder column have been set up byNayfeh (1970) and Lafrance (1975).

The occurrence of satellite drops is described by all non-linear theoriesbut always in a symmetrical position between the main drops because ofthe inherent symmetry of the infinite jet problem. Pimbley and Lee (1977)have shown that the satellite drop position depends on the amplitude ofthe disturbance. The mentioned theoretical approaches consider a nonlineartemporal instability of an infinite liquid cylinder with a spatially harmonicinitial surface displacement. The waves are not regarded as travelling wavebut as a periodic shrinking and bulging of the jet radius. In his review

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article Bogy (1979) pointed out that all previous analysis have shortcomingsespecially when considering satellite drop formation. In a series of articlesChaudhary (Chaudhary and Redekopp, 1980; Chaudhary and Maxworthy,1980a,b) developed a theoretical approach where the initial jet perturbationis induced by an initial velocity field where the initial surface profile is takento be undisturbed. Analogous to previous theories, interface displacementsare regarded up to the third order. Again, the agreement between theoryand experiment is only qualitatively satisfactory.

For an accurate description of the jet profile a complete treatment ofthe nonlinearities in the Navier-Stokes equations (see Eq. 2.7) is required.Because of the complexity of the full Navier-Stokes equations Eggers andDupont (1994) and Eggers (1995) derived a one-dimensional model by ex-panding the radial variable in a Taylor series and keeping only the lowest-order terms. Their model includes viscous body forces and viscous boundaryconditions. They assume that a liquid cylinder initially receives a sinusoidalperturbation (see Eq. 2.16). The problem is solved numerically by an im-plicit centered difference method. Despite the simplifications the computedprofiles agree well with available experimental data. Brenn et al. (2000) andBrenn et al. (2001) derived a dispersion equation for non-Newtonian liquidjets and pointed out the large impact of the viscoelastic properties on thegrowth rate of the perturbations. Detailed reviews on the problem of liq-uid jet disintegration in ambient air are given by Middleman (1995), Eggers(1997) and Lin and Reitz (1998).

When considering a liquid jet surrounded by another immiscible liquidthe effect of interfacial tension and viscous forces acting on the cylindercolumn have to be incorporated. Tomotika (1935) solved the Navier-Stokesequations for an infinite liquid cylinder column in an ambient immiscible fluidat small motions by neglecting effects of inertia and discarding the squaresand products of velocity in Eq. 2.7. As boundary conditions he assumed noslip at the interface, continuous tangential stress at the interface and that thenormal stress difference between inside and outside of the column is solely dueto the interfacial tension. The perturbation is expected to be symmetricalwith respect to the jet axis (see Eq. 2.16). An implicit dispersion equation ofcomplex form is derived. He showed that the wavelength, , correspondingto the maximum instability depends on the viscosity ratio, , and adopts aminimum at = 0.28. For the result coincides with the findings ofRayleigh (1879). The growth rate, , of the jet perturbation in a liquid/liquidsystem is given as:

=

2cont rF (, ). (2.20)

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where is the interfacial tension, cont the viscosity of the continuous phase,rF the jet radius. is calculated by means of a complex equation systemand depends on the viscosity ratio, , and the wavelength, , which again isa function of the jet diameter (see also Elmendorp (1986)). Rumscheidt andMason (1962) performed experiments in a four-roller apparatus with severalrather viscous material systems and found good agreement between their ex-perimentally determined wavelength, , and calculated values according toTomotika (1935). Limiting solutions for jet stability based on Tomotikasgeneral low velocity equations were presented by Meister and Scheele (1967).They considered border cases for the fluid viscosities aiming to derive simpleequations for the wavelength and the wave growth and proved their appli-cability. In consecutive studies (Meister and Scheele, 1969b,a) included theimpact of the velocity of the disperse fluid streaming from a nozzle into theirapproach for predicting the jet length and the size of the separated droplets.In both studies the interfacial velocity plays an important role in determin-ing the velocity of propagation of the disturbance. The growth rate wascalculated according to the limiting solutions of Tomotikas theory. Due todeficiencies in predicting the velocity profile in the jet and the unknown initialperturbation level in experiments the agreement between experimental dataand their theories was rather poor. Another approach in describing limitingsolutions of the general stability theory for a stationary liquid column wascarried out by Lee and Flumerfelt (1981). Governing characteristic groupsinclude the viscosity ratio, , the density ratio of the fluids and the Ohne-sorge numbers. Border cases of the dimensionless groups were consideredand limiting solutions were worked out aiming to improve the applicabilityof the relations in comparison to the theory of Meister and Scheele (1967)with regard to real processes.

Grace (1982) investigated experimentally the breakup of a liquid jet ina flowing environment. He calculated the thread diameter corresponding tothe continuity equation and found that the diameter of the major breakupfragment was approximately 2 to 2.2 times the equilibrium jet diameter for 1. As long as the jet diameter is less than the nozzle diameter, theo-retical data according to Meister and Scheele (1967) agree well with experi-mentally determined drop diameters. Discrepancies were pointed out for jetdiameters exceeding the nozzle size.

Kitamura et al. (1982) overcame the deficiency of unknown flow profilewithin the jet by injecting the disperse phase into a co-flowing continuousphase where the jet velocity relative to the continuous phase was zero. Asa consequence, the jet may be considered as a stationary liquid column andthus fulfilling the prerequisites of the stability analysis. Assuming that thejet breaks up into drops when the disturbance reaches the dimensions of the

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jet radius and that the jet radius and jet velocity do not change after exitingthe nozzle, the breakup length is given as (Kitamura et al., 1982):

l = ln

(rF0

) vjet

. (2.21)

In experiments the initial disturbance amplitude 0 is unknown. For this rea-son Kitamura et al. (1982) determined an empirical equation describing thefirst coefficient in Eq. 2.21, rF/0. For zero relative velocity good agreementbetween measured drop sizes and breakup lengths and calculated values ac-cording to Tomotika (1935) was found and the applicability of the stabilityanalysis to finite jets was proven. Discrepancies between theory and experi-ment were highlighted at nonzero relative velocity.

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