Copyright

by

Matthew Evan Thrasher

2005

A Liquid Stream Bouncing off a Moving Liquid Bath

by

Matthew Evan Thrasher, B.S.

THESIS

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

MASTER OF ARTS

THE UNIVERSITY OF TEXAS AT AUSTIN

August 2005

A Liquid Stream Bouncing off a Moving Liquid Bath

APPROVED BY

SUPERVISING COMMITTEE:

Harry L. Swinney, Supervisor

Jack B. Swift

For my family

Oh! Blessed rage for order, pale Ramon,

The maker’s rage to order words of the sea,

-Wallace Stevens, The Idea of Order at Key West

Acknowledgments

Harry Swinney deserves the lion’s share of the credit and acknowledge-

ment for this work. I am learning from him how to be a scientist and how to

be generous. Jack Swift helps me continuously, whether I am qualifying for

PhD candidacy, learning fluid dynamics, or completing a Master’s degree. Bill

McCormick has offered guiding advice about experiments, ways to understand

the bouncing jet, and where to visit in Italy. Abe Yarbrough and Olivier Praud

helped me to build the initial apparatus to stabilize the bouncing jet and to

record movies of it. Yee Kwong Pang, Chih-Piao Chuu, and Sunghwan Jung

have done large and significant pieces of this work. Yee Kwong and I built the

second generation apparatus and he took much of the first data. Chih-Piao

and Sunghwan developed the most recent incarnation of the experiment by

adding a more versatile pump and collecting data for more viscosities. Mem-

bers of CNLD engaged me in very useful discussions about the phenomena.

Olga Vera and Elena Simmons helped me to stay on track. And John Bush

has suggested fruitful directions to pursue, such as what role the Coanda effect

could play in initiating a bounce.

Matthew Thrasher

The University of Texas at Austin, July 2005

vi

A Liquid Stream Bouncing off a Moving Liquid Bath

Matthew Evan Thrasher, M.A.

The University of Texas at Austin, 2005

Supervisor: Harry L. Swinney

A stream of viscous liquid poured into a horizontally moving bath of

the same liquid can bounce off the surface. Many fluids are shown to bounce,

including silicone oil, which is studied in detail. The bouncing jet is stable for a

large range of the jet’s radius (0.5 to 1.2 mm), the jet’s vertical velocity (40-170

cm/s, corresponding to nozzle heights 1.7 to 14.1 cm), the bath’s horizontal

velocity (1-35 cm/s), and the dynamic viscosity (50.8 to 515 mPa·s). For some

parameters, four different stable configurations can be obtained. A physical

mechanism is proposed for the bouncing stream. When the stream contacts

the surface, a thin layer of air separates the jet and the bath. The jet ramps off

an indentation that it makes in the surface. After bouncing, the jet moves in

a roughly parabolic flight until it hits the surface again, where it may undergo

additional bounces.

vii

Table of Contents

Acknowledgments vi

Abstract vii

List of Tables x

List of Figures xi

Chapter 1. Introduction 1

1.1 The Bouncing Jet . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Overview of each chapter . . . . . . . . . . . . . . . . . . . . . 3

1.3 How to bounce a jet at home . . . . . . . . . . . . . . . . . . . 4

1.4 Units of viscosity . . . . . . . . . . . . . . . . . . . . . . . . . 7

Chapter 2. Background 10

2.1 Fluid mechanics of falling jets . . . . . . . . . . . . . . . . . . 10

2.1.1 Inviscid inertial jet . . . . . . . . . . . . . . . . . . . . . 11

2.1.2 Viscous inertial jet . . . . . . . . . . . . . . . . . . . . . 12

2.1.3 Rayleigh instability for an inviscid jet . . . . . . . . . . 14

2.1.4 Rayleigh instability for a viscous jet . . . . . . . . . . . 16

2.2 Thin layers of air . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Air entrainment . . . . . . . . . . . . . . . . . . . . . . 17

2.2.2 Non-coalescence . . . . . . . . . . . . . . . . . . . . . . 21

2.2.3 Double bubbles and antibubbles . . . . . . . . . . . . . 23

2.3 Bending and rebound of fluid jets . . . . . . . . . . . . . . . . 24

2.3.1 Fluid rope . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.2 Coanda effect . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.3 Kaye effect: a transiently bouncing, non-Newtonian jet . 30

2.3.4 Stone skipping . . . . . . . . . . . . . . . . . . . . . . . 32

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2.4 Trajectory of the jet . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4.1 Trajectory of an inviscid jet . . . . . . . . . . . . . . . . 34

2.4.2 Trajectory of a viscous jet . . . . . . . . . . . . . . . . . 36

Chapter 3. Experiment 38

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Overview of apparatus . . . . . . . . . . . . . . . . . . . . . . 39

3.3 Details of apparatus . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4 Experimental methods . . . . . . . . . . . . . . . . . . . . . . 44

3.5 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Chapter 4. Results 54

4.1 Types of jet behavior . . . . . . . . . . . . . . . . . . . . . . . 54

4.1.1 Entrainment of air . . . . . . . . . . . . . . . . . . . . . 54

4.1.2 Types of bouncing . . . . . . . . . . . . . . . . . . . . . 55

4.2 Parameters for stable bouncing . . . . . . . . . . . . . . . . . . 61

4.2.1 Multi-stability and hysteresis . . . . . . . . . . . . . . . 68

4.3 Bounce initiation . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3.1 By perturbation . . . . . . . . . . . . . . . . . . . . . . 72

4.3.2 By changing Vbath: surface-bound penetration . . . . . . 74

4.3.3 By changing Q, subsurface penetration . . . . . . . . . . 74

4.4 Additional observations . . . . . . . . . . . . . . . . . . . . . . 76

4.4.1 Non-bouncing states . . . . . . . . . . . . . . . . . . . . 76

4.4.2 Jet breakup and dripping . . . . . . . . . . . . . . . . . 81

4.4.3 Non-coalescence: floating drops on a moving liquid bath 81

4.4.4 Antibubble production and extraction . . . . . . . . . . 84

Chapter 5. Discussion and Conclusions 88

5.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2 Implications, significance, and applications . . . . . . . . . . . 89

5.3 Suggestions for future research . . . . . . . . . . . . . . . . . . 90

Bibliography 92

Vita 101

ix

List of Tables

1.1 Fluids that bounce. . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1 Silicone oils used. . . . . . . . . . . . . . . . . . . . . . . . . . 42

x

List of Figures

1.1 The Bouncing Jet. The conditions are as follows: velocity of thebath Vbath=16 cm/s, flow rate Q = 0.35 cm3/s, falling heightH = 5 cm, dynamic viscosity of the silicone oil µ = 103 mPa·s( 100 times that of water). The distance from the initial impactto the last merging is 3.7 cm. The distance that the jet bouncesin the first, larger arc is 1.5 cm. . . . . . . . . . . . . . . . . 2

1.2 The fluid bath can be rotated on either a record player or amanual turntable. The diameter of the dish on the record playeris 19 cm and the diameter of the pan on the manual turntableis 24 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Pour the liquid from a few centimeters above the surface. Thefluid here is laundry detergent. The pouring height is threecentimeters. The bath is moving left to right. See Table 1.1 formore conditions for bouncing. . . . . . . . . . . . . . . . . . . 5

1.4 With the right conditions, the jet will bounce. The jet in thisfigure bounces twice. In the image to the left, the black controlrod has a drop of liquid hanging from its end. Because of thepoor contrast, a sketch of the situation is provided to the right.The jet falls straight down, then bounces off the surface twice.In the image to the right, the black line is in place of the jet. 6

2.1 A typical bouncing jet. The numbers refer to which sectiondeals with that particular aspect. 1) A falling jet 2) A jet im-pinging on a surface 3) A jet bending 4) A bouncing jet. Thedynamic viscosity is 347 mPa·s, the falling height is 4 cm, theflow rate is 0.34 cm3/s, the horizontal bath velocity is 3.3 cm/s. 11

2.2 A viscous jet accelerated by gravity. Left: The white teflonnozzle with diameter 0.52 cm can be seen at the top of theimage. The silicone oil’s dynamic viscosity is 347 mPa·s. Left:The measured shape of the jet compared to predictions of afalling viscous jet and of a solid body falling under gravity. Theerror of the diameter measurements is typically one pixel (0.002cm or 4% of the jet diameter in the worst case). The fluidparameters used in this calculation are: dynamic viscosity 100mPa·s, surface tension 20 mN/m, and density 0.968 g/cm3. . 13

xi

2.3 The Rayleigh instability of a liquid jet. The jet can be seenpinching off into separate drops. Time increases up the page.The pictures above are an artist’s rendition of the break-up ofa jet. The frame is co-moving with the jet. Taken from Powersand Goldstein[52]. . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 The impinging jet. a) No air entrainment at subcritical ve-locities. b) Air entrainment at supercritical velocities. c) Asequence of images showing the transition between subcritical(upper left image) and supercritical velocities (all other im-ages) for air entrainment. The lower right image is the steadytrumpet-shaped air skirt. The fluid is silicone oil of dynamicviscosity µ = 970 mPa·s. The jet diameter is 1.5 mm. The im-ages are ordered left to right by row. The time between imagesis 130 ms. Taken from Lorenceau, Quere, and Eggers [23]. . . 18

2.5 Picture of a floating drop with a cartoon of the drop’s cross-section. The layer of air is on the order of microns thick andit takes seconds or minutes to drain. Taken from Couder et al.[13]. Though the exact conditions of the drop above is not givenin [13], typical values are as follows: the radius of the drop ≈1.2 cm, the air film thickness ≈ 1.5 µm, dynamic viscosity µ≈ 500 mPa·s, the draining time was up to half an hour, andsurface tension ≈ 21 mN/m. . . . . . . . . . . . . . . . . . . 22

2.6 Double bubble and antibubble. a) A double emulsion. Theexact parameters for this double bubble are not given in [63].For other double emulsions, the outer fluid is silicone oil withdynamic viscosity µ = 480 mPa·s, the middle fluid is a glycerol-water mixture with µ = 50 mPa·s, and the inner fluid is siliconeoil with µ = 50 mPa·s. the interfacial tension is ≈ 20 mN/m.b) A sequence of images showing a large antibubble in water(about 1 cm in diameter) burst. After bursting, the colored wa-ter of the inner bubble sinks. The small volume of air from theantibubble’s thin shell rises upward. The images are separatedby 0.1 s. Taken from Utada et al. [63] and Dorbolo et al. [18]. 24

2.7 The liquid-rope coiling effect. (a) Coiling of honey. Fallingheight H = 3.4 cm and kinematic viscosity ν = 60 cm2/s. (b)Coiling of silicone oil. Radius of rope = 0.034 cm, kinematicviscosity ν = 1000 cm2/s, falling height H = 0.36 cm, and flowrate Q=0.0044 cm3/s. (c) Coiling of silicone oil. Radius of therope in the image = 0.04 cm, kinematic viscosity ν = 125 cm2/s,falling height H = 10 cm, and flow rate Q=0.213 cm3/s. Takenfrom Maleki et al.[40]. . . . . . . . . . . . . . . . . . . . . . . 25

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2.8 A liquid stream widening as it merges into a liquid bath withoutentrainment. The bath’s meniscus on the edge of the tank blursthe bottom of the image. The actual liquid surface level isslightly above the bottom of the image. The fluid in this imagehas a dynamic viscosity of 347 mPa·s, a flow rate of Q = 0.16cm3/s, a falling height of H = 4.23 cm. The liquid bath isstationary, Vbath = 0 cm/s. . . . . . . . . . . . . . . . . . . . . 26

2.9 The Coanda effect. a) A laminar thermal plume is attractedto a solid wall 0.71 cm away. Taken from Pera and Gebhart[50]. b) A submerged, horizontal jet is attracted to the top orbottom surface of a channel. Taken from Shimada et al.[59]. . 29

2.10 A sequence of images of the Kaye effect. The bouncing is onlytemporary and the fluid is dilute solution of the polymer poly-isobutylene in dekalin. Each number indicates its frame num-ber from a 250 frame per second cine film. Taken from Collyerand Fisher [10], permission granted by the Nature PublishingGroup. Although no length scale is given in [10], in Kaye’soriginal paper [32], the liquid was poured from 25 cm and thestream diameter was ≈ 1 mm. . . . . . . . . . . . . . . . . . 31

2.11 A labeled schematic of a stone skipping. Taken from Bocquet[4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.12 A sequence of images showing an aluminum disc skipping offwater. The disc’s radius = 2.5 cm, thickness = 2.75 mm, trans-lation velocity v 3.5 m/s, angular velocity = 65 rev/s, attackangle θ = 20 , trajectory angle β = 20 . The time betweenimages is 6.5 ms. Taken from Clanet et al. [7], reprinted withpermission from the Nature Publishing Group. . . . . . . . . 34

2.13 A jet moving under gravity. Taken from Vanden-Broeck andKeller [64]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1 Experimental setup. An annular bath of liquid is rotated. Apump withdraws oil from a central catch and releases it abovethe surface. The outer diameter of the tank is 40.5 cm and thefluid is 7.7 cm deep. . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Calibration of the gear pump head. The manufacturer’s speci-fication for the gear pump head (the dashed line) is inaccurate.The correct value for the volume pumped per gear head revo-lution is 0.1046 cm3 per revolution. . . . . . . . . . . . . . . . 48

3.3 Calibration of the rotating table. The calibration is accurate towithin one percent. . . . . . . . . . . . . . . . . . . . . . . . 49

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3.4 Calibration of the image. A ruler was placed in the focal planeto determine the number of pixels per unit length. The smallestdivision on the ruler is 1/100th of an inch. In this image, theflow rate Q = 0.35 cm3/s, the falling height H = 4.72 cm, thebath’s horizontal velocity Vbath = 0.0 cm/s, and the dynamicviscosity µ = 103 mPa·s. . . . . . . . . . . . . . . . . . . . . 50

3.5 Calibration of the jet diameter and velocity. The jet diameterwas measured by imaging it with a zoom lens (stars). The solidline is the diameter predicted if the jet was treated as a solidbody falling without resistance under gravity. The two dashedlines are piecewise polynomial fits to the experimental data.Polynomial fit #1 has parameters: 0.0032x2−0.0338x+0.1578.Polynomial fit #2 has parameters: 0.0001x2−0.0046x+0.0881.The transition between the two polynomials occurs at 4.72 cm.The solid line with dots is the solution of equation 2.5 with theappropriate parameters. The nozzle is 0.52 cm can be seen atthe top of the image. The oil’s dynamic viscosity is 103 mPa·s. 53

4.1 Entrainment of a thin cylindrical film of air or not. Betweenthe two images, only the flow rate Q changes. a) The jet is notentraining air (Q = 0.35 cm3/s). b) The jet is entraining air (Q= 0.87 cm3/s). For both: height H = 5 cm, horizontal velocityof the bath Vbath = 15.7 cm/s. The viscosity µ = 103 mPa·s.Note: The top image in (a) was taken at a slightly higher incli-nation than (b), so the dark indentation on the surface appearsdifferently. Also, the top and bottom images were not takensimultaneously, so slight differences may be present. The in-dentation of the non-entraining jet is in general larger than theindentation of the entraining jet. The level of the bath’s surfaceis labeled in (b) to avoid ambiguity, because there is a reflectionof the jet’s edges on the bath’s surface. . . . . . . . . . . . . 56

4.2 The bouncing jet. Between the two images, only the flow rateQ changes. a) Q = 0.23 cm3/s. b)Q = 0.44 cm3/s. For both:Height H = 5 cm. The horizontal velocity of the bath Vbath =15.7 cm/s. The dynamic viscosity µ = 103 mPa·s. . . . . . . 57

4.3 The double bouncing jet. Flow rate Q = 0.35 cm3/s, height H= 5 cm, horizontal velocity of the bath Vbath = 15.7 cm/s, anddynamic viscosity µ = 103 mPa·s. The top and bottom imageswere not taken simultaneously, so slight differences exist. . . . 58

4.4 Changing the horizontal velocity of the bath for a fluid withdynamic viscosity µ = 103 mPa·s. a) Horizontal velocity of thebath Vbath = 11.0 cm/s. b) Vbath = 15.7 cm/s. For both: flowrate Q = 0.35 cm3/s, height H = 5 cm. . . . . . . . . . . . . 59

xiv

4.5 A trailing (non-bouncing) jet at high bath velocity. Vbath = 42.3cm/s, Q = 0.35 cm3/s, H = 5 cm, µ = 103 mPa·s. . . . . . . 60

4.6 Changing the horizontal velocity of the bath for a higher vis-cosity fluid viscosity µ = 347 mPa·s. a) Horizontal velocity ofthe bath Vbath = 0.656 cm/s. b) Vbath = 3.28 cm/s. For both:flow rate Q = 0.16 cm3/s, height H = 4 cm. . . . . . . . . . 60

4.7 Phase diagram of flow rate Q vs. horizontal bath velocity Vbath.Dynamic viscosity µ = 50.8 mPa·s, falling height H = 3 cm. . 62

4.8 Phase diagrams of flow rate Q vs. horizontal bath velocity Vbath

for different dynamic viscosities µ = 50.8, 103, 216, and 515mPa·s. Falling height H = 3 cm. . . . . . . . . . . . . . . . . 64

4.9 Phase diagrams of flow rate Q vs. horizontal bath velocity Vbath.Dynamic viscosity µ = 347 mPa·s, falling height H = 4.23 cm. 65

4.10 Phase diagram of Vjet vs. Vbath. Flow rate Q = 0.35 cm3/s,viscosity µ = 103 mPa·s. The large uncertainty around Vjet

= 95 cm/s is due to the transition from a non-entraining jetto an air entraining jet. The stream is very sensitive to smallperturbations. The vertical velocity of the jet corresponds to afalling height H from 1.7 to 14.1 cm. . . . . . . . . . . . . . . 67

4.11 Multi-stability: a sequence of images showing three possiblestates at Vbath = 12.5 cm/s. H = 5.10 cm, µ = 103 mPa·s, Q= 0.35 cm3/s. . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.12 Multi-stability: three states for conditions different than Fig.4.11. a) A plunging jet, b) A surface-bound jet, and c) A bounc-ing jet. All states at Vbath = 15.2 cm/s. H = 6.4 cm, µ = 103mPa·s, Q = 0.35 cm3/s. . . . . . . . . . . . . . . . . . . . . . 71

4.13 A time sequence showing the control rod perturbing the jet,initiating a bounce, and then initiating a double bounce. Vbath

= 16 cm/s, H = 5 cm, µ = 103 mPa·s, Q = 0.35 cm3/s. . . . 73

4.14 This plunging jet is bound on its upper side to the surface.Small air bubbles are streaming off the tip of the air skirt. Theview is from below the surface. The jet impacts the surfaceat the left. The fluid bath is moving right to left. This setof images is a parameter sequence showing a bounce initiatedby changing the bath velocity. The change in horizontal bathvelocity between images is about 0.5 cm/s, starting from Vbath

= 21.2 cm/s. H = 5.10 cm, µ = 103 mPa·s, Q = 0.35 cm3/s. 75

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4.15 A time sequence of images showing a bounce begin by decreas-ing the flow rate from an air-entraining jet. The time betweenimages is one quarter of a second. The change in pump ratebetween images is about 0.02 cm3/s, starting from Q = 0.56cm3/s and ending at Q = 0.45 cm3/s. Vbath = 5.2 cm/s, H =4.23 cm, µ = 347 mPa·s. . . . . . . . . . . . . . . . . . . . . 77

4.16 Non-bouncing conditions for a fall height H and flow rate Qtoo small and too large. a) H = 2.2 cm (corresponding to Vjet

= 46.2 cm/s) is too large for Vbath = 15.7 cm/s. “Too large”may seem counter-intuitive since the falling height is small, butFig. 4.10 confirms that for these conditions, the falling heightis in fact slightly too large. Because the conditions are closeto the boundary of the stable region (with its uncertainty), thejet could possibly bounce if it was perturbed. b) H = 10.2cm (corresponding to Vjet = 144 cm/s) is too small for Vbath =15.7 cm/s. Again, Fig. 4.10 confirms “too small.” For both:horizontal velocity of the bath Vbath = 15.7 cm/s, flow rate Q= 0.35 cm3/s, and viscosity µ = 103 mPa·s. . . . . . . . . . . 78

4.17 A time sequence of a bubble breaking off the bottom of thecylindrical sheath of air. The circle in (a) marks where the aircollects. The bubble in (c) is streaked because of the camera’sexposure time. The time between images is 120 milliseconds.H = 6.4 cm, Q = 0.35 cm3/s, µ = 103 mPa·s, Vbath = 15.2 cm/s. 79

4.18 A time sequence of a bump of air going to the surface alongthe inside of the cylindrical sheath of air. (a) The entrained aircollects at the bottom of the air skirt. (b)-(f) Once enough airhas been collected, a pocket of air rises along the side of the airskirt. The arrows mark the air rising. (g)-(i) The entrained airis disturbed after the air bump reaches the surface. µ = 347mPa·s. This is an early image sequence and the exact conditionswere not recorded. But this state is easily reproduced with amoderate fall height H ≈ 3 cm, a low horizontal bath velocityVbath ¡ 1.0 cm/s, a flow rate large enough to entrain air, butsmall enough so that no bubbles break off the edge of the airskirt Q ≈ 0.4 cm3/s. . . . . . . . . . . . . . . . . . . . . . . . 80

4.19 Non-bouncing conditions, jet breakup. H = 3 cm, Q = 0.06cm3/s, µ = 50.8 mPa·s. Each image is 40 ms apart. The hori-zontal bath velocity (Vbath = 2.0 cm/s) plays a minimal role inthe jet break-up; however, it does carry to the right the non-coalesced drops which are sitting on the surface. . . . . . . . . 81

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4.20 A drop sitting in the trough of a bouncing jet and a bouncingjet with the same parameters without a drop sitting there. Doesthis count as another multiply-stable state? H = 3 cm, Q =0.26 cm3/s, µ = 50.8 mPa·s, Vbath = 4.3 cm/s. The text on eachimage is information about the setting of the camera and theexposure. It is displayed by the camera’s control module. Tokeep the text legible, the liquid bath moves to the left, which isthe opposite of this thesis’s convention. . . . . . . . . . . . . . 82

4.21 A time sequence of a jet intermittently bouncing and producingdrops that rest on the surface for some time before coalescing.H = 3 cm, Q = 0.26 cm3/s, µ = 50.8 mPa·s, Vbath = 4.3 cm/s.The time between images is 56 milliseconds. The bath movesright to left, see the caption of Fig. 4.20 for details. . . . . . . 83

4.22 a) A drop of liquid on the surface following behind the liquidstream. H = 3 cm, Q = 0.26 cm3/s, µ = 50.8 mPa·s, Vbath =4.3 cm/s. b) A bubble below the surface following behind theliquid stream. H = 4.6 cm, Q = 0.24 cm3/s, µ = 347 mPa·s,Vbath = 4.0 cm/s. . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.23 Three antibubbles are formed by an air-entraining jet. In b)the jet is temporarily interrupted by the control rod. This per-turbation starts the jet entraining. The entrained cylindricalsheet of air then closes off to form the antibubbles. The whitearrow in (j) points to the first antibubble formed. Note thatthe antibubbles rise very slowly. H = 3 cm, Q = 0.44 cm3/s, µ= 50.8 mPa·s, Vbath = 5.9 cm/s. The bath moves right to left,see the caption of Fig. 4.20 for details. Two antibubbles slowlypass behind the plane of focus in all images. . . . . . . . . . . 86

4.24 A jet pulling up an antibubble to the surface. The antibub-ble approaches from the left, as the bath moves left to right.Upon reaching the surface, the antibubble simply becomes anon-coalesced drop floating on the bath’s surface. The timebetween each image is 64 milliseconds. H = 3 cm, Q = 0.44cm3/s, µ = 50.8 mPa·s, Vbath = 6.2 cm/s. The bath moves rightto left, see the caption of Fig. 4.20 for details. . . . . . . . . . 87

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Chapter 1

Introduction

1.1 The Bouncing Jet

It’s Sunday afternoon and you’re frying shrimp. The frying pan has

some oil in it, but there’s not enough. You pour a little oil from the bottle,

and you pour it carefully, so as not to splash yourself. You tilt the bottle and

pour a thin stream of oil while moving it in a circular motion around the pan.

To your surprise, the oil bounces off the surface of oil already in the pan! The

stream of oil didn’t merge smoothly with the oil in the pan, as you expected

it too. Instead, the incoming oil rebounded off the surface (see Figure 1.1).

How does this happen? Read on; this thesis investigates this “bouncing jet”

phenomenon.

If examined carefully, a thin jet of fluid impinging on a bath of the same

fluid can behave in many different ways. The jet can coil up like a rope. It

can float on the surface or plunge air into the fluid bath. This thesis examines

a jet which rebounds off a bath of the same fluid.

The behavior of two fluids when they are brought together is industri-

ally very important. Chemical agents are often mixed by plunging a jet of the

1

Figure 1.1: The Bouncing Jet. The conditions are as follows: velocity ofthe bath Vbath=16 cm/s, flow rate Q = 0.35 cm3/s, falling height H = 5 cm,dynamic viscosity of the silicone oil µ = 103 mPa·s ( 100 times that of water).The distance from the initial impact to the last merging is 3.7 cm. The distancethat the jet bounces in the first, larger arc is 1.5 cm.

solution into the solution bath [5] 1 . This plunging jet entrains air, which can

be crucial or devastating depending on the intended reaction. Dam spillways

and chutes can be designed to incorporate air into the water or to exclude

it. The presence of air in the flow decreases the shear stress, increases the

flow momentum, prevents cavitation, and increases the oxygen and nitrogen

content of the water [6]. The behavior of a solid object when it impacts a fluid

surface is also practical. In World War II, the British developed a dam-busting

bomb that skipped over the water, not unlike a skipping stone [68]; conversely,

torpedoes are designed to penetrate the water immediately and without much

loss of velocity. As another example, consider the cooling of metal with oil.

1The references are organized by subject in the bibliography, and thus do not appear inorder in the text.

2

If a jet of oil is sprayed on a surface to cool it, any unintentional deflection

or bouncing of the jet could be detrimental to the process (e.g. heat damage

or an explosion). Understanding the bouncing jet can only benefit the design

and operation of these various fluid applications.

Since the early 1900s, the scientific research of fluid jets has acceler-

ated with the importance of fluid handling in industrial and manufacturing

processes; however, as far as the author knows, the bouncing jet phenomenon

presented in this thesis has not been previously studied systematically. We

have investigated this phenomenon by mapping the range of conditions suffi-

cient for a fluid jet to bounce. To accomplish this, an experimental apparatus

to stabilize a bouncing jet has been designed and built.

1.2 Overview of each chapter

This chapter establishes the broad context of the work and motivates

the experiments. Also, instructions are given below for how the reader might

easily reproduce the bouncing jet using common household items. Chapter

Two introduces the relevant previous work. Chapter Three details the ex-

perimental apparatus, methodology, and calibration. The exact question of

interest is introduced and the approach to this question is outlined. Chapter

Four explains the observations and results of the experiments. Because a jet

impinging on a bath is an exceedingly rich and complex situation, Chapter

Four also gives examples of interesting observations that future research could

elucidate. Chapter Five summarizes the findings, discusses their implications,

3

and indicated a few potential applications.

1.3 How to bounce a jet at home

No expensive equipment is needed to produce a bouncing jet. The phe-

nomenon is ubiquitous in viscous oils, as is shown in the subsequent chapters.

Items readily available in the home can recreate the experiments found in this

thesis. 2 The equipment suitable for a bouncing jet are:

1. Variable speed record turntable (33 and 45 rpm) or a rotating platform

you turn manually (such as a Lazy Susan).

2. A dish, at least 10 cm in diameter and at least 4 cm tall. For better

visibility, clear plastic or glass is preferable. A pie pan works.

3. A beaker or empty can - used to pour liquid. A hole can also be punched

in the bottom of the can to make a fluid jet.

4. A small rod - used to perturb the jet and initiate a bounce.

To reproduce the bouncing jet, first secure the dish on the axis of the

turntable (see Fig. 1.2). Fill the dish with an appropriate depth of fluid,

usually about 3-5 cm. Begin rotating the turntable. The speed of rotation is

about one rotation per 1 or 2 seconds. If no turntable is available, the cup

2In an educational outreach, a web site was created for the public which covers the in-formation provided in this section. See http://chaos.ph.utexas.edu/research/fluids/bouncing_jet.html for movies and this thesis’s associated web site.

4

Figure 1.2: The fluid bath can be rotated on either a record player or a manualturntable. The diameter of the dish on the record player is 19 cm and thediameter of the pan on the manual turntable is 24 cm.

Figure 1.3: Pour the liquid from a few centimeters above the surface. Thefluid here is laundry detergent. The pouring height is three centimeters. Thebath is moving left to right. See Table 1.1 for more conditions for bouncing.

5

Figure 1.4: With the right conditions, the jet will bounce. The jet in thisfigure bounces twice. In the image to the left, the black control rod has a dropof liquid hanging from its end. Because of the poor contrast, a sketch of thesituation is provided to the right. The jet falls straight down, then bouncesoff the surface twice. In the image to the right, the black line is in place of thejet.

can be moved circularly around the pan. Pour a thin stream of fluid into the

pan from 2 to 15 cm above the surface (see Fig. 1.3). Watch for the jet to

bounce while you are pouring (see Fig. 1.4). To encourage the jet to bounce,

intermittently interrupt the jet by passing the rod through the jet or deflect

the jet by touching it with the rod. The thin, black control rod has a drop of

fluid hanging from it in Fig. 1.4. See Figure 4.13 for a demonstration of the

control rod. Also, vary the velocity of the bath with respect to the stream by

changing the spin rate of the turntable and the distance of the jet from the

center of the pan. Vary the pouring height. If the surface is dirty, stirring

the fluid will replace the surface with clean fluid. If the surface is covered in

bubbles, blowing air on the surface will pop the bubbles or move them to the

6

edge of the container. If the liquid stream is bouncing, moving the stream

closer to the center of the container, where the horizontal velocity of the bath

is lower, will increase the angle of the bounce. Table 1.1 lists the conditions

for a bouncing jet in different types of fluids. To achieve a very small pouring

height, hold the small rod close to the surface and pour the liquid down the

small rod. With practice, the jet can bounce stably for tens of seconds at a

time.

1.4 Units of viscosity

Viscosity is a measure of a fluid’s resistance to deform under shear

stress[25]. There are two types of viscosity, dynamic and kinematic. Viscosity

is stated in either cgs or SI units, depending on the conventions of a particular

field. With few exceptions, dynamic viscosity is used in this thesis and it is

given in SI units of mPa·s.

Dynamic viscosity µ has cgs units of Poise (P), which are dyne/cm2

= g cm−1 s−1. The SI units of dynamic viscosity are Pa·s. One Poise is 0.1

Pa·s. The centiPoise (cP) is a common unit, and it is equivalent to 0.01 Poise

or 1 mPa·s. The dynamic viscosity of water is 1.002 mPa·s at 20 C [69]. In

summary,

1 Poise (P) = 100 cP = 1dyne

cm2 = 1g

cm·s = 0.1 Pa·s = 100 mPa·s. (1.1)

Kinematic viscosity ν has cgs units of Stokes (S or St), which are cm2/s.

The SI units of kinematic viscosity are 10−4 m2/s. One centiStokes (cS or cSt)

7

is equivalent to 0.01 cm2/s. Manufacturers often list silicone oils with units

of cS. The kinematic viscosity of water is 0.01004 cm2/s at 20 C [69]. In

summary,

1 Stoke (S) = 100 cS = 1cm2

s= 10−4 m2

s. (1.2)

The two viscosities are related by

ν = µ/ρ. (1.3)

For all the fluids used in this thesis, the values of kinematic viscosity in cS

and dynamic viscosities in cP happen to be similar, because all of the fluids

used in this thesis have a density close to 1 g/cm3.

Both kinematic and dynamic viscosities are used in this thesis. If the

word “viscosity” is used without an adjective, dynamic viscosity is implied.

In the instance that kinematic viscosity is used, the adjective “kinematic” is

present.

8

Table 1.1: Fluids that bounce.

Fluid Measured Height Flow Bath DepthDynamic Rate VelocityViscositymPa·s cm cm3/s cm/s cm

silicone oil 9.6±0.1a 1 0.013 12 350.8±1 0.8 0.8 21 4103±2 3 0.5 24 4216±4 3 0.5 21 4347±4 3 0.5 7 4515±10 3 0.5 3 4

soybean oilb 38.3±0.5 1.5 0.42 17 2.5corn oilb 42±0.5 3 1 14 3canola oilc 64±1 3 0.9 14 410W-40 motor oild 250±4 5 0.5 17 3dish soape 440±10 6.5 0.1 26 280W-90 motor oilf 450±10 8 0.3 14 2laundry detergentg 500±10 5-12 0.67 11 3shampoo h >2000 20 0.2 38 1.5

The errors on the viscosity measurements were calculated as 1% of full scaleon the viscometer[11]. Otherwise, the values given for height, flow rate, bathvelocity, and depth are approximate values to guide the reader’s reproductionof the bouncing jet.a The 9.6 mPa· silicone oil had a very short trailing jet only and very littlebounce, if any.b Fiesta store brandc Capullo de Mazola brandd Mobil Clean 5000 brand (available at Wal-Mart)e Sunlight brandf Super Tech brand gear lubricant GL-5 (available at Wal-Mart)g Seventh Generation laundry detergenth Grisi brand Aloe Vera

9

Chapter 2

Background

No previously published study has examined a liquid jet bouncing off

a moving surface; however, many previous studies of jets can inform an un-

derstanding of this particular phenomenon. Viscous jets behave very differ-

ently depending on numerous conditions. Understanding these behaviors con-

tributes to a comprehensive account of the bouncing jet. To prepare the reader

for the discussion given in the Results section, the following chapter outlines

the topics in fluid dynamics that are pertinent. It should be noted that all of

the different behaviors discussed here occur in our experiments.

Figure 2.1 is an image of a typical bouncing jet viewed from the side.

The sections in this chapter follow a fluid parcel sequentially through (1) the

jet’s initial stages of falling, (2) its separation from the bulk fluid by a thin

layer of air, (3) its bending upwards, and (4) its bounce and subsequent flight

(see Fig. 2.2).

2.1 Fluid mechanics of falling jets

The first feature discussed is a jet falling under gravity. To understand

the condition of the jet prior to hitting the surface, two questions must be

10

Figure 2.1: A typical bouncing jet. The numbers refer to which section dealswith that particular aspect. 1) A falling jet 2) A jet impinging on a surface3) A jet bending 4) A bouncing jet. The dynamic viscosity is 347 mPa·s, thefalling height is 4 cm, the flow rate is 0.34 cm3/s, the horizontal bath velocityis 3.3 cm/s.

addressed. First, how does the stream accelerate? This question is answered

by the first two subsections. Second, does the stream remain intact or break

into drops while it is falling? This question is answered by the second pair of

subsections. This section on falling jets closely follows Anno’s The Mechanics

of Liquid Jets [1].

2.1.1 Inviscid inertial jet

The acceleration of a stream under gravity is analyzed first for the

inviscid falling jet and then for the viscous falling jet.

11

Consider an incompressible fluid jet falling straight down under gravity

g, where viscosity and surface tension are zero. Volume is conserved for an

incompressible fluid, so the volume per unit time falling is

Q = πr2v(z) = πr20v0, (2.1)

where v(z) is the jet’s vertical velocity, z is the vertical coordinate, and r0

and v0 are the jet’s initial radius and initial vertical velocity, respectively. The

Navier-Stokes equations reduce to

v(z)dv(z)

dz= g, (2.2)

where g is gravitational acceleration. Using the initial condition v(z = 0) = v0,

the jet’s radius r is expected to decrease as it falls according to:

r(z) =r0

4

√2gzv20

+ 1, (2.3)

which gives the appropriate limits when z goes to 0 and infinity [1]. This flow

corresponds to a vertical velocity like that of a free falling body:

v =√

2gz + v20. (2.4)

2.1.2 Viscous inertial jet

Now consider an incompressible fluid jet falling under gravity g with

dynamic viscosity µ and surface tension γ, where the air surrounding the jet

is ignored. It is assumed that the jet’s velocity v is only a function of height

12

Figure 2.2: A viscous jet accelerated by gravity. Left: The white teflon nozzlewith diameter 0.52 cm can be seen at the top of the image. The silicone oil’sdynamic viscosity is 347 mPa·s. Left: The measured shape of the jet comparedto predictions of a falling viscous jet and of a solid body falling under gravity.The error of the diameter measurements is typically one pixel (0.002 cm or4% of the jet diameter in the worst case). The fluid parameters used in thiscalculation are: dynamic viscosity 100 mPa·s, surface tension 20 mN/m, anddensity 0.968 g/cm3.

z. The dimensionless velocity W as a function of the dimensionless height ξ

is described by the differential equation [1],

∂2W

∂ξ2=

(W +

1

W

∂W

∂ξ− λ

W

)∂W

∂ξ− 1, (2.5)

where

v =

(3µg

ρ

)1/3

W, z =

(3µg

ρ

)1/31

gξ, and λ =

1

2

(γ

ρr0

)(3µg

ρ

)1/31√W0

,

(2.6)

where v is the jet’s velocity, ρ is the fluid density, and W0 is the initial value of

dimensionless velocity. Surface tension and viscosity both slow the acceleration

of the falling jet; surface tension resists the decreasing radius of curvature

13

around the jet’s circular cross-section and viscosity resists the stretching along

the vertical axis.

With the velocity as a function of height v(z) known, the velocity can

be calculated using Q = πr2v. It is clear how the jet diameter depends on

the fall height, so attention can be turned to the stability of the jet. This

chronology is the standard analysis protocol in fluid dynamics: once the flow

is computed, its stability must be calculated to test whether or not it will

actually be seen in the laboratory.

2.1.3 Rayleigh instability for an inviscid jet

Figure 2.3: The Rayleigh instability of a liquid jet. The jet can be seenpinching off into separate drops. Time increases up the page. The picturesabove are an artist’s rendition of the break-up of a jet. The frame is co-movingwith the jet. Taken from Powers and Goldstein[52].

An inviscid, incompressible cylindrical jet with surface tension γ is

perturbed away from a perfect cylinder. Consider a cylindrical jet that has a

14

sinusoidal disturbance of the form

r(z) = a + α(t)cos(kz), (2.7)

where a is the undisturbed jet’s radius, α(t) is the time-dependent amplitude

of the radial perturbation, k is the disturbance wave number, and z is the

axial dimension. The wavelength λ of the perturbation is λ = 2π/k. The

perturbation grows in time exponentially like α(t) = α0eωt, where ω is the

rate of growth. Rayleigh’s famous result for the growth rate of the instability

ω is

ω2 =γ

ρa3

(η(1− η2)

I1(η)

I0(η)

), (2.8)

where ρ is the fluid density, I0(η) and I1(η) are modified Bessel functions of the

first kind, and η = ka is the ratio of jet’s radius to the perturbation wavelength.

The jet is unstable for η < 1 with the most unstable wave number at η = 0.696.

Figure 2.3 is an artist’s rendition of the Rayleigh instability propagating along

a liquid stream. For η < 1, I1(η)I0(η)

≈ η2

and the growth rate becomes

ω2 ≈ γ

2ρa3η2(1− η2). (2.9)

Thus, the inviscid jet is unstable and the most unstable wavelength is

known. Since the fluid used in the experiments is viscous (ranging from 9.8 to

515 mPa·s), the effect of viscosity must be addressed.

15

2.1.4 Rayleigh instability for a viscous jet

For an incompressible viscous fluid, the growth rate ω is modified to

be:

ω2 +3µk2ω

ρ− γ

2ρa3(1− η2)(η)2 = 0. (2.10)

Viscosity damps the instability by making it more difficult for the jet to break

into drops. When the jet is very viscous, meaning (3µk2/2ρ)2 À (γ/2ρa3), the

growth rate simplifies to

ω ≈ γ

6µa(1− η2). (2.11)

Now the smaller wave numbers are more unstable and the most unstable wave

number is zero. Hence, the longer the wavelength of perturbation, the more

unstable the jet is. For the falling liquid jet to remain intact then, the initial

perturbation α0 and time t must be small enough such that α(t) << a for all

the time that the jet exists.

2.2 Thin layers of air

At this point in the literature review, the fluid parcel has fallen to the

surface. The next feature considered is the thin layer of air that separates the

jet from the bulk fluid as it is bouncing.

When a body impacts another body, they can either merge or bounce

off each other. When a solid hits a solid, either one can happen, depending

on the type of collision (elastic or inelastic). When a liquid hits a liquid, most

often the fluids merge. However, this is not the case with the bouncing jet or

16

with liquid drops floating on the surface. The first subsection examines the

case where the jet becomes part of the bulk liquid. The second subsection

deals with the case where the jet remains a separate entity after collision with

the bulk liquid. The third subsection mentions two curious configurations

involving thin layers of fluid and air: the double bubble and the antibubble.

2.2.1 Air entrainment

A jet incident on a fluid surface is called an impinging jet. The imping-

ing jet is the central exemplar of flows that entrain air into a fluid. At low

velocities, the jet produces a small dip in the surface where the jet merges with

the bath. The dip is caused by the balance of viscous forces pulling downward

and surface tension pulling upward. Above a critical threshold velocity, the

jet breaks through the surface spontaneously and entrains a thin cylindrical

tube of air, because the small tip radius fractures. This is called the plunging

jet. A thin layer of air entrained along with the jet [23]. See Figure 2.4.

The depth D that the air skirt reaches is D ≈√

µVc

ρg, where µ is the

dynamic viscosity of the fluid, Vc is the critical velocity for spontaneous air

entrainment, g is gravitational acceleration, and ρ is the fluid’s density [23].

This depth is determined by a balance of viscous and gravitational forces [24].

To determine the threshold velocity Vc for air entrainment, a balance

must be found between the viscous drag Fd (a force per unit length) and the

force Fγ of surface tension γ (also a force per unit length). Jeong and Mof-

fatt solved the low Reynolds number Navier-Stokes equations for two counter-

17

Figure 2.4: The impinging jet. a) No air entrainment at subcritical velocities.b) Air entrainment at supercritical velocities. c) A sequence of images showingthe transition between subcritical (upper left image) and supercritical veloci-ties (all other images) for air entrainment. The lower right image is the steadytrumpet-shaped air skirt. The fluid is silicone oil of dynamic viscosity µ = 970mPa·s. The jet diameter is 1.5 mm. The images are ordered left to right byrow. The time between images is 130 ms. Taken from Lorenceau, Quere, andEggers [23].

rotating vortices underneath a fluid surface [29]. Above a critical angular

velocity, the surface dips down, forms a cusp, and draws air into the fluid

very similarly to the plunging jet. They found that that the radius of curva-

ture of the air skirt’s edge decreases exponentially with the capillary number

Ca = µV/γ [29],

rc ≈ d exp(−Ca), (2.12)

where d is an external length, and rc is the radius of curvature at the edge of

the air skirt. Jeong and Moffatt [29] described a physical argument proposed

by John Hinch. Hinch presented the argument after Jeong and Moffatt gave a

18

seminar at the University of Cambridge’s Department of Applied Mathematics

and Theoretical Physics. Hinch showed that balancing the viscous drag and

the surface tension yields the same exponential dependence.

But the surrounding fluid (i.e. air) should be taken into account. Using

Eggers’s result that rc should scale as (µ0/µ)4/3 due to the non-zero viscosity

of the surrounding fluid µ0 [21], Lorenceau et al. argue that the threshold

velocity for entrainment occurs at [23]

Vc ≈ γ

µln(

µ

µ0

). (2.13)

This relationship indicates that the viscosity of the surrounding fluid µ0 (i.e.

the air around the bouncing jet) plays an important role in setting Vc.

A similar entrainment situation occurs when fluid is withdrawn from a

tube that is close to a free surface, such as draining a bathtub. In this case,

the suction produces a small dip in the free surface directly above the intake

tube. As the withdrawal rate is increased, eventually the dip breaks and a

very thin stream of the upper fluid is entrained along with the lower fluid [8].

Air entraining flows occur in various situations. The plunging jet (from

Eggers [21]) and the two counter-rotating vortices below a fluid surface (from

Jeong and Moffatt [29]) have been discussed. Additionally, an applied over-

pressure on the surface (i.e. blowing a jet of air on the surface) will generate

entrainment [48]. Air is entrained when a solid body hits a fluid surface with

a high velocity (e.g. tucking into a cannonball and jumping into the pool)

[47]. Entrainment effects are enhanced by perturbations on an otherwise non-

19

entraining, uniform plunging jet and by perturbations on the bath’s surface

[46, 49]. A very small, sudden change in the shape of the jet can create a cavity

around the jet when the non-uniformity hits the liquid surface. The collapse

of this cavity can entrain air.

Air entrainment is extremely important in many industrial processes.

In some applications, this effect is beneficial; in others, it is detrimental. The

plunging liquid jet reactor allows the rate of certain reactions to be enhanced

(by causing turbulent mixing or by increasing the contact area between a liquid

and a reacting gas) [5]. The diffusion and aeration of liquids are also enhanced

by entrainment (e.g. oxygenation of water) [41]. However, coating techniques

are often limited by air entrainment [60]. Above a critical speed, imperfections

and undesired bubbles are introduced into the coating. Entrainment during

the filling of casts and molds not only produces structural and cosmetic flaws,

but it can also deteriorate the entraining fluid itself (e.g. bubbles in molten

glass or metal) [23, 41]. There is also great industrial interest in the break-

up of drops when they impact a fluid surface (e.g. metal drops falling on

liquid slag) [33–35]. For an extensive review of the engineering literature on

air entrainment, the reader is directed to Bin’s article [3].

It has been seen that a thin layer of air can be entrained into the bath

when an impinging jet strikes the surface at a high enough velocity. This

entrainment happens as the jet is merging with the bath. The air separates

the jet’s fluid partially from the bath’s fluid for a short distance. However, a

thin film of air can play a different role; it can separate the jet’s fluid entirely

20

from the bath, as is the case in non-coalescence.

2.2.2 Non-coalescence

Drops can often rest on a surface for several seconds before merging

into the bath. When there is a delay between initial contact and the two fluids

becoming a single body of fluid, this is called non-coalescence. In some cases,

non-coalescence means that the two fluids never merge. The topic of non-

coalescence has recently gained interest due to extensive computer simulations

and theoretical advancements [20, 22, 44]. However, floating drops have been

observed for over a hundred years. In 1881, Osborne Reynolds presented his

paper “On the Floating of Drops on the Surface of Water Depending Only on

the Purity of the Surface [56].” Water drops floating on a bath of water were

also studied by John Tyndall in 1885 [66].

The key element for non-coalescence is the long time required to drain a

very thin layer of air between the drop and the bath. This time is long because

the gap is very narrow (often ∼ 1 micron) and the interstitial fluid (often air)

has a non-zero viscosity. Lubrication theory is certainly applicable in these

circumstances [14]. Even though the viscosity of air is very small compared to

that of water, it has a large effect when the air must be squeezed out of a long,

thin gap. In many cases, the thin layer of air does not breakup under surface

tension and the air below the drop must slowly drain from a small exit at the

surface. The stability of thin films of air was studied by Lezzi and Prosperetti

[36].

21

Figure 2.5: Picture of a floating drop with a cartoon of the drop’s cross-section.The layer of air is on the order of microns thick and it takes seconds or minutesto drain. Taken from Couder et al. [13]. Though the exact conditions of thedrop above is not given in [13], typical values are as follows: the radius of thedrop ≈ 1.2 cm, the air film thickness ≈ 1.5 µm, dynamic viscosity µ ≈ 500mPa·s, the draining time was up to half an hour, and surface tension ≈ 21mN/m.

The time of non-coalescence can be extended by several different meth-

ods, all of which either draw more air under the drop or slow the draining of

the air. Additional air can be drawn under the drop by Marangoni convection

(because of a thermal gradient between the drop and the bath) [15, 16, 58], the

vertical oscillation of the bath [13, 66], or a shear between the drop and the

bath [15, 16]. The time needed for draining is extended when in microgravity

[45] and when air is replaces with a more viscous fluid.

Among its many applications, non-coalescence is especially important

in the study of suspensions of droplets, such as emulsions, fogs, and mists,

as Jearl Walker has noted [66]. The thin layer of air between a drop and a

22

solid surface also determines the amount of splashing that a drop produces

when it hits the surface with a high velocity [70]. The splashing is drastically

decreased when the pressure of the air is below a critical value.

So far in this section, a thin layer of air has been entrained into the

fluid bath, and it has allowed the jet’s fluid to remain apart from the bath for

a considerable amount of time. The last two configurations of thin air layers

are the double bubble and the antibubble.

2.2.3 Double bubbles and antibubbles

A double bubble consists of a bubble of liquid A inside a bubble of liquid

B surrounded by a third liquid C (Fig. 2.6a). For example, a water drop nested

in an oil drop nested in a water drop. Using microfluidic devices, Utada et

al. created double bubbles (also called double emulsions). This technique was

modified to create thin spherical shells by cross-linking an adhesive polymer

solution in UV light. Thus, by utilizing stable thin sheets of fluid inside

another fluid, they made hollow microspheres with potential applications in

drug delivery [63].

An antibubble is a spherical thin sheet of air. It can be thought of as

an air bubble which is filled with a bubble of fluid. Antibubbles are normally

difficult to make reliably, but once formed can be stable for many seconds (Fig.

2.6b) [18].

We have thus reviewed the different roles that thin layers of air can

take: entrainment of air via an impinging jet, non-coalescence of two fluid

23

Figure 2.6: Double bubble and antibubble. a) A double emulsion. The exactparameters for this double bubble are not given in [63]. For other doubleemulsions, the outer fluid is silicone oil with dynamic viscosity µ = 480 mPa·s,the middle fluid is a glycerol-water mixture with µ = 50 mPa·s, and the innerfluid is silicone oil with µ = 50 mPa·s. the interfacial tension is ≈ 20 mN/m.b) A sequence of images showing a large antibubble in water (about 1 cm indiameter) burst. After bursting, the colored water of the inner bubble sinks.The small volume of air from the antibubble’s thin shell rises upward. Theimages are separated by 0.1 s. Taken from Utada et al. [63] and Dorbolo etal. [18].

bodies, the double bubble, and the antibubble.

2.3 Bending and rebound of fluid jets

At this point in our sequence, the fluid jet is separated from the bulk

fluid by a thin layer of air and is being deflected upwards. This section deals

with some different mechanisms of changing the jet’s direction of motion upon

contact with a fluid surface. Situations are listed in which liquid streams are

deflected, bent, or otherwise accelerated. First, the buckling instability called

the liquid rope-coil effect is discussed. The next section describes the deflection

of a jet due to pressure and viscosity called the Coanda effect. Then the largely

unexplained, transient bouncing of a non-Newtonian liquid jet called the Kaye

24

effect is covered. Finally, for the sake of comparison and analogy, the last

section gives an overview of the physical mechanism of skipping a stone on a

water surface.

2.3.1 Fluid rope

The sight of poured honey coiling on piece of toast is familiar to most.

The coiling of a liquid is similar to the coiling of an elastic rope on, say, the

deck of a ship [37], but is in fact more complicated since the liquid stream can

twist, stretch, and bend (Fig. 2.7) [40].

Figure 2.7: The liquid-rope coiling effect. (a) Coiling of honey. Falling heightH = 3.4 cm and kinematic viscosity ν = 60 cm2/s. (b) Coiling of siliconeoil. Radius of rope = 0.034 cm, kinematic viscosity ν = 1000 cm2/s, fallingheight H = 0.36 cm, and flow rate Q=0.0044 cm3/s. (c) Coiling of siliconeoil. Radius of the rope in the image = 0.04 cm, kinematic viscosity ν = 125cm2/s, falling height H = 10 cm, and flow rate Q=0.213 cm3/s. Taken fromMaleki et al.[40].

25

Liquid coiling was first studied by Barnes and Woodcock in 1958 [2]. In

1959, G. I. Taylor proposed a mechanism for the coiling. As the stream falls

toward the surface, it accelerates under gravity. As it accelerates, the stream

becomes narrower. However, as the stream touches the surface, it slows down

and the jet’s diameter increases (Fig. 2.8). This stress in the viscous fluid is

communicated upstream. If the stream is arriving at the surface more quickly

than it can be smoothly absorbed, the stress builds up and the stream buckles

at its narrowest point. This buckling moves the jet to one side and this begins

the coiling [61]. Sheets of liquid can also fold with regularity [57]. For a good

overview of the liquid rope-coils, see Jearl Walker’s “Amateur Scientist” article

in Scientific American [67].

Figure 2.8: A liquid stream widening as it merges into a liquid bath withoutentrainment. The bath’s meniscus on the edge of the tank blurs the bottomof the image. The actual liquid surface level is slightly above the bottom ofthe image. The fluid in this image has a dynamic viscosity of 347 mPa·s, aflow rate of Q = 0.16 cm3/s, a falling height of H = 4.23 cm. The liquid bathis stationary, Vbath = 0 cm/s.

26

There are three regimes in liquid rope coiling. Each regime involves an

interplay between viscous, gravitational, and inertial forces [40]. Viscous coil-

ing occurs when gravitational and inertial forces are negligible. The frequency

of viscous coiling Ωv is

Ων =Q

Ha2, (2.14)

where Q is the flow rate, H is the height from which the fluid is poured, and a

is the radius of the liquid rope in the coil. Gravitational coiling occurs when a

balance between viscous forces and gravitational forces exists, and the liquid’s

inertia is negligible. The frequency of gravitational coiling Ωg is

Ωg =

(gQ3

νa8

)1/4

, (2.15)

where g is the acceleration due to gravity and ν is the kinematic viscosity.

Finally, inertial coiling occurs when a balance between the liquid stream’s

inertial forces are balanced by viscous forces (in the form of shear along the

coil), and gravity is negligible. The frequency of inertial coiling Ωi is

Ωi =

(Q4

νa10

)1/3

. (2.16)

This last regime is described by Mahadevan, Ryu and Samuel [38, 39].

A jet can be bent upon impact with a fluid surface by a buckling in-

stability called the liquid rope-coiling effect. The bouncing jet could possibly

be buckling at its point of first contact with the fluid bath, although it is

undetermined in which of the three regimes it buckling.

27

2.3.2 Coanda effect

A burning candle creates a heated, buoyant plume of air rising from

its flame. If a wall is near, this plume of air will bend toward the wall and

hug the wall as it rises [19, 50] (see Figure 2.9). If you pour water from a

tall straight glass, the water will bend at the rim and follow the outside edge

of the water before falling off into a stream [55]. This often makes a mess

and is why beakers have indented lips and why teapots have spouts. These

two situations are examples of the Coanda effect. A third, more pertinent

example of the effect is that of a horizontal jet of water. If the horizontal jet

is injected below the fluid surface, it will be attracted to the fluid surface or

to the bottom of the tank [30, 59]. This is sketched in Figure 2.9. This kind of

situation occurs frequently in waste pipes that drain into a river. The ejected

fluid forms a neutrally buoyant, swiftly moving, horizontal jet, which either

rises to the surface and forms a surface boil or sinks to the river bed and hugs

the bottom.

The Romanian engineer Henri Coanda observed this effect when trying

to deflect hot exhaust fumes away from the fuselage of his primitive jet aircraft

in 1910, predating other devices by thirty years[55]. While testing the plane in

a field one day, Coanda did not notice that he was approaching the Paris walls

very quickly. At the last moment, he pulled back on the controls. The plane

flew over the walls and crashed. Coanda was not seriously injured. This was

the first recorded flight of a jet aircraft. The exhaust plume was pointed away

from the fuselage. However, Coanda noticed when the plane was in motion,

28

Figure 2.9: The Coanda effect. a) A laminar thermal plume is attracted to asolid wall 0.71 cm away. Taken from Pera and Gebhart [50]. b) A submerged,horizontal jet is attracted to the top or bottom surface of a channel. Takenfrom Shimada et al.[59].

instead of the exhaust being ejected away from the body of the aircraft, the

exhaust was sucked toward it[55].

The Coanda effect bends a liquid jet because either the jet is attached

to the wall or a relative vacuum attracts the jet to the wall. The pressure is

lower near the wall, which entrains the surrounding fluid into the jet. This

entrained fluid and the jet are restricted to the wall by a viscous boundary

layer. So rather than separate, the jet follows the contour of the nearby surface.

The Coanda effect is how a beach ball stays suspended in the air by a

household fan in department stores across America. It is also why a stream of

water touched by a fingertip is deflected to one side [25]; the water jet follows

the outer surface of the fingertip, even though the curvature is convex. The

Coanda effect is used to increase the lift/drag ratio of airfoils and hydrofoils.

29

It can also increase the efficiency of combustion in a flame [55].

Along with the liquid rope-coiling effect, the Coanda effect could be the

mechanism that starts a jet bouncing. This, too, will have to be determined.

The next way a jet can bend upwards for a rebound is called the Kaye effect.

2.3.3 Kaye effect: a transiently bouncing, non-Newtonian jet

The Kaye effect is a bouncing liquid stream, which is visually very sim-

ilar to the bouncing jet presented in this thesis [32]. The Kaye effect occurs

in non-Newtonian fluids, such as a solution of polyisobutylene in decalin. The

solution is poured straight down onto a stationary plate. Because the fluid is

non-Newtonian, a small mound of fluid forms on the surface where the fluid

reaches the surface faster than the surface can absorb it. Sporadically, a thin

streamer will emerge from the mound (Fig. 2.10). Because the streamer is

bouncing off the mound and not replenishing the mound, the mound’s height

decreases and the stream’s inclination increases until the streamer stops bounc-

ing altogether. Subsequently, the mound begins to be grow again and soon

enough the bouncing streamer reappears.

There exist several theories about this spurting of fluid, dealing with

the polymer and shear-thinning nature of the solution, but none have been

conclusively confirmed [9, 10, 26]. In comparison, this thesis’s bouncing jet is

present in Newtonian and non-Newtonian fluids and is stable for minutes but

must occur over a bath moving horizontally with respect to the falling jet. It

seems plausible that the bouncing mechanism is the same for both the Kaye

30

Figure 2.10: A sequence of images of the Kaye effect. The bouncing is onlytemporary and the fluid is dilute solution of the polymer polyisobutylene indekalin. Each number indicates its frame number from a 250 frame per secondcine film. Taken from Collyer and Fisher [10], permission granted by theNature Publishing Group. Although no length scale is given in [10], in Kaye’soriginal paper [32], the liquid was poured from 25 cm and the stream diameterwas ≈ 1 mm.

31

effect and the bouncing jet. The non-Newtonian nature of the Kaye effect is

only necessary to create a mound of fluid to bounce off of; the mound could

serve the same function as the moving bulk surface in the bouncing jet of this

thesis.

Another transient bounce of a viscous fluid was observed by Jearl

Walker in his study of the liquid rope-coiling effect [67]. He was describ-

ing what happens when a glob of Karo dark corn syrup breaks away from a

suspended container. The glob drags with it a thin stream of syrup which is

attached to the top of the glob. In his words:

The glob hit the bottom first and sent a shock through the thin

stream, which stretched, twisted and turned, even occasionally

bouncing from the pool.

This “bouncing” was temporary and only mentioned in passing. It is not clear

whether this is more similar to the Kaye effect or the bouncing jet. There is

also the chance that all three phenomena are different manifestations of the

same effect.

2.3.4 Stone skipping

The similarity of a rock and a fluid jet both rebounding off a surface

might suggest an analogous situation. A brief description of the physics of

skipping rocks across liquid surfaces is provided.

The mechanism that rebounds a stone off of a water surface is a reaction

32

Figure 2.11: A labeled schematic of a stone skipping. Taken from Bocquet [4].

lift force−→F · −→n due to the collision (see Figure 2.11):

−→F =

1

2ρv2Sim

(Cl−→n + Cf

−→t

), (2.17)

where Cl and Cf are lift and friction coefficients, ρ is the fluid density, v is the

stone’s velocity relative to the surface, Sim is the surface area of the immersed

part of the stone. There exists a “magic angle” of about 20 between the

surface and the stone’s bottom surface where the speed needed for a bounce

is minimized (Fig. 2.12) [43]. This formulation of the problem is by Bocquet

[4]. The spin does not appear in the reaction force. In this model, the stone’s

spin stabilizes its orientation, despite the repeated collisions that the rock has

with the surface.

Stone skipping concludes this section on the different ways a fluid jet

can be bent for rebounding. In following some fluid parcel during the bouncing

process, the parcel has now bounced successively and is entering its projectile

trajectory. The shape of this trajectory is discussed next.

33

Figure 2.12: A sequence of images showing an aluminum disc skipping offwater. The disc’s radius = 2.5 cm, thickness = 2.75 mm, translation velocityv 3.5 m/s, angular velocity = 65 rev/s, attack angle θ = 20 , trajectory angleβ = 20 . The time between images is 6.5 ms. Taken from Clanet et al. [7],reprinted with permission from the Nature Publishing Group.

2.4 Trajectory of the jet

Following the template of the early discussions, this section will come

in two parts. The first part will cover the inviscid flight of a fluid stream, and

the second part will cover the viscous flight of a fluid stream.

2.4.1 Trajectory of an inviscid jet

The trajectory of the jet after bouncing is very close to parabolic. The

most basic treatment of the problem assumes the trajectory is simply that of

34

a free-body. The maximum height of the bounce H obtained is thus

H =v2

0sin2θ

2, (2.18)

where v0 is the initial velocity and θ is the angle the velocity makes with the

horizontal. The horizontal range R is

R =v2

0sin(2θ)

g, (2.19)

and g is gravitational acceleration.

Figure 2.13: A jet moving under gravity. Taken from Vanden-Broeck andKeller [64].

Dias and Vanden-Broeck computed the shape of a viscous fluid flowing

from an angled nozzle at velocities low enough that the liquid attached itself to

35

the underside of the nozzle and dribbled down [17]. Later Vanden-Broeck and

Keller performed a sophisticated analysis in addition to a numerical calculation

in which the velocities were high enough that the jet rose and fell under gravity

in an arc reminiscent of a free-body trajectory [64]. Consider a steady, two-

dimensional, incompressible, inviscid fluid jet. The jet is ejected from a nozzle

aimed at an angle θ from the horizontal (Fig. 2.13). Ignore surface tension

and air resistance. The flow is determined only by the Froude number Fr =

vc(Qg)−1/3, where Q is the flux, g is gravitational acceleration, and vc is the

velocity at point B in Figure 2.13 (the top of the jet at the peak of its flight).

Vanden-Broeck and Keller obtained an outer solution of the jet’s trajectory,

which is good far from its crest B, and an inner solution, which is good near

B. The inner and outer solutions each have a parameterized asymptotic curve

for the upper and lower surface of the jet, ABC and DEF respectively. These

asymptotic solutions match the numerical solutions well for Fr > 1. These

solutions are for the inviscid case and the viscous case is much more difficult.

Interestingly, at Fr = 0, there is a 120 angle on the jet’s upper surface where

the jet reaches its highest point (point B in Fig. 2.13). Stokes reached the

same conclusion about the 120 angle in the case of water waves [64].

2.4.2 Trajectory of a viscous jet

The viscosity of the jet introduces great difficulty in describing the exact

shape of the arc. When calculating the shape, velocity profile, and trajectory

of a jet usually the jet falls straight down [31]. Flows with two free surfaces

36

are even more difficult to solve than flows with one free surface (e.g. pouring

or flow of a weir [65]).

Goodwin and Scholwalter [27] used a finite element method to simulate

the liquid streams oriented at an angle with respect to gravity and under the

action of surface tension, gravity and viscous. The shape and trajectory of the

viscous jet for several different inclination angles were computed. However, no

functional form or other general result was presented that could be compared

quantitatively with images of the bouncing jet.

Some theoretical and numerical work has also been done for the tra-

jectory of a liquid jet spun from a rotating orifice, but to leading order, the

steady-state solution is independent of viscosity. The next order term is de-

pendent of viscosity. Thus far there are only a few trajectories plotted for

different values of viscosity [54].

A similar problem, on which substantial progress has been made, is the

viscous catenary [62]. An initially horizontal, very viscous liquid bridge droops

under gravity and eventually breaks via pinch-off.

Our literature review is complete. The jet fluid has fallen to the surface,

has entrained a thin layer of air, has been bent upwards, has completed its

projectile flight, and has either bounced again or merged with the bulk liquid.

All of the major topics relevant to the bouncing jet have been introduced. The

specific experiments that have been conducted for this thesis are described in

Chapter Three.

37

Chapter 3

Experiment

3.1 Introduction

When fluid is poured by hand over a rotating bath, the bouncing jet

is stable for a few seconds at most. Small changes in the jet’s uniformity

destabilizes the bouncing. A more controlled apparatus is necessary for close

and systematic study of the bouncing phenomenon.

An experimental apparatus was developed to reliably produce stable

bouncing jets. This chapter describes the equipment and the methods used to

collect data. The data consist of observations and images. The compilation of

the observations and the processing of the images are outlined.

The purpose of the experiments were two-fold. First, the experiments

were designed to find the range of parameters where a bouncing jet is stable.

Second, the experiments were to allow a close inspection of the geometry and

behavior of the bouncing jet to help deduce the force balance and bouncing

mechanism.

The parameters varied in this set of experiments were the viscosity of

the fluid, the velocity and radius of the jet, and the velocity of the fluid bath.

38

3.2 Overview of apparatus

The fluid was contained on a rotating table in an acrylic tank. A Kodak

high speed CCD camera imaged the bouncing jet. The images were read out

to a computer. The fluid flow rate was controlled with a pump.

Figure 3.1: Experimental setup. An annular bath of liquid is rotated. A pumpwithdraws oil from a central catch and releases it above the surface. The outerdiameter of the tank is 40.5 cm and the fluid is 7.7 cm deep.

3.3 Details of apparatus

A clear acrylic tank held the fluid on a rotating table. The tank was

clear to allow visualization and imaging through the side of the tank while it

was rotating. The tank’s circular base plate was 0.5 cm thick and 61 cm in

39

diameter. The sides of the tank were cylindrical and acrylic with a 40.5 cm

outer diameter, 20.4 cm tall and 0.7 cm thick. The sides of the tank were

secured to this base with angle brackets and sealed with water-based caulking

(Sta-Kool 390 Elastomeric Roof Patch). Sealant should not be silicone-based,

since it will dissolve in silicone oil and cause the tank to leak. The inner

cylinder was made of aluminum and had 27.4 cm outer diameter, was 7.7 cm

tall, and 0.9 cm thick. The tank was designed such that the fluid overflowed

into the inner cylinder where it could be pumped again. This kept the distance

between the nozzle and the surface constant (see Figure 3.1).

The different fluids used in the experiment are listed in Table 3.1. Sili-

cone oil was chosen as a pure, well-characterized, readily available Newtonian

fluid that comes in a wide range of viscosities. Silicone oil is polydimethylsilox-

ane (PDMS). The longer the polymer, the more viscous the fluid. A rotary

viscometer (Cannon LV2000 Series II with the low centipoise, small sample

adapter) was used to measure the viscosities of each fluid. Viscosity and sur-

face tension and density are given at 25C [12, 53]. These quantities vary

with temperature about 0.09 mPa·s/K, 0.07 mN m−1 K−1, and 0.00105 cm3

cm−3 K−1[12, 42]. By temperature variations, the density changes less than

1% percent. The largest contribution in determining the viscosity is the error

associated with the viscometer’s measurement error of 1% of full range[11].

From measurements made by Dr. Christopher Rulison at ThetaDyne Corpo-

ration, Kruss USA, and Augustine Scientific[42], the surface tension typically

varies from the nominal value by ± 0.5 mN/m. Viscosity was measured at 24

40

C. The silicone oils are sold by kinematic viscosity ν. The rotary viscometer

measures dynamic viscosity µ. See Section 1.4 for further information and

conventions about viscosity.

Ideally, the surface tension could be varied in as a controlled manner as

the viscosity. However, most fluids have a surface tension close to that of water

(σ= 72 dyne/cm at 25 C) or close to that of silicone oil (σ = 21 dyne/cm

at 25 C)[69]. Glycerine has a surface tension of 63 dyne/cm at 20 C [69].

Surfactants can decrease the surface tension, but to what degree is difficult to

control. The bouncing phenomenon was not successfully observed in water,

glycerine, or several different mixtures of the two liquids. However, an exhaus-

tive study was not performed and these fluids could produce a stable bouncing

jet. The surface tension of a water/glycerine mixture is approximately four

times more than the surface tension of silicone oil. Thus, for the quantitative

experiments performed, only different viscosity silicone oils were used and the

surface tension was approximately constant.

Caution: Dust and surfactants effect the stability and behavior of ex-

periment. Dust and bubbles sitting on the surface will deflect the jet and often

destabilize a bouncing jet. It is best to keep the fluid clean and uncontami-

nated. Reynolds observed this in 1881 [56].

The fluid tank was rotated by a Rate of Turn Table (Model C-181

Ser. 846, Mfg. by Genisco, Los Angeles, CA). The table could rotate at

angular speeds from 3x10−5 to 3.33 Hz. When changing the rotation rate,

adequate time was given for the change to be communicated to all of the fluid,

41

Table 3.1: Silicone oils used.Nominal Measured Surface Density Source†Dynamic Kinematic TensionViscosity ViscositycS* mPa·s mN/m g/cm3

10 9.6±0.1 20.1±0.5 0.935±0.01 Dow Corning50 50.8±1 20.8 0.960 Dow Corning100 103±2 20.9 0.968 Clearco200 216±4 21.0 0.967 Clearco350 347±4 21.1 0.968 Dow Corning500 515±10‡ 21.1 0.973 Clearco

*Companies that produce silicone oils most often list the fluids in units ofcentistokes(cS), which is a measure of kinematic viscosity(1 cS = 0.01 cm2/s).† Dow Corning fluids are from the 200 series. Clearco Products is located inBensalem, PA.‡ Calculated from nominal kinematic viscosity and density, not measured. Thefluid was contaminated before a viscosity measurement was made.See Section 1.4 for a discussion of the error.

42

such that the fluid was in solid-body rotation. Only a relative velocity was

needed between the jet and the bath, so other geometries are conceivable. For

example, a rectangular channel of fluid pumped at a constant velocity could

have been built. Also, the jet could have been rotated instead of the bath.

But, as mentioned before, the bouncing and entrainment is sensitive to any

mechanical disturbances of the jet. To complicate matters, the surrounding

air would have a velocity with respect to the jet. In this case, rotation is the

easiest and best way to get a constant relative velocity between the jet and

the bath.

The fluid flow rate was controlled with a digital gear pump (Cole-

Parmer Model 75211-30, Mfg. by Barnant Co., Barrington, IL). The pump

drive was capable of rotating from 60 to 3600 revolutions per minute (rpm).

The pump drive was fitted with a gear pump head that moved 0.092 mL

per revolution (mL/rev) (Cole-Parmer Suction Shoe Model No. 73003-14).

Calibration revealed that the actual volume per revolution was 0.1046 ± 0.0004

cm3/rev. The error was estimated by using a bootstrap method (removing the

data for each viscosity separately and refitting the reduced data set) [51].

Calibration will be discussed in a later section. On the pump line was an air

reservoir that trapped air bubbles and pulsations in flow rate (Cole-Parmer

MasterFlex L/S Pulse Dampener Model No. 07596-20). The connecting tubes

were 1/4” polyethylene with 1/4” NPT, Swagelok, and push-in compression

fittings.

The height of the nozzle and the fluid surface were measured using a

43

one meter cathetometer.1 It measured the height of the bath surface and of the

nozzle to within ± 0.03 cm. The accuracy of the cathetometer was confirmed

within its precision by imaging a ruler, the nozzle, and the surface in the same

image. For more details on the calibration of the image, see Section s:cali:

Calibration.

The jet was perturbed, stopped, and deflected with a control rod. The

control rod was a piece of plastic tubing held horizontal by a vertical sup-

port which could rotate. This manipulation was often necessary to initiate a

bouncing jet.

The bouncing jet was imaged with a high-speed charged coupled device

(CCD) camera (Motion Corder Analyzer Model SR-1000, made by Eastman

Kodak, San Diego, CA). The camera’s frame rate was from 30 to 1000 frames

per second (fps) with a maximum resolution of 512 × 480 pixels. The camera

was in the lab frame, as was the jet’s nozzle. The images were read out to

a computer via a standard SCSI card or a video capture card (AVerMedia

DVDEZMaker PCI card).

3.4 Experimental methods

There are many parameters that could be varied in this experiment

including, the viscosity of the bath, the viscosity of the surrounding fluid (air

1The instrument’s make and model are not known. The only visible label on the instru-ment is “P-1551.” Its construction and history suggest the instrument was made in the 19thcentury.

44

in our experiments), the interfacial tension between the two fluids, the density

difference between the two fluids, the radius of the jet, the vertical velocity of

the jet, the horizontal velocity of the bath, and the angle of incidence of the

jet.

The parameters directly controlled and varied were the viscosity of

the fluid µ, the flow rate Q of the jet, the fall height H of the jet, and the

velocity Vbath of the fluid bath. The velocity of the bath was controlled via the

rotational velocity of the bath and maintaining the jet at a given radius. Five

cuts were taken in Q vs. Vbath parameter space, each at a different viscosity

(50.8, 103, 216, 347, 515 mPa·s) and with a constant falling height (H = 3

cm). One cut was made in H vs. Vbath at a single viscosity (103 mPa·s) and

constant flow rate (Q= 0.35 cm3/s). Changing the fall height H changes both

the radius of the jet and the velocity of the jet, according to the relation in

Equation (2.5).

For consistency, the criterion for a bouncing jet to be deemed “stable”

at a certain point in parameter space is defined as: a bouncing jet must be

initiated by perturbing the liquid jet and must last for at least 5 seconds.

The control rod which perturbed the jet was controlled by hand. Though

the control rod changed the instantaneous radius, velocity, and shape of the

jet in a complicated, time-dependent manner, the range of stable conditions

in parameter space was reproducible on different occasions and by different

experimenters.

The most common type of experiment was a parameter sweep to deter-

45

mine the extent of parameter space where the bouncing jet was stable. The

steps required to do an experiment sweeping Q vs. Vbath are as follows:

1. Measure the fluid height and adjust nozzle height to the desired value of

H.

2. Start the pump at rate Q, giving ample time for the flow rate to equili-

brate to the correct value, since the pulse dampener and tubing act as a

RC low-pass filter. One minute is sufficient.

3. Set the table spinning at rate ω, such that Vbath = ωR0, where R0 is the

distance from the center axis of the tank to the jet. Allow ample time

for the fluid to reach solid-body rotation. Depending on the viscosity of

the fluid, this time can be from 5 seconds to 30 seconds.

4. Manipulate the jet with the control rod until a judgment can be made

whether the jet bounces stably or not.

5. Record an image or movie of the experiment, if desired. Calibrate the

image by taking a picture of a ruler by the jet, if necessary. See the

Section s:cali: Calibration for more details.

6. Stop jet from bouncing by interrupting jet with the control rod.

7. Repeat from Step 3 until a sweep in ω is complete for a single value of

Q.

8. Repeat from Step 2 for a new value of Q.

46

Because of the meniscus at the edge of the tank, the jet could not be

imaged above and below the fluid surface simultaneously. Different camera

angles were necessary to view different parts of the jet. If the shape of the

interface below the surface was desired, the camera was lowered below the

fluid level and angled upwards. If the shape of the jet above the surface was

desired, the camera was raised above the fluid level and angled downwards.

These were the two primary camera angles. If an ambiguity existed in the

geometry or topology of the jet and bath, images could be taken at other

angles and positions, such as vertically from the above.

3.5 Calibration

Several instruments required calibration: the gear pump, the rotat-

ing table, and the camera images. The methods of calibration for each are

described here.

The gear pump head had a specified pump volume of 0.092 mL/rev. As

Figure 3.2 shows, the actual pump volume per revolution is 0.1046 ± 0.0004

cm3. Within uncertainty, the gear pump performs identically for all of the

viscosities that were used in these experiments.

The Rate of Turn Table is over 30 years old, but the calibration is still

good to within a percent, as Figure 3.3 demonstrates. This data was collected

by recording the time of 10 turns with a stopwatch. The majority of the

error came from the reaction time involved in starting and stopping the watch

by hand. The point at 1.8 rad/s was timed for 30 full rotations and is very

47

0 200 400 6000

20

40

60

Pump Drive Speed ωdrive

(rpm)

Mea

sure

d Fl

ow R

ate

Q (

mL

/min

)

kin. visco. ν = 1 cm2/skin. visco. ν = 2 cm2/skin. visco. ν = 3.5 cm2/skin. visco. ν = 5 cm2/sBest Fit Line 0.1046 mL/revMfg. Specs 0.092 mL/rev

Figure 3.2: Calibration of the gear pump head. The manufacturer’s specifica-tion for the gear pump head (the dashed line) is inaccurate. The correct valuefor the volume pumped per gear head revolution is 0.1046 cm3 per revolution.

48

0 1 20

1

2

Dial Indication of Rotation Speed ωdial

(rad/s)

Mea

sure

d R

otat

ion

Spee

d ω

exp (

rad/

s)

ωexp

= 0.99336 ωdial

Measured PointsBest Fit Line

Figure 3.3: Calibration of the rotating table. The calibration is accurate towithin one percent.

49

close to the expected value. The rotation rate along with the distance of the

jet from the axis of the turntable was used to calculate the horizontal bath

velocity. A typical distance was 16 cm. The measurements of the distance

were reproducible within 0.5 mm. The measurements were made by hand and

accurate to within 1 mm. So the total uncertainty was 1% for the horizontal

bath velocity.

Figure 3.4: Calibration of the image. A ruler was placed in the focal plane todetermine the number of pixels per unit length. The smallest division on theruler is 1/100th of an inch. In this image, the flow rate Q = 0.35 cm3/s, thefalling height H = 4.72 cm, the bath’s horizontal velocity Vbath = 0.0 cm/s,and the dynamic viscosity µ = 103 mPa·s.

The camera position and angle was changed often to get the best shot

50

possible. This necessitated calibrating the image frequently. A ruler was

placed in the camera’s focused depth of field at the same distance as the jet is

from the camera lens. From this image, the number of pixels per unit length

was measured. The error for measurements made in this way is approximately

one pixel.

The calibrated images were used to measure the velocity of the fluid

jet. The velocity was then computed by using the flow rate and the continuity

condition. Given a pump rate Q and a measured jet diameter d, the velocity

of the jet is simply

Vjet =4Q

πd2. (3.1)

Because the jet’s diameter as a function of height is difficult to calcu-

late, a polynomial fit to the experimental data was used over two separate

intervals. One quadratic function fit the data from 0 < H < 4.72 cm, while

a second quadratic function fit the data for values of H ≥ 4.72. The error of

the diameter measurements is typically one pixel (0.002 cm or 4% of the jet

diameter in the worst case).

There was a choice in the method of measuring the velocity of the jet.

Small tracers particles or bubbles could have been tracked in the jet at two

successive times to compute the velocity. This was difficult to realize though,

because of the relatively long exposure time of the CCD camera compared to

the fast motion of the tracers. Since the fluid is to a very good approximation

incompressible and the cross-section of the jet is approximately circular, the

velocity of the jet was measured by imaging the diameter of the fluid jet.

51

With this apparatus built and calibrated, the experimental data were

collected. The results of the experiments follow in the next chapter.

52

0 5 10 150

0.05

0.1

0.15

Jet D

iam

eter

(cm

)

0 5 10 150

100

200

Height (cm)

Jet V

eloc

ty (

cm/s

)

ExperimentalPolynomial fit #1Polynomial fit #2

Figure 3.5: Calibration of the jet diameter and velocity. The jet diameter wasmeasured by imaging it with a zoom lens (stars). The solid line is the diameterpredicted if the jet was treated as a solid body falling without resistance undergravity. The two dashed lines are piecewise polynomial fits to the experimentaldata. Polynomial fit #1 has parameters: 0.0032x2 − 0.0338x + 0.1578. Poly-nomial fit #2 has parameters: 0.0001x2 − 0.0046x + 0.0881. The transitionbetween the two polynomials occurs at 4.72 cm. The solid line with dots isthe solution of equation 2.5 with the appropriate parameters. The nozzle is0.52 cm can be seen at the top of the image. The oil’s dynamic viscosity is103 mPa·s.

53

Chapter 4

Results

The results of the experiments are presented here. The impinging jet

behaves in many different ways when it hits the surface, depending on the

exact conditions. The first section outlines the various observed behaviors of

the jet. The second section gives the range of parameters where the bouncing

jet is stable. The third section describes the different methods of initiating

a bouncing jet. The fourth section presents some additional observations for

future work.

4.1 Types of jet behavior

This section enumerates the various outcomes of a jet impinging on a

fluid surface.

4.1.1 Entrainment of air

Figure 4.1 shows two impinging jets, one is plunging and one is not

plunging. For a jet to plunge, it must overcome the capillary force, break

through the surface, and entrain air into the liquid bulk. The top image in

Fig. 4.1a was taken from slightly above the fluid surface, and it shows the

54

jet falling. Upon impact with the surface, an indentation is made, which can

be seen in the top and bottom images of Fig. 4.1a. The conditions are not

sufficient for plunging. The bottom two images in Fig. 4.1 were taken from

slightly below the fluid surface, and it shows the critical difference between

entraining air and not entraining air. In Fig. 4.1b, a cylindrical sheet of air is

entrained into the liquid bulk and the surface dent extends in the horizontal

direction less. Because of the meniscus at the edge of the tank, the bouncing

jet could not be imaged both above and below the surface simultaneously in

most cases. This is why there are separate top and bottom images at angles

slightly above and below the liquid bulk’s surface.

These two states, plunging and not plunging, have been studied before

[23]. The new phenomenon that this thesis concentrates on is shown in the

next subsection.

4.1.2 Types of bouncing

The liquid stream impinging on a liquid bath does more than simply

entrain air, it can bounce off the surface (Fig. 4.2). To observe the full range

of bouncing behavior, four parameters were varied experimentally: fall height

H, flow rate Q, horizontal bath velocity Vbath, and dynamic viscosity µ. Each

parameter’s effect on the bouncing jet is examined here.

In Fig. 4.2, the bouncing jet is shown for two different flow rates. For

both Fig. 4.2a and b, the liquid stream falls to the surface, bends around

approximately semicircularly on top of the surface, and proceeds to bounce off

55

Figure 4.1: Entrainment of a thin cylindrical film of air or not. Between thetwo images, only the flow rate Q changes. a) The jet is not entraining air(Q = 0.35 cm3/s). b) The jet is entraining air (Q = 0.87 cm3/s). For both:height H = 5 cm, horizontal velocity of the bath Vbath = 15.7 cm/s. Theviscosity µ = 103 mPa·s. Note: The top image in (a) was taken at a slightlyhigher inclination than (b), so the dark indentation on the surface appearsdifferently. Also, the top and bottom images were not taken simultaneously,so slight differences may be present. The indentation of the non-entraining jetis in general larger than the indentation of the entraining jet. The level of thebath’s surface is labeled in (b) to avoid ambiguity, because there is a reflectionof the jet’s edges on the bath’s surface.

56

Figure 4.2: The bouncing jet. Between the two images, only the flow rate Qchanges. a) Q = 0.23 cm3/s. b)Q = 0.44 cm3/s. For both: Height H = 5 cm.The horizontal velocity of the bath Vbath = 15.7 cm/s. The dynamic viscosityµ = 103 mPa·s.

the surface. The convention in this thesis is that the bath moves left to right.

Any exception to this convention is noted. In Fig. 4.2a, the stream’s diameter

is smaller because of the smaller flow rate. Because the thinner stream has less

momentum to reverse, the semicircular indentation on the surface is smaller

compared to the jet with a thicker diameter in Fig. 4.2b. The thinner jet in

4.2a contacts the surface of the bath less than in 4.2b, and thus undergoes

less viscous drag. This is made apparent by the greater velocity of the liquid

stream after bouncing.

Upon returning to the bulk surface after bouncing, the jet can bounce

again (Fig. 4.3). This simply means that the conditions are adequate for the

jet to bounce again as it approaches the fluid surface for the second time. On

the second bounce, the sufficient conditions for a bounce are slightly different

than those for a jet impinging vertically. This is because on the second bounce,

57

Figure 4.3: The double bouncing jet. Flow rate Q = 0.35 cm3/s, height H =5 cm, horizontal velocity of the bath Vbath = 15.7 cm/s, and dynamic viscosityµ = 103 mPa·s. The top and bottom images were not taken simultaneously,so slight differences exist.

the jet hits the surface at an angle. A third bounce was difficult to observe.

If a jet did bounce three times, it did so only transiently. It was unclear from

observations whether the jet was rebounding or simply lying on the surface.

The latter case is called the “trailing jet,” and is discussed in detail later.

The horizontal velocity of the bath can exaggerate or diminish the

bouncing, as demonstrated in Fig. 4.4. At a low bath velocity, the jet picks

up only a little momentum in the horizontal direction, and it bounces almost

normally to the bath’s surface (Figure 4.4b). The small horizontal velocity

only serves to insure the stream does not collide with itself, which destabilizes

the bouncing. The horizontal distance that the jet bounces from the point

of impact is similar to the standoff distance in viscous jets flowing upwards

then downwards under gravity[27]. At a higher horizontal bath velocity, the

jet gains a great amount of horizontal momentum, and the jet bounces at an

angle inclined more horizontally (Fig. 4.4b). The angle of the jet’s rebound

58

Figure 4.4: Changing the horizontal velocity of the bath for a fluid with dy-namic viscosity µ = 103 mPa·s. a) Horizontal velocity of the bath Vbath = 11.0cm/s. b) Vbath = 15.7 cm/s. For both: flow rate Q = 0.35 cm3/s, height H =5 cm.

can be changed continuously by changing the bath’s horizontal velocity.

At a critical horizontal bath velocity, the bouncing is prevented entirely

(Fig. 4.5). We call this a trailing jet. If the horizontal bath velocity is in-

creased, the jet bounces at a more horizontal angle. At a critical horizontal

bath velocity, the jet no longer bounces, but simply impacts the surface nor-

mally, bends 90 degrees, and lays on the surface. The jet does not coalesce

with the bulk fluid immediately; rather, the stream is bent horizontally and

rests on the surface until the thin layer of air between it and the bulk liquid

drains. Once the layer of air becomes thin enough, the jet and the bulk coa-

lesce. In Fig. 4.5, this coalescence occurs roughly 4 cm after first contact with

the surface.

59

Figure 4.5: A trailing (non-bouncing) jet at high bath velocity. Vbath = 42.3cm/s, Q = 0.35 cm3/s, H = 5 cm, µ = 103 mPa·s.

Figure 4.6: Changing the horizontal velocity of the bath for a higher viscosityfluid viscosity µ = 347 mPa·s. a) Horizontal velocity of the bath Vbath = 0.656cm/s. b) Vbath = 3.28 cm/s. For both: flow rate Q = 0.16 cm3/s, height H =4 cm.

60

All of the previous figures in this chapter have used the same fluid of

dynamic viscosity, µ = 103 mPa·s. This is about 100 times more viscous than

water. However, the two images in Fig. 4.6 have a viscosity about 3.5 times

higher, µ = 347 mPa·s. Compare Fig. 4.6 with Fig. 4.4. The main difference

is that the radius of the semicircular indentation that the jet makes is much

smaller for the higher viscosity fluid. Since the contact with the bulk fluid is

less, the more viscous jets rebound with more of their original momentum. To

achieve a comparable bounce (such as Fig. 4.6b and Fig. 4.4a, the flow rate,

the height, and the horizontal bath velocity are less.

Now that the basic types of bouncing have been described, the range

of parameters that stable bouncing occurs is presented.

4.2 Parameters for stable bouncing

In mapping the range of parameters where a liquid stream bounces, four

parameters were varied: flow rate Q, horizontal bath velocity Vbath, dynamic

viscosity µ, and the height of the nozzle H above the fluid surface. Phase

diagrams were then constructed showing the region of stable bouncing. In

actuality, there are more parameters that could have been varied (see Section

3.4). The four parameters which were varied here were chosen as the most

fundamental parameters that could be controlled with high accuracy. Because

there are many parameters in the problem, the phase diagrams presented here

are low-dimensional cross-sections of the high-dimensional bubble in parameter

space of stable bouncing. Each independent parameter is counted as a separate

61

dimension.

0 0.5 10

15

30

Q (cm3/s)

Vba

th (

cm/s

)50.8 mPa×s Phase Diagram

No stable bounce

Stable bounce

No stable bounceJe

t bre

ak−

up

Figure 4.7: Phase diagram of flow rate Q vs. horizontal bath velocity Vbath.Dynamic viscosity µ = 50.8 mPa·s, falling height H = 3 cm.

Figure 4.7 shows the region of stable bouncing for a constant falling

height H of 3 cm and a dynamic viscosity of µ = 50.8 mPa·s. The flow rate Q

and the horizontal bath velocity Vbath were varied. The area between the lines

is the range of parameters that a stable bouncing jet can be initiated using

the control rod. See Section 3.4 for the criterion of stable bouncing. The

upper bound of the phase diagram denotes a transition from stable bouncing

to no stable bouncing. In the region below the lower bound of the phase

diagram, there is also no stable bouncing. The region of stable bouncing is

62

singly connected for the range of parameters tested in this thesis. For values

of Q below the dashed vertical line, the liquid jet broke up into drops before

reaching the bath surface or the liquid dripped from the nozzle (See Section

4.4.2 for more details.).

The typical error in the horizontal bath velocity was ± 0.5 cm/s. This

corresponds to an error of ± 0.03 rad/s in the rotation rate at a distance of

16.72 cm from the turn table’s axis. The typical error in the flow rate was ±0.0013 cm3/s. A typical error bar is shown on the lower left most point in Fig.

4.7.

Figure 4.8 shows the region of stable bouncing for four viscosities. The

maximum flow rate found to create a jet that stably bounced was approxi-

mately the same for all viscosities, around Q = 1 cm3/s. Changing viscosity

makes a dramatic difference in the range of Q and Vbath where stable bouncing

occurs. Of the five viscosities tested, µ = 50.8 mPa·s had the largest range.

The jet could bounce twice for some subset of conditions at most viscosities,

but the exact region of double bouncing was not mapped.

For each viscosity, the lower bound of the phase diagram demarcates

a transition from no stable bouncing to stable bouncing. For every viscosity

except 50.8 mPa·s, the upper bound of the stable region denotes a transition

from a stably bouncing jet to a trailing jet. For 50.8 mPa·s, the trailing jet

was not observed at high horizontal bath velocities (see Fig. 4.7). At this

viscosity, there was not the same smooth transition from bouncing to trailing

as observed at the higher viscosities.

63

0 0.5 10

15

30

Q (cm3/s)

Vba

th (

cm/s

)

50.8 mPa×s Phase Diagram

No stable bounce

Stable bounce

No stable bounce

Jet b

reak

−up

0 0.5 1 0

10

20

103 mPa s Phase Diagram

Q (cm3/s)

Vba

th (

cm/s

)

No stable bounce

Stable bounce

Trailing Jet

0 0.5 10

10

20

Q (cm3/s)

Vba

th (

cm/s

)

216 mPa×s Phase Diagram

Trailing jet

Stable bounce

No stable bounce

0 0.5 10

2

4

6

Q (cm3/s)

Vba

th (

cm/s

)

515 mPa×s Phase Diagram

Trailing jet

Stable bounce

No stable bounce

Figure 4.8: Phase diagrams of flow rate Q vs. horizontal bath velocity Vbath

for different dynamic viscosities µ = 50.8, 103, 216, and 515 mPa·s. Fallingheight H = 3 cm.

64

0 0.5 10

5

10

15

Q (cm3/s)

Vba

th (

cm/s

)

347 mPa×s Phase Diagram

Trailing jet

Stable bounce

No stable bounce

Figure 4.9: Phase diagrams of flow rate Q vs. horizontal bath velocity Vbath.Dynamic viscosity µ = 347 mPa·s, falling height H = 4.23 cm.

65

Because the gear pump could drive a minimum flow rate of 0.105 cm3/s,

the phase bubble does not close at low Q for three of the viscosities in Fig. 4.8.

This is a limitation of the experimental setup; the sharp cut-off in the stable

region is not physical for µ = 103, 216, or 515 mPa·s. The sharp cut-off could

possibly be physical for µ = 50.8 mPa·s, because of the onset of the Rayleigh

instability or a dripping nozzle. The existence of a sharp cut-off is unresolved.

An earlier version of the pump system could produce small flow rates to close

the phase diagrams at low flow rates, but it could not produce large flow rates

to close the stable region at high flow rates. Figure 4.9 displays data collected

with the earlier pump system using a fluid with a viscosity of 347 mPa·s. The

stable region closes at low Q, but is left unexplored at high Q. This data

for the range of stable bouncing cannot be compared to the other viscosities,

because the falling height for the data in Fig. 4.9 was H = 4.23 cm and not

H = 3 cm, as it is for the data in Fig. 4.8.

Figure 4.10 shows the range of vertical jet velocities Vjet and horizontal

bath velocities Vbath where stable bouncing was possible. Between the two lines,

the jet bounces. Above both lines, the jet trails. Below both lines, the jet does

not bounce. The flow rate and viscosity were kept constant at Q = 0.35 cm3/s

and µ = 103 mPa·s, respectively. The jet velocity was measured at the level

of the bulk surface. A rough trend is that, for higher vertical jet velocities, a

higher horizontal bath velocity was needed for stable bouncing. The vertical

velocity of the jet ranges from 37.5 to 170 cm/s. This corresponds to a falling

height (from the bath’s surface to the nozzle) of 1.7 to 14.1 cm. There is a

66

1000

20

40

60

Vjet

(cm/s)

Vba

th (

cm/s

)

No stable bounce

Stable bounce

Trailing jet

Figure 4.10: Phase diagram of Vjet vs. Vbath. Flow rate Q = 0.35 cm3/s,viscosity µ = 103 mPa·s. The large uncertainty around Vjet = 95 cm/s isdue to the transition from a non-entraining jet to an air entraining jet. Thestream is very sensitive to small perturbations. The vertical velocity of the jetcorresponds to a falling height H from 1.7 to 14.1 cm.

67

greater uncertainty in the range of stability around Vjet = 95 cm/s. At this

velocity, the jet transitions between smoothly merging without entrainment

for lower jet velocities and a plunging jet entraining air at higher jet velocities.

Near this transition, small perturbations in the jet uniformity can induce or

abate entrainment. The value of the horizontal bath velocity shown for each

vertical velocity between 90 and 105 cm/s is the mean of repeated trails. The

uncertainty is the standard deviation. These data points were individually not

reproducible, but the mean was stationary. For the other values, the individual

data points were reproducible to within 3-5% of the horizontal bath velocity.

4.2.1 Multi-stability and hysteresis

Two comments are in order here; one is about multi-stability and one

is about hysteresis.

At the same conditions, the jet can simply merge smoothly into the

bulk, it can plunge and entrain air, it can bounce once, and it can bounce

twice. This means that for the same conditions, there are multiple stable

states, depending on the history and method of preparation.

There are many fluids problems which exhibit multiply stable states.

One standard example is the infinitely long Taylor-Couette cell flow[28]. In the

case of a liquid stream impacting a moving liquid bath, at least four different

behaviors can be observed for the same conditions. Figures 4.1, 4.2, and

4.3 are good examples of these four different stable configurations. For some

conditions, the number could be greater than four if the different configurations

68

in Fig. 4.11 and Fig. 4.12 are counted. The ways of preparing each state is

addressed in the next section.

Figure 4.11 shows three states possible for the same conditions. The im-

ages were taken from below the bath’s surface. Figure 4.11a shows a smoothly

merging jet. Figure 4.11b shows a jet that is entraining air only at its front

lip, which protrudes into the bulk liquid. Small bubbles can be seen breaking

from the tip of the entraining air. The bubbles, which are moving with the

bath, are streaked because of the camera’s exposure time. Otherwise, there is

very little streaking in the images, because the other features do not change

in time. Figure 4.11c shows a bouncing jet. The conditions are identical in all

three cases, only the history of the stream is different.

Figure 4.12 shows three other possible states for another set of condi-

tions. Figure 4.12a shows a plunging jet. The jet curves to the right because

the liquid bath is moving left to right. Figure 4.12b shows a plunging jet that

is bound to the surface on its upper edge. Note the large depth and horizontal

extent of the surface indentation made behind the jet. Figure 4.12c shows a

bouncing jet from below the surface. These images were also taken from an

angle below the bath level and show the bottom of the jet.

The second comment is about hysteresis. Because of the multi-stability

of the situation, the parameter range shown in Fig. 4.8 and Fig. 4.10 is the

range over which a stably bouncing jet can be initiated by only perturbing the

stream with a control rod. If, say, the horizontal bath velocity could also be

manipulated to initiate a bounce, the range of stable bouncing would change.

69

Figure 4.11: Multi-stability: a sequence of images showing three possible statesat Vbath = 12.5 cm/s. H = 5.10 cm, µ = 103 mPa·s, Q = 0.35 cm3/s.

70

Figure 4.12: Multi-stability: three states for conditions different than Fig.4.11. a) A plunging jet, b) A surface-bound jet, and c) A bouncing jet. Allstates at Vbath = 15.2 cm/s. H = 6.4 cm, µ = 103 mPa·s, Q = 0.35 cm3/s.

71

This is because the jet could be made to bounce at a high bath velocity, then

the bath velocity could be slowly lowered. Often using this method, a jet can

bounce at a bath velocity lower than the bath velocity at which a jet can be

made to bounce by only perturbing the falling liquid stream with a control

rod.

4.3 Bounce initiation

A falling liquid stream can be made to bounce by using three different

methods, which are outlined here. A bouncing jet can be initiated by per-

turbing the stream as it falls, by changing the horizontal bath velocity, or by

changing the flow rate.

4.3.1 By perturbation

Figure 4.13 shows how a control rod can perturb the falling stream such

that different states are produced. Each different state is individually stable

for periods longer than a minute.

Figure 4.13 starts with a smoothly merging stream in (a). If a smoothly

merging jet is perturbed slightly, it can start to bounce (Fig. 4.13b-f). “Per-

turb” here means to temporarily interrupt the stream as it falls, as seen in Fig.

4.13b. This temporarily blocks the stream and a change in jet diameter and

velocity is created. So the diameter and velocity varies along the length of the

stream. This disturbance falls and strikes the surface, often entraining air or

changing the state of the jet. The jet is beginning to bounce and stabilize in

72

Figure 4.13: A time sequence showing the control rod perturbing the jet,initiating a bounce, and then initiating a double bounce. Vbath = 16 cm/s, H= 5 cm, µ = 103 mPa·s, Q = 0.35 cm3/s.

73

Fig. 4.13d-e. The bouncing jet is stable in Fig. 4.13f. If the jet is perturbed as

it falls toward the liquid surface after bouncing a first time, the jet can bounce

a second time(Fig. 4.13g-j).

Perturbation can also make the jet entrain air. If the control rod is

plunged under the surface where the stream hits the surface, some fluid from

the stream is carried below the bath surface and air is entrained. This method

often changes a non-entraining jet (Fig. 4.1a) to an entraining jet (Fig. 4.1b).

How a non-uniform jet behaves as it collides with the surface of a bath has

been previously studied [46, 49]. An otherwise non-entraining jet can be made

to entrain air by these disturbances [46, 49].

4.3.2 By changing Vbath: surface-bound penetration

The liquid stream can also be made to bounce by changing the hori-

zontal velocity of the liquid bath. Figure 4.14a-d shows how increasing the

horizontal bath velocity effects the interaction of the impinging jet with the

bath. The upper surface of the jet is attached to the bath’s surface (Fig. 4.14a-

d). As the horizontal velocity of the bath increases, the jet is curved around

the surface indentation more (Fig. 4.14b-c). At a high enough horizontal bath

velocity, the jet bounces (Fig. 4.14d).

4.3.3 By changing Q, subsurface penetration

Finally, the bouncing of a jet can be initiated by changing the flow

rate Q. Figure 4.15 shows a time sequence of this process. In Fig. 4.15a-d,

74

Figure 4.14: This plunging jet is bound on its upper side to the surface. Smallair bubbles are streaming off the tip of the air skirt. The view is from below thesurface. The jet impacts the surface at the left. The fluid bath is moving rightto left. This set of images is a parameter sequence showing a bounce initiatedby changing the bath velocity. The change in horizontal bath velocity betweenimages is about 0.5 cm/s, starting from Vbath = 21.2 cm/s. H = 5.10 cm, µ =103 mPa·s, Q = 0.35 cm3/s.

75

the stream is penetrating the surface and entraining a cylinder of air. The

cylinder of air rises toward the surface. John Bush (personal communication,

Nov. 2004) pointed out that this attraction to the surface is reminiscent of

the Coanda effect, where a thermal plume is attracted to a wall like in Fig.

2.9. In Fig. 4.15e-f, the jet begins to break through the surface and bounce.

The exact mechanism for initiating a bounce is unresolved.

Thus, the falling liquid stream can behave in many different ways. The

range of parameters for which the bouncing jet is stable has been mapped.

And at some conditions, four or more different states are stable.

4.4 Additional observations

A liquid stream impinging on a moving liquid bath behaves in more

ways than can be studied in depth in this thesis. In this section, a collection

of observations are presented to demonstrate this complexity and to encourage

future research on this topic. There is a great deal still to be studied.

4.4.1 Non-bouncing states

If the falling height and flow rate are outside of the region of stable

bouncing, several different non-bouncing states can be observed. In the first

state (Fig. 4.16a), the jet smoothly merges with the bath without entrainment.

In the second state (Fig. 4.16b), the jet entrains a cylindrical skirt of air into

the fluid. As the air collects in the skirt, bubbles detach from the air skirt

(Fig. 4.17). In a third state, the air is entrained into a thin cylindrical skirt,

76

Figure 4.15: A time sequence of images showing a bounce begin by decreasingthe flow rate from an air-entraining jet. The time between images is onequarter of a second. The change in pump rate between images is about 0.02cm3/s, starting from Q = 0.56 cm3/s and ending at Q = 0.45 cm3/s. Vbath =5.2 cm/s, H = 4.23 cm, µ = 347 mPa·s.

77

but bubbles are not created. In this case, the air that builds up in the air skirt

forms an air lump on one side of the air skirt and travels upward. Once the air

bump reaches the surface and releases its air, the jet can either be dramatically

disturbed or not. Figure 4.18 shows the case where the jet is disturbed after

the air pocket reaches the surface.

Figure 4.16: Non-bouncing conditions for a fall height H and flow rate Q toosmall and too large. a) H = 2.2 cm (corresponding to Vjet = 46.2 cm/s) is toolarge for Vbath = 15.7 cm/s. “Too large” may seem counter-intuitive since thefalling height is small, but Fig. 4.10 confirms that for these conditions, thefalling height is in fact slightly too large. Because the conditions are close tothe boundary of the stable region (with its uncertainty), the jet could possiblybounce if it was perturbed. b) H = 10.2 cm (corresponding to Vjet = 144 cm/s)is too small for Vbath = 15.7 cm/s. Again, Fig. 4.10 confirms “too small.” Forboth: horizontal velocity of the bath Vbath = 15.7 cm/s, flow rate Q = 0.35cm3/s, and viscosity µ = 103 mPa·s.

78

Figure 4.17: A time sequence of a bubble breaking off the bottom of thecylindrical sheath of air. The circle in (a) marks where the air collects. Thebubble in (c) is streaked because of the camera’s exposure time. The timebetween images is 120 milliseconds. H = 6.4 cm, Q = 0.35 cm3/s, µ = 103mPa·s, Vbath = 15.2 cm/s.

79

Figure 4.18: A time sequence of a bump of air going to the surface along theinside of the cylindrical sheath of air. (a) The entrained air collects at thebottom of the air skirt. (b)-(f) Once enough air has been collected, a pocketof air rises along the side of the air skirt. The arrows mark the air rising.(g)-(i) The entrained air is disturbed after the air bump reaches the surface. µ= 347 mPa·s. This is an early image sequence and the exact conditions werenot recorded. But this state is easily reproduced with a moderate fall heightH ≈ 3 cm, a low horizontal bath velocity Vbath ¡ 1.0 cm/s, a flow rate largeenough to entrain air, but small enough so that no bubbles break off the edgeof the air skirt Q ≈ 0.4 cm3/s.

80

4.4.2 Jet breakup and dripping

If the falling height is too large and the liquid stream is not viscous

enough, the stream can break into drops before it reaches the bath surface

(Fig. 4.19). The jet was observed to break up for the 50.8 mPa·s fluid at

very low flow rates. The dashed vertical line in Figure 4.7 denotes the flow

rate at which break-up was observed. After the jet has broken into drops,

the nozzle drips from that time on. If the stream is prepared carefully by

slowly decreasing the flow rate, this breakup can be delayed. The jet was not

observed bouncing if the jet had already broken into drops.

Figure 4.19: Non-bouncing conditions, jet breakup. H = 3 cm, Q = 0.06cm3/s, µ = 50.8 mPa·s. Each image is 40 ms apart. The horizontal bathvelocity (Vbath = 2.0 cm/s) plays a minimal role in the jet break-up; however,it does carry to the right the non-coalesced drops which are sitting on thesurface.

4.4.3 Non-coalescence: floating drops on a moving liquid bath

Figure 4.20a is a jet bouncing at a low horizontal bath velocity. Figure

4.20b shows a drop sitting in the trough in the surface made by the jet. This

observation demonstrates that the jet is not surrounded by fluid on all sides.

81

Only the bottom and sides of the jet contact the bath; the top of the jet is

open to the air. This confirms the previous interpretation of the images, which

is complicated because of the diffraction of light around the interfaces of clear

silicone oil.

Figure 4.20: A drop sitting in the trough of a bouncing jet and a bouncingjet with the same parameters without a drop sitting there. Does this count asanother multiply-stable state? H = 3 cm, Q = 0.26 cm3/s, µ = 50.8 mPa·s,Vbath = 4.3 cm/s. The text on each image is information about the setting ofthe camera and the exposure. It is displayed by the camera’s control module.To keep the text legible, the liquid bath moves to the left, which is the oppositeof this thesis’s convention.

Drops can often rest on a surface for several seconds before merging

into the bath (Fig. 4.21). The time of non-coalescence can be extended by a

shear between the drop and the bath [15]. In Fig. 4.21, the horizontal bath

velocity is slightly too slow. The jet bounces only intermittently, because the

liquid that hits the surface after bouncing distorts the shape of the interface

and destabilize the bouncing jet. The jet then forms drops that rest on the

surface.

82

Figure 4.21: A time sequence of a jet intermittently bouncing and producingdrops that rest on the surface for some time before coalescing. H = 3 cm, Q= 0.26 cm3/s, µ = 50.8 mPa·s, Vbath = 4.3 cm/s. The time between images is56 milliseconds. The bath moves right to left, see the caption of Fig. 4.20 fordetails.

83

Figure 4.22a displays a bubble of air caught behind a jet that is not

entraining air. The convection driven by the impinging jet creates an eddy in

which the bubble is stable. Figure 4.22b shows a complementary case. A drop

of liquid rests in the indentation formed by the impinging drop. Drops like this

were observed before and called satellite drops [45]. The drop is stabilized by

the continual influx of air between it and the bath supplied by the impinging

jet.

Figure 4.22: a) A drop of liquid on the surface following behind the liquidstream. H = 3 cm, Q = 0.26 cm3/s, µ = 50.8 mPa·s, Vbath = 4.3 cm/s. b) Abubble below the surface following behind the liquid stream. H = 4.6 cm, Q= 0.24 cm3/s, µ = 347 mPa·s, Vbath = 4.0 cm/s.

4.4.4 Antibubble production and extraction

In Section 2.2.3, antibubbles were introduced. In the course of our ex-

periments, antibubbles were observed. When the fluid jet was entraining a thin

84

cylindrical skirt of air into the fluid bath, often the thin sheet of air closed on

itself to create an antibubble. This process is shown in Fig. 4.23. Antibubbles

are normally difficult to make reliably. However, many antibubbles are formed

when the conditions are such that the jet is at the threshold of air entrainment.

At this threshold, the jet will not entrain air unless it is perturbed externally.

After the jet is perturbed, it begins to entrain a cylindrical sheath of air. This

thin layer of air pinches off such that it forms a stable, spherical layer of air

(an antibubble). The spherical layer of air contains silicone oil in its interior

and is surrounded by silicone oil. The size distribution of antibubbles created

by this method was broad.

Once an antibubble is formed, it rises very slowly to the surface. Its

upward velocity is slow enough that the tank could complete several full ro-

tations before the antibubble came to the surface. In Fig. 4.24, one complete

rotation of the tank takes 2.7 seconds. As the tank rotated, the antibubble

would come back underneath the jet. When the antibubble was directly under

the jet, the jet could pull the antibubble to the surface. Figure 4.24 shows

this process. The antibubble is confirmed to be a drop of oil surrounded by

a sheath of air, because after it is brought to the surface, it is simply a non-

coalesced drop of liquid. The mechanism by which a jet pulls an antibubble

to the surface is undetermined.

These observations indicate that a liquid stream impinging on a moving

liquid bath is capable of many complex and dynamic behaviors.

85

Figure 4.23: Three antibubbles are formed by an air-entraining jet. In b) thejet is temporarily interrupted by the control rod. This perturbation starts thejet entraining. The entrained cylindrical sheet of air then closes off to formthe antibubbles. The white arrow in (j) points to the first antibubble formed.Note that the antibubbles rise very slowly. H = 3 cm, Q = 0.44 cm3/s, µ =50.8 mPa·s, Vbath = 5.9 cm/s. The bath moves right to left, see the caption ofFig. 4.20 for details. Two antibubbles slowly pass behind the plane of focusin all images.

86

Figure 4.24: A jet pulling up an antibubble to the surface. The antibubbleapproaches from the left, as the bath moves left to right. Upon reaching thesurface, the antibubble simply becomes a non-coalesced drop floating on thebath’s surface. The time between each image is 64 milliseconds. H = 3 cm,Q = 0.44 cm3/s, µ = 50.8 mPa·s, Vbath = 6.2 cm/s. The bath moves right toleft, see the caption of Fig. 4.20 for details.

87

Chapter 5

Discussion and Conclusions

This research studied the phenomenon of a bouncing jet, that is, the

rebound of a viscous stream off of the surface of a moving liquid bath of the

same viscous liquid. Chapter One introduced the phenomenon and described

how to reproduce it at home. Chapter Two reviewed the relevant background

material. Chapter Three detailed the experimental apparatus. And Chapter

Four presented the results. Now, Chapter Five summarizes the findings of this

study.

5.1 Main results

This study showed that a fluid jet can bounce off of a moving fluid

surface stably for a wide range of parameters: dynamic viscosities from 38 to

>2000 mPa·s (Table 1.1), velocities of the bath’s surface from 1 to 35 cm/s

(Fig. 4.8), velocities of the jet from 40 to 170 cm/s (Fig. 4.10), and jet

diameters from 0.5 to 1.2 mm (Fig. 3.5).

Many aspects of the jet can be understood by applying what is known

already about fluid jets. These aspects include: the narrowing of the jet as

it falls (Section 2.1), the entrainment of air and non-coalescence during the

88

collision with the surface (Section 2.2), the bending of the jet upwards to

bounce (Section 2.3), and the trajectory of the viscous jet (Section 2.4).

5.2 Implications, significance, and applications

This research offers several interesting and potentially useful results

for physics and industry. The bouncing jet is a new example of a multiply

stable fluid configuration. It is also a new example of steady non-coalescence.

Previous examples of steady non-coalescence include a pendant drop hanging

over a moving surface [15], two drops with different temperatures [15], and

drops bouncing on a surface [13]. Viscous liquids that bounce are common,

so the bouncing jet can be used in the classroom or teaching laboratory to

demonstrate and explain fluid dynamics, force balances, and other physics

principles.

Because it is easy to reproduce and it exhibits many varied behaviors,

the bouncing jet can be a playground for the study of fluid jets, their different

behaviors, and their stability and instability. Hobbyists, science teachers, and

students can study the phenomenon. Students can research one of the many

open topics as a science fair project, as a undergrad research project, or as a

graduate research project.

The bouncing phenomenon could be a new technique available for in-

dustry for controlling a fluid jet or the entrainment of the fluid surrounding

the jet.

89

5.3 Suggestions for future research

Two fluids, each with their own surface, capable of colliding and de-

forming each other is a very complicated situation. In the course of conducting

this work, there were many exciting and intriguing observations that could not

be given the close attention they deserve. The details of these observations

are collected in the Section 4.4: Additional Observations.

Many behaviors of the impinging jet remain to be investigated. There

are many more parameters to vary, such as the viscosity of the surrounding

fluid (which is air in these experiments), the interfacial tension between the

two fluids, the density difference between the two fluids, the angle of incidence

of the jet, and the viscosity of the jet compared to that of the bath. In these

experiments, the two fluids were the same in the jet and the bath.

In addition to these parameters, there are several dynamic processes

which are worthy of study. For instance, the process of antibubble formation

(Fig. 4.23), the pockets of air running along the entrained air trumpet (Fig.

4.18), how a drop is stable sitting inside the surface trough made by the jet

(Fig. 4.20 and Fig. 4.22), surface-bound air entrainment (Fig. 4.12), the

trajectory of a viscous jet rising and falling under gravity (Section 2.4.2), and

the physics of the trailing jet (Fig. 4.5). Around the trailing jet, air is squeezed

out between a moldable bath surface and a moldable jet cylinder.

The experimental apparatus is fairly simple to construct. This, com-

bined with the plethora of unexplained phenomena, should inspire others to

90

pursue research similar to the work presented here. Also, the phenomenon is

easily reproduced and it is hoped that many hobbyists, teachers, and demon-

strators can find the bouncing jet interesting and useful in exploring fluid

dynamics. Hopefully, this research is only a beginning and will be continued

by others.

91

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Vita

Matthew Evan Thrasher was born on March 13, 1981 in Tulsa, Okla-

homa to Cheryl Lea and Thomas Stephen Thrasher. He started high school

at the Bartlesville Mid-High School, Bartlesville, Oklahoma (1995-1997) and

finished at the Oklahoma School of Science and Mathematics, Oklahoma City,

Oklahoma (1997-1999). During September 1999 to June 2003, he earned

a Bachelor of Arts in Physics from Harvard College in Cambridge, Mas-

sachusetts. During the summer of 2003, he started working at the Center

for Nonlinear Dynamics at the University of Texas at Austin. In September

2003, he began his studies in the Department of Physics at the University of

Texas at Austin.

Permanent address: PO Box 3065Bartlesville, OK 74006

This thesis was typeset with LATEX† by the author.

†LATEX is a document preparation system developed by Leslie Lamport as a specialversion of Donald Knuth’s TEX Program.

101