Investigation of Drop Generation from Low Velocity Liquid Jets and Its Impact Dynamics on Thin Liquid Films

Investigation of Drop Generation from Low Velocity Liquid Jets and its Impact Dynamics on Thin Liquid Films

A dissertation submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY (Ph.D.) in Mechanical Engineering of the College of Engineering and Applied Sciences by Sucharitha Rajendran

September 2017

B.Tech., National Institute of Technology-Durgapur, 2010 M.S., University of Cincinnati, 2012

Committee Chair: Dr. Milind A. Jog Committee Member: Dr. Raj M. Manglik Committee Member: Dr. Kumar Vemaganti Committee Member: Dr. Woo Kyun Kim

ABSTRACT

Drop generation from jet breakup and drop-impact on thin films are two key processes in many spray, coating and deposition applications which are experimentally and computationally investigated in this dissertation. A high-speed camera is used to capture the drop formation and its impingement on thin films and a modified volume of fluid (VOF) method is used to accurately simulate jet/film breakup. It is shown that the proposed modifications in interfacial property evaluations for VOF method provide accurate predictions of moving interface for highly viscous liquids.

To understand the parameters governing uniform drop generation, a numerical study of the jet disintegration process, supplemented by experimental observations, is conducted. The simulations show that while an increase in viscous force offers greater resistance to propagation of the surface undulations thus resulting in stretching of the liquid jet, surface tension determines the radius of curvature of the pinch-off location. The interplay of these two forces leads to deviations from uniform drop generation and two different modes of liquid jet pinch-off that lead to satellite drop creation are observed. It is shown that the occurrence of these modes is governed by the liquid

Morton number. A semi-empirical relation is proposed to predict the onset of the two modes.

In most spray applications, the drops formed from a jet are incident on a target surface usually with a thin layer of liquid film. While viscous resistance to impact of the drop is offered by the film thus controlling drop spreading and crown formation, surface tension is noted to primarily govern the growth and eventual break-up of the ejected crown. An empirical relation for the onset of splash is obtained from high-speed image analyses of the drop-film interactions. Numerical simulation of the process indicates that the contribution of the target film and impacting drop in the creation of ejected crown depends on the film thickness. At low liquid film thickness, the liquid ii from the impacting drop primarily forms a jet that is ejected post-impact. For thicker films, the liquid layer absorbs a part of the energy from the impacting drop and thus the ejected jet is formed from both the impacting drop as well as the liquid layer. The temporal dynamics of deposition of

Newtonian drops on thin films indicates that while surface tension and inertial forces govern resultant crown height, their contribution on crown diameter is not as significant. Viscosity is noted as the key factor that governs the crown diameter. In addition, experimental investigation of drop impact dynamics of shear thinning polymeric solution is carried out to understand the effects of solution rheology on the temporal dynamics. For aqueous solution of HEC 250 HHR, the apparent viscosity is low during initial impact (high shear rate) and this increases the likelihood of splashing compared to a Newtonian liquid with the same zero shear viscosity. High-speed images of drop impact on thin film of aqueous solution of PEO WSR 303 indicate that extensional viscosity also plays an important role in governing the post-impact behavior.

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ACKNOWLEDGMENTS

I wish to express my most sincere gratitude and appreciation to those who have contributed to this work and to those who have been my support system throughout my academic journey.

First, I am extremely grateful to my academic advisor, Dr. Milind A. Jog, and co-advisor Dr. Raj

M. Manglik for their guidance and all the helpful brainstorming sessions. These discussions always helped me gain new insights into the problem and often I would come out looking at the problem in new light. Apart from this, I have often found myself being inspired by Dr. Jog and Dr.

Manglik’s clarity in thinking – be it to resolve some tough numerical road blocks or to device better means of conceptual analysis. I would like to also thank my committee members, Dr. Kumar

Vemaganti and Dr. Woo Kyun Kim for taking interest in my research and providing helpful suggestions and important insights.

During my course of research, I have gotten to know some university staff members that I wish to express my gratitude to. Bo Westheider has helped me with setting up experiments and fixing multiple equipment for the different needs of the testing. Larry Schartman has helped me innumerable times to set-up and fix (and repeat) workstations for the different needs that arose.

Luree Blythe and Barbara Carter have always been very approachable and helpful. I would also like to thank the very knowledgeable staff and consultants at OCC who helped with the many software and printing issues I’ve had over the years.

Apart from working on my research, I was a part of the CEEMS (Cincinnati Engineering Enhanced

Math and Science) fellowship and had the lovely opportunity to interact with students and teachers in their daily classroom lessons. I would like to thank Dr. Anant Kukreti, Julie Steimle, Debora

Liberi and all the teachers for this experience. I have also been a part of the UC Simulation center and my interactions with Proctor and Gamble mentors and the projects I worked on have taught

v me a lot. Working with both Dr. Joe Grolmes and Rakesh Gummalla has been a lovely learning experience. Dr. Brent Rudd and Fred Murrell’s enthusiasm to maintain the great working environment made it the experience marvelous. Apart from these two organizations, I would also like to thank the Education and Research Center’s (ERC) Pilot Research Program (PRP) for providing the initial funding for this project.

The interest to pursue a PhD degree in Fluid and thermal science stems from Dr. Amarnath

Mullick’s always cheerful motivation. I take this opportunity to thank him for encouraging me to pursue summer research fellowships during my under-graduation at NIT Durgapur, in addition to providing guidance to a very challenging and interesting senior year project.

I appreciate the help and support provided by my colleagues at Thermal Fluid and Thermal

Processing Laboratory. This was always a fun work environment. During my stay at Cincinnati, I have had the fortune of the company of some very good friends that have made this duration even more memorable: Deepak Saagar, An Fu, Wilma Lam, Sai Deogekar, Raghunandan Chilkuri,

Srikirti Velaga, Amruta Dongonkar, Gautam Krothapalli, Santhosh Dungi, Suryanarayana Pappu,

Anup Srikumar, Aditya Mantri, Sruti Jagabatulla, Vibhu Gautam, and many others. I would like to express my most sincere gratitude to Karthik Remella for not only being a pillar of support though everything, but also for being a constant source of inspiration and pushing me to always try harder.

And finally, I am forever indebted to my parents, Rajendran and Jadila and to my brother

Shashank, and to all my family for their unconditional love and support. Through times that have seemed discouraging and through those that were joyous, they have always shared an important part. They are my backbone. This work would not be a reality without them and I would like to dedicate this dissertation to them.

TABLE OF CONTENTS

Abstract ...... ii

Acknowledgments...... v

Table of Contents ...... vii

Nomenclature ...... x

List of Figures ...... xii

List of Tables ...... xvii

1 Chapter 1: Introduction ...... 1

1.1 Jet breakup and the resultant drop distribution ...... 1

1.2 Drop impact dynamics ...... 5

1.3 Scope of the current research ...... 9

2 Chapter 2: Modified Volume of Fluid method for two-phase flows ...... 12

1.4 Introduction ...... 12

1.5 Numerical Procedure ...... 14

1.5.1 Governing Equations ...... 15

1.5.2 Modifications to the VoF solver...... 18

1.5.3 Solution Procedure ...... 22

1.5.4 Numerical domain and mesh discretization ...... 23

1.6 Performance tests on modified VoF solver ...... 24

1.6.1 Stationary drop in stagnant zero-gravity ambience ...... 25

1.6.2 Drop impact in liquid pool ...... 28

1.6.3 Liquid jet breakup in stagnant ambience...... 31

1.6.4 Drop impact on thin liquid film ...... 33

1.7 Sensitivity of value of C in the modified property averaging equation (2-9) ...... 36

1.8 Conclusions ...... 37

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3 Chapter 3: Numerical analysis of low-velocity jet disintegration and subsequent drop generation ...... 39

1.9 Introduction ...... 39

1.10 Experimental procedure ...... 43

1.11 Numerical procedure and problem formulation ...... 45

1.12 Results and discussion ...... 53

1.13 Conclusions ...... 70

4 Chapter 4: Experimental study on drop impact on a thin film ...... 72

1.14 Introduction ...... 72

1.15 Experimental procedure ...... 77

1.16 Results and discussion ...... 80

1.17 Conclusions ...... 91

5 Chapter 5: Numerical analysis of drop impact of thin liquid films and the effect of film thickness on impact dynamics ...... 93

1.18 Introduction ...... 93

1.19 Numerical procedure and problem formulation ...... 96

1.19.1 Modified VoF numerical scheme ...... 96

1.19.2 Problem definition and the computational domain ...... 102

1.20 Results and discussion ...... 105

1.20.1 Validation of modified VoF Scheme ...... 105

1.20.2 Effect of liquid property on drop impact on thin films ...... 107

1.20.3 Effect of thin film height ...... 114

1.21 Conclusion ...... 122

6 Chapter 6: Secondary drops formation on impact on a thin film ...... 124

1.22 Introduction ...... 124

1.23 Experimental Procedure ...... 127 viii

1.23.1 Experimental Setup ...... 127

1.23.2 Image Processing...... 130

1.24 Results and discussion ...... 132

1.24.1 Secondary drop characteristics ...... 133

1.24.2 Crown characteristics during splash ...... 139

1.25 Conclusions ...... 142

7 Chapter 7: Non-Newtonian drop impact on thin liquid films...... 144

1.26 Introduction ...... 144

1.27 Experimental Procedure ...... 147

1.27.1 Experimental Setup ...... 147

1.27.2 Characteristics of the working liquids tested ...... 148

1.28 Results and Discussion ...... 151

1.28.1 Newtonian liquids ...... 152

1.28.2 Effect of non-Newtonian viscous behavior ...... 161

1.29 Conclusions ...... 169

8 Chapter 8: Conclusions and applicability of current research ...... 172

9 Chapter 9: Recommendations to further current work ...... 177

10 References ...... 179

11 Appendix ...... 190

Appendix A.1. Experimental Uncertainty ...... 190

Appendix A.2. Process flow for image processing of jet disintegration ...... 192

Appendix A.3. Process flow for image processing of drop impact ...... 193

Appendix A.4. MATLAB code for image processing of secondary drops ...... 195

NOMENCLATURE

Parameters:

d [m] Diameter

r [m] Radius

lc [m] Capillary length

L [m] Liquid jet break-up length

g [m/s2] Gravitational acceleration

t [s] Time

v [m/s] Velocity

H [m] Height of thin film

p [N/m2] Pressure

Greek symbols:

ρ [kg/m3] Density of the fluid

μ [kg/m-s] Dynamic viscosity of the fluid

훾 [m2/s] Kinematic viscosity of the fluid

θ [rad] Angle

α Phase fraction

σ [N/m] Surface Tension

Dimensionless parameters:

𝜌vd Reynolds number 푅푒 = 휇

𝜌v2d Weber number 푊푒 = 𝜎

g휇4 Morton’s number 푀표 = 𝜌𝜎3

gd2 Bond number 퐵표 = σ 휇 푂ℎ = Ohnesorge number √𝜌𝜎d

v2 Froude number 퐹푟 = 푔푑

v. (time step) Courant number 퐶표 = grid spacing

∗ H = H/ddrop Dimensionless film height

t Dimensionless time 휏 = d⁄v

Subscripts:

jet Liquid jet

drop Liquid drop

l Liquid

a Air

LIST OF FIGURES

Schematic Representation of the jet breakup regime. [4] ...... 4

Stability curve showing the typical variation of breakup length with jet velocity[3] ...... 5

Basic factors influencing drop impact ...... 6

Equation for the constant C in the property averaging equation (4.5) derived from the different liquids tested...... 20

Variation of fluid property (density) at the interface for arithmetic weighted averaging (green) and the current method (red)...... 21

(a) the wedge-shaped numerical domain (b) typical mesh structure ...... 24

Initial and boundary conditions for numerical solution of stationary drop under no external forces...... 26

Comparison of the two VoF solvers: spurious currents around the interface of a stationary drop in ambient air in the absence of gravity ...... 27

Comparing kinetic energy around the viscous (휈푙𝑖푞푢𝑖푑/휈푎𝑖푟 = 0.7) stationary drop interface for interFoam and the modified VoF solver ...... 28

Experimental data showing dimensionless penetration depth (L/R) as a function of time of impact of drops with same impacting velocity (0.55 m/s) for 3 different viscosities [35]...... 29

Initial and boundary conditions for numerical solution of drop impact on a deep pool...... 29

Performance of interFoam and modified solver against experimental findings [35] for entrapped air rupture time during viscous drop impact on a deep pool ...... 30

Initial and boundary conditions for numerical solution of jet breakup...... 32

Comparison between (a) experimental, and numerical predictions of (b) interFoam and (c) modified VoF for evolution of surface instabilities during circular jet breakup of a propylene glycol jet of 1.499 mm diameter and We

= 5. The time difference between these two images is 2.5 ms...... 33

Initial and boundary conditions for numerical solution of drop impact dynamics...... 35

Comparison of numerical prediction from the two solvers with experimental time progression of drop impact dynamics (Successive image are 2 ms apart) ...... 35

Experimental observation show the significance of liquid properties on drop size distribution at We = 5 and jet diameter = 0.84 mm for three different liquids...... 42

Schematic of the experimental set-up ...... 44 xii

Comparing the effectiveness of arithmetic weighted averaging and new averaging technique in computing density as a function of VoF phase indicators...... 48

(a) Wedge-shaped computational domain (b) Pictorial representation of the boundary conditions in the Y-X plane

...... 51

Mesh at Section A (Figure 3-4 (b)) has uniform spacing through nozzle width and region of liquid jet stream along the axis of symmetry; mesh is more relaxed further away...... 52

Validation of numerical method, by comparing, experimental results of Propylene Glycol jet of diameter 0.84 mm and We = 10 (in the white panels), with the numerical results (in the dark panel) ...... 54

(a and b) Numerical results showing Mode A and Mode B ligament stretching for differing liquid viscosities. (c and d) Predicted numerical velocity profiles for primary and satellite drops for liquid jets of different working liquid viscosities. The legend for velocities for each of the liquids is shown above it...... 56

(a and b) Influence of surface tension on liquid jet breakup studied numerically by varying the surface tension of the working liquid while keeping the viscosity the same. (c)Velocity profiles of liquid drops as they breakup from liquid jets of liquids varying in their surface tension. The legend for the velocity profile is provided...... 59

Strain Rate development for Propylene Glycol (diameter 0.84mm We 10) during breakup as a progression of time.

The top half of each computational image shows the strain rate while the bottom half shows the alpha variation

(white depicting the liquid and black the ambient air). For each time step, corresponding images from experiment have been shown for comparison...... 62

Strain Rate development for Propylene Glycol (diameter 1.5 mm We 10) during breakup as a progression of time.

The top half of each image shows the strain rate while the bottom half shows process of breakup (white depicting the liquid and black the ambient air)...... 65

Breakup length correlation for different Newtonian liquids showing deviation at low Weber numbers and large diameters ...... 68

Jet breakup mode changing with Mo. Transition from Mode A to Mode B pinch-off during satellite drop formation is seen to occur around Mo = 1x 10-5 ...... 69

A famous Harold Egerton photograph, Milk Drop Coronet, 1957, on an American postcard. © Harold and Esther

Edgerton Foundation, 2012, Courtesy of Palm Press, Inc...... 73

Sketch of the experimental set-up ...... 78

xiii

Comparing past correlations with experimental data and with data from other studies for H* ≤ 0.1 (a) Linear scale

(b) Logarithmic scale ...... 82

Comparing current experimental data with past correlation for (a) Water and (b) Ethylene Glycol ...... 84

Time progression of drops of water ...... 86

A comparison of impact behavior of EG and 50% PG drops ...... 88

A comparison of impact behavior of 25% PG and 75% PG drops ...... 89

Correlation (equation (4-2)) performance with (a) current experimental data and (b) data from past literature ...... 90

Equation for the constant C in the property averaging equation (4.5) derived from the different liquids tested...... 100

Variation of fluid property (density) at the interface for arithmetic weighted averaging (green) and the current method (red)...... 101

Boundary conditions and initializations for the computational domain ...... 104

The mesh in the wedge-shaped computational domain ...... 105

Verifying computational results with experimental results ...... 107

Effect of changing viscosity on drop impact (We = 429.85, Bo = 4.69)...... 108

Effect of changing the surface tension on drop impact (Re =638.09)...... 109

Pressure profiles showing the effect of change in viscosity ...... 111

Pressure profiles showing effect of changing surface tension on drop impact ...... 113

Velocity vectors – as surface tension is changed...... 114

Log-log plot of spread factor r/Ddrop as a function of dimensionless time, τ, for We = 8000 for data from Josserand and Zaleski [52] and the current numerical simulation...... 116

Parameters of the ejected jet that are used for numerical comparison at different dimensionless film heights...... 117

The location of the base (a.k.a. spread factor r/Ddrop) as a function of dimensionless time ...... 118

Variation of ejected jet parameters (a) ejected angle (b)height (c) base thickness (d) radius of curvature with respect to dimensionless time...... 119

Vorticity within the liquid layer for different liquid film heights (H*)...... 121

Movement of thin film liquid for different film heights (H*) ...... 122

Water drop impacting on thin film at different Re to form secondary drops from the crown rim (a) secondary drops deposit on thin film (b) secondary drops are air-borne...... 126

xiv

MATLAB image processing to identify and measure secondary drops ejected from an ethylene glycol impacting drop of diameter 2.9 mm at 4.17 m/s on a stagnant thin target film...... 131

Current experimental data comparing dimensionless maximum crown height, Hmax /Ddrop , as a function of impact drop Weber number, We with data from Cossali et al. [65]...... 133

Sauter Mean Diameter (D32) of the secondary drops as a function of liquid property and impact drop Weber number (We) ...... 135

Ejected velocities for secondary drops from an ethylene glycol impact drop of Ddrop = 4.49 mm Vimpact =2.99 m/s . 138

Ejected velocities for secondary drops from an ethylene glycol impact drop of Ddrop = 4.46 mm Vimpact =2.25 m/s . 138

Computational images highlighting the change in ejected drop trajectory for the above two ethylene glycol impacts.

...... 139

Dimensionless crown height (Hmax/Ddrop) as a function of liquid property (Morton number, Mo) and impact Weber number, We ...... 140

Crown growth velocity (Hmax/tHmax), as a function of liquid property (Morton number, Mo) and impact velocity

(Vimpact) ...... 141

Dimensionless maximum crown diameter (Dcrown/Ddrop) as a function of impact Weber number, We ...... 141

Measured apparent viscosity vs shear rate for different concentrations of HEC 250 HHR and PEO WSR 303.

Concentrations used in the current study are highlighted in red...... 150

Effect of surface tension on deposition as a drop of similar inertia and viscosity but varying surface tension impact on a thin liquid film ...... 154

Dimensionless crown parameters as a function of dimensionless time for the two Newtonian liquids (ethylene glycol and 63% glycerin water) to show the effect of impacting inertia as a drop impacts a thin liquid film...... 155

Effect of inertia on deposition as a drop of similar viscosity and impacting We but varying surface tension, impacts on a thin liquid film ...... 156

Effect of viscosity on deposition as a drop of similar inertia and zero-shear viscosity but varying surface tension impact on a thin liquid film ...... 159

Dimensionless crown parameters as a function of dimensionless time for the two Newtonian liquids (1200 wppm of

HEC 250 HHR and 63% glycerin water) to show significance of viscosity as a drop impacts a thin liquid film. .... 160

Effect of shear rheology on deposition as a drop of similar zero-shear Re and We as they on a thin liquid film of similar liquid ...... 162

Dimensionless crown height as a function of dimensionless time for the two liquids (2000 wppm HEC 250 HHR and

63% glycerin water) to show significance of shear rheology as a drop impacts a thin liquid film...... 163

Dimensionless crown diameter as a function of dimensionless time for the two liquids (2000 wppm HEC 250 HHR and 63% glycerin water) to show significance of shear rheology as a drop impacts a thin liquid film...... 165

Comparing shear rheology of two shear thinning aqueous polymer solutions (2000 wppm of HEC 250 HHR and

1200 of PEO WSR 303) with similar zero-shear Re and We as they on a thin liquid film of similar liquid ...... 166

Dimensionless crown height and diameter as a function of dimensionless time for the two liquids polymeric solutions (2000 wppm HEC 250 HHR and 1200 wppm PEO WSR 303) and the Newtonian 63% glycerin water to show significance of shear rheology as a drop impact on a thin liquid film...... 167

Shear and Extensional viscosity for HEC 250 HHR with molecular weight of 450 kg/mol as tested by Meadows et al.

[95] ...... 169

Shear and Extensional viscosity for PEO WSR 301 with molecular weight of 4000 kg/mol as tested by Lindner et al.

[96] ...... 169

xvi

LIST OF TABLES

Classification of breakup regimes[5, 6] ...... 3

Newtonian properties of the liquids used to determine the ...... 19

Percentage difference between the experimental and numerical crown diameter ...... 36

Testing sensitivity of the value of C in equation (2-9) w.r.t experimental drop impact of an ethylene glycol drop of

4.56 mm impacting at 2.05 m/s ...... 37

Properties of the experimentally tested liquids ...... 45

Grid Independence test: relative percentage error in numerical breakup length between consecutive meshes is checked...... 52

Validity of the numerical solution: percentage difference between the experimental and numerical breakup length.54

Images showing (a) deposition (no splash), (b) prompt splash and (c) crown (delayed) splash along with respective properties...... 74

Propertied of the liquids used in the experiment ...... 79

Grid independence test: Relative % difference between consecutive meshes ...... 104

Validating the numerical scheme: difference between the computational and experimental crown diameters (EG : ethylene glycol, x PG: x % by volume propylene glycol water mixture) ...... 106

Properties of working liquids used in the experiment...... 129

Range of experimental input test parameters ...... 130

Liquid characteristics of the five liquids tested ...... 148

Properties of the polymers used in the current experiment ...... 150

xvii

1 CHAPTER 1: INTRODUCTION

The disintegration of liquid jets due to their hydrodynamic instability is an integral part of many industrial applications. The ability to control this disintegration enables precise drop size distribution where it is desired, e.g. in chemical deposition, aerosol science, combustion, and spray coating and cooling. The impact of these generated drops on the target surface is again of great significance in these industries. Depending on the technological purpose, drop splashing on impact at the surface may or may not be desired. This current research work aims at investigating the disintegration and subsequent drop generation from liquid jets followed by inspecting the dynamics of droplet impact. Thus, this endeavor can be broadly divided into two components: jet disintegration, and drop impact dynamics. In this chapter, a brief introduction to both the components is provided with the aim to present a small background and a succinct description of the current understanding of the physical phenomena.

1.1 Jet breakup and the resultant drop distribution

As an uninterrupted cylindrical column of liquid exits from a nozzle tip, the continuous liquid jet develops some surface disturbances as a result of the forces acting on it. The induced perturbations on the jet could either grow or be repressed based on the growth rate of the initial perturbation. Under appropriate conditions, the initial disturbance gets amplified and this eventually leads to the breakup of the continuous column of liquid. The breakup is facilitated by the system trying to attain a minimum energy state as the liquid column grows. This leads to the breakup of the long jet into smaller liquid sections thereby lowering the surface energy (by lowering the surface area) of the total system. The properties of the liquid jet breakup that are of most relevance are the amplitude of surface disturbance, the jet breakup length (the length of the

continuous column of liquid prior to disintegration), the breakup pattern, and the resultant drop size. Generally, the outcome of liquid jet breakup is primarily dependent on some key parameters such as the surface tension, viscosity, density of the liquid, and, the velocity and diameter of the incoming jet.

To understand the effect of axisymmetric disturbance on the liquid jet, linear stability theory was first proposed by Rayleigh [1]. Further investigations have consequently modified this to include effects of viscosity and aerodynamic drag [2]. These considerations of the linear stability theory are very helpful in qualitatively determining the pattern of break-up and thus, characterizing the different breakup regimes. The four regimes of break-up are well-outlined by Lin and Reitz [3].

For low incoming jet velocities, long wavelength and small amplitude disturbance primarily cause the surface disturbance that eventually break up the liquid jet. This occurs in the Rayleigh and the first wind-induced regimes of break-up. The second wind-induced and atomization regimes are characterized by short unstable waves. Figure 1-1 shows a schematic representation of the four modes of breakup as was depicted by Faeth in 1991 [4]. For the Rayleigh regime, the resultant primary drops have diameters of about twice the incoming jet; while for the first wind-induced regime these are of the order of the diameter of the jet. The other two regimes result in drops much smaller than the jet diameter. While linear theory helps estimate and predict the primary drop breakup from the jets, the secondary drops formed (if any) are not well understood by this theory.

Secondary drops are sometimes formed during breakup in addition to the primary larger (main) drop. This typically occurs, when the main drop leaves behind a small filament of liquid as it pinches off from the main column. This filament, in turn, forms a smaller drop known as satellite or secondary drop. Nonlinear analysis of the breakup phenomenon could provide some insight into satellite droplet features of the jet breakup process [3]. The criterion for the transition between

the breakup regimes has been investigated by a few researchers. A summary of these criteria is presented in Table 1-1.

Table 1-1: Classification of breakup regimes[5, 6]

Breakup Regime Formation Drop Criteria

Mechanism diameter

characteristics

Rayleigh Surface Tension Ddrop ~ 2Djet 푊푒퐿 > 8 and

푊푒퐴 < 0.4 or

< 12 + 3.41 푂ℎ0.9

0.9 First wind-induced Surface Tension and Ddrop ~ Djet 푊푒퐴 < 12 + 3.41 푂ℎ

relative motion of jet 푊푒퐴 < 13

and ambient air

Second wind-induced Relative motion of jet Ddrop < Djet 13 < 푊푒퐴 < 40.3

and ambient air

Atomization Not clearly identified Ddrop << Djet 푊푒퐴 > 40.3

2 𝜌푎푉 퐷푗푒푡 ∗ 푊푒 = 퐴 𝜎

Rayleigh First wind- Second wind- Atomization induced induced

Figure 1-1: Schematic Representation of the jet breakup regime. [4]

Another discernible means to distinguish the different regimes of breakup is provided by the nature of the curve in the breakup length vs jet velocity plot (Stability curve – Figure 1-2).

Several experimental and numerical investigations have helped establish the general nature of this curve. At low velocities, the breakup length increases linearly with the velocity. This is typical of the Rayleigh breakup regime. Beyond the maxima of the curve (where maximum breakup length is obtained – section A of Figure 1-2 ), the breakup length decreases with increasing velocity (first wind-induced breakup regime section B of Figure 1-2). At higher velocities, second wind-induced regime (section C of Figure 1-2) is observed where the drops are much smaller than the jet diameter. Beyond this, as the velocity increases, the jet disintegrates very close to the nozzle exit and atomization takes place (section D of Figure 1-2).

Rayleigh First wind- induced Second wind- induced

Atomization Jet breakup length (m) length Jetbreakup

Velocity (m/s)

Figure 1-2: Stability curve showing the typical variation of breakup length with jet velocity[3]

In this current work, we are essentially interested in low-velocity liquid jets and this corresponds to the Rayleigh-like regime of break-up described above.

1.2 Drop impact dynamics

In many applications (coating, spraying) the resulting drops from a drop generator (e.g. controlled laminar jet break-up – Rayleigh regime) impinge on the target surface of the application.

The consequence of such an impact depends on a number of elements: thermodynamic state of the drop, the liquid property of the drop (viscosity, density, and surface tension), velocity and diameter of impacting drop, properties of the surrounding ambiance, and the target surface characteristics.

The generic possible aftermath of drop impact can be classified into deposition and splash.

Deposition occurs typically for low-velocity drop impacts on a wetting surface. The drop spreads on the target surface and as it spreads the drop does not break up into smaller fragments. In the

event of a splash, however, the drop may spread on the target, but as this occurs, parts of the drop not in contact with the surface begin to break off from the main liquid resulting in smaller drops ejected as a result on impact and thus causing a splash. This typically is seen to occur for high- velocity drops and impact on hydrophobic surfaces and rough surfaces. As the drop spreads on the target surface, the liquid not attached to the target surface forms a corona or a crown. If the splash, i.e. drops breaking off from the main drop, occur before the formation of this corona, the phenomena is termed as prompt splash, otherwise, it is termed as corona or crown splash. Rein[7] and Yarin[8] have in-depth literature reviews on the multiple aspects and considerations in the field of drop impact dynamics. The different considerations in drop impact studies are schematically represented in Figure 1-3.

Liquid drop shape

Drop impact direction

Target Surface

Solid Surface Thin Liquid Film Deep Liquid Pool Rough

Smooth

Figure 1-3: Basic factors influencing drop impact

Usually, most numerical studies and experimental analyzes treat the falling drop as a sphere. This, however, is inaccurate due to the aerodynamic effects acting on the drop as it travels

through another fluid which tends to make the drop more elliptical than spherical. Different generation techniques can be used to produce the impacting drop. When impact studies are conducted, they can be associated with a single drop impact or with a stream of drops impacting the target surface. In the latter case, caution is taken to ensure that the drops are produced at a low frequency so as to ensure that generated drops are not affecting the impact of the preceding drops.

Drop generation techniques, such as laminar jet break-up, are a function of surface oscillations.

These oscillations could also transfer to the drop thereby causing some changes to the movements and the internal flow within the drop. While there are studies that look at these effects on the impact[9–11], the phenomena of drop impact is so intricate and has a multitude of contributing considerations that as an approximation, a majority of the available literature consider the impacting drop to be spherical and steady.

Based on the application and the drop generation technique (sprays, atomizers, laminar jet breakup, etc.), the drop could impact the target surface either obliquely or normal to the surface. The angle of impact heavily influences the post impact dynamics. In addition, especially in experimental studies, velocity measurement techniques too can influence the outcome of impact.

While some studies consider the direct measurement of drop velocity a few instances prior to impact, some others rely on the height of release of the drop. In the latter case, terminal velocity is calculated based on various available literature and correlations. The drop, however, may not always attain terminal velocity prior to impact and this could be a significant contributor to the uncertainty in the experimental measurements.

Target surface characteristics are another major factor for consideration in classifying drop impact dynamics. Typically, the velocity, as well as the mass of the impacting drop, is small enough that the elastic response of the solid surface is neglected. The influence of surface

roughness and the shape of the surface have been studied extensively by many studies [7, 11–15].

Generally, rough surfaces tend to enhance splashing of the drop while a smoother surface tends to curb splashing to some extent. The onset of the corona splash and prompt splash, the angle of the ejected fingers of the corona (crown), the dynamics of the ejected splash droplets are some of the areas that have been well examined in these studies.

When a drop impacts in a deep stagnant pool, in addition to splash and deposition, there are other possible outcomes such as floating or bouncing drops. Also, post impact dynamics of the receiving liquid pool are extremely diverse and interesting. The instability caused by the impact could lead to propagating surface waves around the entrainment region. Another interesting phenomenon that occurs when deep craters are formed by the impacting drop in the liquid pool is the Worthington Jet. When the liquid crater formed collapse back towards the center, a single long column of liquid called the Worthington Jet could form. Examining the characteristics of this is another vast field of study.

Drop impact studies on solid targets and deep pools have received a lot of attention. In many industrial applications that involve coatings and depositions, typical drop impact occurs on a thin layer of liquid formed from prior drops. This current research work concentrates on analyzing this form of impacting drops with the desire to contribute more to its understanding. Few researchers have examined drops impacting thin liquid films. While the outcomes are similar to the other drop impact characteristics, such as deposition, crown formation and splashing in general, the factors dictating the post-impact dynamics are not well understood. In the current work, experimental studies along with numerical modeling of the impact dynamics are pursued with the aim to gain a better insight into the mechanism of development of drop impact on thin liquid films.

1.3 Scope of the current research

This current research work aims at investigating the disintegration and subsequent drop generation from liquid jets followed by inspecting the dynamics of droplet impact, using both experimental and computational methods.

1. The numerical study of these complex phenomena requires accurate capture of the interface

and the related forces acting on it. Chapter 2 explores the numerical modification to VoF

method. The original volume of fluid (VoF) formulation is noticed to cause spurious currents

and leads to incorrect interface calculations for highly viscous liquid. A method to resolve this

issue by modifying the property averaging at the interface is proposed. This modified

numerical scheme is observed to provide better numerical solution for jet disintegration and

for drop impacts on thin liquid films. The validity of the proposed modifications are tested

with four test case studies.

2. Chapter 3 concentrates on reporting analyzes of the liquid jet disintegration and the consequent

drop generation. Prior experimental observations have shown the generation of satellite drops

during low-velocity Newtonian jet break-up. The underlying factors governing this are found

to be a combination of liquid properties and issuing jet parameters. A numerical study to

identify and quantify the influence of these parameters is undertaken. Two modes of pinch-off

of the low-velocity liquid jet disintegration leading to non-uniform drop generation are

identified. These two modes are also noted to affect the jet breakup length. The flow domain

analysis in combination with experimental measurements helps delineate the variation in

breakup length for these observed modes of pinch-off.

3. An experimental investigation to identify the factors governing post-impact dynamics of the

phenomena is explored in Chapter 4. Experiments are carried out with varying drop sizes

ranging from 3.5 mm to 5.2 mm and impact velocities ranging from 1 m/s to 3 m/s. Four

different liquids are used to study the effect of liquid properties on the phenomena of splashing.

A high-speed digital camera Hi-D cam – II version 3.0 – (NAC Image technology) is used to

capture the phenomena of splashing. The threshold of splashing is found to be related to drop

size, impact velocity, liquid properties and thin film thickness. Experimental analysis shows

the significance of inertial, viscous and capillary forces in determining the splash/no-splash

limit and a predictive empirical relation for the same is established.

4. Chapter 5 outlines a numerical investigation on the factors controlling the propagation of the

drop as it impacts of a film of liquid. The influence of viscous and surface tension forces is

isolated and studied using the modified numerical technique. These observations augment the

empirical relation established in Chapter 4. The influence of the liquid film height is analyzed

using the numerical formulation as experimental control of the same can be challenging. The

flow domain examination suggests a variation in the influencing factors governing the

development of the ejected jet (the crown) with the change in liquid film height. This study

offers a basis to define “thin-films” based on the height of the film.

5. Characterization of secondary drops formed as a result of splashing are experimentally

explored in Chapter 6. Analyses with high-speed camera imaging of drop impact dynamics of

different Newtonian liquids with varying impact conditions is used to characterize the impact.

Sauter mean diameter (SMD), velocity of the resultant secondary drops, the crown height and

development time are some of the characteristics that are investigated. Empirical correlations

to predict some of these parameters of interest are found to be dependent on the impacting

Weber number (We) and the liquid property.

6. Chapter 7 explores the temporal characteristics of Newtonian drop deposition as they impact a

thin liquid film. Experimental study using high-speed camera imaging attempts to isolate the

influence of surface tension, inertia and viscous forces on the crown growth and deposition.

These finding pave the way for understanding non-Newtonian (shear-thinning) influence on

drop impact dynamics. Shear thinning rheology for two polymeric solutions (HEC 250 HHR

and PEO WSR 303) are characterized and their influence on the temporal development of the

impact is studied. Current experimental study along with observations in past literature

highlights the significance of extensional rheology in addition to shear rheology.

2 CHAPTER 2: MODIFIED VOLUME OF FLUID METHOD FOR TWO-PHASE

FLOWS

1.4 Introduction

Numerical work on multiphase phenomena has steadily advanced to provide insights into analysis of the underlying physics. While the significance of experimental analysis cannot be undermined, to ensure reliable and accurate measurements in these multiphase flows, the multiple scales involved need to be quantified. In addition to high speed measurements, this demands the use of highly sensitive and non-invasive techniques to completely resolve different contributing spatial and temporal aspects. The capability of numerical methods to observe detailed hydrodynamics in addition to providing fast highly resolved solutions has provided tremendous contribution to advancement of research and technology. Often numerical and experimental techniques are required to form a complete understanding. Berberovic et al. [16] used numerical techniques along with experimental observations to understand dynamics of drop impact of liquid surfaces and have characterized formation and propagation of the surface wave during impact.

Extensive investigation of dynamics of confined bubbles in laminar micro channel flow by

Khodaparast et al. [17] too required integration between experimental and numerical analysis.

Similarly, research in jets and atomization characterization [18–20] has advanced with the help of numerical techniques. In addition to flow characterization, coupled heat transfer solution has been implemented by Trujillo et al. [21] to study drop impact heat transfer. These studies and many other similar ones have used an underlying fluid flow multiphase (or two-phase) numerical method and have added required addition to incorporate other factors such as temperature, concentration, porous flows, etc. In these studies, the treatment of the flow physics forms the foundation for the

numerical solution. For accurate numerical study of such phenomena that are found to be of particular benefit in applications associated with liquid breakup, atomization, liquid collisions and entrainment; precise capture of the liquid and gas interface is essential. Interfacial tracking methods in incompressible fluids include front tracking[22, 23], boundary integral[24], volume of fluid

(VoF)[25] and level set (LS)[26, 27] methods. In the past decade, the techniques that have received the most attention are the last two – VoF and LS – due to their robustness and ease of implementation. As the interface evolves in a fixed computational mesh, both these methods use a separate parameter to identify the different phases in each cell in the domain. The primary difference in these methods comes in their treatment of this phase parameter. In the VoF method, an advection equation governs the fraction of volume of the primary phase as the interface moves in time and space. This makes the interface discontinuous and therefore, reconstruction of the calculated interface is required to obtain a smooth curve for curvature and normal estimations at each interface location. The interface is typically smeared over a few cells owing to the change in volume fraction as it goes from one phase to the other. In contrast, the LS method defines the interface by a smooth continuous function and therefore eliminated the requirement for reconstruction. As the method identifies the interface as the zero level of a signed distance function, the interface solution is sharp unlike in the VoF method. However, though accurate estimations of the interface topology are achievable, the LS method does not inherently conserve mass by the advection step unlike the VoF method. This makes the VoF method more favorable to use and has been found use in many commercial solver codes. This is the method under consideration for the current study. As interface tracking is very sensitive to the reconstruction,

VoF technique is seen to yield acceptable results only for certain applications and working liquids and some efforts have been taken to improve the interface tracking for this method. One such

scenario where this VoF method was found to produce errors in numerical solution is for viscous

Newtonian liquids in drop generation and impact studies. Prior numerical studies that use VoF techniques have primarily looked at fluids with viscosities similar to water. As the viscosity if the working fluid is increased, the error in numerical prediction has been observed to be greater. This current study aims to provide a modification to the interface reconstruction thus helping to reduce this error. Validation studies to test the suggested modification to the VoF method are conducted with available experimental evidence. In this study, the VoF method used is based on the open source two-phase solver present in OpenFOAM®. This package is well suited to handle complex geometries and is easily parallelizable without the limits of licensing. It is based on C++, where equations can be in a form that has a close resemblance to its mathematical equivalent. This VoF interface tracking solver was first implemented in 1999 by Ubbink & Issa [28]. A detailed summary of the original solver has been presented by Deshpande et al.[29]. Modification to the interface reconstruction is implemented to ensure better predictions for viscous working liquids.

1.5 Numerical Procedure

The numerical procedure to solve two-phase incompressible flow as used in OpenFOAM 2.2.1 is discussed here. This is the version of the open-source solver that has been used throughout this study. The original two-phase incompressible solver available in this is interFoam, and as noted by Deshpande et al. [29], this can be employed in a number of problems reliably. The performance of this original solver was noticed to drop when the viscosity of the working fluid is high. A correction to help with this concern is proposed here and the validity of the same is tested in the next section. This VoF solver is an eulerian solution that uses finite volume discretization of the governing equations on a fixed grid in the domain. All the flow variables are stored in the cell center in this finite volume technique.

1.5.1 Governing Equations

The current numerical technique considers two immiscible fluids (a gas and a liquid) that are separated by a sharp interface. However, the two fluids will not be solved individually with their different properties. Instead, a single set of equations are solved considering one fluid media with properties and variables differing continuously from one fluid to the other. The concerned fluid media are distinguished by a phase fraction property that has values between 0 and 1. In the current solver, 1 is identified as liquid and 0 as gas. The interface is defined by the region where the value of this phase fraction transitions from 0 to 1. Thus, the interface is smeared here and is not sharp.

The definition for this phase fraction (α) is as follows.

Volume of primary phase (2-1) 훼 = Total control volume

1 in primary fluid (liquid) where, 훼 = {0 < 훼 < 1 in transition region (interface) 0 in secondary fluid (air)

As this phase fraction identifies the fluid, it is used to determine the property of the fluid in each computation cell as under.

휑 = 훼 휑1 + (1 − 훼)휑2 (2-2) where, 휑 is the property of the fluid and 훼 is the phase fraction in the cell and subscripts 1 and 2 represents liquid and air respectively.

Therefore, the properties in each cell are computed using the above equation at each time and space. The transport of this newly defined fluid bulk property (phase fraction, α) is defined by an advection equation:

휕훼 (2-3) + 훻. (v̅훼) = 0 휕푡

In addition to this, this one-fluid finite volume solver uses the continuity and momentum equations to resolve the flow field. For the sake of this study, incompressible flow solution is interest and hence energy equation is not added. The properties such as density and viscosity in these two equations (are evaluated based on the phase fraction (α).

continuity: ∇. (v̅) = 0 (2-4)

∂(𝜌v̅) momentum : + ∇. (𝜌v̅ v̅) = −∇푝 + ∇. T̅ + 푓 + 𝜌g̅ ∂t 푠 (2-5) where, T̅: viscous stress tensor and

푓푠: volumetric force due to surface tension

It is to be noted that the three equations here are solved on a control unit volume. Hence all the components are volumetric components. As the current modulation ignores fluid elasticity, the stress tensor in the momentum equation comes from the macroscopic friction between adjacent fluid moving at different velocities and therefore, is the viscous stress component. The surface tension force term in the momentum equation calls for special consideration as it is not inherently a volumetric term like the other components in the equation. One way to resolve this would be to use this force as a boundary condition on the free surface and the surface pressure is obtained by linear interpolation between the surface pressure required and the fluid pressure inside the interface. This requires multiple iterations to obtain surface pressures within some tolerance interval with respect to that at previous time step. After the correction of the surface pressure, the momentum and continuity equations are re-solved before advancing to the next time step. This was the original VoF method suggested by Hirt and Nichols [25] in 1981. In addition to higher computational time due to the iterative surface pressure correction, this required approximate prior

knowledge of the interface shape that was to be obtained from upstream and downstream cells.

Brackbill et al.[30] in 1992, proposed a means to resolve the problems associated with this surface treatment by converting the surface tension forces to an equivalent volume force that can be added to the Navier-stokes equation as an additional body force term. Their method called continuum surface force (CSF) calculates this equivalent volumetric surface tension force as:

푓푠 = 𝜎 휅 풏 훿푠 (2-6) where, σ is the surface tension coefficient, κ is the interface curvature, 퐧 is the surface normal and δs is a Dirac delta function that assists with concentrating this calculation on the interface.

In this model, the interface curvature κ is determined as a function of local gradients of the surface normal, 퐧 which in turn is a function of the phase fraction (α).

휅 = 훻. 풏̂ and 풏 = 훻. 훼 (2-7)

Tang and Wrobel [31] show how this surface tension model can be written in terms of pressure drop across the interface and expressed in terms of a volume force in the momentum equation. They normalize the interface curvature by using the volume averaged density (ρ).

𝜌 (2-8) 푓푠 = 𝜎 휅 훻. 훼 0.5(𝜌1 + 𝜌2)

This method, being easy to implement, is used in most multiphase research work. It must be however noted, that the numerical solution would depend of the estimation of the interface normal,

퐧̂ which in turn, is a function of the phase fraction. Thus, correct estimation of these factors determines the accuracy and performance of the numerical solution.

1.5.2 Modifications to the VoF solver

One significant cause for incorrect estimation of interface curvature and normal, is from deviations in estimations of fluid properties across the smeared interface. Inaccurate interface calculations, in turn, would result in surface tension force calculations that are unreliable and thus cause failure in predicting correct interface pressure gradients. To resolve this difficulty in interface tracking, there have been past studies that have either looked at correcting the Courant number estimation, like

Beerners et al.’s [32] work, or some others that have proposed alternative methods to estimate interface curvature and therefore surface tension force [33], while still others have looked at better meshing techniques to resolve the computational domain in a dynamic manner [34]. In our preliminary analysis of the complication in correct interface estimation, it was found that the relative viscosity of the two phases in question was an underlying source of inaccuracy. Thus, an attempt to correct this is made by proposing a modification to the interface property averaging technique. Multiple experimental studies are tested with a generic exponential property averaging equation as defined by equation (2-9). The experiment used for this analysis is the dynamic development of the crown on impact of a liquid drop on a thin film of the same liquid (Chapter 4 and 5). Numerical solutions duplicating the experimental results obtained using different working liquid Newtonian properties (Table 2-1) are used to identify the correct expression for the constant

C in this exponential property averaging. The results obtained are then correlated with their relative viscosity ratios and this is shown in Figure 2-1. Thus, a property estimation equation that depends on the viscosity of the working fluids is determined and is given in equation (2-10).

퐂 휑 = 휑2 + (휑1 − 휑2)훼 (2-9)

휸ퟏ ퟏ.ퟓ( ⁄휸 )+ퟎ.ퟕퟓ (2-10) 휑 = 휑2 + (휑1 − 휑2)훼 ퟐ

where, 훾 is the kinematic viscosity of the fluid in consideration.

Table 2-1: Newtonian properties of the liquids used to determine the

Density Viscosity Surface Tension Liquids (kg/m3) (Pa.s) (N/m)

Water 998.0 0.001 0.0728

25% by volume of Propylene Glycol 1007.5 0.00255 0.0541 (25% PG)

50% by volume of Propylene Glycol 1017.0 0.005 0.0452 (50% PG)

75% by volume of Propylene Glycol 1026.5 0.012 0.0411 (75% PG)

Ethylene Glycol (EG) 1113.2 0.0161 0.0484

Though the current modification to the VoF method was derived based on an experimental study of drop impact on thin films, this is tested with other two-phase flows as discussed later. An example of the significance of property estimation equation used is shown in Figure 2-2. The plot shows the variation in interface density in the x-axis, between air, whose density is 1.225 kg/m3 to that of a viscous liquid (ethylene glycol), whose density is 1113.2 kg/m3. As the phase fraction (α) in the y-axis goes from 0 to 1, the interface moves from air to that of the liquid. As seen, the variation for the exponential averaging equation (red line) is not as gradual as the arithmetic averaging (green line). This ensures a sharper interface that is not smeared over multiple cells, thus

providing better estimates of the interface curvature and therefore, surface tension force and interface pressure drop.

where, x % PG: x % by volume of propylene glycol and water mixtures EG: ethylene glycol

훾 훾퐿푖푞푢푖푑 1 = 훾2 훾퐴푖푟

Figure 2-1: Equation for the constant C in the property averaging equation (2-9) derived from

the different liquids tested.

)

(kg/m

Density

Phase fraction, 휶

Figure 2-2: Variation of fluid property (density) at the interface for arithmetic weighted

averaging (green) and the current method (red).

In addition to altering the property estimation, the curvature field is smoothened in accordance with Ubbinks et al.’s [28] work where they resolved to correct the smoothness in the phase fraction field in order to smoothen the interface curvature as the two are related by equation

(2-7). Their proposed equation involves averaging the phase fraction in each computational cell volume by area averaging with respect to the values at each face as shown in equation (2-11).

∑ 훼[푓푎푐푒] 퐴푟푒푎[푓푎푐푒] 훼[푐푒푙푙] = (2-11) ∑ 퐴푟푒푎[푓푎푐푒]

Thus, the final equations solved in the modified VoF solver are the continuity, momentum and phase transport equations with the surface tension force determined by equation (2-8). The properties in this one fluid numerical approach (VoF) are estimated as per equation (2-10) and the

interface curvature is further smoothened by area averaged smoothing of the phase fraction as in equation (2-11).

1.5.3 Solution Procedure

The equation set described in the previous section require to be converted into equivalent partial differential equations (PDEs) to be solved iteratively on the computational grid as no analytical solutions exist for these. The process of converting each of the spatial and temporal term in the equation to make them PDEs is called discretization. The spatial terms are discretized using upwind scheme. These variables are calculated and the average values are stored in the cell center.

Reconstruction from these averaged values is required to evaluate the cell interface values which need to be used for the next iteration in the solution process. Limited linear piecewise reconstruction is used to diminish large variations during this reconstruction. The PIMPLE algorithm (combination of PISO and SIMPLE) is used as the iterative procedure for coupling mass and momentum conservation. For this coupling, within each time solution, this algorithm solves the pressure equation while invoking an explicit correction to velocity. Optionally, each iteration step can begin with a solution to the momentum equation. This is called the momentum predictor loop and is set to loop 2 times for each step. The linear solvers used in the PIMPLE method are preconditioned Conjugate Gradient for pressure and a preconditioned Bi-Conjugate Gradient for velocity components. As a transient solution is the goal of these numerical solutions, an implicit

Euler time discretization is used. This first order implicit scheme is preferred to an explicit scheme as this guarantees boundedness and is unconditionally stable. For providing good numerical stability, adaptive time stepping is used where at the beginning of each time loop, the time step Δt is calculated dynamically using the following criteria:

Co Co ∆푡 = min { max ∆푡표; (1 + 휆 max) ∆푡표; 휆 ∆푡표; ∆푡 } (2-12) Coo 1 Coo 2 푚푎푥

In the above criteria, ∆푡푚푎푥and Comax are predefined values for maximum permissible time step and maximum Courant number. The damping factors, λ1 and, λ2, are defined as 0.1 and 1.2 respectively. The superscript o refers to values at the previous time-step. The maximum Courant number is set to be 0.1 for the current numerical study.

1.5.4 Numerical domain and mesh discretization

The numerical problems used in this study to test the two-phase solvers are assumed to be axisymmetric. This assumption is an approximation based on experimental evidence and helps reduce the computational domain and therefore makes the numerical solution faster. The entire physical domain of the problem can be considered a cylinder with the axis representing the axis of symmetry. The numerical domain used for these problems is a wedge-shaped thin sliver (with a small angle, < 5o) of this cylinder. There is a small thickness in the z-direction that is typically one cell thick. This wedge straddles the X-Y plane and runs along the central axis of symmetry as shown in Figure 2-3 (a). The typical orthogonal structured non-uniform mesh for this domain is shown in Figure 2-3 (b). The significant phenomena under examination is typically around the axis of symmetry in all the problems considered. Therefore, the mesh is denser in this region than farther away. Likewise, the mesh is compact near the target surface (as in cases discussed in sections 2.3.2 and 2.3.4), to ensure accurate capture of the phenomena in these regions. Prior to analysis of the numerical solution, the density of the grid is varied till a grid independent solution is obtained.

Y Nozzle Nozzle Wall X

Y Atmospheric Outlet X

Axis of Symmetry

Axis of symmetry of Axis

(a) (b)

Figure 2-3: (a) the wedge-shaped numerical domain (b) typical mesh structure

1.6 Performance tests on modified VoF solver

All implicit multi-fluid numerical methods rely on interface capturing techniques that are known to be susceptible to generation of numerical instabilities and result in unrealistic interface flows.

When the phenomena in consideration is dominated by inertia, these numerically generated interface flows would not lead to unrealistic results. However, when capillary effects (surface tension forces) are significant in the phenomena, these unrealistic flows can lead to seriously flawed numerical analysis. To compare and test the efficiency of the two numerical VoF solvers - interFoam and the modified solver, four tests are performed and the results obtained are outlined in the following section. In all these four tests, surface tension forces are significance and dictate the phenomena in consideration. Therefore, the difference between the two solvers is more pronounced.

1.6.1 Stationary drop in stagnant zero-gravity ambience

The basic VoF scheme (in interFoam solver), provides an inaccurate approximation of interface curvature owing to large gradients and a relatively larger spread of the interface. Consequently, the error in curvature results in an imbalance between the pressure and surface tension forces that lead to formation of spurious currents in the domain. The proposed modified VoF solver aims at creating a sharper interface and thereby improves the curvature estimation. To evaluate the influence of the modified treatment of interface, a stationary 2D drop under absence of gravity in a medium of air is studied. The initial and boundary conditions for this numerical analysis is shown in Figure 2-4. A 4 mm liquid drop with surface tension of 72.8 mN/m and density 1000 kg/m3 is placed in stagnant air. The domain has a uniform grid distribution with a grid spacing of 0.001 mm. Two different drop viscosities (0.001 Pa.s and 0.01 Pa.s) are tested to see the influence of viscosity on the numerical generated current. These currents, also called parasitic currents, appear as vortices around the interface and, as shown in Figure 2-5, are also present when there are not external forces acting in the physical domain. This numerical error is not eliminated by grid refinement. In fact, as noted by Brackbill [30], the magnitude of these spurious currents could get amplified by a finer grid. Figure 2-5 shows the spurious currents around a stationary 2D drop in air in the absence of gravity. The observed currents for both interFoam and the modified VoF solver is shown for water and a more viscous (10 times that of water) liquid drop. It is observed increasing viscosity of the liquid, increases the intensity of the parasitic currents around interface of the stationary drop. The modified VoF solver, does not eliminate these currents, but the intensity of the same is noted to be greatly reduced. Thus, it is noted that a more accurate sharper approximation of the interface in the modified VoF solver can help reduce the spurious currents in the two-phase problem. To estimate influence of these numerically induced currents on the

domain, the kinetic energy around the drop interface is presented in Figure 2-6. As is observed, though the modified VoF solver does not eliminate the kinetic energy at the interface of this viscous drop, the numerically generated kinetic energy is an order of magnitude lower than that generated by the interFoam solver.

Y Atmosphere

X zero gradient velocity, zero gradient zero pressure and zero gradient zero velocity,

pressure and zero phase zerogradient phase

Zero gravity fraction

Outlet fraction Initial drop with zero velocity and d diameter

Axis drop

Target Surface zero velocity, zero gradient pressure and zero gradient phase fraction

Figure 2-4: Initial and boundary conditions for numerical solution of stationary drop under no

external forces.

휈푙푖푞푢푖푑 = 0.07 휈푎푖푟 (water)

휈푙푖푞푢푖푑 = 0.7 휈푎푖푟

interFoam VoF solver Modified VoF solver

Figure 2-5: Comparison of the two VoF solvers: spurious currents around the interface of a

stationary drop in ambient air in the absence of gravity

2 2 Kinetic Energy (kg.m /s )

interFoam VoF solver Modified VoF solver

휈 Figure 2-6: Comparing kinetic energy around the viscous ( 푙푖푞푢푖푑 = 0.7) stationary drop 휈푎푖푟

interface for interFoam and the modified VoF solver

1.6.2 Drop impact in liquid pool

For the second test, experimental findings of Tran et al. [35] on viscous drop impact on a deep pool with attention to the entrapped air layer is considered. The initial temporal dynamics is studied for these impact conditions with the different viscous liquids used in the experimental work. At low impacting velocities, the air layer underneath the impacting drop prevents merger of the drop and pool surface and the drop could bounce off [10]. Increasing the impact velocity was observed to aid entrapment of the air layer. The centerline depth as the drop merges with the pool (penetration depth) was studied as a function of time taken for rupture and was found to be linearly increasing with time. In addition, viscosity was noted to delay rupture time as higher the viscosity, more the

resistance to allow entrapped air to rise and rupture the interface. These experimental findings are presented in Figure 2-7.

20 cSt 10 cSt

5 cSt

Figure 2-7: Experimental data showing dimensionless penetration depth (L/R) as a function of

time of impact of drops with same impacting velocity (0.55 m/s) for 3 different viscosities [35].

Y Atmosphere

X zero gradient velocity, zero gradient zero pressure and zero gradient zero velocity,

pressure and zero phase gradient zero phase

gravity fraction

Outlet fraction Initial drop with velocity

vimpact and diameter ddrop

Axis

Deep Pool

Target Surface zero velocity, zero gradient pressure and zero gradient phase fraction

Figure 2-8: Initial and boundary conditions for numerical solution of drop impact on a deep

pool.

5 cSt

20 cSt

10 cSt

Figure 2-9: Performance of interFoam and modified solver against experimental findings [35] for

entrapped air rupture time during viscous drop impact on a deep pool

To test the effectiveness of the modified VoF solver against the data, a drop impact on a pool in similar conditions is numerically simulated using both the solvers (interFoam and modified VoF).

The initial and boundary conditions for this problem are shown in the schematic in Figure 2-8. A drop with diameter 1.9 mm and velocity of 0.55 m/s as provided by Tran et al. [35] impacts a pool of like liquid (10 mm deep) as shown in Figure 2-8. The mesh for this domain under consideration contains approximately 390000 cells. While analyzing the effectiveness of the two solvers, it is observed that interFoam predicts a much slower evolution of air entrapment and rupture than the modified solver. The findings from both the solvers are compared with experimental data provided by Tran et al.[35] in Figure 2-9. For all the three viscosities tested, the modified VoF solver is better

at predicating the entrapped air dynamics. At lower viscosities (5 cSt), the modified solver is more precise than at higher viscosities (20 cSt), though it is still similar to the experimental data. For all the three viscosities, penetration depth at rupture (τ = 0) are comparable with experimental data.

However, for the 20 cSt drop impact, time for rupture is shorter than that physically observed. As observed in the previous case study with numerically induced spurious currents in stationary drops, higher the viscosity more are the parasitic currents. Thus, with increasing viscosity, there is relatively more error in predicting the pressure and surface tension balance and thus the interface.

It is speculated that this causes the numerical analysis to be more accurate when viscosities are lower. However, comparing with the experimental data, the deviation is more acceptable than the original VoF solver.

1.6.3 Liquid jet breakup in stagnant ambience

Precise numerical characterization of capillary driven phenomena like liquid jet breakup requires correct estimation of surface tension and pressure forces on the interface. This becomes even more significant when low velocity liquid jets are taken into consideration. A circular jet breaking up in stagnant ambience, is governed by the interplay of multiple forces such as inertial, gravitational, aerodynamic, capillary and viscous. When the jet is fast moving, though surface tension (capillary forces) are a prominent cause for creation and propagation of surface instabilities, inertial and aerodynamic forces are most dominant in the interplay [3, 36, 37]. This balance however changes when the jet is slow moving (smaller, lower weber numbers). To investigate the accuracy of the two solvers in capturing the temporal growth of the surface instabilities, experimental data for

Newtonian low velocity viscous liquid jet breakup is used. A propylene glycol jet of initial diameter 1.499 mm emerges from a circular nozzle at We = 5 and disintegrates to generate drops in stagnant ambience under the presence of gravity. Experimental observations show the

phenomena to be symmetric about the axis and hence an axisymmetric wedge shaped mesh with non-uniform structured mesh with about 160000 cells is used for this study. The initial and boundary conditions for this problem are shown in Figure 2-10.

Y Atmosphere

X zero gradient velocity, zero gradient zero velocity, zero Nozzle pressureandphase zero inlet pressure and zero phase

fraction Atmosphere

Initial jet with velocity fraction

gravity vjet and diameter djet

Axis

Outlet zero gradient velocity, zero gradient pressure and zero gradient phase fraction

Figure 2-10: Initial and boundary conditions for numerical solution of jet breakup.

The nozzle wall has a no-slip boundary condition imposed on it. To emulate the experimental test conditions, the inlet has a uniform velocity inlet condition defined based on the prescribed inlet velocity from the syringe pump in the experimental study. At the outlets, the pressure is assumed to be atmospheric and zero gradient pressure is the prescribed boundary. Breakup length calculations and other numerical analyses are carried out after 0.2 s of computed real time. This provides time for numerical instabilities, if any, to deteriorate. Figure 2-11 show the time progression of surface instability on the propylene glycol jet considered. High-speed experimental images are compared with predictions from the numerical solvers tested. It is observed that while the original solver (interFoam) is unable to capture the generation and propagation of surface instabilities unlike the modified VoF solver. For water, this is however not the case and both the solvers agree with the experimental observation. As propylene glycol is 40 times as viscous as

water, the inaccuracies in the spurious currents generated are much larger for this liquid. These cause a longer unperturbed jet surface and therefore predict a much larger breakup length numerically. Better interface tracking methods help with this problem, and in this current study, the modification on the property averaging helps improve the interface.

(a) (b) (c) (a) (b) (c)

Figure 2-11: Comparison between (a) experimental, and numerical predictions of (b) interFoam

and (c) modified VoF for evolution of surface instabilities during circular jet breakup of a

propylene glycol jet of 1.499 mm diameter and We = 5. The time difference between these two

images is 2.5 ms.

1.6.4 Drop impact on thin liquid film

During drop impact on a thin liquid film of the same liquid, it is observed that viscosity primarily governs the spread on the thin film and surface tension governs the crown growth. These studies find applications in several industries ranging from coatings to pesticides. The correct interplay of

these two forces can give insights into the growth and spread of the impacting drop as well as help enhance or curb splashing as per the requirement of the application. Thus, determining the current numerical solution for these problems becomes critical. A similar wedge shaped axisymmetric domain (Figure 2-3 (a)) with structured mesh is used for numerical analysis of this problem. In the stagnant ambience, the drop of liquid with known impact velocity and diameter are initialized along with the stagnant thin film height and a schematic of the same is shown in Figure 2-12.

Figure 2-13 shows the time progressions of an ethylene glycol drop of diameter 4.56 mm impacting at a velocity of 1.89 m/s on a thin film. Successive images are at a gap of 2 ms and the temporal progression of the impact dynamics as predicted by the two solvers are compared with the experimental observations. While the initial spread of the drop on the film surface looks similar

(at 2 ms and 4 ms), the crown tips are noticed to be thinner in the numerical solution using interFoam. This thinner crown tip leads to a faster moving crown rim that predicts the pinch-off secondary drop from the crown rim at 6, 8 and 10 ms after the impact. The interFoam solver enhanced the magnitude of significance of the surface tension force unlike what is observed in the experiment. The modified VoF solver is able to correct this error and identifies the correct temporal and spatial dynamics of the drop impact dynamics as is seen in Figure 2-13. Comparison of the spatial dynamics is done by measurement of the crown diameter as the drop merges with the thin film (at 10 ms in Figure 2-13). The maximum difference between the solver predictions and that from the experiment for different Newtonian viscous liquids are compared in Table 2-2. The properties of the working liquid and the impacting drop are also provided in the table. The modified solver is in agreement with both the final crown diameter as well as with the temporal development of the crown growth as shown in Figure 2-13.

Y Atmosphere zerogradient velocity, zero X zero gradient velocity, zero gradient pressure and zero

pressure and zero phase fraction gradientfraction phase Outlet

gravity Axis Initial drop with velocity vimpact and diameter ddrop

Initial thin film with height H Target Surface zero velocity, zero gradient pressure and zero gradient phase fraction

Figure 2-12: Initial and boundary conditions for numerical solution of drop impact dynamics.

Experiment interFoam Modified VoF

Figure 2-13: Comparison of numerical prediction from the two solvers with experimental time

progression of drop impact dynamics (Successive image are 2 ms apart)

Table 2-2: Percentage difference between the experimental and numerical crown diameter

% % Surface Density Viscosity Vimpact Ddrop difference difference Liquid Tension (kg/m3) (Pa.s) (m/s) (mm) (mN/m) (interFoam) (Modified)

Ethylene 1113.2 0.0161 48.4 1.89 4.56 7.67 1.26 Glycol 25% Propylene 1007.5 0.00255 54.1 1.87 3.79 13.53 1.92 Glycol and Water 50% Propylene 1017.0 0.005 45.2 1.91 4.68 9.61 0.89 Glycol and Water 75% Propylene 1026.5 0.012 41.1 2.38 4.93 8.12 1.50 Glycol and Water

1.7 Sensitivity of value of C in the modified property averaging equation (2-9)

To test the sensitivity of the value of C in two-phase numerical solutions, the progression of an ethylene glycol drop of diameter 4.56 mm and impacting velocity of 2.05 m/s impacting on a target thin film surface is considered. The value of C in equation (2-9) is varied by 10% and the resultant crown diameter at 10 ms after impact is considered. In the experimental result of this drop impact, at 10 ms, the crown was observed to have almost merged with the thin film surface and this condition is taken for comparison. The below table 2-3 shows that a variation by about ±10% in the value of C increases the deviation from the observed crown diameter value.

Table 2-3: Testing sensitivity of the value of C in equation (2-9) w.r.t experimental drop impact

of an ethylene glycol drop of 4.56 mm impacting at 2.05 m/s

% difference from experimental Solver observation interFoam 7.67

Modified (equation (2-10)) 1.27

C + 10% 4.90

C – 10% 5.11

1.8 Conclusions

Two phase numerical analysis using the open source VoF solver, interFoam, available as a part of the OpenFOAM computational tool is tested for multiple two-phase flow problems. Though this solver has been used in many inertia dominated as well as surface tension dominated numerical analysis in literature, this current study finds that high viscous working fluids could lead to inaccurate numerical solutions. A modification to the property averaging technique used in the solver is proposed and this is observed to improve numerical solutions for high viscous liquids. In addition, curvature smoothing technique available in established literature have further helped the improve the numerical solution in high viscous, surface tension dominated phenomena. The governing equations in the solver along with the algorithm used for solution have been provided.

As has been observed in previous numerical studies, spurious currents are noted to be a key source of numerical error. For the original interFoam solver, the origin of spurious currents in noted to significantly increase for viscous working liquids. Though the proposed modification too sees this increase in spurious currents with viscosity, their magnitude is seen to be reasonably lower than in

the original solver. The numerical solution of a stationary drop with no external forces is used for this analysis. In addition, experimental data from available literature and past studies have been used to test the validity and applicability of the two solvers in consideration here. For all the tests performed the modified VoF solver is observed to perform better. For the cases of drop impacting on a pool and during drop generation from liquid jets, the time progression of the surface tension governed dynamics is observed to be slower in the interFoam solver. However, in the case of drop impacting on a thin liquid film, intricate interplay between the surface tension and viscous force is critical and here, the effect of surface tension is exaggerated in the original solver.

The proposed modification to the VoF solver is noted to provide better numerical solutions in the tested cases in this study. However, these validation tests have been conducted only for Newtonian liquids in ambient air. Further studies with non-Newtonian liquids are required to create a better understanding of the behavior of numerical solution with liquid viscosity. In addition, the current study concentrates on two-phase numerical solutions. Having observed the difficulty in computational predictions with the interplay of viscous and surface tension forces in two fluids, numerical solutions involving interaction between multiple viscous liquids would require more careful numerical treatment. Furthering the understanding of numerical solution techniques will help provide a more stable universal numerical solver.

3 CHAPTER 3: NUMERICAL ANALYSIS OF LOW-VELOCITY JET

DISINTEGRATION AND SUBSEQUENT DROP GENERATION

1.9 Introduction

A liquid jet is a column of liquid that splits or breaks up eventually, due to the inherent instability of the column. The scales on which these jets could occur, range from astronomical scales[38] to nanoscales[39]. This subject has attracted attention both for its applicability in diverse industries and because of the lack of a complete physical understanding of the phenomena. The incentive for study in this field stems from two related aspects of liquid jets: jet breakup and the resulting drop distribution. The practical applications of the process of jet breakup include, but is not limited to, biological sprays, agriculture, fuel combustion, ink-jet printing, firefighting, medicine, and cosmetics.

As a liquid stream develops in a quiescent ambiance, hydrodynamic instabilities lead to disturbances on the surface of the continuous column. When these disturbances grow, they tend to break up the column into drops. Liquid discharge along with its subsequent pinch-off is of great significance to a number of studies in meteorology, aerosol science, chemical deposition, and to the medical community. Most of these require homogeneous discharge that is dictated by the uniformity of the size distribution of the resultant drops while some others require an understanding of the expected drop size distribution. Many applications usually operate liquid jets at very low Weber numbers (We) to maintain strict control of the process. A uniform distribution of drop sizes can be obtained by dripping, drop-on-demand type techniques, or stable disintegration of continuous low velocity laminar liquid jets. For instance, while Berkland et al.

[40] coupled a carrier stream with acoustic excitation to exercise precise control on polymeric

sprays, piezo-ceramics was used by Wu et al. [41] in producing continuous monodisperse drops from a glass nozzle as a function of disturbance frequency and Reynolds number (Re). Another method to control drop distribution is seen in the work by Xu et al. [42] who used voltage control techniques. Most applications have non-Newtonian liquids as the working fluid and to understand their implications better, Ruo et al. [43] numerically examine the effect of viscoelastic stress, surface tension and electric field on the created surface disturbance and its propagation on the liquid column. This current study concentrates on understanding break-up and the consequent drop distribution from the stable disintegration of laminar low-velocity liquid jets. Though most engineering applications could involve complex geometries and liquids with varying rheological properties, quantitative analysis of a simple system will help provide a better understanding of the fundamental process of jet break-up. Hence, our current study investigates liquid jets, ejected from circular nozzle orifice into a quiescent ambient atmosphere.

Our earlier experimental investigation [44], focused on understanding the interplay of forces that act on a low velocity circular liquid jet as it breaks up in quiescent ambiance. The study uses different Newtonian liquids (pure and mixtures) and a range of diameters to understand the significance of liquid properties and liquid jet diameters on the breakup of the ensuing jet. Two modes of liquid jet disintegration were observed: drops pinching off from the continuous stream, and, liquid ligaments that pinch off from the main jet and further break-up to form drops. While the former mode of breakup is observed to occur at lower ensuing jet Re, the latter occurs at larger

Re. The existence of two modes of disintegration of liquid jets, suggests a transformation in the inertial balances within the system, and thus these two modes are termed: inertia-balanced and inertia-dominated regimes respectively. A semi-empirical correlation to predict the breakup length of the liquid jet has been suggested in this study. The inertia-balanced regime of the break-up was

found to be in accordance with the Rayleigh-type breakup regime that has been identified in past literature (resultant drops are uniform and twice the diameter of the issuing jet). A deviation from this characteristic Rayleigh-type breakup was observed for the viscous liquids tested at low We

(We < 30) and large jet diameters. Sample images of this observed distinctive behavior are shown in Figure 3-1 for a We of 5 and an issuing jet diameter of 0.84 mm. For water, uniform drops that have a diameter of about twice the issuing jet, are pinched off from the continuous liquid column, while, for the other two liquids (ethylene and propylene glycol) the breakup of the jet does not result in uniform drop generation. Intermittent satellite or secondary drops are seen to be generated by both these liquids. Ethylene glycol jets are observed to have a lower breakup length than water and the satellite drops generated merge with the oncoming drop, ultimately producing uniformly sized drops. Going by the same trend, propylene glycol, being more viscous, should ideally have a shorter breakup length than ethylene glycol. However, what is observed is quite peculiar. Not only is the breakup length longer, but also, the continuous stream is seen to stretch before breaking up. Also, the satellite drops formed do not merge with the oncoming drops thus resulting in an uneven drop size distribution. The current study seeks to create a better understanding of the physics of liquid jet breakup and the consequential drop generation. As experimental analyzes are unable to observe small-time dynamics as well as dynamic features of the breakup within the liquid

(e.g. pressure drop, vorticity), it becomes apparent that the current experimental findings need to be integrated with a thorough numerical investigation of the process.

Water Ethylene Glycol Propylene Glycol

0s 0.002s 0.004s 0s 0.001s 0.004s

Figure 3-1: Experimental observation show the significance of liquid properties on drop size

distribution at We = 5 and jet diameter = 0.84 mm for three different liquids

The numerical solution involves resolving the following two conservative equations: conservation of mass and momentum. These equations together, form the Navier-Stokes equations that describe the hydrodynamics of the flow under study while taking into consideration the inertial, pressure, surface tension and viscous forces. In general, these equations cannot be solved analytically and the need for numerical methods emerge. Since the flow considered here involves two fluids (air and liquid), the surface tension effects needs to be accounted for. Therefore, the interface between these two fluids needs to be tracked accurately. Two methods to numerically study and capture the interface that has received a lot of attention are the volume of fluid (VoF) method [25, 34] and the level set method [45]. Farvardin et al. [46], performed three-dimensional simulations using VoF techniques to study the breakup length variation with increasing We.

Srinivasan et al. [18] too used the VoF methodology to study unperturbed cylindrical liquid jets, to capture the non-linear dynamics of the process. While Farvardin et al. [46] used water as their working liquid, in Srinivasan et al.’s work [18], water and ethanol have been used. Some others [26,

27, 47] have applied the level set method in a Eulerian domain to capture the two-phase interface dynamics during primary breakup. In this current work, the VoF method used is based on the open source two-phase solver present in OpenFOAM®. This package is well suited to handle complex geometries and is easily parallelizable without the limits of licensing. It based on C++, where equations can be in a form that has a close resemblance to its mathematical equivalent. This solver uses the VoF method for interface tracking. This was first implemented in 1999 by Ubbink & Issa

[28]. A detailed summary of the solver has been presented by Deshpande et al. [29].

The next section briefly describes the experimental procedure that was used to further the investigation that was done in our earlier endeavor [44]. Following that, a summary of the VoF interface tracking method used in the numerical study is outlined. The evaluation of the findings of this work is presented subsequently.

1.10 Experimental procedure

Liquid jet is generated from a stainless steel needle that is attached to a constant mass flow rate syringe pump. The ejected liquid jet breaks up at some distance from the needle exit. To maintain the ambiance at stagnant conditions as well as to collect the liquid jet following its breakup, a closed collecting tank is provided. This also functions as a reservoir of the pumped liquid and is recirculated to the syringe pump. A schematic of the experimental setup is provided in Figure 3-2. A high-speed camera (Hi-D cam – II version 3.0 – NAC Image Technology) is used to capture the very fast process of jet breakup. The white screen is placed at the rear of the collecting tank and it is illuminated from the edge (using single bulb focusing light system (ARRI)

with glossy aluminum reflectors). This lighting set-up helps to obtain clear and high-contrast images that are then further processed in the computerized data acquisition system. Image analyzes and measurements are done by the image processing software – IMAGEPRO PLUS 4.0 (Media

Cybernetics).

Syringe White Screen Pump Lighting Needle

Computerized data acquisition system

Collecting Tank High-speed camera

Figure 3-2: Schematic of the experimental set-up

In addition to different stainless steel needles to obtain multiple liquid jet diameter, six different liquids were used in the experiment to understand the consequence of changing liquid properties on the liquid breakup process. The three pure Newtonian liquids used are water, ethylene glycol and propylene glycol. To obtain a range of properties in Newtonian liquids, three mixtures of water and glycerin by volume are used. To obtain correct values of viscosity and surface tension of these mixtures, measurements in a rheometer (TA instruments AR 2000), and a tensiometer (SensaDyne QC 6000) are taken both before and after the experiment. The values of

the reported properties of these liquids tested are tabulated in Table 3-1. The experimental parameters of testing are discussed in detail in our earlier study [44].

Table 3-1: Properties of the experimentally tested liquids

80% (v/v) 75% (v/v) 65 % (v/v) Ethylene Propylene Properties Water Glycerin Glycerin Glycerin Glycol Glycol Water Water Water

Density, ρ(kg/m3) 998 1113.2 1036 1219 1194.85 1153.8

Surface Tension, 72.8 48.4 36.6 67.2 64.3 69.5 σ(mN/m)

Viscosity, μ(mPa.s) 1 16.1 40.4 58 38 17.5

Capillary length, 2.728 2.096 2.153 2.371 2.342 2.478 lc(mm)

Morton Number, Mo 2.55×10-11 5.22×10-6 2.41×10-4 3.00×10-4 6.44×10-5 2.38×10-6

1.11 Numerical procedure and problem formulation

In the VoF interface tracking method, an indicator function is defined to identify the fraction of liquid in each cell. In the macroscopic physical world, the interface between two fluids is a discontinuous boundary and therefore the properties of the fluids undergo a sudden jump across the interface. In numerical schemes, a discontinuity in property jump is not particularly well handled. It is therefore not defined by a sharp limit and a transition region of some finite thickness exists. This transition region gets smeared over a few cells (2-3) in the mesh and the properties change continuously along this region. The source terms in the mass and momentum equations are

appropriately modified to allow for this gradual transition. Also, the surface force acting on the interface (surface tension) has to be incorporated as a volumetric term in the momentum equation.

As the solver is essentially based on cell volumes, this conversion of a surface force to an appropriate volumetric term becomes necessary. This is accomplished by the continuous surface force model proposed by Brackbill et al. [30]. This method too relies on the VoF treatment of the interface as a smeared region where fluid properties are varied continuously as given by the advective transport equation (3-1).

휕훼 (3-1) + 훻. (푣̅훼) = 0 휕푡

Here, the quantity α is known as the indicator function that helps identify the two phases of fluids present in the domain. It is defined as:

푉표푙푢푚푒 표푓 푝푟𝑖푚푎푟푦 푝ℎ푎푠푒 (3-2) 훼 = 푇표푡푎푙 푐표푛푡푟표푙 푣표푙푢푚푒

1 in primary fluid (liquid) where, 훼 = {0 < 훼 < 1 in transition region (interface) 0 in secondary fluid (air)

The local values of fluid properties such as density and viscosity are calculated using this indicator function. Typically, a weighted averaging of properties is used. If φ represents some property of the fluid, the weighted average used is as given in equation (3-3).

휑 = 훼 휑1 + (1 − 훼)휑2 (3-3) where, 휑 is the property of the fluid and 훼 is the phase fraction in the cell and subscripts 1 and 2 represents liquid and air respectively.

This form of averaging of fluid properties is appropriate for fluids whose properties are comparable. When one of the fluids has a larger fluid property value than the other, this averaging

of properties becomes biased towards the fluid property with larger value. This is seen by the green line in Figure 3-3. Figure 3-3, shows the variation of density of the fluid at the transition interface region as a function of the indicator function. The left-hand side of the axis depicts the volume fraction of liquid and the right-hand side shows that of air. The viscosity must transition from

1.225 kg/m3 for air to 1113.2kg/m3 for the liquid in a short interface of a few cells. The green line representing the weighted average in the interface region is seen to be highly biased towards the liquid (higher density). Some past studies have looked at different means to overcome this problem associated with the biased fluid property averaging. Beerens et al. [32] use a numerical scheme for heavy oils and include a viscous time scale during the estimation of the courant number (Co) to overcome the complication at the interface. Bohacek [33] studied high capillary number fluids (large viscous force versus surface tension) and they propose a new surface tension model to calculate interface normal and curvature using the geometry of the cell where the surface tension force was not imposed on the interface but rather on only cells containing the denser fluid. Other studies such as Mooney et.al [34], look at dynamic meshing techniques to help capture the interface in a better manner. In the current study, as viscosity is identified to be the property causing the difficulty in the estimation of interface property averaging, a means to include the viscosity of the two media in consideration is suggested as provided equation (3-4). This method to estimate the properties in the interface is reached at from the numerical study of drop impact on a thin liquid film (Chapter

5). The same formulation is observed to provide agreeable results in this current numerical examination. 휸 ퟏ.ퟓ( ퟏ ⁄ )+ퟎ.ퟕퟓ 휑 = 휑 + (휑 − 휑 )훼 휸ퟐ 2 1 2 (3-4) where, 훾 is the kinematic viscosity of the fluid in consideration.

The biased change in properties across the interface as in the case of arithmetic weighted averaging, as shown in Figure 3-3, is speculated to lead to incorrect numerical predictions. When 47

the relative viscosity between the two liquids is moderate, in the case of water, the arithmetic weighted averaging is seen to yield good predictions. For viscous fluids, however, this posed a problem and property estimation using equation (3-4) helps in better numerical estimations. This modified property estimation (for viscosity μ, surface tension σ, and density ρ) is used in the mass and momentum conservation equations along with the phase advection equation (3-1) to develop

the numerical scheme for the two-phase problem in consideration.

)

(kg/m

Density

Phase fraction, 휶

Figure 3-3: Comparing the effectiveness of arithmetic weighted averaging (equation (3-3)) and

new averaging technique (equation (3-4)) in computing density as a function of VoF phase

indicators.

mass continuity: ∇. (𝜌v̅) = 0 (3-5)

∂(𝜌v̅) (3-6) momentum continuity: + ∇. (𝜌v̅ v̅) = −∇푝 + ∇. T̅ + 푓 + 𝜌g̅ ∂t 푠

where, T̅: stress tensor and

fs:volumetric force due to surface tension

The exact description of solution procedure has been outlined by Deshpande et al. [29].To obtain and preserve the sharp interface during the VoF method, most other solvers use geometric reconstruction. OpenFOAM® algorithm uses a compressive flux model called interface compression which was introduced by Rusche [48], which helps limit the diffusion of the interface.

The PIMPLE algorithm of Patankar and Spalding [49] has been used as the solution method. Default discretization schemes are used: backward Euler in time, limited linear for velocity components, van Leer for the scalar quantities. The pressure-velocity coupling is done using the PIMPLE scheme, with four corrector loops. The linear solvers used in the PIMPLE method are preconditioned Conjugate Gradient for pressure and a preconditioned Bi-Conjugate Gradient for velocity components. For providing good numerical stability, the time step Δt at each instant was calculated dynamically using the following criteria:

Co Co ∆푡 = min { max ∆푡표; (1 + 휆 max) ∆푡표; 휆 ∆푡표; ∆푡 } (3-7) Coo 1 Coo 2 푚푎푥

The numerical domain consisting of structured orthogonal grids is employed on a wedge- shaped numerical domain shown Figure 3-4 (a). The flow is assumed to be axisymmetric and as

OpenFOAM® handles asymmetry as a quasi-3D case, the wedge-shaped computational domain is used. The boundary conditions for the axisymmetric wedge-shaped computational domain are

shown in Figure 3-4 (b). The nozzle wall has a no-slip boundary condition imposed on it. The length of the nozzle is the same as that used in the experimental study[44]. To emulate the experimental test conditions, the inlet has a uniform velocity inlet condition defined based on the prescribed inlet velocity from the syringe pump in the experimental study. At the outlets, the pressure is assumed to be atmospheric and zero gradient of pressure is the prescribed boundary.

Experimental observations show the phenomena to be symmetric about the axis and hence an axisymmetric wedge-shaped domain is assumed. Breakup length calculations and other measurements are taken after 0.2 seconds of computed time. This provides time for numerical instabilities, if any, to deteriorate.

(a)

Section A

(b)

Figure 3-4: (a) Wedge-shaped computational domain (b) Pictorial representation of the

boundary conditions in the Y-X plane

The mesh in the Section A (red box in Figure 3-4 (b)) is depicted in Figure 3-5. The grid near the axis of symmetry is uniform and, to lower the computational time cost, the grid density is higher than that further away. With the same grid density structure, multiple grid sizes were tested to check for grid independence and is tabulated in Table 3-2. This also provides information on the number of cells (# cells) in the computational domain for each of the tested mesh sizes. The relative error between two consecutive mesh refinements was used to test grid independence. As break-up length of the jet was the primary experimental parameter measured, the same is used to check for grid independence. The domain is decomposed and the simulation is parallelized, to help reduce the computational time. As the mesh size is increased, the computational time increases and the relative error is observed to decrease. For the lowest grid density, the relative error was about 10%, while for the highest mesh density, it was around 2.5%. Mesh 3 consisting of about 160,000 cells is utilized for obtaining results in the current study as this mesh was found to provide numerically accurate grid-independent results within an optimal computational time.

Atmospheric Outlet Atmospheric

Nozzle Wall Outlet Atmospheric Nozzle Wall Y

X AxisAxis of symmetryof Symmetry

Figure 3-5: Mesh at Section A (Figure 3-4 (b)) has uniform spacing through nozzle width and

region of liquid jet stream along the axis of symmetry; mesh is more relaxed further away.

Table 3-2: Grid Independence test: relative percentage error in numerical breakup length

between consecutive meshes is checked.

Diameter Weber Mesh 1 Mesh 2 Mesh 3 Mesh 4 Liquid (mm) Number % error % error % error % error

Ethylene Glycol N/A 8 5.13 2.30 0.64 20 # cells 38196 60144 94505 136596

Propylene Glycol N/A 6.72 4.60 2.49 0.84 10 # cells 31170 67485 105751 157547

Propylene Glycol N/A 6.81 4.26 2.78 0.84 20 # cells 31170 67485 105751 260000

Propylene Glycol N/A 8.61 5.64 2.45 0.64 30 # cells 38196 60144 94505 136596

1.12 Results and discussion

Prior to utilizing the numerical method to understand the complex behavior of the jet breakup and the consequent drop generation process, the accuracy of the obtained solution needs to be within agreeable limits. The experimental observations carried out in Rajendran et al. [44], are used to evaluate to the validity of the numerical solution used in this study. Breakup length is the length of the continuous column of the liquid prior to breakup and is a parameter that is a direct consequence of the manner of break-up. This parameter is therefore used as the parameter to test the efficiency of the numerical solution. Table 3-3 shows the observed percentage difference between the experimental and numerical observation of the breakup length for four different numerical simulations for the mesh size under consideration (Mesh 3 is chosen after the grid independence test - Table 3-2). The average uncertainty in the experimental measurements is

11.8%. The maximum percentage difference between the experimental and numerical breakup length is 15.19 % for a 0.84 mm diameter, We 20 propylene glycol jet. As this percentage difference is within limits of the experimental uncertainty, it is concluded that the numerical solution of viscous liquids is in agreement with the experimental observation. As an additional check, the time progression of the computational breakup process is compared with the experiment. Figure 3-6 shows the experimental observation in the white panels and the numerical solution in the dark panels. The progression of the surface undulations, as well as the time for pinch-off and the structure of pinch-off, are seen to be consistent. It is intriguing to note the shape of the pinched drops in the two cases (experiment and computation). It is felt that the shape of the pinched drops are affected by the treatment of the surface tension forces (as a volumetric force instead of a surface force) in the numerical method and hence do not match with the shape observed

in the experiment. However, the average diameter of the drops pinching off are in agreement with those observed experimentally.

Table 3-3: Validity of the numerical solution: percentage difference between the experimental

and numerical breakup length.

Mesh 3 % Liquid Diameter (mm) Weber Number difference

Ethylene Glycol 0.64 20 9.05

Propylene Glycol 0.84 10 12.2

Propylene Glycol 0.84 20 15.19

Propylene Glycol 0.64 30 12.06

Time 0 2 ms 3 ms

Figure 3-6: Validation of numerical method, by comparing, experimental results of Propylene

Glycol jet of diameter 0.84 mm and We = 10 (in the white panels), with the numerical results (in

the dark panel)

The underlying motivation to further the understanding of the process of breakup developed from the experimental observation highlighted in Figure 3-1 where the variation in liquid property is seen to alter breakup pattern. With to view to understand the significance of the key properties of the liquid, namely viscosity and surface tension, on the manner of jet breakup, numerical studies isolating the implication of each property parameter are carried out. Figure 3-7 depicts the influence of viscosity on the pattern of breakup and the drop generation method that follows from it. For this comparison, We is maintained a constant and Re is altered by altering the viscosity of the liquid in consideration. The left dark panels (Figure 3-7 a and b) show the time progression of the liquid jet breakup while the images on the right (Figure 3-7 c and d) show the

Time 0 0.002 s 0.004 s 0 0.4 0.8 1.2 1.5 Velocity Magnitude

Time 0 0.002 s 0.004 s

Re = 58.77 Re = 58.77

We = 10 We = 10

(a) (c) Time 0 0.001 s 0.002 s 0 1 2 2.5

Velocity Magnitude

Time 0 0.002 s 0.004 s

Re = 18.08

We = 10

Re = 18.08

We = 10

(d)

(b)

Figure 3-7: (a and b) Numerical results showing Mode A and Mode B ligament stretching for

differing liquid viscosities. (c and d) Predicted numerical velocity profiles for primary and satellite drops for liquid jets of different working liquid viscosities. The legend for velocities for

each of the liquids is shown above it. velocity contours for the two cases considered. Liquid jet breakup occurs when the surface perturbations grow to form a swell at the end of the continuous stream. The portion of the liquid jet adjoining the swell is narrow and this is where, ideally, pinch-off of the swelling would occur and result in the formation of the drop. The narrow end of the continuous stream would continue to grow till another sizable swell is formed and the process is repeated. For the lower viscosity considered (higher Re, Re = 58.77), prior to the large primary drop being pinched-off, the section at the end of the continuous jet develops a small swollen tip that is pinched-off to form a smaller secondary/satellite droplet. The formation of this smaller swollen bead is seen at the 0.004 s in

Figure 3-7 (a). This presents the first observed mode of pinch-off – Mode A. Here, the primary drops formed have a higher velocity than the smaller satellite drops and hence the satellite drops

merge backward with the oncoming drop (Figure 3-7 (c)). For the more viscous case considered

(lower Re, Re = 18.08), instead of drops, segments of the liquids containing multiple swells are pinched-off. This detached liquid segment has a sizable swell downstream and this eventually forms the primary drop while the thinner end of the liquid segment forms the smaller satellite drop.

This presents the second observed mode of pinch-off – Mode B. Here, as seen in the corresponding velocity contour in Figure 3-7 (d), the smaller drops have a larger velocity than the primary drops and merge forwards. As the viscosity is increased for the liquid, the surface perturbations have to work against a larger resistance to grow on the jet surface and ultimately pinch off the liquid segments from the main continuous jet. This leads to stretching of the liquid segments as they pinch-off. Thus, two modes of liquid breakup at low We that generate satellite drops have been identified: Mode A, where, the satellite drops formed are downstream of the primary drops and merge backward, and, Mode B, where, stretching of the liquid leads to faster satellite drop upstream of the primary drops and these merge forwards.

While the viscosity of the liquid is observed to restrain the growth of the surface perturbations, the preliminary recognizable effect of surface tension is in controlling the number of surface undulations or swellings in the continuous liquid column. Figure 3-9 shows the effect of modifying the surface tension of the liquid while keeping the other liquid parameters a constant.

For lower surface tensions (higher We, We = 25.43 (Figure 3-9 (a))), the liquid jet is noted to develop fewer swells in comparison to the higher surface tension (We = 20.34 (Figure 3-9 (b))).

During pinch-off of the liquid jet, the growing surface perturbations result in a swelling at the end of the liquid column. As the swelling grows, the liquid adjoining this would thin down. As the radius of curvature of this thinning section connected to the swelling decreases, the pressure gradient due to the surface tension increases (∝ σ and ∝ R-1). This pressure gradient along the axis

of the liquid jet facilitates the movement of liquid from the thinner section into the swell thus providing for the growing bulge. A drop finally pinches off from the thin section when the section becomes too thin to sustain the growing swelling. When surface tension of the liquid is increased

(Figure 3-9 (b)), it facilitates for a larger pressure gradient thereby creating more movement of fluid from the continuous stream to the swelling. This promotes larger ligaments containing multiple swelling to be pinched off from the main liquid jet. Thus, in addition to promoting more undulations on the liquid jet, surface tension also provides for thinning of the ligaments. These

Time 0 0.003 s 0.005 s

Re = 26.12 Velocity Magnitude We = 25.43

(a)

Time 0 0.003 s 0.005 s

Re = 26.12 Re = 26.12

We = 25.43 We = 20.34

Re = 26.12

We = 20.34

(b)

(c)

Figure 3-8: (a and b) Influence of surface tension on liquid jet breakup studied numerically by

varying the surface tension of the working liquid while keeping the viscosity the same.

(c)Velocity profiles of liquid drops as they breakup from liquid jets of liquids varying in their

surface tension. The legend for the velocity profile is provided. thin segments undergo pinching at both ends (Figure 3-9 (b) at 0.003s) and ultimately lead to the formation of satellite drops. When the surface tension is lower (We = 25.43), the segment pinching off results in the formation of one primary and one satellite drop. As seen in the corresponding velocity contour (Figure 3-9 (c)), the satellite drop is much faster than the primary drop and merges forwards (Mode B). For Figure 3-9 (b), a ligament with multiple swellings is pinched off. This further breaks up into multiple primary and satellite drops whose velocities are comparable (Figure

3-9 (c)) and thus do not merge within the computational domain, thus resulting in a non-uniform

drop distribution. As the satellite drops pinch off at the end of the ligaments, the wavelengths at which this occurs is much smaller than that of the primary drops. While Rayleigh’s linear stability theory [1] provides a means to understand the physics of the formation of these drops, the formation of the satellite drop at these small We is not explained. Pimbley and Lee [50] propose that while the linear stability theory is capable of explaining the primary drop formation, the secondary drop formation can be explained by considering a non-linear break-up.

Given the specific analysis of the individual contribution of viscous and surface tension forces, the source of the distinctive deviation from the Rayleigh-like breakup observed experimentally (Figure 3-1) is investigated. The surface tension of the liquid provides a stress on the surface of the liquid jet. A material with higher viscosity will strain lesser for a given applied stress. Consequently, observing the development of the strain rate of the liquid at the interface will help further the understanding of the distinctive deviation observed experimentally. Figure 3-9 and

Figure 3-10 depict breakup for a propylene glycol jet for two different diameters. The computational result along with the corresponding experimental images is provided. The experimental images were obtained at a frame rate of 2000 fps. Therefore, the time interval between two consecutive images is 0.0005s. The computational images shown in these figures are presented at the same interval and they comprise of two components: the top half of these images represents the shear rate in the domain under consideration while the bottom half (dark panels)

푡 depicts the breakup of the liquid in the ambient stagnant air. Dimensionless time (휏 = ) is (푑⁄v)푗푒푡 used to aid an unambiguous comparison. The images on top of these computational images are the experimental observations are the same corresponding time and is shown for comparison. Figure

3-9 showcases the generic process of breakup as observed. As the swell at the tip of the continuous jet is developing, the strain rate is seen to be higher on the surface of the swelling than within the

liquid (Figure 3-9 at τ = 0). As the swelling propagates and grows, the strain rate increase on the surface of thin liquid element connecting the swelling to the unbroken liquid jet and at the trailing edge of the swell (τ = 1.55). As time progresses, the radius of curvature of this section (thin liquid connected to the swelling at the tip) decreases thus increasing surface tension force and thereby, the strain rate increase begins to be reflected within the liquid (τ = 2.33). Eventually, the strain rate at the tip of the thin bridge becomes very high and this leads to pinch-off of the swelling thus producing the drops (τ = 3.1). This process described is the general observed mechanism of the breakup of low-velocity liquid jets.

Time Progression

τ = 0

Image from Experiment

t/τ = 1.55

Image from Experiment

t/τ = 2.33

Image from Experiment

t/τ = 3.1

Image from Experiment

Figure 3-9: Strain Rate development for Propylene Glycol (diameter 0.84mm We 10) during

breakup as a progression of time. The top half of each computational image shows the strain

rate while the bottom half shows the alpha variation (white depicting the liquid and black the

ambient air). For each time step, corresponding images from experiment have been shown for

comparison.

For the case depicted in Figure 3-10, the diameter of the ensuing jet in increased. Thus, the same surface tension as in the previous case has to now act on a larger diameter and a large surface area during the drop pinch-off and generation. Also, as the drop diameter is larger while the We is maintained the same, the significance of the viscous force is higher. As observed earlier, the consequence of a larger viscous factor could result in stretching of the liquid bridge thus slowing the propagation of the generated surface undulations. This makes the liquid ligament connecting the surface undulations (swelling) longer than in the previous case. The strain rate here too begins to increase on the surface of the thin liquid ligament connected to the swelling. But here, the ligament has a swelling on either of its sides, thus, the pinching off process and the corresponding increase in the strain rate is observed on both ends. The rates at which the strain rates increase is not similar at both ends, thereby leading to different pinch-off times at the two ends of the ligament.

The process of pinch-off at the rear end of the thin ligament is shown in Figure 3-10. As the strain rate along the ligament increases at the rear end, that on the surface of the succeeding drop too increases (Figure 3-10 at τ = 1.29). At pinch-off, the liquid segment is pinched off as the shear rates at its rear end become very high. As the radius of curvature decrease at this end of the segment, the axial pressure gradient drains liquid in the opposite direction to the flow thereby thinning the ligament at this location while increasing the strain rate. As the pinch-off occurs the tip of the ligament forms a bead to minimize surface energy (Figure 3-10 at τ =1.29). Notice that the strain rate increase remains on the surface here rather than increasing inside the liquid (Figure

2 10 at τ =1.29 and 2.58). Thus, the interplay of surface tension and viscosity can cause intriguing results of jet breakup.

Time Progression

τ = 0

Image from Experiment

t/τ = 0.65

Image from Experiment

t/τ = 1.29

Image from Experiment

t/τ = 2.58

Image from Experiment

Figure 3-10: Strain Rate development for Propylene Glycol (diameter 1.5 mm We 10) during

breakup as a progression of time. The top half of each image shows the strain rate while the

bottom half shows process of breakup (white depicting the liquid and black the ambient air).

The identified distinctive modes of break-up (Mode A and Mode B) also results in a change in the observed breakup length trend as shown in the plots in Figure 3-11. The plots show the experimentally observed breakup data (data points) along with the predicted correlation [44] (thin gray lines). It is observed that at low We and high jet diameters, the breakup length does not conform with the established recognized behavior. In addition, while for certain liquids, the observed breakup length is lower than expected (below the thin gray lines), for others it is higher.

Experiments with glycerin water mixtures were used to generate more information with regard to this observed variability. Mixture by volume of water and glycerin help modify the properties of the liquid and the relative significance of the surface tension and viscosity with ease. As pure glycerin is known to be highly hygroscopic, the properties of the mixtures were checked both before and after the experiments to test for consistency. All the liquids tested (pure and mixtures) are Newtonian in their behavior. To first demarcate the occurrence of this deviation in the breakup length, the parameters Re and Bo are considered. Re dictates the contest between the inertial and viscous force, thus representing the ability of a highly significant viscous force to slow the propagation of surface perturbation and thus stretch the liquid, while Bo, and represents the contest between the gravitational and capillary forces, thus indicating the ability of a high significant surface tension force to cause more surface undulations. These dimensionless parameters in combination with Morton’s number ( Mo, indicative of the property of the liquid in consideration) aids in identifying an empirical relation dictating the occurrence of the deviation in the breakup length (thick black line in Figure 3-11 plots). This relation is defined as:

퐵표 (3-8) = 0.2 푀표0.25 푅푒

The data points that occur above this relation are observed to deviate from the breakup length predictions as shown in the plots in Figure 3-11.

Water, Mo = 2.55 x 10-11 Ethylene Glycol, Mo = 5.22 x 10-6

65% (v/v) Glycerin + Water, Mo = 2.38 x 10-6 75% (v/v) Glycerin + Water, Mo = 6.44 x 10-5

80% (v/v) Glycerin + Water, Mo = 3.00x 10-4 Propylene Glycol, Mo = 5.15 x 10-4

Figure 3-11: Breakup length correlation for different Newtonian liquids showing deviation at

low Weber numbers and large diameters

To test the impact of the properties of the liquid on underestimating (water, ethylene glycol, 65% glycerin-water) or overestimating (propylene glycol, 75% glycerin-water, 80 % glycerin-water) the breakup length from the prediction, experimental study were first attempted. From the experimental observations, Mo seems indicative of the deviation from the prediction as seen in

Figure 3-11. All the liquids with Mo ≤ 65% glycerin-water indicate breakup lengths lesser than the prediction while all the liquids with Mo ≥75% glycerin-water indicate larger breakup lengths than predicted. To narrow down and validate the hypothesis that Mo is indicative of the pattern of deviation of breakup length, a numerical study was carried out for different Mo liquids with a constant diameter of the jet (1.5 mm) and constant We (We = 5), and the results are summarized in Figure 3-12. For low Mo fluids (Mo = 5 x 10-6), with high diameter and low We, pinch-off occurs in accordance with the Mode A breakup mechanism that has been described before. For higher Mo fluids (Mo = 5 x 10-5), Mode B seems to indicate the pattern of break-up with a drop attached to a thin ligament pinching off from the unbroken liquid column. Numerical simulations

suggest that the transition from Mode A to Mode B occurs around Mo of 1 x 10-5, beyond which

Mode B breakup is seen to consistently occur.

Thus numerical simulations in combination with experimental observations have helped identify the significance of surface tension and viscosity in determining the pattern of pinch-off and subsequent drop generation. Mo, being a characteristic of the liquid, is indicative of the relative significance of the viscous and surface tension force of the particular liquid and therefore helps to categorize the mode of break-up of the liquid jet.

Mo = 5 x 10-6 Mo = 1 x 10-5 Mo = 5 x 10-5

Figure 3-12: Jet breakup mode changing with Mo. Transition from Mode A to Mode B pinch-off

during satellite drop formation is seen to occur around Mo = 1x 10-5

1.13 Conclusions

A numerical technique to model viscous liquid jets breaking up in stagnant ambiance has been discussed in this current study. The weighted arithmetic property averaging at the interface is recognized to cause erroneous numerical solution and therefore, a modification to the same is suggested. Validation of the suggested interface property averaging technique is provided using experimental observations. Liquid jet break-up, in general, is influenced by the interplay of surface tension, viscous, gravitational and inertial forces. The significance of viscous and surface tension forces are isolated and analyzed using the computational solution. Viscous forces are noted to offer more resistance to the propagation of the developed surface perturbations thus stretching the liquid ligaments near the pinch-off location. As the radius of curvature at this location changes, surface tension changes aids in thinning the ligaments and thus generating the pinched off drops. In addition surface tension primarily influence the creation of surface perturbations on the continuous liquid column. Two modes of breakup pattern that are different from the classic Rayleigh-breakup pattern and that generate satellite drops at low We, are identified: Mode A- where both primary and satellite drops are pinched off and the satellite drops are observed to merge backwards with the oncoming primary drops, and, Mode B- where ligaments of liquid (drop(s) with a tail) is pinched off from the liquid jet which further break-up to form primary drops and satellite drop. In

Mode B, the satellite drops are observed to merge forwards. To understand the influence of these two modes of breakup in relation to the breakup length predictions from our previous experimental study[44] , further experimental analysis in combination with numerical examination is carried out.

An empirical relation to estimate the deviation from the experimental relation prediction is presented. Data points that are higher than the suggested relation in equation (3-8) are observed to depart from the observed experimental trend for low We Rayleigh breakup. Liquids with Mo > ~

1 x 10-5 are seen to follow Mode A of breakup while those with lower Mo adhere to Mode B of pinch off. Mode A leads to breakup lengths lower than the prediction while Mode B is indicative of larger breakup lengths.

At these low weber numbers, it is intriguing to note the alterations in jet breakup and the resultant drop generation patterns that is caused by the variation in properties of Newtonian liquids.

In most applications, however, the liquids in use are not purely Newtonian. They typically contain either polymeric or surfactant additives or both. The significance of understanding how these additives can alter the behavior of breakup and drop generation is heightened from the current study. Further analysis, both experimental and computational, can help begin to formulate a better perspective of the effect of non-Newtonian rheology on the phenomena in consideration.

4 CHAPTER 4: EXPERIMENTAL STUDY ON DROP IMPACT ON A THIN FILM

1.14 Introduction

The significance of drop impact and its subsequent splashing was recognized in studies as early as 1876 by Worthington[51]. Since then, various components of this phenomena have gained attention in a number of experimental, numerical and analytical work. The phenomena is also very captivating, as is demonstrated by the numerous television and print commercials that picture this phenomenon. One instance is the Harold Egerton photograph of a drop of milk falling on a thin film of milk on an American postcard in 1957 (Figure 4-1). This process of drop impact is of immense research interest not only because of its wide range of engineering and industrial impact but also because of the academic challenge it presents. Spray impingement is a common tool used in a multitude of industries to enhance heat and mass transfer. Workers and operators in such industries, that involve spray coating, metal annealing, and fertilizers, are in constant danger of contact and/or inhalation of these small drops of toxic chemicals. A means to predict and control the splash occurring while maintaining the functionality of the process, becomes crucial. As the phenomenon of drop impact is inherently complex, its physical understanding is far from complete.

With the exception of a few applications, most spray impacts occur on a thin film accumulated after the impact from previous droplets. The interaction of a liquid drop impinging on a layer of thin film is addressed in the present work.

Figure 4-1: A famous Harold Egerton photograph, Milk Drop Coronet, 1957, on an American

postcard. © Harold and Esther Edgerton Foundation, 2012, Courtesy of Palm Press, Inc.

When a drop impinges on a thin liquid film, it leads to different regimes of liquid movement that can broadly be classified under deposition or splashing. Illustrations of these regimes are provided in Table 4-1. Deposition, post impact of a drop, involves the drop merging with the liquid film without forming secondary drops. Deposition may or may not include capillary surface waves that form a crown of liquid, before merging with the liquid film. Splashing happens when secondary drops are discharged as a drop impacts the thin film. At high Weber numbers (We) and high Reynolds numbers (Re), on impact, an ejected jet is formed at the neck (small region between the drop and the thin film). The ejected jet spreads out and tilts upwards, forming a crown whose rims are unstable and break-up into secondary drops before merging into the thin film. This was termed crown splashing by Deegan et al. [52] and by Josserand and Zaleski [53]. Prompt splash is a supplementary phenomenon that could occur when secondary drops are formed at the instant of drop impact on the thin film. Deegan et al. [52] studied the complexities of splashing and found that during a prompt splash, the ejected jet shot out from underneath the drop parallel to the fluid layer.

Their study delineated the different regimes of drop impact based on two parameters: We and Re.

They present a correlation to demarcate occurrence of prompt splash from crown splash. Delayed splash or crown splash, occurs when the liquid crown breaks up at or beyond the expansion of the crown.

Table 4-1: Images showing (a) deposition (no splash), (b) prompt splash and (c) crown (delayed)

splash along with respective properties.

(a) Liquid: 50% by volume (b) Liquid: Water (c) Liquid: 50% by volume

Propylene Glycol Propylene Glycol

Re = 1899 Re = 8055.4 Re = 2070.89

We = 458.08 We = 277.16 We = 493.97

Another extensive review of drop impact dynamics was provided by Yarin [8]. Drop impact dynamics depends on a number of parameters: drop size and impact velocity, drop and film liquid properties (density, viscosity, rheology), interfacial tension, surface properties (roughness, contact angle, wettability), and thermal properties. Both experimental and analytical methods have been used to find the splashing threshold. Yarin and Weiss [54] experimentally defined a velocity threshold for drop splashing that is based on the frequency of falling drops, the viscosity, surface tension and diameter of the impacting liquid drop.

Three dimensionless parameters have been commonly used in scaling drop impact and

𝜌푣2푑 휇 splash threshold: Weber number(푊푒 = ), Ohnesorge number(푂ℎ = ) and dimensional 𝜎 √𝜌𝜎푑

퐻 thin film height(퐻∗ = ). Cossali et al.[55] tested the splashing threshold by analyzing drop impact 푑 experiments that involved a wide range of liquids on different target surface conditions. The H* in these experiments were maintained between 0 and 1. They proposed a correlation to estimate splashing limit; 푊푒 푂ℎ−0.4 = K = constant, where K depends on the non-dimensional film thickness to predict the deposition-splashing limit. The value of K was found to increase with increasing H*. This parameter K was initially developed to describe drop impact on dry solid surfaces [54, 56] and has been adapted by some researchers to describe drop impact on thin films.

Multiple viscous liquids impact on thin films was tested by Rioboo et al.[57]. They defined two thresholds: the deposition-crown and crown-splash limits or criteria. Both these thresholds were seen to be influenced by the film height H* when the film was very thin (H* < 0.06) but were seen to be a constant at larger film thicknesses (0.06 < H* < 0.15). For each of these thresholds, values of K at different film thicknesses have been defined in their study. Okawa et al. [58] find a similar relation in their experimental study for the two thresholds as defined by Rioboo et al. [57], though their value of K for the two thresholds are slightly different. Though their film height ranges were higher (H* = 0.48 – 68) than these studies with H* < 0.1, their findings for the onset of the different regimes of drop impact are very similar to that of Rioboo et al. [57]. In addition to defining the onset of the regimes of drop impact dynamics, this study also looks at secondary drop characteristics and its dependence on the K parameter. The difficulty in producing and maintaining small film thicknesses motivated Wang and Chen [59] to explore reliable and repeatable means to produce thin films of a surface. With the help of their new technique of maintaining thin films with higher confidence, they found that the critical splash level was insensitive to film height when the film is

sufficiently thin (H* ≤ 0.1). Different concentrations of glycerol water mixtures were used to experimentally find that this criterion can be expressed as a constant We that changed with changing viscosity of the liquid in consideration. Similar observations were made by Vander Wal et al. [15] where We number was found to be the determining parameter to establish the onset of splash for thin films (H* ~ 0.1). However, this study finds that the splash criterion to be independent of the viscosity of the liquid. Experimental examinations using multiple viscous liquids were used to understand the onset of splashing both on dry surfaces and on thin liquid films.

Their work also highlights the effect of surface roughness when a drop impacts a dry surface.

Apart from experimental studies, there have been a few numerical modeling techniques that have helped understand the onset of splashing from an analytical perspective. Based on the concept of kinematic discontinuity proposed by Yarin and Weiss [54] a model for the propagation of the crown formed during drop impact was proposed by Trujillo et al. [60]. Their solution resulted in a defining time scale beyond which it was found that the crown characteristics were independent of upstream conditions. The model also attempted to resolve the effect of surface film characteristics and was validated with experimental data. A 2D flow model was numerically simulated using VoF method by Coppola et al. [61]. Their work concentrated on validating the proposed kinetic discontinuity theory numerically.

In this current work, the onset of delayed splash is studied experimentally. While there are a few studies that outline the onset of splashing, there is variability in their predictions based on their proposed correlations. In this study, we attempt to understand the underlying reason for this disparity. Many studies have concentrated on using water as the working liquid and the proposed correlations do not truly take into account the significance of viscosity of the liquid. In this

experimental study, five different liquids with varying liquid properties are tested with the aim to create an improved understanding of the phenomena for Newtonian liquids.

1.15 Experimental procedure

Single drop impingement is studied as it impacts a thin film formed from prior impinging drops. The thickness of this thin film formed was maintained at H* ~ 0.1. Figure 4-2 provides a sketch of the experimental apparatus used. A container made of acrylic sheets was used as the main observation chamber and also functioned as the reservoir/collecting tank. The container had a raised acrylic platform inside it where the copper target surface was installed. The walls of the container were made to be large enough to avoid interference with the splashing droplets and to provide for a clear viewing of the phenomena of impingement and splash. Impacting drops were generated at a steady rate from a NEXUS 3000 syringe pump. The flow rate was varied from 0.5 ml/min to 8 ml/min for the different cases tested experimentally. This resulted in a range of velocities from 1 m/s to 3 m/s for drop diameters ranging from 3.5 mm to 5.2 mm. The drops were released from a prescribed height above the target surface, through a stainless steel circular needle.

The needle was maintained perpendicular to the target surface using a bubble level. Since the target surface was much larger than the impacting drop, the increase in film thickness with each impacting drop was found to be negligible. The film thickness (H*) was periodically verified to be ~ 0.1 by capturing images using the high-speed camera. The height, from which the drop was released, was adjusted while keeping a constant flow rate of liquid from the syringe pump. With the change in height, the impacting velocity is dramatically altered. As a single drop impacts the thin liquid film, it may either result in a splash or in deposition of the drop.

Syringe White Screen Pump

Lighting

Needle Needle

Target

surface

Computerized data acquisition system Collecting Tank

High-speed camera

Figure 4-2: Sketch of the experimental set-up

To study the phenomena of drop impact, a high-speed digital camera (Hi-D cam – II version 3.0 –

NAC Image Technology) was used. The camera system was kept at an appropriate angle to view the impact without causing hindrance to the phenomena. Images were recorded at a shutter speed of 1/2000 s and a frame rate of 500 fps with the camera placed at about 1.5 ft. from the target surface. A single bulb focusing light system (ARRI) with glossy aluminum reflectors was used to obtain clear and high contrast images. The light system was focused on a white screen that was placed parallel to the plane of viewing. The high-quality images thus obtained were analyzed using the image processing software, IMAGEPRO PLUS 4.0 (Media Cybernetics). The impacting diameter of the drop is calculated based on the area occupied by the drop, assuming the drop to be spherical and therefore, a circle in the image plane. Drop impact velocity is determined by

comparing the position of drops, prior to impact, in successive frames with respect to a fixed point in the frame. The impact velocity is not a constant due to gravitational acceleration and drag force acting on the drop. Therefore, only three appropriate images before impact are considered for calculations. For each set of experiment, images are calibrated using the outer diameter of the stainless steel needle diameter. Each experiment is repeated to check for repeatability and the average measurements are recorded.

To help understand the influence of liquid properties on the splashing phenomena, different liquids with varying properties were used for the experiment. Table 4-2 presents the properties of the five liquids used. Mixtures of water and propylene glycol by volume were used as working fluids to obtain a range of liquid properties for the experiment. The values of viscosity and surface tension of these mixtures were found by using the AR 2000 controlled stress/controlled shear rate

Rheometer and the SensaDyne QC6000 Tensiometer respectively.

Table 4-2: Propertied of the liquids used in the experiment

Density Viscosity Surface Tension Liquids (kg/m3) (Pa.s) (N/m)

Water 998.0 0.001 0.0728

25% by volume of

Propylene Glycol 1007.5 0.00255 0.0541

(25% PG)

50% by volume of

Propylene Glycol 1017.0 0.005 0.0452

(50% PG)

75% by volume of

Propylene Glycol 1026.5 0.012 0.0411

(75% PG)

Ethylene Glycol (EG) 1113.2 0.0161 0.0484

1.16 Results and discussion

To understand the splashing behavior of drops impacting a thin film of the same liquid, controlled experiments were performed. Analyzes of the high-speed images obtained during the experiment are used to establish the outcome of each drop impact. Correlations from prior thin film drop impact studies are compared with current and past experimental data in Figure 4-3. The suggested correlations for determining the onset of splashing are primarily of two forms. While Cossali et al.

[55] [57] −0.4 0.8 0.4 and Rioboo et al. have correlations of the form of 푊푒푂ℎ = 푊푒 푅푒 = C1, Vander

[15] [59] Wal et al. and Wang and Chen propose that 푊푒 = C2 provides the demarcating line beyond which splashing would occur. In the former correlation form, both the studies propose different values for the constant C1: Rioboo et al. (2003) observe C1 to be a constant for larger film thicknesses (0.06 < H* < 0.15) and was seen to be a function of the film thickness only for very

[55] thin films (H* < 0.06), while Cossali et al. observed C1 to always be a function of the film thickness. As the data depicted in Figure 4-3 is presented only for H* ~ 0.1, the curves shown for these two correlations are seen to be parallel. For the second suggested relation (푊푒 = C2), while

[15] [59] Vander Wal et al. identifies C2 to be invariant of the liquid property, Wang and Chen note that the constant value of We is dependent on the Oh number of the liquid. Therefore, the constant

We line for Wang and Chen [59],would ideally be parallel to Vander Wal et al.’s [15] but different for each liquid shown in the plots (fig. 3-3). The plots in Figure 4-3 show the performance of these

correlations with experimental data from the current study as well as those in past literature for H*

~ 0.1. The plots are shown in both the linear scale (Figure 4-3 a) and the logarithmic scale (Figure

4-3 b) to understand the effectiveness of these correlations. While it is noted that most splashing

−0.4 0.8 0.4 events occur as predicted by the 푊푒푂ℎ = 푊푒 푅푒 = C1 curves, deposition events are noted to occur well above these curves for most liquids taken in consideration. The 푊푒 = C2 on the other hand, is seen to heavily under-predict the splashing threshold for certain liquids while over-predicting the same for some others. Current experimental data for water and ethylene glycol are shown on Oh versus Re plots in Figure 4-4. These too show predictive curve for the correlations discussed above with Rioboo et al. [57] and Wang and Chen [59] curves conceivably being parallel to those exhibited in the plots.

(a)

(b)

Figure 4-3: Comparing past correlations with experimental data and with data from other

studies for H* ≤ 0.1 (a) Linear scale (b) Logarithmic scale

As there is an overlay between the splash and deposition events in the plot of We versus

Re, constant We lines will be unable to demarcate the onset of splashing effectively. In addition, from the Oh vs Re curves it is observed that, for larger diameter drops for water (dotted red in

Figure 4-4 (a)), the trend seems to differ significantly from that of smaller diameters. It is therefore felt that considering only Re and We (and/or Oh = We0.5Re-1) as the governing dimensionless parameters could, therefore, lead to neglecting some other important forces that may be significant in governing the drop impact outcomes. To identify the relevant significant parameters, the effect of different working liquid properties is considered in this current experimental study. The analysis ensues from the examination of the high-speed images captured during drop impact of a thin film in stagnant ambiance.

Vander Wal et al. (2006), Wang & Chen (2000)

Rioboo et al. (2003), Cossali et al. (1997)

(a)

Vander Wal et al. (2006), Wang & Chen (2000)

Rioboo et al. (2003), Cossali et al. (1997)

(b)

Figure 4-4: Comparing current experimental data with past correlation for (a) Water and (b)

Ethylene Glycol

The impact of water drops on a thin water film is shown in Figure 4-5. The occurrence of four different phenomena are captured and presented here. Figure 4-5 (a) and (b) show splashing occurring, while Figure 4-5 (c) and (d) show deposition. Images are captured for each 0.002 s. As the drop impacts on the thin film surface, at impact, a splash could occur. This is the prompt splash and is seen to occur in Figure 4-5(a). The capillary surfaces waves formed on the crown create fingers of the crown that extend upwards. These surface waves could then grow into fingers of the crown that could then brea-up into droplets. In Figure 4-5(a), the drops formed from the prompt splash are still seen at 0.01 s. The fingers of the crown are seen to be growing and this ultimately breaks up at 0.14 s, while falling back into the thin film layer. Figure 4-5(b) shows the case of a delayed or crown splash, where, at impact, no prompt splash occurs. However, at impact, the

formation of capillary waves can be seen on the surface. These propagate to the fingers of the crown that breakup at 0.012 s. In the crown splash, the droplets formed are noticed to be much larger and uniform in comparison to the prompt splash in Figure 4-5(a). Figure 4-5(c) and (d) present deposition of the drops. In Figure 4-5(c) the capillary surface wave formed at impact at

0.002 s is seen to propagate and form fingers of the crown. These fingers, however, do not further breakup into drops before deposition on the thin liquid film. The drop in Figure 4-5 (d) does not form capillary waves on impact at 0.002 s. Minor surface undulations are observed at 0.01 s. These undulations, however, do not form finger-like structures before deposition. Accordingly, there can be two types of deposition: fingered and non-fingered. By inspecting Figure 4-5(c) with (a) and

(b), it is noted that both an increase in diameter and an increase in impact velocity affects droplet splashing on a thin film. The diameter of the resulting crown on the thin film is seen be a function of the drop diameter. So, for the same velocity of impact, a smaller drop results is a smaller crown diameter (Figure 4-5 (a) and (c)).

0 s 0.002 s 0.01 s 0.014 s

(a) Time progression of water drop impact on a thin film. rd 2.3 mm V 2.2 m/s

Re = 10100 We = 305

0 s 0.002 s 0.006 s 0.012 s

(b) Time progression of water drop impact on a thin film. rd 1.8 mm V 2.5 m/s

Re = 8982 We = 308

0 s 0.002 s 0.008 s 0.0012 s

Re =7904 We = 239

0 0.002 s 0.01 s 0.014 s

(d) Time progression of water drop impact on a thin film. rd 2.6 mm V 1.1 m/s

Re = 5709 We = 86

Figure 4-5: Time progression of drops of water

To understand the influence of different forces on the phenomena of splashing, the effect of liquid properties is studied in addition to the drop size and drop velocity. As the drop impacts the thin film, the base of the crown has a velocity comparable to the impact velocity. This velocity of increment of the crown base, decreases with crown development until the crown collapses on the thin film. The impact of the drop primarily receives resistance from the thin film. The decrease in velocity of the crown base is therefore based on the resistance offered by the liquid. This resistance

is the viscous property of the liquid. The drop impact of EG and 50% PG is shown in Figure 4-6.

To understand the role of the viscosity of the liquid, drops of two different liquids impacting on a thin film with nearly the same velocity are shown in Figure 4-6. The We of these two cases shown in Figure 4-6 is about the same. For the EG drop, at 0.004 s, the rim of the crown is still uniform and no undulations are observed. Instability on the crown rim is not observed till deposition occurs at 0.018 s. For 50% PG water solution, though the drop diameter is smaller than that of EG in

Figure 4-6 (a), minor surface waves are noticed at 0.006 s. These disturbances grow and ultimately result in a modest splash at 0.018 s. Though the diameter of the EG drop is higher than the 50%

PG drop, the EG drop does not splash. As the viscosity of EG is higher than 50%PG, EG offers more resistance to the growing crown base and thus suppresses splashing by a greater measure than 50% PG solution. Consequently, it can be concluded that viscosity of the liquid affects the spread of the liquid crown layer and restrains the phenomena of drop splash on a thin liquid film.

0 s 0.004 s 0.01 s 0.018 s

(a) Time progression of EG drop impact on a thin film. rd 2.27 mm V 2.17 m/s

Re = 681.2 We = 491.7

0 s 0.006 s 0.014 s 0.018 s

(b) Time progression of 50% PG drop impact on a thin film. rd 2.2 mm V 2.18 m/s

Re = 1951 We = 470.5

Figure 4-6: A comparison of impact behavior of EG and 50% PG drops

Figure 4-7 illustrates two liquid drops, of 25% PG and of 75% PG, impacting of a thin liquid film of same liquid. The two cases considered have similar velocities but different drop diameter, Re and We. For 25% PG, at 0.002 s, the impact is seen to generate capillary waves. The formation of capillary waves is not observed at impact for 75% PG. For both the liquids shown, the capillary waves lead to the formation of fingers of the crown that eventually break-up to form droplets. For

25% PG (Figure 4-7 (a)), this breakup occurs more easily than for the 75% PG drop. The number of splash drops formed is more for 25% PG. As the liquid crown is spreading, the instability on the rim leads to crown formation and possible breakup leading to the formation of droplets. In addition to this instability driven crown breakup, the crown edges are constantly looking to minimize their surface energy. This surface energy is lower for a unit volume of a drop than for the same unit volume of the crown. As seen in Table 4-2, the surface tension of 25% PG is higher than that of 75% PG. As surface tension governs the surface energy, the 25% PG drop splashes more violently than the 75% PG. It can thus be concluded that a higher surface tension helps in breaking up the crown rim hence forming splashed droplets.

0 s 0.002 s 0.014 s 0.022 s

(a) Time progression of 25% PG drop impact on a thin film. rd 2.25 mm V 2.5 m/s

Re = 5051.2 We = 525.2

0 s 0.004 s 0.02 s 0.026 s

(a) Time progression of 75% PG drop impact on a thin film. rd 2.5 mm V 2.45 m/s

Re = 1048 We = 749.6

Figure 4-7: A comparison of impact behavior of 25% PG and 75% PG drops

Thus, it is hypothesized that while the crown rim formation and its subsequent growth is resisted by the viscous force of the liquid in the stagnant thin film, the breakup of the crown rim is dictated by the surface tension force acting to minimize surface energy at the rim edge. These two interactions can be expressed in terms of dimensionless parameters of Re and Bo

𝜌 푔 푑2 (퐵표 = ), where Re denotes the inertial to viscous force interaction and Bo denotes the 𝜎 gravitational to capillary (surface tension) force interplay. Thus, these two parameters are used to define the splashing threshold. A plot of Re vs. Bo with the current experimental data is shown in

Figure 4-8 (a). Experimental analysis provides the following relation between Re and Bo to predict the onset of splash:

-0.2 0.1 Re = 60.95 Mo Bo *Correlation to demarcate splashing from deposition

(a)

*Correlation to demarcate splashing from deposition

(b) Figure 4-8: Correlation (equation (4-2)) performance with (a) current experimental data and (b)

data from past literature

0.1 (4-1) 푅푒 = 퐶3퐵표

It is observed that the demarcation curve is dependent on the property of the liquid in consideration.

Further experimental evaluation shows that C3 is a function of Morton’s number (Mo) as: 퐶3 =

60.95 푀표−0.2. Thus, the semi-empirical relation to predict the onset of splash of drops in thin films can be expressed as in equation (4-2).

푅푒 = 60.95 푀표−0.2퐵푂0.1 (4-2)

The curves shown in Figure 4-8 (a) and (b) correspond to the above identified relation. To examine and validate the accuracy of this relation, the experimental data from past studies that were used in Figure 4-3 are tested against this relation in equation (4-2). The plot in Figure 4-8 (b) shows that the current expression is able to predict the onset of splash with good accuracy for all the experimental data tested from past literature. It must be noted that the validity of this expression is for thin films with dimensionless heights (H*) ~ 0.1.

1.17 Conclusions

An experimental investigation of drop impact on thin liquid films is carried out. In order to obtain a comprehensive basic understanding of the phenomena, multiple Newtonian liquids are experimentally tested. A range of drop diameters and impact velocities are tested for a dimensionless film height (H*) of 0.1. High-speed imaging is utilized to capture the phenomena and the analysis has been discussed. The effectiveness of the predictive correlations for determining the onset of splashing in a few studies that have considered drop impact on thin films

(H* ~ 0.1) has been tested with the past and current experimental data. These correlations are found to have some inaccuracies in their predictions, thus indicating the presence of potential significant considerations that have previously been overlooked. The experimental analysis of the

phenomenon under consideration signifies the importance of the liquid properties, specifically viscosity and surface tension, apart from the indisputable contribution of the drop diameter and impact velocity. These observations lead to two important hypotheses: the drop impacting a thin film encounters viscous resistance from the thin film as the drop spreads along the film forming a crown, and, the surface tension of the liquid in the crown rim aims to minimize the surface energy of the rim thus governing its breakup and production of secondary drops. Consequently, these leads to two governing dimensionless parameters: Re and Bo respectively. Based on these two parameters, a predictive semi-empirical correlation for demarcating the onset of splash when a

Newtonian drop impacts a thin liquid film is developed. In addition to the current experimental data, the effectiveness of this predictive relation is tested against data from past experimental studies.

A numerical study of the phenomena is desired to scrutinize the effectiveness of the proposed hypotheses. The experimental analysis does not describe the early impact-times physics of the phenomenon. This will be a significant study to understand the development of the post-impact dynamics. Also, as most engineering applications utilize polymer additives to Newtonian liquids, an in-depth study and analysis of the effect of non-Newtonian liquid characteristics is needed. In an effort to provide a better understanding of the phenomenon under consideration, both numerical analysis and the effect of non-Newtonian characteristics, are currently under progress.

5 CHAPTER 5: NUMERICAL ANALYSIS OF DROP IMPACT OF THIN LIQUID

FILMS AND THE EFFECT OF FILM THICKNESS ON IMPACT DYNAMICS

1.18 Introduction

Drop impacting on a target surface is a common phenomenon that finds considerable utilization and relevance in multiple fields. Spray paintings and coatings, fuel injection in IC engines, seed and microorganism dispersion, soil erosion, pesticide sprays on agriculture, forensics and other aerosol spray applications are some everyday areas that encounter drop impact and splashing. Many studies have primarily focused on determining a criterion to demarcate the onset of the occurrence of different kinds of splash (prompt and crown splash) and/or deposition as the drop impacts the target surface. For drop impacting a dry surface, some experimental studies have established semi-empirical relations between Reynolds number (Re) and Weber number (We) to correlate the splashing/deposition limit [7, 12, 15, 62]. These relations have been found to be sensitive to the surface roughness. Similarly, in understanding the complexities of splashing in deep pools, some earlier research investigated the influence of viscous liquid in addition to water to find that

Froude number (Fr) and Re were the characterizing parameters[63, 64]. Sivakumar and Tropea [65] point out that excluding some definitive applications, most drop impacts on an initially dry and/or heated surface will eventually result in the formation of a thin film of impinging liquid on which all further impact would occur. The significance of drop impact on thin liquid layers of identical liquid has resulted in an increasing interest in recent experimental and numerical research work.

Yarin and Weiss [54] found the splashing limit to depend on viscosity, surface tension, density and drop impacting frequency. Following this, a few similar studies have attempted to establish a deposition/splashing limit. While Cossali et al. [66] and Rioboo et al.[57] find that We and Ohnesorge

number(Oh) determine this limit, Vander Wal et al. [15] and Wang and Chen [59] note the limit to be a function of only We. Though many of the studies have utilized water as the working fluid, there is some variability in their observations. To address this, our prior experimental work

(Chapter 4) focused on using different viscous Newtonian liquids to understand the phenomena and its mechanism. The other major contributor to the variability in the existing literature may arise from the defined thickness of the liquid film layer. Though, prior semi-empirical correlations contain similar parameters, their functionality and the constants in the established relations are seen to vary. This indicates possible unaccounted variables in these investigations of onset of splash. Theoretical analysis of the phenomena has presented kinematic discontinuity as the primary source of splashing [67]. Trujillo and Lee [60] find that the crown dynamics were heavily dependent on liquid film depth and surface roughness from their numerical analysis of wet and dry targets.

Wang and Chen[59], identified the difficulty in measurement of the film thickness. They note that the measurement uncertainty is very large for smaller film thicknesses and thus propose a method to form films with higher accuracy and better repeatability. Due to the difficulty in maintaining film thickness, most experimental studies have used H* ≥ 1, where H* is the dimensionless film height/thickness defined as the ratio of the film height to the diameter of the impacting drop.

Vander Wal et al. [15, 68] experimentally studied the influence of varying target conditions on drop impact outcomes. They find that dimensionless film height/thickness, H*, can be used to categorize the type of impact. They primarily categorize the film thickness into three categories:

0. 1 ≤ H* ≤ 1 (thin films), 1 ≤ H* ≤ 10 (thick films/shallow pools) and H* ≥ 10 (deep pools). This categorization is used by certain other researchers to define thin film thickness in their experimental study [52, 58] where the H* ≥ 0.1. Fewer studies have looked at H* ≤ 0.1 [15, 54, 55, 57,

59] and have found that H* ~ 0.1 has a limiting effect on the type of drop impact outcome, though the understanding for this demarcation is not clear.

Experimental control of film height is not only complex but also has a large uncertainty [59].

To interpret the impact of the film height in dictating the post-impact dynamics, a numerical investigation of the phenomena is required. The potential of numerical techniques to provide a coherent interpretation of the drop impact dynamics has led a number of studies to develop numerical modeling technique to achieve the same. Numerical methods also aid to examine early time dynamics of this impact as they are very difficult to observe and study experimentally. Many numerical studies have been able to isolate the key influence of certain governing parameters and fluid properties as well as the effect of heat transfer in drop impingement [69, 70] on a dry target.

Weiss and Yarin [71] used the boundary-integral method to understand the early time dynamics of drop impact liquid films at low and high Weber numbers. Their study ignores liquid compressibility and viscosity while surface tension and gravity are taken into consideration. While, the compressibility of the liquid may be a negligible factor, experimental evidence shows that viscosity of the liquid does play a crucial role in drop impact dynamics. Their numerical work gives an insight into the dynamics at the neck region of impact and on crown formation. To study the effect of viscosity, Trujillo and Lee [60] study both inviscid and viscid cases with their numerical scheme. The crown dynamics is studied with respect to film height, wall friction, and impact velocity. More recently, volume of fluid (VoF) based numerical techniques are being used to model characteristics of a number of two-phase flows. Josserand and Zaleski [53] were one of the first to use the VoF numerical technique to observe ejected sheet at the next when a drop impacts a thin liquid film. Their axisymmetric incompressible, zero gravity solution provides a theoretical understanding of the transition from deposition to splashing. The evolution of the drop impact on

thin films is studied in detail by the two-dimensional numerical VoF scheme employed by Coppola et al. [61]. They note that, three-dimensional consideration would be required to accurately represent the crown behavior but the two-dimensional assumption is effective in studying some peculiar characteristics such as the neck dynamics. Other recent numerical studies [72, 73] have also used

VoF methods to capture large time-scale characteristics of the impact, such as crown behavior as well as bubble entrainment at high impact velocities.

This current study aims at using the open source CFD software, OpenFOAM® to understand typical characteristics of drop impact on thin films and to identify the influence of the film thickness/height. Axisymmetric spherical drop impact with surface tension, gravity and viscosity is considered. The results are compared with established relations in past numerical studies as well as with current experimental analysis (Chapter 4). VoF technique is well known to induce spurious currents, especially in two-phase phenomena. In order to limit the spurious flows, the Oh number is required to be greater than 0.01 [61]. This, however, hinders investigation of the true significance of liquid properties on the phenomena. To counter this, the existing VoF technique is modified and the results are seen to be consistent with experimental observations. Thus, this study is primarily divided into two key sections: (1) modified VoF scheme and analysis of viscous liquid drop impacts on thin films (2) effect of liquid film thickness and its influence on early time dynamics of the phenomenon.

1.19 Numerical procedure and problem formulation

1.19.1 Modified VoF numerical scheme

All computations are performed using the open source code OpenFOAM® which uses a cell-center based volume method. The VoF interfacial tracking method was first introduced by

Hirt and Nicholas [25] and they suggested the use of an indicator function to identify the different

phases in the computation. This indicator function is also called the phase fraction as it indicates the fraction of a particular phase in each cell in the domain. The phase fraction thus has a maximum value of 1 inside one phase (in this study – liquid) and a value of 0 in the other (in this study – air).

It is defined as follows:

Volume of primary phase 훼 = Total control volume

1 in primary fluid (liquid) where, 훼 = {0 < 훼 < 1 in transition region (interface) 0 in secondary fluid (air)

In between the two phases, at the interface, the value of this phase fraction would switch values abruptly. Though such a sudden change in value for this phase fraction is observed in nature, this would cause a discontinuity in the numerical scheme. Therefore, the interface has some small thickness (about 2 / 3 cells) in which the value of the phase fraction changes gradually. Transport equation for this indicator function (equation (5-1)) is solved simultaneously with the continuity

(equation (5-2)) and momentum equations (equation (5-3)) to facilitate this change.

휕훼 (5-1) transport of 훼: + 훻. (푣̅훼) = 0 휕푡

mass continuity: ∇. (𝜌v̅) = 0 (5-2)

∂(𝜌v̅) (5-3) momentum continuity: + ∇. (𝜌v̅ v̅) = −∇푝 + ∇. T̅ + 푓 + 𝜌g̅ ∂t 푠 where,

v̅ is the velocity of the field

𝜌 is the density of the field

푝 is the pressure

훼 is the indicator function a. k. a the phase fraction

T̅: stress tensor and

푓푠: volumetric force due to surface tension

Thus, the solver considers the domain to contain effectively one fluid whose physical properties are varied in accordance with the phase fraction in each cell in the computational domain.

Typically, a weighted average distribution of the phase fraction is used to calculate the properties of the fluid at each location. The variation created by the averaging determines the location of the interface and the number of cells that could contain the (smeared) interface. The more gradual the phase fraction distribution, the more the smearing of the interface. The weighted averaging property distribution is as defined in equation (5-4) and is depicted by the green line in Figure 5-2.

In the plot, the x-axis depicts the property (density) and the y-axes define the phase fraction of the liquid (left axis) and of air (right axis) of the cell in consideration.

휑 = 훼 휑1 + (1 − 훼)휑2 where 휑 is the property of the fluid and 훼 is the phase fraction in the cell and (5-4) subscripts 1 and 2 represents liquid and air respectively.

One important issue in free surface numerical solutions like that in the current study is that small deviations in the estimation of the fluid properties can lead to considerable changes in the location and behavior of the interface and, therefore, present inaccurate computational solutions.

Inaccurate interface calculations, in turn, would result in surface tension force calculations that are unreliable caused by failure in predicting the correct pressure gradients across the interface. This issue is observed to become more pronounced for high viscous liquids.

To resolve the difficulty in interface tracking, there have been past studies that have either looked at correcting the Courant number estimation, like Beerners et al.’s [32] work, or some others

that have proposed alternative methods to estimate curvature of the interface and therefore the surface tension force [33], while still others have looked at better meshing techniques to resolve the computational domain in a dynamic manner [34]. In our preliminary analysis, of the complication of correct interface estimation, it was found that the relative viscosity of the two phases in question was the underlying source of inaccuracy. Thus, in an attempt to correlate the property averaging method to the ratio of viscosities of the phases involved, a modification to the existing technique is attempted. Multiple experimental cases are tested with a generic property averaging equation as defined by equation (5-5). The time for development of the crown, as well as the crown diameter, are used to validate the computational solution with the experimental results (Chapter 4). The value of constant C to obtain valid solutions for these simulations is then correlated with their relative viscosity ratios to obtain an equation for the estimation of the constant C as is shown in Figure

5-1.

퐂 휑 = 휑2 + (휑1 − 휑2)훼 (5-5)

휸 ퟏ.ퟓ( ퟏ ⁄ )+ퟎ.ퟕퟓ 휑 = 휑 + (휑 − 휑 )훼 휸ퟐ 2 1 2 (5-6) where, 훾 is the kinematic viscosity of the fluid in consideration.

Thus, the resultant modified property averaging is dictated by equation (5-6). Apart from the current study of drop impact on thin films, this modification to property estimation in the interface is tested with other two-phase flows such as jet breakup and drop generation (Chapter 2) and is seen to provide reliable computational solutions. The variation of the property across the interface is shown by the red line in Figure 5-2. The plot shows the variation in interface density from air, whose density is 1.225 kg/m3 to that of the liquid (ethylene glycol), whose density is 1113.2 kg/m3.

As seen, the variation is not as gradual as the arithmetic averaging technique (green line – Figure

5-2) but is sharper thus producing lesser smearing of the interface.

where, x % PG: x % by volume of propylene glycol and water mixtures

EG: ethylene glycol

훾 훾퐿푖푞푢푖푑 1 = 훾2 훾퐴푖푟

Figure 5-1: Equation for the constant C in the property averaging equation (4.5) derived from

the different liquids tested.

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)

(kg/m

Density

Phase fraction, 휶

Figure 5-2: Variation of fluid property (density) at the interface for arithmetic weighted

averaging (green) and the current method (red).

In addition to altering the property estimation in the interface, the curvature field is smoothened in accordance with Ubbinks et al.’s [28] phase fraction smoother. In their work,

Ubbinks et al. [28] propose that non-smoothness in the curvature field can be resolved by correcting the smoothness in the phase fraction field. The proposed equation involves averaging the phase fraction in each computational cell volume by area averaging with respect to the values at the faces of the volume as shown in equation (5-7).

∑ 훼[푓푎푐푒] 퐴푟푒푎[푓푎푐푒] 훼[푐푒푙푙] = (5-7) ∑ 퐴푟푒푎[푓푎푐푒]

In OpenFOAM®, the sharp interface is provided for by using a compressive flux model called interface compression which was introduced by Rusche [48] and helps to limit the diffusion of the interface. The pressure-velocity coupling is carried out by the PIMPLE scheme with four corrector

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loops. The SIMPLE algorithm is used as the solution technique. The default discretization scheme used in this solver are backward Euler for time, limited linear for velocity and van Leer for scalar quantities. The courant number criterion is defined by an equation to provide good numerical stability.

Co Co (5-8) maxo max o o t minoo  t ; 1 1  t ; 2  t ;  t max Co Co

where, Δtmax and Comax (= 0.1) are predefined values for maximum permissible time step and maximum Courant number. The damping factors, λ1 and λ2, are defined as 0.1 and 1.2 respectively.

Detailed solution procedure for the OpenFOAM® solver is provided by Deshpande et al. [29] whose study evaluates the performance of the solver for different two-phase flows.

1.19.2 Problem definition and the computational domain

The impact of an axisymmetric spherical drop of liquid with viscosity μ, surface tension σ, density ρ, diameter d, and velocity v, on a thin film of height, h, of the same liquid is considered.

As the problem is considered to be axisymmetric, and OpenFOAM ® handles asymmetry as a quasi-3D case, a small wedge (< 5o) of the actual three-dimensional domains is taken into consideration as shown in the domain Figure 5-4. A thin film of height, H, is initialized on the target surface. The boundary conditions for the domain under consideration are as depicted in

Figure 5-3. The top edge of the domain is defined as atmospheric where pressure is fixed at zero gage and no liquid is expected at this location (thereby the volume fraction is set to zero). As this domain is ideally very far away from the impact location, there would no velocity gradient across this boundary. The right boundary is defined as an outlet where velocity, pressure, and the phase fraction would have no gradients. The bottom plane is set as the target plate and therefore, a no-

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slip zero velocity condition is imposed along with zero gradients in pressure and phase fraction.

Since the domain considered is wedge-shaped, and the domain is axisymmetric, only a hemispherical part of the spherical drop is defined in the 5o wedge-domain. As no inlet condition is provided in the domain, to initialize the simulation, a drop with the defined diameter d, and velocity v, is patched onto the domain. The thin film with the defined thickness or height H, is also patched onto the domain. The drop is placed at a small distance from the thin film, such that it can develop numerically

As the solution complexity would be highest near the drop impact and near the thin film layer, the computational domain consists of different resolutions of grid density. As seen in the sample grid shown in Figure 5-4, the domain is broadly divided into two parts along the axis of symmetry. The top portion has a gradual decrease in mesh density with increasing distance from the impact zone and the bottom portion has a uniform grid distribution. Maintaining the same basic structure of the grid is the domain, multiple mesh sizes were tested to conduct a grid independence study and to choose the best mesh to analyze the computational results. To perform the grid independence study, a particular case of drop impact was considered and the relative error between consecutive increments in overall grid density in the diameter of the crown developed during impact was compared. The relative error in the crown diameter at a particular instance after impact between consecutive meshes is shown in Table 5-1. The dimensionless time τ is expressed as

푡⁄ where refers to t is the time in seconds after drop impacts the thin film. The number (푑⁄v)푑푟표푝 of cells in these meshes is provided. With increase in grid density, the computational time cost increases. The domain is parallelized to help minimize the computational cost. Though Mesh 4 provides the least relative error, Mesh 3 is seen to be the most optimum as it provides good computational results within a reasonable computational time.

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Atmosphere

zero gradient velocity, zero

zero gradientvelocity,zero zero gradient and pressuregradient zero

pressure and zero phase fraction fraction phase gradient

Outlet

Axis Initial drop with velocity v and diameter d Y Initial thin film with height H X Target Surface zero velocity, zero gradient pressure and zero gradient phase fraction

Figure 5-3: Boundary conditions and initializations for the computational domain

Table 5-1: Grid independence test: Relative % difference between consecutive meshes

Mesh 1 Mesh 2 Mesh 3 Mesh 4

Number of Cells 26160 59040 104640 163800

% difference at τ 4.44 N/A 8.22 3.09 0.36

% difference at τ 4.0 N/A 7.25 1.24 0.31

% difference at τ 3.11 N/A 13.52 1.45 0.13

% difference at τ 1.33 N/A 14.52 2.88 0.81

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< 5o wedge patch patch wedge 1

Axis of patch wedge 1

symmetry

Figure 5-4: The mesh in the wedge-shaped computational domain

1.20 Results and discussion

1.20.1 Validation of modified VoF Scheme

The modified VoF scheme developed in this study is validated with respect to the experimentally measured crown diameters for the different viscous liquids used. At specified dimensionless time, the crown diameters for the computational solution and the experiment are

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compared and the resulting difference is shown in Table 5-2. Mesh 3 is used for this comparison as it has been identified to be an optimum computational choice. The experimental uncertainty in the measurement of the crown diameter arises from the uncertainty in the image calibration measurement and is 0.7%. The percentage difference between the computational and the experimental measurement of the crown diameters is seen to be within reasonable limit of the experimental uncertainty as seen in Table 5-2 and therefore, the computational solutions in this study can be said to represent the physical process adequately.

Table 5-2: Validating the numerical scheme: difference between the computational and

experimental crown diameters (EG : ethylene glycol, x PG: x % by volume propylene glycol

water mixture)

Diameter of the % difference with Liquid τ drop (mm) experiment

EG 4.44 4.56 1.26

25 PG 4.43 3.79 1.92

50 PG 2.86 4.68 0.89

75 PG 4.82 4.93 1.50

Liquid Time (ms) Expt image Computational image

50 PG 4

D = 4.62 mm

V = 1.91m/s

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EG 4

D = 4.25 mm

V= 2.73 m/s

Figure 5-5: Verifying computational results with experimental results

1.20.2 Effect of liquid property on drop impact on thin films

Experimental analysis of the impact of drops on thin liquid films has helped to understand the underlying parameters that govern the phenomena. The governing dimensionless parameters have been identified as Re and Bond number (Bo) in a semi-empirical relation that was defined to identify the onset of splash for a given Newtonian drop (Chapter 4). Though certain hypothesis can be made, the experimental study, however, does not provide the knowledge of the process of the dynamic process of drop impact. To understand the underlying physics behind this phenomena, the current suggested numerical method is used. As Re and Bo had been identified as the governing dimensionless parameters, the effect of altering these factors should help in better understanding of this complex phenomena under study.

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Re 1276.18

Re 638.09

Re 319.04

Figure 5-6: Effect of changing viscosity on drop impact (We = 429.85, Bo = 4.69).

Effect of viscosity: To study the manner in which the viscous force acts and contributes to the drop impact dynamics, the viscosity of the liquid is modified while keeping the other factors a constant.

Figure 5-6 shows the development of the drop as it impacts the thin film of liquid. The images shown are captured at the same time instance. The colored portion of the image represents the liquid in the thin film while the white space represents the liquid in the drop. Though the properties of both the liquids are the same, the film liquid is colored to help identify the movement of each of the components (drop and thin film). It is observed that the position of the necking (outlined in the blue circle) is closer to the impact location for larger viscosity of the liquid. In addition, the radius of curvature of the thin film (outlined in green circle) is larger when viscosity is larger

(smaller Re). This indicates viscous resistance offered by the liquid in the thin film to the

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advancement of the impacting drop. As the progression of the drop is curbed by the viscous resistance thin film, the radius of curvature of the developing crown is altered.

We = 214.92 Bo = 2.34

We = 429.85 Bo = 4.69

We = 859.70 Bo = 9.37

Figure 5-7: Effect of changing the surface tension on drop impact (Re =638.09).

Effect of surface tension: Similar to isolating the response of viscous force on the drop impact phenomena, the reaction of surface tension force is evaluated in Figure 5-7. Identical to the previous comparison, the blue circles point to the position of necking while the green circles point to the curvature of the thin film layer. In these comparisons, the viscosity is maintained the same.

The location of the necking is observed to remain the same as surface tension is changed. The change in the radius of curvature of the thin film liquid layer is not pronounced as it was on changing the viscosity of the liquid. This further lays emphasis on the conjecture that the viscous resistance of the thin liquid film governs the spreading of the drop along the thin liquid layer. As

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surface tension is not seen to affect the spreading of the drop on the surface of the thin film but rather noticed to influence the splashing behavior of the crown formed, it is hypothesized that just as viscosity of the thin film governs the spreading of the drop along the thin liquid layer, surface tension of the liquid governs the splashing characteristic of the crown formed as a result of drop impact.

Observing Figure 5-6, it is seen that splashing is indirectly affected by the viscosity of the liquid as this affects the formation of the crown. As has been observed, the liquid viscosity affects the radius of curvature and the location of the neck formed at the origin of the crown. A smaller radius of curvature would contribute to higher surface tension forces (𝜎 ∝ 1/푅) and therefore, in turn, affect splashing. This would contribute to a larger pressure gradient in the direction along this change in curvature, i.e. along the direction of crown formation, and therefore force the flow of more liquid from the drop into the crown formation region thus increasing the height of the crown. Figure 5-8 shows the pressure profiles in the three cases where viscosity is changed for the liquid. The pressure along the curvature at the onset of the crown is seen to be highest for the low viscosity case (Re =1276.18) and therefore the crown height is largest for this and lowest for Re =

319.04 (high viscosity).

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0 100 Pressure [m2s−2] Density -100 200

Re 1276.18

Re 638.09

Re 319.04

Figure 5-8: Pressure profiles showing the effect of change in viscosity

Following the comparison of pressure profiles for changing viscosity that for variations in surface tension is compared next in Figure 5-9. The pressure variation along the radius of curvature at the neck of crown formation is seen to be similar to that in the previous case though the location of the neck does not change with change in surface tension. The pressures at the peak (tip) of the crown are seen to be higher with higher surface tension (though the viscosity is the same for these

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cases). As surface tension tries to minimize the surface energy, the development of a larger crown is prohibited by the higher surface tension forces. The velocity vectors depicted in

Figure 5-10 are flatter and smaller for higher surface tensions displaying that the surface tension force is countering the incoming liquid into the crown.

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0 100 Pressure [m2s−2] Density -100 200

We 859.70 Bo 9.37

We 429.85 Bo 4.69

We 214.92 Bo 2.34

Figure 5-9: Pressure profiles showing effect of changing surface tension on drop impact

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1 Velocity v [ms−1]

0 2

We = 214.92 Bo = 2.34

We = 429.85 Bo = 4.69

We = 859.70 Bo = 9.37

Figure 5-10: Velocity vectors – as surface tension is changed.

1.20.3 Effect of thin film height

In addition to the property of the Newtonian liquid (viscosity and surface tension), the dimensionless thin film height (H* = H/Ddrop) is another important parameter that contributes to post-impact dynamics. Typically, studies, both experimental and computational, categorize the post-impact dynamics broadly on the basis of the dimensionless film height. Based on these defined categories, the target surface is characterized as dry surface, thin-films, thick-films or deep pool. However, the definition for these categories are ambiguous. Therefore, while some studies define thin films as those with H* ≤ 0.1 [15, 57, 59] some other have considered a higher limit [52, 66].

As the current study is interested in drop impact dynamics on thin-films, an attempt to interpret

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the consequence of the dimensionless height of the liquid film is pursued. Study of early-time dynamics of the drop impact on a liquid film will help perceive the influence film thickness on development of the drop. The current numerical method is seen to agree well with the experimental observations. These experimental observations are however taken at an interval of 2 ms. To observe early-time dynamics, the interval of observation is required to be of the order of 10 μs.

Experimental observations at this pace are difficult to capture and hence numerical analysis at these time-scales is attempted. Josserand and Zaleski [53] use similar early-time numerical anaysis to formulate a theory predicting transistion between splashing and deposition for impacting drops.

They use a water-like liquid to help eliminate the strong numerical instabilities that lead to spurious currents at the interface. In their study, they find that the jet base location (r/Ddrop) is a function of time (τ = t.v/ Ddrop) and had negligible dependence of the Re of the impacting drop. In order to verify the early-time predictions of our current numerical mehtod, the liquid considered by

Josserand and Zaleski [53] has been used and this functional dependence of the jet base location with time is tested. Figure 5-11 contains data points from Josserand and Zaleski [53] as well as the current numerical study and is seen to be in good agreement. The solid line in the plot indicates the square root scaling at short times for the spreading behavior [53] and is in agreement with the current numerical study. In their examination of the development of the ejected sheet at early times,

Coppola et al. [61], however, find a deviation from this square root behavior that does not agree witht eh solid line in Figure 5-11. It is felt that as their analysis was essentially two-dimensional,

Coppola et al. [61] observe a deviation from this consideration that is treated as a quasi-threee- dimensional simulation. As the spreading of the drop (r/Ddrop) is seen to be independent of the

Reynolds number at early times, it can be concluded that a universal tragectory exists for the jet

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base at early propogation when it can be considered inviscid, though viscous effects are very noticable at later times as seen in Figure 5-6.

Figure 5-11: Log-log plot of spread factor r/Ddrop as a function of dimensionless time, τ, for We

= 8000 for data from Josserand and Zaleski [53] and the current numerical simulation.

In order to test the consequence of changing film thickness, different parameters of drop impact are tested with respect to the film thickness. As the drop impacts on the liquid layer, an ejected “jet” is developed at the interface of the impact. This ejected jet develops into the familiar crown. The properties of this ejected jet are compared for the different liquid film heights considered. The definition of these parameters used in the current study is provided in Figure 5-12.

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Ddrop – Diameter of drop r – Location of base of jet R –Radius of curvature JT – Jet Thickness JH – Jet Height D JH θ drop θ – Ejected angle JT r H* H* = dimensionless film height

Figure 5-12: Parameters of the ejected jet that are used for numerical comparison at different

dimensionless film heights.

The location of the base of the ejected jet (a.k.a spread factor (r/Ddrop)) is found to be independent of the film height at early times (τ < 1 Figure 5-13) and some variation is seen at τ >1. The variation of the other parameters of the ejected jet are provided in the plots in Figure 5-14. The ejected angle for lower H* is seen to decrease with time indicating that the ejected jet forms a small acute angle with respect to the target film and this decreases with time as the ejected jet merges with the liquid film. In contrast, higher H* results in a slow decrease of ejected angle with time and when the ejected jet begins to merge with the liquid film, the angle is close to 90o (τ > 1 Figure 5-14 (a)).

The height of ejected jet in all the four cases considered is seen to reach a constant value around τ

= 2 (Figure 5-14 (b)) where the jet would begin merging with the liquid film. The thickness at the base of the ejected jet is noticed to increase at early times (τ ≤ 1) and is seen to attain an almost constant value later (Figure 5-14 (c)) indicating that the ejected jet begins to merge with the thin liquid film. The radius of curvature at the outer end of the ejected jet (Figure 5-12 – circle) is seen to decrease with time. The decrease is more pronounced when the film height (H*) is smaller

(Figure 5-14 (d)). For larger H*, this radius of curvature almost remains a constant throughout.

One common theme in all these plots is that the height ratios, H*, of 0.2 and 0.3 have a different variation from that of 0.08 and 0.09. To understand there reason for this and to detect if this could

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indicate a possible definition of “thin-film thickness”, further analysis of the flow pattern in these cases is considered.

Figure 5-13: The location of the base (a.k.a. spread factor r/Ddrop) as a function of dimensionless

time

(a) (b)

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Figure 5-14: Variation of ejected jet parameters (a) ejected angle (b)height (c) base thickness

(d) radius of curvature with respect to dimensionless time.

As the drop impacts the stationary thin film, the inherent energy of the impinging drop as well as the energy induced on the thin liquid will, in combination, influence the formation and development of the ejected jet. To study the contribution of these two energies (one from the impacting drop and the other of the thin liquid film), vorticity magnitudes within the liquid layer are analyzed and are presented for four representative cases in Figure 5-15. The range for the vorticity magnitude is the same for all the cases for convenient comparisons. For H* < 0.1 (the first two cases in Figure 5-15), the vorticity magnitudes are very large at the base of the ejected jet. These arise from the impacting drop and the contours are seen to converge into the ejected jet.

The vorticity magnitudes on the thin film beyond the ejected jet are very small and do not participate in the ejected jet. For higher film thicknesses (H* > 0.1 – last two contours in Figure

5-15), this is not the case. The high magnitude vorticities from the impacting drop meet the lower magnitude vorticities from the thin film beyond the ejected jet. This indicates that energy from the

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thin film contribute to the formation and propagation of the ejected jet. Also, at the location where these two energies are seen to meet, there is a local minima at the mid-point of the jet base (red box in Figure 5-15 for H* 0.2 and H* 0.3). Interestingly, this location is seen to be at H* ~ 0.1.

This suggests that at small liquid film heights, the energy from the impacting drop overcomes that of the liquid in the thin film and as the liquid film thickness increases, the contribution of the liquid film in the ejected jet increases. Moreover, the location of the minima in the vorticity magnitudes when the two energies meet at the base of the ejected jet is seen to be at H* = 0.1. This intriguing observation could help categorize thin film drop dynamics as impacts on liquid films of H* ≤ 0.1.

Further, observing the movement of liquid in the film as the ejected jet is formed (Figure 5-16), it is noted that for thicker films (H* > 0.1) the liquid in ejected jet formed is primarily from the thin film while for thinner films, it is from the impacting drop. In all of these computational solutions, the impacting drop has the same property (liquid property and impacting diameter and velocity).

Therefore, it can be concluded that for H* > 0.1, the impacting drop has a larger depth to which its energy can penetrate and the thin film is capable of supplementing the energy to form the ejected jet. Whereas, when H* < 0.1, the energy of the impacting drop is directed to create the ejected jet as it does not have the liquid film depth to absorb some of the impacting energy.

Vorticity 0.01 2500 5000 7500 10000 Magnitude (s-1)

H* = 0.04

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H* = 0.08

H* = 0.2

H* = 0.3

Figure 5-15: Vorticity within the liquid layer for different liquid film heights (H*)

H* = 0.08

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H* = 0.2

H* = 0.3

Figure 5-16: Movement of thin film liquid for different film heights (H*)

1.21 Conclusion

A numerical model to evaluate the consequence of viscous and surface tension forces during a Newtonian drop impact on a thin liquid film is presented. Typical numerical modeling techniques are found to produce spurious currents at the interface for high viscosity liquids and thus, leads to incorrect evaluation of pressure gradient and consequently the movement of the interface. A method to overcome this by modifying the property averaging technique is suggested in this study.

This modification to the numerical scheme is also seen to improve computational predictions for viscous liquid jet breakup (Chapter 2). This modified numerical technique is validated with the experimental study of drop impact on thin films (Chapter 4). Isolated numerical analysis of response of liquid viscosity and surface tension is conducted. While both these forces act in

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combination to determine the post-impact dynamics of the phenomena under consideration, the viscous resistance of the liquid film is seen to resist the propagation of the drop and thus the formation of the crown, while the surface tension force primarily opposes the development of the crown as it tries to minimize the surface energy.

The aforementioned numerical technique is also used to analyze the contribution of the thin film thickness on the development of the drop impact. Early-time analysis of the phenomena is validated with past numerical work of Josserand and Zaleski [53]. A distinction on the basis of dimensionless film thickness (H*) is perceived by comparing the ejected jet characteristics at early-times. It is concluded that for small liquid film thicknesses (H* < 0.1), the energy of the impacting drop goes towards creating the ejected jet as the drop does not have a liquid layer to absorb its impacting energy. For larger film heights (H* > 0.1), the thin film layer has sufficient thickness to absorb a part of the energy from the impacting drop and thus the ejected jet is formed due to the combined action of both the impacting drop as well as the liquid layer. Interestingly, when both the liquid layer and the impacting drop contribute to the formation of the ejected jet, the location of the minima of the vorticity magnitude at the base of the ejected jet is seen to be at

H* ~ 0.1. This could therefore, provide an approach to classify the impact dynamics on the basis of liquid film thickness and its influence on determining post-impact dynamics.

Though arduous, experimental evidence of the behavior of thin film liquid relative to that of the impacting drop will provide immense insight into the suggested mechanism of drop impact and the consequent post-impact dynamics. Some experiments as well as computational studies have captured air entrainment at early impact times. This would be of great significance in many applications and a further study to understand the cause and effect of this entrainment is required.

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6 CHAPTER 6: SECONDARY DROPS FORMATION ON IMPACT ON A THIN

FILM

1.22 Introduction

The significance of spray impact on a target surfaces gained a lot of attention in the early

1980s when many experimental studies were placing emphasis on understanding this complex phenomenon. It was largely accepted that to comprehend the phenomena from a fundamental manner, single drop impacts must be investigated [7, 12, 74]. Amongst the many possible scenarios of impacting drops, those that impact solid targets or deep pools have received the most attention.

The primary focus of most studies has been on understanding the process of impact and the post- impact dynamics such as crown formation and splash. The impact of drops and sprays on diverse target surfaces is of high significance and thus has demands efforts to provide accurate physical understanding of the outcomes. The possible outcomes can very broadly be categorized into splash and deposition. Splash has traditionally been defined as the production of secondary drops from the impacting event. While in some scenarios, such as nasal sprays, pesticides and fuel injectors, production of secondary drops is critical to provide a small distribution of uniform drop sizes, in other cases such as coatings and harmful chemical applications, this is to be curtailed. This has led to a number of studies, both experimental and analytical, to examine the splash-deposition limit in order to propose well-established correlations to predict the onset of splash given specific operation limits. Since, most engineering applications that involve sprays of individual droplets, typically have them impacting on a thin layer of liquid formed by prior drops, some recent studies have started studying impacts of drops on thin liquid films [7, 15, 55, 65]. While some of these studies have attempted to predict the onset of splashing, there is some variability in their work. This

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variability in the findings, despite use of similar working fluid (water), points to an incomplete understanding of the impact phenomena. In our endeavor to develop a better physical understanding, normal impact of viscous Newtonian liquids were tested experimentally to establish a correlation for the onset of splashing on thin liquid films which is seen to agree well with past experimental data.

Apart from being able to predict the onset of splash, the characterization of these secondary drops that result from splash becomes significant in many engineering applications. Analysis of both experimental and numerical methods used in the different mass and momentum interactions during drop impact on thin films can be found in recent review by Liang and Mudawar [75]. The key events during splashing from a drop impact on thin films are formation of crown and/or a central jet (column of liquid rising at the center of impact). When splashing occurs, secondary drops can either be created at the rim edge of the crown and/or at the tip of the central jet as these features are typically conducive to growth of instabilities. The secondary drops thus produced can deposit on the thin film or can get airborne. Examples of these behaviors of secondary droplets generated from splashing of the crown rim are presented in Figure 6-1. These drops typically contaminate neighboring targets during spray coating, or, pollute the ambiance environment during chemical spraying. The knowledge of spreading behavior of these secondary drops can thus help establish tighter control on such consequences and has been the central focus of a few past studies.

(a) We = 308 Re = 8982 (b) We = 305 Re = 10100

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Figure 6-1: Water drop impacting on thin film at different Re to form secondary drops from the

crown rim (a) secondary drops deposit on thin film (b) secondary drops are air-borne.

Stow and Stainer [76] used water drops on smooth and roughened surfaces to study resultant secondary drop size distribution. They were interested in identifying the significance of governing operational variables on secondary drop diameter and velocity. For impacting drops on wetted surface, they find that the mean diameter of secondary drop decreases with increasing height of liquid film while the number of secondary drops increases. In contrast, Cossali’s study [66], which was one of the first to perform a temporal study on the evolution of secondary drops formed on impact with a wetted target, find that the film thickness has no effect on secondary drop diameter.

Another study that aimed to use experimental techniques to characterize the secondary drop size was by Okawa [58] and they found this to be dependent on the dimensionless parameter K = We

Oh-0.4 which has been reported by a few prior studies to characterize the onset of splash. Though some dependence of secondary drop size on film thickness was detected, its contribution was determined to be minuscule. Vander Wal et al. [68] determined that the influence of surface tension and viscosity was to increase the size of generated secondary drops while decreasing the number produced. Viscosity of the impacting drop is also observed to determine the origin of the secondary drops. While low viscosity enables secondary drop generation in the early stages of crown growth, higher viscosities is seen to inhibit this till the complete growth of the crown [55].

To understand the basis of secondary drop generation, in addition to experimental studies, analytical modeling approaches have also been employed to describe this impact dynamics. These modeling approaches mainly rely on the development of instability during the process. Deegan et al. [52] identify three sources of instability as prompt instability that occurs at impact; crown instability that leads to formation of jets at the rim; and crown instability that leads to formation

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of drops. They determine that the type of instability dictate the resultant secondary drop distribution. One suggested mechanism for production of secondary drops is through Rayleigh-

Plateau capillary instability. Endorsements for this theory comes from numerical simulations and experimental validations that study and observe the significance of initial perturbance on impact behavior [77–79]. However, Yarin and Weiss [54] conclude their experimentally observed number of crown rim jets was inconsistent with the Rayleigh-Plateau instability and suggest a different nonlinear perturbation mechanism where the rim propagates normal to the crown expansion. Other research work has suggested a combination of different instabilities that include as Rayleigh-

Taylor [67], Richtmyer-Meshkov [80] (where a shock wave interacts with the free surface) and

Marangoni driven flow [81] (where surface tension difference promotes the crown breakup).

Many of the aforementioned experimental work have employed similar working fluids, and yet there is some divergence in their observations as well as in the proposed mechanism for the formation of secondary drops during splashing. This necessitates a more in-depth investigation of the splashing during drop impact on thin films. In addition, our previous experimental study on the onset of splashing, has identified the relevance of viscous working fluids. Therefore, to provide a more comprehensive experimental characterization of secondary drops generated during drop impact on thin films, an experimental investigation involving multiple viscous Newtonian liquids is conducted.

1.23 Experimental Procedure

1.23.1 Experimental Setup

Though photography was invented in mid-1820, when Worthington [82] first examined the phenomena of splashes it had not developed enough to capture these fast events. Therefore, he used electric flash of very short duration which was timed suitability to illuminate stages of

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splashing as desired. He notes that based on repetitions, a drawing of the splash phenomena was made for further analysis. This was necessary as photographic plates were not sufficiently sensitive to respond to short exposures. With advancements in high speed photography, image capture of these fast phenomena became more accessible for analysis. In addition to attracting scientific attention, this phenomena of splash gained a lot of artistic interest and one such well known iconic image was that of a milk drop splashing captured by Edgerton [83] in 1954 at 10000 fps. Thoroddsen et al. [84] review a number of recent technological advances in high speed image cameras and their use in analysis of fast events focusing on drop and bubble coalescence, impact and pinch-off. The

X-ray technique is another interesting means of image capture used to observe features that would otherwise be obstructed in visible light. This has been used by Zhang et al. [85] to analyze formation of jets from impact of single drop with a deep layer of the same liquid. Ninomiya and Iwamoto [86] use Particle Image Velocimetry (PIV) in addition to high-speed imaging to measure motions of the fluid layer and the liquid film during the development of a milk drop crown. A recent development in the field of image capture has been Digital on-line holography (DIH), which is an optical technique to measure particle sizes and their three-dimensional position and velocities. This is found to be very helpful in understanding the complex trajectory of fast impact of drops on thin liquid films by Guildenbecher et al. [87].

For the current study, high-speed camera imaging, using the Photron Mini UX100 camera, was the chosen technique for image capture. The range of frame rates used for capturing the generation and trajectory of secondary drops was 4000 to 6000 fps: higher frame rates for higher impacting velocities. To provide proper flicker-free illumination, Nila Zaila Single Tungsten LED lights were used. The light is incident on a white screen placed parallel to the plane of viewing. With this set- up, images were captured using a computerized data acquisition system connected to the high-

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speed camera system. The high-quality images thus obtained were analyzed using an in-house

MATLAB code as well as IMAGEPRO PLUS 4.0 software.

To enable an in-depth understanding of secondary drop formation from drop impact on thin films, certain input parameters are varied. These include, liquid properties (viscosity, density, surface tension) and input drop diameter and velocity. To vary the liquid properties, three different liquids are used as working fluids. The working liquid is pumped at a constant flow rate using a NEXUS

3000 syringe pump from stainless steel needles of varying diameter mounted vertically at varying heights above the target thin film surface. A schematic of the experimental set-up is provided in

Figure 4-2 and a detailed account of the set-up is provided in Chapter 3. The range of working liquid properties, and input parameters are provided in Table 6-1 and Table 6-2.

Table 6-1: Properties of working liquids used in the experiment

63% Glycerin

Properties Water Ethylene Glycol Water Mixture

Density, ρ (kg/m3) 998 1113.2 1036

Surface Tension, σ 72.8 48.4 68 (mN/m)

Viscosity, μ (mPa.s) 1 16.1 15.5

Morton number, Mo 2.55 x 10-11 1.55 x 10-6 5.09 x 10-6

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Table 6-2: Range of experimental input test parameters

63% Glycerin Properties Water Ethylene Glycol Water Mixture

Drop Diameter, d (mm) 3.1 – 4.2 3.8 – 4.3 2.4 – 3.1

Impact Velocity, u (m/s) 1.7 – 2.6 2.5 – 3.3 4.7 – 5.0

Weber number, We 174 – 313 616 – 961 502 – 793

Reynolds number, Re 6894 – 9322 741 – 912 915 – 1135

1.23.2 Image Processing

The high-speed images obtained have multiple characteristics that are required to be systematically measured. The sequence of images captured consist in progression, the drop impacting on the thin film, growth of the crown, generation of secondary drops and the deposition of the remaining crown onto the thin liquid film. The key parameters required to be measured are the impacting drop diameter and velocity, the crown characteristics (height and width) with time and secondary drop characteristics (diameter and velocity). The crown characteristics (height and width) with time are measured using the image processing software IMAGEPRO PLUS. To ensure correct measurements, the images are calibrated with the outer diameter of a stainless steel needle.

Using the IMAGEPRO PLUS software for measurements of image drop diameter and velocity, and secondary drop characteristics becomes tedious. Therefore, to enable efficient image processing, a MATLAB code was developed. Prior to making measurements with the code, the images are correctly calibrated as before. Figure 6-2 shows the process of image processing using the MATLAB code to identify the secondary drops generated. To correctly characterize the impacting drop, two/three images prior to impact are considered. The velocity and the drop

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diameter are calculated from these images. Next, to characterize the secondary drops, consecutive splash images are converted into gray scale to aid in easier extraction of splash characteristics.

Moreover, the background in these images is bright, thus helping detection of the smaller characteristics such as the secondary drops. The sensitivity of the circle detector is adjusted to capture the secondary drops in the focal plane. The process is repeated for more consecutive splash image pairs to obtain the correct secondary drop velocity and diameter.

Figure 6-2: MATLAB image processing to identify and measure secondary drops ejected from an

ethylene glycol impacting drop of diameter 2.9 mm at 4.17 m/s on a stagnant thin target film.

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1.24 Results and discussion

Drops impacting on thin films can lead to different outcomes such as prompt splash, crown splash or deposition. In the first case, at impact secondary air-borne drops are generated well before a crown begins to form. Crown splash occurs after the crown has formed. The instabilities on the crown surface grown and when these reach the crown rim, secondary drops can get pinched off from the crown. The last event, deposition, occurs when the instabilities on the crown are not large enough to aid drop formation. This typically occurs at low impacting drop velocities. Section 3 discusses a means to predict if a drop would deposit or produce a crown splash as it impacts a thin liquid film. This current study attempts to gain a deeper understanding of crown splash as a

Newtonian drop impacts a thin film target. Three Newtonian liquids are considered for this experimental analysis as they impact a thin film with non-dimensional height (H* = H/Ddrop) of

0.1. Cossali et al. [66] were one of the first to experimentally study the evolution of impact of water drops on thin liquid films. Their study shows that maximum crown height of a splashing drop after impact was dependent on impact drop velocity, but showed weak influence of the film thickness.

To aid comparison of the current experimental study with that of Cossali et al. [66], the conditions for comparison are outlined here. Water data from the current experiment are compared with

Cossali et al. [66]. In the current study, the drop diameters range from 3.08 – 4.17 mm and the impacting velocities range from 1.74 – 2.56 m/s, while Cossali et al. [66] used an initial drop diameter of 3.8 mm and impacting velocity range of 1.35 – 3.25 m/s. Figure 6-3 plots the maximum crown height versus the impacting Weber number for both sets of data. The current experimental data is seen to be congruent to Cossali et al. [66]’s findings. Similar to Cossali et al. [66]’s study, here, results are obtained from 2D images and do not resolve the out-of-plane direction. Symmetry

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is the post-impact dynamics is assumed and therefore, the mean results discussed here would remain valid.

Figure 6-3: Current experimental data comparing dimensionless maximum crown height, Hmax

[66] /Ddrop , as a function of impact drop Weber number, We with data from Cossali et al. .

The process of a crown developing post-impact and the growth of surface instabilities leading to ejection of secondary drops (splashing) is a complex phenomenon and involves multiple elements that need consideration. These elements can be broadly grouped into secondary drop characteristics and crown characteristics. These following sections discuss these specific characteristics in further detail.

1.24.1 Secondary drop characteristics

Section 3 provides a convenient means to predict occurrence of splash as a given drop impacts on a thin film target. During splash, as secondary drops are ejected from the crown rim due to Rayleigh-Taylor instabilities, it could be valuable to understand the expected drop

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distribution and their trajectory. A crown splash typically produces a range of droplet sizes. Any measurement made to quantify these, needs to represent some kind of average measurement along with the distribution. There are different methods used for estimating average drop sizes and sprays, and these can produce greatly different results. The choice of averaging method largely depends on the application of interest. In the current scenario, the secondary drops generated can indicate the resultant active surface area. These could be important in some application to increase the effectiveness of mass distribution (for e.g. pesticide sprays) and could be undesirable in some others (for e.g. in coating these secondary drops can contaminate nearby equipment/products). As the overall surface area is the critical factor, the Sauter Mean Diameter (SMD or D32) is used in this study to characterize the secondary drop size. This is defined as the equivalent volume to surface area ratio that is representative of the ensemble of ejected drops.

푁 3 ∑푖=1 퐷푖 푑푟표푝 (6-1) 푆푀퐷 = 퐷32 = 푁 2 ∑푖=1 퐷푖 푑푟표푝

It is observed that while the D32 distinctly increases with increase in impact velocities, the increase with impact drop diameter is not as pronounced. Also, in Section 3 and 4, it was established that surface tension of the liquid is primarily responsible for creation and propagation of the instabilities on the crown surface that ultimately lead to formation of the secondary drops.

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Figure 6-4: Sauter Mean Diameter (D32) of the secondary drops as a function of liquid property

and impact drop Weber number (We)

Therefore, an empirical correlation relating the secondary drop diameter (D32) to the impacting

Weber number (We) is formulated as:

퐷32 = 4.4 x 10−4(푀표−0.1푊푒0.75) (6-2) 퐷푑푟표푝

This relation is found to predict the Sauter Mean Diameter for the experimental data for all the three Newtonian liquids in consideration. Morton number (Mo) in the above relation is a dimensionless number that is a function of liquid property and provides a way to characterize each liquid. The plot of D32 as a function of We and Mo is provided in Figure 6-4. The bars in this plot represent the distribution in secondary drop diameters in the observation plane. The maximum deviation of the experimental data from the predictive correlation is found to be 13.2 %. The experimental uncertainty arises from the methods used to estimate the measurements. This

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includes the pixel count error, the Vernier caliper error and the error in the syringe pump flow control. This experimental uncertainty in measurement of drop diameter is estimated to be 8.5%.

Similar to measuring the secondary drop diameter distribution and Sauter Mean Diameter, knowledge of the ejected velocity would be helpful to predict the trajectory of the drop. The velocity of the ejected drops is calculated from the planar change in drop location between consecutive 2D images. While the distribution of the secondary drop diameters (bars in Figure

6-4), is not too wide, that in ejected velocity was seen to be very wide in certain cases. Therefore, a definition to represent the mean velocity, similar to the Sauter Mean Diameter for diameter distribution, would not be favorable. A study to understand the origin of this wide ejected velocity distribution is therefore undertaken. As a drop impacts a thin film, a crown is formed and with the spread of the crown, at the rim, for appropriate condition, secondary drops are pinched off from the rim. It is found that for some impact conditions, drops begin to be pinched off as the crown is still growing. For certain other conditions, secondary drop generation is only noticed to occur as the crown recedes (falls back) to merge with the thin film layer. Two such cases of secondary drop pinch off are shown in Figure 6-5 and Figure 6-6 for Ethylene glycol. In Figure 6-5, two different drop sizes with a large variation in ejected velocity is shown. In these figures, the time difference between consecutive images is the same (0.6 ms) and the yellow line in these images are provided to help aid easy comparison between velocities. The drop of 0.74 mm diameter is very fast moving in comparison to the 0.94 mm diameter drop. The first drop (0.74 mm) is noticed to be ejected before the crown reaches its maximum height, while the second one (0.94 mm) is ejected as the crown recedes. The case shown in Figure 6-6, the impacting ethylene glycol drop is slower moving.

Secondary drops of similar size and velocity are seen to be ejected from the crown rim here. The drops are seen to pinch off as the crown recedes into the thin liquid film layer. As before, the time

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between consecutive images is a constant (1.2 ms) and the yellow lines are provided to aid easy comparison between velocities of the highlighted ejected drops. It is noticed that with decrease in impact velocity (and therefore, We), there is more uniformity in ejected velocity distribution.

To further see the exact change in trajectory of the ejected drops, the numerical study in

Section 4 is used. Figure 6-7 shows the ejected drops for the two experimental case discussed. For the first impact (Ddrop = 4.49 mm Vimpact =2.99 m/s), the first ejected drop is pinched off as the crown is still growing and expanding. From the velocity vectors, it is observed that the trajectory is in the direction of crown rim growth. As the second ejected drop is pinched off, the crown has started to recede and therefore, as the velocity vectors indicate, there are two components of velocity here: one in the direction of rim stretching and the other in the direction of the receding crown. Thus, the resultant velocity is slower and these drops are the ones that are most likely to fall back onto the target thin film at some distance from the impact location. The second case of impact (Ddrop = 4.46 mm Vimpact =2.25 m/s) is slower moving and as the ejected drop is only pinched-off when the crown begins to recede, this secondary drop is slower moving are more likely to fall back onto the thin film layer at some distance from the impact location. In contrast, the fast moving ejected drops are more likely to be air borne and would not affect the surrounding target film. From an application point of view, these faster moving drops could affect the operator or nearby equipment, while the slower moving ones affect the nearby target surface.

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9.8 ms 14 ms

10.4 ms 14.6 ms

11 ms 15.2 ms

V = 0.58 m/s V = 1.59 m/s 0.94 mm 0.74 mm

Figure 6-5: Ejected velocities for secondary drops from an ethylene glycol impact drop of Ddrop = 4.49 mm Vimpact =2.99 m/s

12.4 ms

13.6 ms

14.8 ms

16 ms

V = 0.29 m/s ; V = 0.29 m/s 0.86 mm 0.81 mm

Figure 6-6: Ejected velocities for secondary drops from an ethylene glycol impact drop of Ddrop

= 4.46 mm Vimpact =2.25 m/s

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fast moving drop Velocity Magnitude

slow moving drop

D = 4.49 mm V =2.99 m/s D = 4.46 mm V =2.25 m/s drop impact drop impact

Figure 6-7: Computational images highlighting the change in ejected drop trajectory for the

above two ethylene glycol impacts.

1.24.2 Crown characteristics during splash

In addition to understanding the secondary drop characteristics during splash, the characteristics of the crown are of significance. As is observed above, the velocity of the ejected secondary drops depends on when the pinch off occurs during the crown dynamics. It is noted that the fast moving drops that are ejected before the crown attains its maximum height have a different trajectory than the slower moving drops ejected as the crown recedes to merge with the thin film layer. The key to anticipating the production of air-borne faster moving secondary drops, relies on estimating the maximum crown height (Hmax) and the time for crown growth (tHmax). These parameters are correlated with impact drop conditions to provide reliable correlations based on the experimental data for the three Newtonian liquids used. To estimate the time for crown growth, the velocity of crown growth is correlated with the impact velocity. The correlation to predict the

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maximum crown height and the time to reach this height are as provided in equations (6-3) and below(6-4). These correlations are seen to be in good agreement with the experimental data as has been plotted in Figure 6-8 and Figure 6-9. The maximum percentage deviation from the correlation for maximum crown height is 13.9 % and that for crown growth velocity id 13.8 %. Also, while studying the maximum crown diameter or width (Dcrown) for these experimental data, it was found to be about five times the impact drop diameter (Dimpact) and is plotted in Figure 6-10.

퐻 푚푎푥 = 4.4 x 10−2(푀표−0.02푊푒0.5) (6-3) 퐷푑푟표푝

퐻푚푎푥 0.01 0.55 (6-4) = 0.574 (푀표 푉푖푚푝푎푐푡 ) 푡퐻푚푎푥

Figure 6-8: Dimensionless crown height (Hmax/Ddrop) as a function of liquid property (Morton

number, Mo) and impact Weber number, We

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Figure 6-9: Crown growth velocity (Hmax/tHmax), as a function of liquid property (Morton

number, Mo) and impact velocity (Vimpact)

Figure 6-10: Dimensionless maximum crown diameter (Dcrown/Ddrop) as a function of impact

Weber number, We

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1.25 Conclusions

The characteristics of secondary drops that are formed as a drop impacts a thin target film are studied in this current work. Three different Newtonian liquids are tested in this experimental work to identify and characterize the secondary drops and the crown formed during the impact. The

Sauter Mean Diameter (SMD or D32) is used to characterize the secondary drop size as it represents the equivalent volume to surface area ratio that is representative of the ensemble of ejected drops. It is found that the SMD is dependent of the liquid characteristic property and the impacting Weber number. A predictive correlation for this relation is constructed and is given by equation (6-2). While tracking the trajectory of these ejected drops, it was found that there are two types of ejected drops which vary vastly in their ejected velocities. Both experimental and numerical analysis were used to understand the basis of formation of these two types of ejected drops. The drop that pinches off as while the crown is still growing and expanding is faster moving and its trajectory is in the direction of crown rim growth. The slower moving ejected drop is pinched off as the crown starts to recede and therefore, as the velocity vectors indicate, there are two components of velocity here: one in the direction of rim stretching and the other in the direction of the receding crown. Thus, the resultant velocity is slower and these drops are the ones that are most likely to fall back onto the target thin film at some distance from the impact location.

The air-borne faster moving secondary drops, depend on the crown characteristics and thus it becomes important to estimating the maximum crown height and the time for crown growth. While the maximum crown height is found to be a function of the liquid properties and the impacting

Weber number (equation (6-3)), the time for impact can be correlated with the impacting velocity

(equation (6-4)). These observations show a means to predict the secondary drop characteristics

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during drop impact on a thin film. This knowledge would help estimate the degree of splashing and contamination to the nearby surroundings.

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7 CHAPTER 7: NON-NEWTONIAN DROP IMPACT ON THIN LIQUID FILMS.

1.26 Introduction

Drop impacts on varying target surfaces occur in systems as diverse as pesticide sprays, pharmaceutical coatings, and material processing. Control of the impact dynamics becomes critical for the coating, deposition and material transfer processes. With the advancement of image capturing capabilities in addition to technical advances in numerical simulations, new details on drop spreading, splashing, and depositing dynamics have been brought forward. This has aided in providing models and scaling laws that describe the complex physics involved. Literature on impact behavior of drops on dry surface and on deep pools has been of interest for over 100 years.

Apart from a few applications, most impact on dry surfaces are essentially on a wetted surface with a thin film formed from prior drop impacts. This has therefore gained attention and been the subject of interest over the past few decades and experimental and numerical analysis have been conducted to provide scaling laws to predict impact outcomes [15, 55, 61, 65]. As a drop impacts on a thin film of liquid, an enormous pressure gradient in created at impact. The drop then spreads on the film forming a crown. The deformation, spreading and collapse of the impacting drop are governed by inertial, viscous and surface tension forces. Our prior research has noted the substantial influence of viscous properties on dictating post impact dynamics and an experimental study coupled with numerical investigation has helped understand the governing physics that dictates the phenomenon. Most studies on the impact of drops on liquid surfaces have looked at

Newtonian liquids – more specifically water. The theoretical understanding of the process of impingement on thin films is therefore constructed on Newtonian basis. Non-Newtonian drop impact increase the rheological complexity of the working fluid and therefore could differ widely

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from Newtonian drop impact behavior. At the minimum, for a process requiring the use of non-

Newtonian working fluids, more parametric definition is necessary to include the viscous and elastic behavior of the fluid. Non-Newtonian effects such as viscoelasticity and shear thinning behavior have been studied in the past for impacts on dry surface and, as briefly outlined here, have a substantial influence.

Certain studies have looked at the influence of non-Newtonian properties on contact line dynamics for impacts on dry targets, heated surface and deep pools. It has been noted that the singularity of the viscous stress at the contact line (point of impact) poses a difficulty even in the analysis of Newtonian fluid behavior [52]. Non-Newtonian flow properties would largely alter this singularity and will, therefore, play an important role in determining the development of the drop, post impact. Bartolo et al.[88] found that polymer additives present in a drop can generate strong normal stresses and, at the point of impact, there is a competition between the surface tension and the polymeric normal stress, which can slow down the spreading and retracting of a polymeric drop as it impacts a dry target surface. This theory is found to be in line with prior studies that find suppression of recoil of drops on addition of polymers [89]. However, some other studies comprising of examinations of drop and sphere impacts on dry, heat and pool surfaces such as

Cheny and Walters[90] ,Crooks et al.[91], and Bergeron et al. [92], attribute this to the extensional viscosity introduced due to the non-Newtonian property of the test fluid. Therefore, though the observed effect of non-Newtonian additives in drop impact dynamics have been consistent, the physical understanding of the effect is still developing. These observations on the influence of non-Newtonian rheology during drop impact on dry surfaces and pools, have encouraged explorations on exploiting this to improve performance during application. For example, Bergeron et al. [92] observes that non-Newtonian elongational viscosity of the polymeric solution provides a

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large resistance to drop retraction after impact thus increasing the amount of pesticide spray retained by a plant. Thus, though typical efficiency of pesticide treatments can be lesser than 50 percent for plants with non-wetting leaves a good improvement was be provided by addition of compatible polymer additives to the pesticide spray.

These studies on drop impact on dry surface and deep pools were motivated by the need to suppress splashing and drop rebounding and have successfully demonstrated that the addition of even small amounts of polymeric substances could alter the impact behavior favorably. Seeking encouragement from these studies, recently Guildenbecher et al. [93] varied drop viscosity using

CMC-Na additive to obtain shear-thinning rheological properties, to study drop impact on liquid films. Though the concentrations of the additive used were very small, they did notice suppression of crown formation and deviation in the size of secondary drops formed on splashing. As the low concentrations of additive were used, they note that the effects shown in their experiment may not be purely due to the non-Newtonian rheology. This suppression of crown growth and secondary drop creation during polymeric drop impact on thin films was attributed to the elasticity of the thin film fluid by Lampe et al. [94]. They experimentally studied water drop impacts on films of viscoelastic micelle solutions and postulated a new form of splashing threshold that accounts for the viscoelastic effects of the fluid.

The identification of the influence of non-Newtonian rheology on drop impact on thin liquid films is still very new. The immense utilization of polymers in a multitude of industries from spray coating to agriculture and medicine demands a comprehensive understanding of its influence on drop impact dynamics. This study aims to further the current understanding of the influence of polymeric additives on drop impact on thin films (of non-dimensional height ≤ 0.1) is using experimental analysis. The shear thinning polymeric liquids that are used for this study are low

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concentrations of HEC-HHR and PEO-WSR 303. The concentrations of the polymer chosen for the study will be compared with equivalent Newtonian liquid to help isolate the non-Newtonian effect. As a drop impacts on a thin film, it spreads along with the film surface to form a crown.

The crown grows outward and upwards and, if the capillary surface waves are considerable, secondary drops could be released from the rim of the crown, forming a splash. The crown itself loses energy as it grows and would merge with the thin film liquid. When secondary drops are not released, the drop liquid completely merges with the thin film target and this is called deposition.

In the current experimental study, deposition is the phenomena investigated, with the aim to identify the influence of non-Newtonian rheology on temporal crown growth dynamics.

1.27 Experimental Procedure

1.27.1 Experimental Setup

The experimental setup used for this study is as shown in Figure 4-2. A drop was generated at the tip of a stainless-steel needle that was mounted on a stand which could be lowered or raised to adjust the height of drop release. In addition to adjusting the height of drop release, the velocity of the drop was further regulated by a syringe pump that supplied working liquid at a constant flow rate. The first few drops impacted a dry copper substrate to form a thin film layer. The ratio of the thin film height to the incident drop diameter is verified to be ≤ 0.1. After obtaining the desired thin film layer, further drops are incident on this layer and the impact dynamics of these drops are captured using a hi-speed camera system. The target copper plate is much larger than the incident drop diameter (20 – 50 times larger) to ensure that the change in thin film height is very slow with each incident drop. The target surface and the stainless-steel needle are maintained perpendicular with the help of a level. High-speed camera imaging, using the Photron Mini UX100 camera, was the chosen technique for image capture. The range of frame rates used for capturing the generation

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and trajectory of secondary drops was 4000 to 6000 fps: higher frame rates for higher impacting velocities. To provide proper flicker-free illumination, Nila Zaila Single Tungsten LED lights were used. The light is incident on a white screen placed parallel to the plane of viewing. With this set- up, images were captured using a computerized data acquisition system connected to the high- speed camera system. The high-quality images thus obtained were analyzed using the IMAGEPRO

PLUS 4.0 software.

1.27.2 Characteristics of the working liquids tested

Prior to analyzing the crown growth dynamics to identify the significance of non-Newtonian rheology, the effect of Newtonian characteristic (surface tension and viscosity) are isolated. The liquid characteristics of the five liquids used in the experiment are shown in Table 7-1. The polymeric solutions are prepared by dissolving the required quantity of polymer in water. As the concentrations of polymer added are very small, they do not alter the density of the water-based solutions. However, they dramatically affect the viscosity and the surface tension of the solution.

To measure the dynamic viscosity and surface tension of the prepared solutions, a rheometer and a tensiometer are used respectively. The next section describes the basic measurement principle of these two measurement methods briefly.

Table 7-1: Liquid characteristics of the five liquids tested

PEO WSR 303 HEC 250 HHR HEC 250 HHR Ethylene 63% Glycerin

1200 wppm 2000 wppm 1200 wppm Glycol Water Mixture

ρ (kg/m3) 998 998 998 1113.2 1163

μo (Pa.s) 0.0136 0.0157 0.0056 0.016 0.0155

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σ (N/m) 0.064 0.0665 0.0675 0.0484 0.068

Viscosity Measurement

The viscosity of the Newtonian and polymeric solutions (with water) were measured using a rheometer. This is a type of viscometer instrument that is used to characterize liquids whose viscosities vary with flow conditions. The flow of fluid in a particular geometry determines the strain rate experienced by the fluid. The corresponding stresses are a measure of the fluid resistance i.e. viscosity of the liquid at that specific flow condition. For this experiment, concentric cylinder geometry is used in a AR 2000 rheometer. In this geometry, the fluid is filled in the annulus between two cylinders and the inner cylinder is rotated while the outer is fixed. Shear stress is measure from the torque on the stationary cylinder and shear rate is determined from the rotational velocity of the moving cylinder. The ratio of the shear stress to the corresponding shear rate would provide the apparent viscosity of the fluid. By determining the fluid viscosity for a range of shear rates, the rheological characteristics of the sample fluid in the domain of application, can be determined. For Newtonian liquids, this apparent viscosity vs. shear rate curve is a constant as the fluid properties do not change with shear. However, for the polymeric solutions, the viscosity is noted to decrease with shear rate and thus, they exhibit shear-thinning characteristics as shown in

Figure 7-1. It should be noted that the rheometer only determines the shear viscosity behavior of the polymeric solutions used and no data on extensional viscosity can be determined. The two polymers used are HEC 250 HHR and PEO WSR 303 and their properties are provided in Table

7-2. The shear rheology at different concentrations for these fluids are shown in Figure 7-1. In order to effectively compare the effect of rheology on drop impact dynamics, the concentrations

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of HEC 250 HHR and PEO WSR 303 that have zero-shear viscosities comparable to the tested

Newtonian liquids are chosen. These are 1.548 x 10-9 mol/cc (2000 wppm) and 9.242 x 10-10 mol/cc

(1200 wppm) of HEC 250 HHR and 1.720 x 10-10 mol/cc (1200 wppm) of PEO WSR 303 properties of these liquids that are finally used for this comparative study are summarized in Table

7-1.

Table 7-2: Properties of the polymers used in the current experiment

Polymer Molecular weight Ionic Nature Appearance

HEC 250 HHR 1300 kg/mol Non-ionic White powder

PEO WSR 303 7000 kg/mol Non-ionic Off-white powder

Figure 7-1: Measured apparent viscosity vs shear rate for different concentrations of HEC 250

HHR and PEO WSR 303. Concentrations used in the current study are highlighted in red.

Surface Tension Measurement

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The surface tension of the liquids used are measured using a bubble pressure tensiometer –

Sensadyne Model QC6000. Two probes with different diameters are immersed into the sampling liquid and air at a constant pressure is supplied through these probes to produce bubbles. The difference in pressures at the orifice exits is proportional to the surface tension of the liquid and this is the principle of operation of the device. The measurements are calibrated on the basis of two known liquids (water and ethanol) before further measurements are made on the five liquids used for this study. Surface tension of a given liquid changes with surface age, which is the time the process allows the molecules in solution to migrate to the air-liquid interface and lower the surface tension. This changing surface tension with surface age is called dynamic surface tension.

Equilibrium surface tension represents the maximum possible reduction in surface tension and this limit is achieved when dynamic surface tension value does not change with surface age. This typically happens at larger surface ages. In the Sensadyne tensiometer, surface age is measured as the time interval between start of bubble formation to its break-off from the orifice edge. As equilibrium surface tension is the measurement of interest, large surface ages (lower bubble frequencies) are used to verify the constant equilibrium surface tension of the sampling liquid. The values for surface tension for the five liquids tested are summarized in Table 7-1.

1.28 Results and Discussion

Polymer additives that are often used in aqueous solutions tend to implement non-Newtonian characteristics to the flow. The rheology of these polymer solutions is dependent on properties of the additive such as concentration, molecular weight and temperature [95]. Polymers typically consist of large-chain molecules that are formed from clusters of smaller units called monomers.

Though most polymers do not show significant surface-active behavior like surfactants, in some cases, the two polymers used in the current experiment, hydroxyethyl cellulose (HEC) and

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polyethylene oxide (PEO), have shown decrease in surface tension with increasing concentration of the polymer. The surface tension and rheology of each liquid solution used in the current study is carefully determined using a tensiometer and rheometer respectively. The rheology curves for the two polymers for the different concentration used is shown in Figure 7-1. The zero-shear viscosity for these polymers is a function of their concentration. Also, as seen for HEC 250 HHR, the concentration also dictates the shear-thinning behavior. Lower concentration of HEC 250 HHR

(1200 wppm = 9.242 x 10-10 mol/cc ) has very low shear thinning behavior and its rheology can be considered Newtonian.

1.28.1 Newtonian liquids

Prior to analyzing the effect of shear-thinning rheology on drop impact and deposition on a thin film target, it is necessary to identify the significance of surface tension and viscosity in

Newtonian liquid drop impacts. As has been earlier identified while attempting to demarcate splashing and depositing drops as they impact a thin liquid film, surface tension governs the formation of surface instabilities on the crown rim, while viscosity provides resistance to the growth of the crown base. To further understand these implications on the crown growth rate, temporal analysis of crown height and the crown base diameter is conducted. The three Newtonian liquids considered for this purpose are ethylene glycol, 63% glycerin water mixture and HEC 250

HHR at 1200 wppm. While first two liquids have similar viscosities but varying surface tensions, the glycerin water mixture and 1200 wppm HEC 250 HHR have similar surface tension and varying viscosities.

Effect of surface tension

The two Newtonian liquids compared to understand the significance on surface tension are ethylene glycol and 63% glycerin water mixture. While the viscosity of both the liquids are similar,

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ethylene glycol has a lower surface tension than aqueous glycerin. Figure 7-2 shows the high- speed images of drops of these two Newtonian liquids with similar inertia (velocity and diameter)

푉 impacting of a thin liquid film of the same liquid. The dimensionless time (휏 = 푡. ) for these 퐷푑푟표푝 images is provided alongside for easy comparison of the temporal behavior of these impacts. At first glance, temporal growth of crown height and crown diameter seem to be similar. For ethylene glycol, as the grown crown begins to merge with the thin liquid layer, capillary instabilities are seen to form small fingers of the crown. This is attributed to the lower surface tension of ethylene glycol, thus aiding the formation and propagation of surface instabilities. For the aqueous glycerin, slight surface instabilities are detectable only at τ = 7.5, when the crown has almost merged with the thin film layer. In addition, during initial temporal progression, the base of the crown for both the liquids are similar. However, the shape of the crown is different and the crown rim is observed to be wider when surface tension is lower in ethylene glycol.

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Figure 7-2: Effect of surface tension on deposition as a drop of similar inertia and viscosity but

varying surface tension impact on a thin liquid film

For further temporal analysis, the evolution of the non-dimensional crown height and diameter are compared. When the impacting drop Re is the same for the two liquids, as their viscosities are similar, the impacting inertia will be equal. Thus, the effect of surface tension on drop deposition and crown growth dynamics can be deduced. It is observed that though the time for growth of the crown, to reach the maximum height and to decay are identical for both the liquids, surface tension being lower for ethylene glycol allows the crown to grow to a larger height. Surface tension offers an opposing force to the crown growth as it tries to minimize the surface energy of the crown rim.

Though the final values of the crown diameter (Figure 7-3) are the same for both the liquids when

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the drop has merged with the thin liquid layer, the development of this is slower for aqueous glycerin with higher surface tension. This is an interesting observation, as it would be expected that the crown base development is affected by the resistance offered by the thin film liquid.

However, here this resistance, viscosity, is the same for the two cases compared and yet a difference in crown base growth and the shape of the base is observed. Therefore, the crown development is indeed a complex function of the surface tension and viscosity of the interacting fluids.

Dimensionless crown height Dimensionless crown diameter

Figure 7-3: Dimensionless crown parameters as a function of dimensionless time for the two

Newtonian liquids (ethylene glycol and 63% glycerin water) to show the effect of impacting

inertia as a drop impacts a thin liquid film.

Effect of impacting inertia

Three different impacts of the same liquid (aqueous glycerin) are compared in Figure 7-4 to ascertain the influence of impacting inertia on the crown formation and development. The lower the impacting inertia, the crown height is lower. This leads to shorter time periods for growth of

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the crown diameter and merger with the target liquid film. This difference in crown height is apparent even at early times. However, the crown shape and the overall dyamics of growth are similar for the three cases shown.

Figure 7-4: Effect of inertia on deposition as a drop of similar viscosity and impacting We but

varying surface tension, impacts on a thin liquid film

The temporal development of the crown height and diameter are provided in Figure 7-5 for different impacting drop inertias for the Newtonian aqueous 63% Glycerin water. The plot for the crown height development shows that the initial growth is independent of the impacting inertia.

With increasing inertia (increasing We and Re), the crown maximum height and the time for development increases. The crown diameter is another characteristic that is studied and it is observed that the shape of the curve (the development of the crown diameter) shown in Figure 7-5

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is very similar for all the three impacts considered here. The crown diameter deviates from this pattern only in the last few time steps before the crown merges with the target liquid. With increasing development time, the crown diameter too is larger. From these observations, it can be concluded that impacting drop inertia plays a significant role is dictating the height to which the crown grows and the therefore development time. It however has very less significance on the initial growth dynamics and on the crown diameter.

Dimensionless crown height Dimensionless crown diameter

Figure 7-5: Dimensionless crown parameters as a function of dimensionless time for the two

Newtonian liquids (ethylene glycol and 63% glycerin water) to show the effect of impacting

inertia as a drop impacts a thin liquid film.

Effect of viscosity

To interpret the effect if viscous force in the process of drop impact, Newtonian liquids drops with similar impacting drop diameter, velocity and surface tension are considered. A Newtonian solution of HEC 250 HHR polymer at very low concentrations (1200 wppm) and an aqueous solution of glycerin with similar surface tension property are the two liquids used for this

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comparison. As the densities of these two liquids are quite different, obtaining similar Re impacts is difficult and thus, this analysis compares two impacts of similar inertial (diameter and velocity) and surface tension forces. Figure 7-6 shows the temporal development of this drop impact as they deposit on a target thin film of similar liquid. Though the impacting inertia and surface tension for both the liquids are similar, the crown development for 1200 wppm HEC 250 HHR drop is slower and the crown is noted to be larger in this case. This can be attributed to the lower resistance offered by the thin film layer as the drop impacts and spreads along this. As was seen in Figure

7-4, higher inertial forces contribute to larger time for crown development. This current scenario

(Figure 7-6) can be thought of as lesser fraction of inertial force competing with the viscous resistance for the Newtonian HEC solution and thus, more is available to contribute to the development and growth of the crown.

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Figure 7-6: Effect of viscosity on deposition as a drop of similar inertia and zero-shear viscosity

but varying surface tension impact on a thin liquid film

Plot of the temporal development of the crown for the liquids considered in the discussion above are provided in Figure 7-7. It is observed that while the crown height for the liquid with higher viscous resistance is more (63% glycerin water), the time taken to deposit on the film layer and the final crown diameter are more for the Newtonian polymer solution (1200 wppm HEC 250

HHR). This indicates that when there is more viscous resistance, the impacting drop spreads lesser on the target liquid layer. In relation to the above finding, the effect of viscosity of the liquid on the crown height is not as significant as the effect of inertia and surface tension.

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Dimensionless crown height Dimensionless crown diameter

Figure 7-7: Dimensionless crown parameters as a function of dimensionless time for the two

Newtonian liquids (1200 wppm of HEC 250 HHR and 63% glycerin water) to show significance

of viscosity as a drop impacts a thin liquid film.

From our observations on Newtonian drop impact deposition, the individual contributions from surface tension, inertia and viscous resistant forces are noted. While surface tension and inertia primarily govern the crown height, their contribution on the crown diameter is not too significant.

While surface tension predominantly offers resistance to grown of the crown, the drop’s impacting inertia contributes to the development time, that in turn dictates the crown diameter. The current observations suggest that viscous resistance mainly governs the spread of the crown base (i.e. crown diameter) and that in turn dictates the time for crown development. For Newtonian liquids, it is possible to isolate the influence of each of these governing parameters and understand their individual contributions. However, for non-Newtonian liquids, this is not as straight-forward. In the current study two shear-thinning polymeric solutions are considered and their shear-thinning

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behavior is calibrated with the AR 2000 rheometer. The following section describes our findings on the same.

1.28.2 Effect of non-Newtonian viscous behavior

Crown characteristics of HEC 250 HHR

Drop impact behavior of low shear-thinning 2000 wppm HEC 250 HHR is compared against 63%

Glycol water impact with similar impacting (zero-shear) Re and We. The shear-thinning behavior of this polymeric solution is shown in Figure 7-1 and it is noted that as the shear rate increases beyond 2 s-1, the viscosity of the liquid begins to drop. As the drop impacts on the target liquid surface, there is a sudden change in velocity and this translates to increasing shear rate. The shear rate continuously changes with the development of the crown as the drop merges with the target liquid surface. From the shear rheology plot in Figure 7-1, this implies that shear viscosity of the liquid would lower continuously as the shear rate increases. The significance of this effect can be seen in the temporal development of similar impacting drops of the non-Newtonian and Newtonian liquid taken into consideration in Figure 7-8. Though the impacting Re and We are similar, the non-Newtonian HEC 250 HHR solution is seen to spread on the target film for a longer time creating a larger final crown diameter as it merges with the film layer. In addition, the lowered viscosity of the liquid during the development is seen to enable surface waves on the crown rim.

The dimensionless crown height and diameter are shown in Figure 7-9 and Figure 7-10 respectively. It is observed that the crown heights develop similarly initially, but as the shear- thinning behavior of the polymer solution starts to take effect, the significance of inertia begins to dominate over the viscous effect (as viscosity decreases) and therefore, as seen in Figure 7-4, the crown height is higher when impacting inertia is higher. The temporal development of the dimensionless crown diameter is very similar for the liquids compared. The lowering of viscosity

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further enables longer growth period of the crown thus leading to a larger final diameter as the drop finally merges with the target film. Figure 7-9 and Figure 7-10 show these observations for three different Re and the observations are seen to be consistent for this shear-thinning liquid.

Figure 7-8: Effect of shear rheology on deposition as a drop of similar zero-shear Re and We as

they on a thin liquid film of similar liquid

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