One Dimensional Model of Thermo-Capillary Driven Liquid Jet Break-Up with Drop Merging

ONE DIMENSIONAL MODEL OF THERMO-CAPILLARY DRIVEN LIQUID JET BREAK-UP WITH DROP MERGING

Dissertation

Submitted to

The School of Engineering of the

UNIVERSITY OF DAYTON

In Partial Fulfillment of the Requirements for

The Degree

Doctor of Philosophy in Mechanical Engineering

Michael Stephen Hanchak

UNIVERSITY OF DAYTON

Dayton, Ohio

December, 2009

ONE DIMENSIONAL MODEL OF THERMO-CAPILLARY DRIVEN LIQUID JET

BREAK-UP WITH DROP MERGING

APPROVED BY:

______Kevin P. Hallinan, Ph.D. Jamie S. Ervin, Ph.D. Advisory Committee Chairman Committee Member Chairperson and Professor, Professor, Mechanical and Aerospace Engineering Mechanical and Aerospace Engineering

______John C. Petrykowski, Ph.D. Edward P. Furlani, Ph.D. Committee Member Committee Member Associate Professor, Senior Research Scientist, Mechanical and Aerospace Engineering Eastman Kodak Company

______Steven W. Benintendi, Ph.D. Tony E. Saliba, Ph.D. Committee Member Dean, School of Engineering Associate, Wood, Herron & Evans L.L.P.

______Malcolm W. Daniels, Ph.D. Associate Dean of Graduate Studies School of Engineering

ABSTRACT

ONE DIMENSIONAL MODEL OF THERMO-CAPILLARY DRIVEN LIQUID JET

BREAK-UP WITH DROP MERGING

Name: Hanchak, Michael Stephen University of Dayton

Advisor: Dr. Kevin P. Hallinan

A numerical model is presented for predicting the instability and breakup of liquid jets of

Newtonian fluid driven by thermo-capillary perturbations. The model uses a one-dimensional slender-jet approximation to obtain the equations of motion in the form of a set of coupled nonlinear partial differential equations (PDEs). These equations are solved using the method-of- lines (MOL), wherein the PDEs are transformed to a system of ordinary differential equations

(ODEs) for the nodal values of the jet variables on a uniform staggered grid. The model predicts instability and satellite formation in infinite threads of fluid and continuous jets that emanate from an orifice. The model is validated using established computational data, as well as axisymmetric, volume of fluid (VOF) computational fluid dynamic (CFD) simulations. The key advantages of the model are its ease of implementation and speed of computation that is several orders of magnitude faster than the VOF CFD simulations. The model enables rapid parametric analysis of jet breakup and satellite formation as a function of jet dimensions, modulation parameters, and fluid rheology.

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The model also incorporates post break-up behavior by providing methods for the fission of the jet into discrete ligaments and drop. Each separate ligament is assigned its own computational domain that is passed to the ODE solver in lockstep. Furthermore, some drops merge downstream as a results of their velocity differentials at the point of break-up; the model implements drop merging by blending discrete computational domains into one, using cubic interpolation and third-order polynomials to create a liquid bridge between the two. The study of merging behavior has application in the field of inkjet printing, wherein the thermal pulse to the surface of a jet is modulated to create different drop volumes. The model reveals that larger than usual drops do not spontaneously form at the end of the filament; they first break into smaller pieces which coalesce downstream.

ACKNOWLEDGEMENTS

I would like to thank Dr. Randy Fagerquist for providing the micrographs of drop formation.

I would also like to thank Eastman Kodak Company for its continued development and exploration of continuous inkjet technologies to which this research is applicable.

Dr. Ed Furlani has been a great mentor to me. He has provided me with guidance and knowledge; for this, I am extremely grateful.

Most importantly, I thank my family, whose support made this work possible.

TABLE OF CONTENTS

ABSTRACT...... iii ACKNOWLEDGEMENTS...... v TABLE OF CONTENTS...... vi LIST OF FIGURES ...... viii LIST OF TABLES...... xiii CHAPTER I: INTRODUCTION...... 1 LITERATURE SURVEY...... 6 OVERVIEW ...... 11 CHAPTER II: THERMAL-CAPILLARY DRIVEN LIQUID JET ...... 13 INTRODUCTION ...... 13 THEORY ...... 15 Navier-Stokes ...... 15 Continuity...... 15 Boundary Conditions...... 16 SOLUTION METHOD...... 18 RESULTS ...... 22 CHAPTER III: BEYOND BREAK-UP: DROP TRACKING AND MERGING ...... 35 INTRODUCTION ...... 35 MATHEMATICAL FORMULATION ...... 36 Moving Mesh ...... 36 Curvature...... 38 Drop Break-up...... 39 Drop Merging...... 41 ALGORITHM...... 43 RESULTS ...... 45 CHAPTER IV: CONCLUSIONS...... 67 FUTURE WORK...... 68

APPENDIX A: AXISYMMETRIC EQUATIONS OF MOTION ...... 70 VECTOR CALCULUS...... 70 Axisymmetric Coordinates...... 71 FLUID KINEMATICS ...... 72 CONTINUITY EQUATION ...... 73 MOMENTUM EQUATION...... 74 ENERGY EQUATION...... 75 APPENDIX B: LINEAR THEORY OF JET BREAKUP: RAYLEIGH ANALYSIS...... 77 EQUATIONS OF MOTION...... 77 LINEAR STABILITY ANALYSIS...... 78 BOUNDARY CONDITIONS...... 79 BIBLIOGRAPHY...... 83

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LIST OF FIGURES

Figure 1.1. Illustration showing a single nozzle with an integrated heater at the orifice and a thermal modulation pulse used to induce Marangoni instability and drop formation...... 2

Figure 1.2: Two diameters of drops originating from the same jet are shown deflecting differentially in an air stream. The filament diameter is approximately ten micrometers (figure not to scale)...... 2

Figure 1.3: A prototypical thermal device is shown in this micrograph. The orifice is surrounded by a ring-shaped heater and various circuitry. This orifice is approximately nine micrometers in diameter...... 4

Figure 1.4: This micrograph illustrates the formation of both large and small drops from single jets. Downstream there are two small drops close to merging. Note that while this image shows several jets being driven with same pattern, these devices allow all jets to be driven independently. Also illustrated is an example waveform that may produce such a drop train...... 5

Figure 1.5: This micrograph illustrates the formation of only large drops. Notice they do not form until some time after the ligament breaks up...... 5

Figure 1.6: Shown here is a series of waveform modulations (top to bottom) that affect the position of a small drop. The transition between small drop mode and large drop mode could prove to be troublesome. The bottom jet illustrates the desired behavior...... 11

Figure 2.1. Slender jet geometry and reference frame...... 17

Figure 2.2. The staggered computational grid employed in the model is shown for (a) an infinite cylinder at pinch-off with periodic boundary conditions, and (b) a nozzle driven jet...... 22

Figure 2.3. The shape of the free-surface at pinch-off (clock-wise from top left): Re = 200, k = 0.7; Re = 200, k = 0.45; Re = 0.1, k = 0.45; Re = 0.1, k = 0.7...... 25

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Figure 2.4. The shape of the free-surface as a function of scaled time (t* = t /t c) for Re = 100: Wavenumbers (from left): k = 0.45, k = 0.7 and k = 0.9...... 27

Figure 2.5. The shape of the free-surface as a function of scaled time (t* = t /t c) for Re = 1: Wavenumbers (from left): k = 0.45, k = 0.7 and k = 0.9...... 27

Figure 2.6. The shape of the free surface at pinch-off is shown for the 1-D model (top row) compared to an axisymmetric CFD simulation (bottom row) for Re = 100. Wavenumbers are (from left) k= 0.45, 0.7, and 0.9...... 29

Figure 2.7. Scaled break-up time vs. peak-to-peak surface tension variation for Re = 100 and k = 0.7...... 31

Figure 2.8. A plot of the break-up time versus wavenumber for Re= 50, 20, 10, 5, 3, 1, 0.5. The

inset shows the wavenumber at the maximum growth rate, kMGR , versus Reynolds number...... 32

Figure 2.9. The surface tension at the orifice varies with time for the nozzle driven jet...... 33

Figure 2.10. Predicted jet profiles at pinch-off are shown for the nozzle driven jet: (a) 1-D analysis, (b) VOF CFD...... 34

Figure 3.1. A fluid ligament is tracked by its beginning position and length, both a function of time...... 37

Figure 3.2. The end of a jet is rendered to illustrate its curvature. The small ellipse represents the in-plane osculating circle at a given point. The large ellipse represents the orthogonal plane osculating circle through the same point...... 39

Figure 3.3. When the smallest minimum of the filament reaches a given threshold, it is broken into two separate computational domains...... 40

Figure 3.4. When two drops or ligaments approach each other within a given threshold, their computational domains are joined by a smooth liquid neck...... 42

Figure 3.5. The height of the free surface for the merging drops is blended using two third-order polynomials that meet in the center at a prescribed minimum height. The newly added surface is the dashed line...... 43

Figure 3.6. Program flow chart is shown...... 44

Figure 3.7. Shown is a typical input temperature pulse train to the model at the nozzle. The solid line is the user’s desired waveform. The dashed line is the result of a convolution with a Gaussian profile to improve numerical stability. Time flows from left to right...... 45

Figure 3.8. A long sequence of drop generation from a single jet is shown. The nozzle is at the left end. Time is increasing from bottom to top...... 51

Figure 3.9. This figure shows a detail view of the satellite formation and subsequent merging shown in the previous figure...... 52

Figure 3.10. Detail of the recoil of a fluid ligament into a satellite drop is shown. Time is increasing from bottom to top. The bulk motion of the ligament is from left to right...... 53

Figure 3.11. Here is a plot of a satellite filament length versus time for a similar case as shown in Figure 3.10. The satellite starts as a ligament and then coalesces into a drop with a decaying sinusoidal oscillation...... 54

Figure 3.12. A main drop engulfing a satellite drop is shown in (a). The smaller drop is overtaken due to the relative velocity of the two. A detail of the first moments of merging are shown in (b). Time is increasing from bottom to top. The bulk motion of the drops is from left to right...... 55

Figure 3.13. Shown is the formation of a large drop that is twice the volume of the fundamental drop. The larger drop is created by a temporary absence of one period of thermal pulsing on the nozzle boundary condition. Notice how a larger drop does not spontaneously form at the end of the filament; it first breaks into smaller pieces which coalesce downstream because of their relative velocities. Time is increasing from bottom to top. The bulk motion of the drops is from left to right. Iteration of the main filament is halted after some time to increase the speed of the computation...... 56

Figure 3.14. This figure shows the detail of two main drops merging to create a 2x drop. The top of the first column continues at the bottom of the second. Time is increasing from bottom to top...... 57

Figure 3.15. Shown is the formation of a large drop that is three times the volume of the fundamental drop. The larger drop is created by a temporary absence of two periods of thermal pulsing on the nozzle boundary condition...... 58

Figure 3.16. Shown is the formation of a large drop that is four times the volume of the fundamental drop. The larger drop is created by a temporary absence of three periods of thermal pulsing on the nozzle boundary condition...... 59

Figure 3.17. Illustrated is a plot of the main filament length versus time for the program run given in Figure 3.12. The break-up length was steady at 240 micrometers until a large drop was commanded. Then the break-up length fell to about 165 micrometers before increasing to the previous value...... 60

Figure 3.18. This plot shows the break-up length of the jet as a function of the dimensionless wavenumber. The model achieves minimum break-up length at a wavenumber of 0.74. The model closely matches the theory of Furlani (2005) except at higher wavenumbers where the slender jet approximation is strictly not valid...... 61

Figure 3.19. This plot illustrates the logarithmic dependence of break-up length on the perturbation amplitude. For this reason, it is often difficult to match perturbation amplitudes to experimental data...... 62

Figure 3.20. This plot shows different break-up behavior based on the perturbation amplitude. The amplitudes are a) 4.5ºC and b) 18ºC. Notice how the first point of break-up occurs on different sides of the satellite ligament for the two different cases...... 63

Figure 3.21. A plot of the free surface of the jet with a curve fit is shown. The free surface closely matches an exponentially increasing sinusoid until approximately two wavelengths from break-up, where non-linear effects dominate...... 64

Figure 3.22. The equivalent radial perturbation shows a linear dependence on the actual temperature perturbation...... 65

Figure 3.23. Shown is a comparison of the model to micrographs of physical hardware during fundamental drop generation. While the filament shape is well-matched, the satellite behavior differs...... 66

Figure 3.24. Shown is a comparison of the model to micrographs of physical hardware during a 4x drop formation for two different pulse profiles. While the filament shape is well-matched, the satellite behavior differs...... 66

Figure B.1. The dispersion equation relates growth rate to the dimensionless wavenumber...... 81

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LIST OF TABLES

Table 2.1. This table compares the scaled break-up times from the 1-D model (bold font) with the model of Ashgriz and Mashayek (1995) (regular font)...... 25

Table 2.2. A comparison of the scaled time to pinch-off from our model (bold-face) with an axisymmetric VOF CFD simulation (regular font) is presented...... 29

Table 2.3. A comparison of main drop and satellite volumes from our model (bold-face) with an axisymmetric CFD model is presented for the example presented in Figure 2.5...... 30

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CHAPTER I: INTRODUCTION

BACKGROUND AND MOTIVATION

The instability of liquid jets and subsequent formation of droplets have numerous industrial applications, not the least of which is continuous ink-jet printing. This type of printing consists of

a fluid manifold containing many microscopic orifices under pressure. The jets of ink created

are forced to break into drops by subjecting the manifold to a periodic perturbation, usually by

piezoelectric actuators, at the rate of hundreds of thousands of drops per second per orifice.

Drops are selected for printing by modulating their electric charge at the point of break-off with a

nearby electrode. An electric field then deflects the non-printing drops to a gutter; the un-

deflected drops travel to the print media to form the image. Typically, drop volumes range

between one and 100 picoliters.

Recently, a novel approach to stimulating liquid jets was invented whereby micro-heaters are

fabricated within each individual orifice using CMOS/MEMS technology (Chwalek et al., 2002;

Delametter et al., 2002; Anagnostopoulos et al., 2003; Furlani et al., 2006). To modulate a jet, a

periodic voltage is applied to the heater, which causes a periodic diffusion of thermal energy from

the heater into the fluid near the orifice (Figure 1.1). Thus the temperature of fluid, and hence the

temperature-dependent fluid properties, density, viscosity and surface tension, are modulated near

the orifice.

Figure 1.1. Illustration showing a single nozzle with an integrated heater at the orifice and a thermal modulation pulse used to induce Marangoni instability and drop formation.

The heaters can be driven independently of neighboring orifices; this permits the controlled introduction of aperiodic perturbations that lead to the formation of larger drops than is normal.

The presence of small and large drops is exploited in a printing device by directing steady, smooth air normal to the array of jets; the differential drag force between the large and small drops separates them. Either of the drop streams may be guttered; the other travels to the printed media (Figure 1.2).

Air Stream

Thermal Device Filament Large Drop Small Drop

Figure 1.2: Two diameters of drops originating from the same jet are shown deflecting differentially in an air stream. The filament diameter is approximately ten micrometers (figure not to scale).

The benefits of this system over traditional methods are many. First, older methods use high voltage electric fields to defect the drops. This voltage can be dangerous and could destroy hardware if a path to ground was unintentionally created. The new method uses only low voltage signals to drive the orifice heaters. Second, older methods are limited in printing speed due to the physics of the vibration of the orifice manifold. The new method can achieve drop generation rates of one million drops per second per orifice; older methods run at one fifth this speed.

Finally, these devices are made using silicon micro-technology and can achieve higher spatial resolution than their predecessors. These advantages translate into better image quality and greater throughput.

The thermal device that supplies the perturbation to the jet consists of a ring-shaped heater

buried in and surrounding the orifice (Figure 1.3). Voltage waveforms are applied to this ring and

heat is generated through electrical resistance. The waveforms are created with arbitrary function

generators and are numerous in number and variety (One example is seen at the top of Figure

1.4). The heat generated must diffuse through the remainder of the orifice before it can affect the

jet of liquid. Initial heat transfer models of the device show there is significant dispersion of

sharp-edged waveforms to more sinusoidal temperature profiles (Sasmal, 2006).

The transition between the formation of small drops and large drops can lead to undesirable behavior. For example, much smaller drops called satellite drops can form; this is also a problem with the more traditional, periodically stimulated liquid jets and is usually addressed by modulating the perturbation amplitude. However, a more critical issue to contend with is the merging of the small, or fundamental, drops as they travel downrange (see Figure 1.4). If two or more small drops are allowed to merge, the air deflection scheme is no longer binary and the final location of the unwanted drops may interfere with the printing process. Even if the perturbation scheme is devised to create large drops only twice the volume of the small drops, small drop merging would create extra print drops.

In most instances the desired large drops do not form spontaneously from the end of the liquid filament (see Figure 1.5). Often the large wavelength section of the filament will break up before reforming into the large drop. It is important to understand how the large drop forms in order to produce perturbation pattern that minimizes the time before it forms; the air deflection portion of the printing scheme cannot begin until the large drops form completely. The predictable generation of both small and large drops without defects is critical to producing inkjet systems that utilize this technology.

Figure 1.5: This micrograph illustrates the formation of only large drops. Notice they do not form until some time after the ligament breaks up.

LITERATURE SURVEY

Classically, the study of the break up of liquid jets dates to 1833 with the experimental work by Savart. In 1849, Plateau was the first to postulate that surface tension was the cause of break up. However, it took the work of Lord Rayleigh (1878, 1879) to put this hypothesis on solid theoretical ground (Eggers, 1997). Rayleigh considered the behavior of an infinite, stationary liquid cylinder in a vacuum, with an initial infinitesimal sinusoidal displacement of wavelength λ imposed along the free-surface (for details, see Appendix B). He obtained a dispersion relation for the disturbance growth rate as a function of wavenumber ( k=2 π/λ), and found that the fastest growing disturbance is the one whose wavelength, λ, is approximately 4.5 times the diameter of the jet, D. Furthermore, those wavelengths that are less than π times the diameter do not cause

the jet to break up (Rayleigh, 1878). Rayleigh’s analysis is still applicable today where, for

example, one can predict both the break-off length (BOL) and the break-off time (BOT) given the

perturbation amplitude or vice versa. In practice, these perturbations are sub-micrometer in

amplitude.

Following Rayleigh, there has been sustained and growing interest in jet instability, as this

process plays a critical role in a wide range of fundamental phenomena with practical applications

spanning multiple disciplines and length scales (Ashgriz and Mashayek, 1995). A comprehensive

overview of this topic has recently been given by Eggers and Villermaux (2008). Lee examined

the inviscid case of the one-dimensional equations and his dispersion relation predicts the fastest

growing wavelength to be 4.45 times the jet diameter, a minor deviation from Rayleigh’s three-

dimensional case (Lee, 1974). Pimbley (1976) extended Lee’s model to explore the roots of the

dispersion relation when jet velocities are low compared to Rayleigh theory; he also mapped

regions of permitted drop formation. Pimbley and Lee (1977) together extended their previous

analyses to include second order effects in the linearization scheme in order to predict the

formation of satellite drops. Green (1976) formulated the nonlinear 1-D problem using Cosserat

theory, a set of one-dimensional postulates. Finally, Eggers (1997) provided a 64-page treatise on

nonlinearity of drop formation and similarity solutions near break-off.

Interfacial instabilities, like the one that causes jet break-up, are quite common. The

Rayleigh-Taylor instability describes the instability occurring at the interface between a dense fluid and a less dense fluid. While the plane interface is stable, any disturbance will cause it to deform continuously until the fluids mix. The Kelvin-Helmholtz instability relates to two stratified fluids in relative motion. If certain criteria are met, the interface can no longer absorb minute fluctuations and eventually it decays into many vortices; this phenomenon is often observed in clouds. Incidentally, the jet instability of interest here is often called the Plateau-

Rayleigh instability. These phenomena are described in detail in the text by Chandrasekhar

(1961).

Research into the phenomenon of jet breakup and drop formation has increased dramatically over the last several years, due in part to rapid advances in microfluidic, biomedical, and nanoscale technologies. Novel applications are proliferating, especially in fields that benefit from high-speed and low-cost patterned deposition of discrete samples (droplets) of micro- or nanoscale materials. A wide range of materials can potentially be jetted using modern microfluidic devices including liquid metals, dispersions of nanoparticles, electrical, and optical polymers, myriad biomaterials, sealants and adhesives. Emerging applications in this field include printing functional materials for flexible electronics, microdispensing of biochemicals, printing biomaterials (e.g. cells, genetic material), and 3D rapid prototyping (Ashley, 2002; Basaran 2002;

Sirringhaus and Shimoda, 2003; Telin et al., 2008; Roth et al., 2004; Calvert, 2001).

The development of reliable high-throughput micro-scale droplet-generating devices requires considerable fluidic modeling in advance of device fabrication. Such modeling is essential in

8 order to obtain sufficient understanding of jetting and drop generation to enable device optimization taking into account critical system parameters as well as the fluid rheology. A rigorous analysis of microfluidic drop generators requires multiphase free-surface computational fluid dynamic (CFD) simulations, often with coupled thermal and structural analysis. While various numerical techniques have been developed for simulating free-surface flows, each has advantages and drawbacks, and all such methods tend to be computationally intensive. The computational methods can be broadly classified as Lagrangian or Eulerian, or hybrid combinations of the two (Scardovelli and Zaleski, 1999; Maronnier et al., 1999; Sellens, 1992;

Lundgren and Mansour, 1988; Hilbing and Heister, 1996; Setiawan and Heister, 1997). In

Lagrangian methods, the fluid interface is tracked with computational nodes that move with the fluid velocity. While this provides an accurate description of the free-surface, its main disadvantage is that the mesh can become severely distorted over time, and the careful monitoring of mesh quality is required with frequent remeshing that significantly adds to computational overhead. Arbitrary Lagrangian–Eulerian methods (ALE) can remedy this problem by allowing the nodes to move independent of the fluid velocity thereby maintaining mesh quality and minimizing remeshing, but implementation can be nontrivial for complex flows. In the Eulerian approach, the computational mesh is fixed, and an unknown function is introduced and solved for whose values define the volume fraction of fluid in each computational cell. The most common implementation of this approach is the volume of fluid (VOF) method (Hirt and Nichols, 1981).

While VOF has advantages in terms of implementation and computational speed, its main disadvantage is that the free-surface is reconstructed via interpolation, which gives rise to errors in surface curvature and hence fluid pressure. Other methods finding increasing use for free- surface analysis include the level set and phase field techniques (Sethian, 1996; Anderson et al.,

1998).

As an alternative or supplement to computationally intensive CFD analysis, one can simplify

the problem and study the dynamics of the jet instability and drop formation using a simplified

one-dimensional slender jet analysis. In the slender jet approximation, the free-surface is

represented by a shape function h( z , t ) where z is the axial coordinate and t is time. The Navier-

Stokes (NS) equations, mass conservation, and appropriate boundary conditions are simplified

and combined to obtain coupled PDEs that govern the fluid velocity ν (z , t ) along the jet, and free-surface function, h( z , t ) . Various methods have been used to solve the slender jet equations.

Eggers and Dupont (1994) and Brenner et al. (1997) used a finite difference approach with a non-

uniform, graded spatial mesh that was adaptively refined and an adaptive fully implicit θ - weighted time-integration scheme. Ambravaneswaran et al. (2002) used the Galerkin finite element method for the spatial discretization and an adaptive, implicit finite difference method for time discretization. Brenner et al. (1997) gives some details of their numerical recipe, specifically the difficulty with modeling a jet with a closed end; the actual domain is mapped to a computational domain that remains fixed as the filament changes length. Eggers (1997) used the same model to observe many break-off events of a continuous jet subject to sinusoidal perturbation. These authors found that 1-D analyses can provide reasonably accurate predictions of jet breakup for a range of practical applications. Moreover, 1-D models are relatively easy to implement and modify, and can reduce simulation time by orders of magnitude relative to axisymmetric CFD analysis. However, the basic foundation of these codes must be greatly expanded to include waveform input, thermo-capillary perturbation, drop tracking, and experimental correlation.

For the present model, the dominant cause of jet instability is the modulation of surface

tension. To first order, the temperature dependence of σ is given by σ()T= σ0 − β ( T − T 0 ) ,

where σ (T ) and σ 0 are the surface tension at temperatures T and T0 , respectively. The pulsed

heating modulates σ at a wavelength λ = v 0τ, where v0 is the jet velocity and τ is the period of

the heat pulse as shown in Figure 1.1. The downstream advection of thermal energy gives rise to

a spatial variation (gradient) of surface tension along the jet. This produces a shear stress at the

free-surface, which is balanced by inertial forces in the fluid, thereby inducing a Marangoni flow

towards regions of higher surface tension (from warmer regions towards cooler regions). This

causes a deformation of the free-surface (slight necking in the warmer regions and ballooning in

the cooler regions) that ultimately leads to instability and drop formation (Furlani, 2005a). The

2 drop volume can be adjusted on demand by varying the period, τ, i.e., Vdrop=π r 0 v 0 τ . Thus, longer pulses produce larger drops, shorter pulses produce smaller drops, and different sized drops can be produced from each orifice as desired. A linear stability analysis of the thermo- capillary driven liquid jet shows this ability to generate drops based on thermal modulation

(Furlani, 2005a, 2005b). This work describes how surface tension gradients create a surface traction that drives fluid convection. Since the cylindrical jet is unstable, it ultimately breaks up.

Gao (2008) extends this analysis to non-Newtonian jets.

Currently missing in the present literature is a rigorous examination of multiple sized drops

forming from a single filament. While research is active in this field, it is experimental in nature.

Such work is illustrated in Figure 1.6 whereby different waveform modulations influence the

position of a small drop adjacent to a large drop (Fagerquist, 2006). A computational model,

short of CFD simulation, would help distill candidate waveforms to a manageable set of

experiments. While full-blown CFD simulations of drop break up have been implemented, the

level of detail needed for this study would require such fine discretization to render them very

computationally expensive. Also, since they rely on volume-of-fluid methods, the free surface is

interpolated which may fail to capture fine-scale curvature and thus affect the internal jet

pressure. The 1-D equations mentioned here, however, model the free surface directly.

Also, it is not currently understood how the modulation of other fluid properties, such as density and viscosity, affect this system. The computational model could provide means to explore the theoretical strength of these other perturbations. Incidentally, the exact nature of the initial perturbation need not be fully understood to design useful ink-jet printing systems.

OVERVIEW

The formulation of a liquid jet driven by a thermal-capillary perturbation is presented in

Chapter II. Starting with the equations of motion, we apply boundary conditions and expand the resulting equations in a Taylor series, truncating the higher-order terms. This results in the so- called slender jet equations with surface tension modulation. The results of the model are compared with existing results of a more complicated model as well as commercial CFD code.

In Chapter III, the model of thermal-capillary driven jet is extended beyond the point of break-up by discretizing the individual filaments and letting the code evolve them separately in

time. Further break-ups or drop merge events are recognized by the code; in the case of a drop

merge, the two drops are allowed to coalesce and thus evolved together. The goal of this work is

a predictive model of the drop merge process, not necessarily the fine-scale detail of the pinch or

merge point, but rather the larger scale momentum and drop volume details.

In Appendix A, the axisymmetric Navier-Stokes equations are derived, as they are the governing equations of all processes herein.

Finally, the linear analysis of jet break-up is given in detail in Appendix B. This work was first established mathematically by Rayleigh. It is presented here not as a direct prerequisite but rather for information and insight into the behavior of liquid jets.

CHAPTER II: THERMAL-CAPILLARY DRIVEN LIQUID JET

INTRODUCTION

We develop a 1-D slender-jet model to predict the nonlinear deformation and breakup of

infinite microthreads and continuous nozzle driven jets of Newtonian fluid. We take into account

Marangoni instability by allowing for a variation in surface tension in our derivation of the

equations of motion. The slender-jet equations reduce to a pair of coupled PDEs for the free-

surface h and velocity v of the jet. We solve these equations using the method-of-lines (MOL),

wherein the PDEs are transformed to a system of ODEs that define the behavior of h and v at the nodes of a uniform staggered computational grid. We integrate the ODE system using explicit forward time-stepping, and track the behavior of the free-surface and the velocity to pinch-off. A key advantage of the MOL approach is that it enables the use of well-established and robust numerical methods for solving the coupled ODEs. The use of explicit time stepping and a fixed uniform spatial grid provide additional advantages in that they facilitate implementation of the model with less complexity than other numerical approaches.

The underlying process followed here is detailed in Eggers and Dupont (1994). There the authors derive a set of one-dimensional equations, one for momentum and one for the height of the free surface, namely,

2 2 ∂=−−tv vvp zzρ +3 ν ( hv zz ) hg −   (2.1) 1 hzz p =σ 21/2 − 23/2  h(1+ hz ) (1 + h z ) 

for the momentum and

∂th =− vh z − v z h 2 (2.2)

for the free surface height; the z subscript denotes differentiation with respect to the z direction

(jet axis). These are the so-called slender jet equations. The authors derived these equations by expanding pertinent variables in Taylor series and substituting the expansions into the axisymmetric Navier-Stokes and boundary condition equations. The resulting equations were solved to lowest order. The inviscid dispersion relation based on these equations predicts the same most unstable wavelength as Lee (1974).

We first study the breakup of micrometer-scale threads of fluid and show that a typical analysis can be completed within a few seconds on a modern workstation. This is four orders of magnitude faster than corresponding VOF CFD simulations. Moreover, it is important to note that while it is easy to impose initial perturbations to a fluid microthread in our model, it can be very difficult to impose such perturbations, especially infinitesimal free-surface displacements, using the VOF CFD approach.

We perform similar analysis of a continuous nozzle driven jet wherein the jet disturbance is

imposed as a time-dependent boundary condition at the nozzle orifice. The jet instability and

pinch-off predictions require a few minutes to complete, which is between one to two orders

faster than VOF CFD simulations. Again, not only is the analysis faster, but the time-dependent

boundary conditions are much easier to implement in our model as compared to the VOF CFD

analysis.

We demonstrate the model via application to practical examples, and characterize its

accuracy using both established computational data from the literature as well as axisymmetric

VOF CFD simulations. In this regard, while other authors have used slender jet analysis to study

jet breakup, relatively few have compared their predictions with independent VOF CFD

simulations. We choose VOF CFD to test our model because of its availability in commercial

fluid simulators, i.e. it is the most commonly implemented approach for free-surface analysis.

THEORY

In this section we derive the equations of motion for an isothermal axisymmetric viscous jet

of incompressible Newtonian fluid with surface tension σ , viscosity µ and density ρ . We

neglect gravity, and take into account Marangoni instability by allowing for a spatial variation of

surface tension along the jet. We solve the following equations:

Navier-Stokes

Dv ρ=−∇p + µ ∇ 2 v , (2.3) Dt

D ∂ where = +v ⋅∇ is the material derivative, ∇ (del or nabla) is the gradient operator, and Dt∂ t

∇2 is the Laplacian. In the absence of viscous dissipation, the energy equation is given by

DT ρc= k ∇ 2 T . (2.4) p Dt

Continuity

The continuity equation for an incompressible fluid is

∇ ⋅v = 0 . (2.5)

Boundary Conditions

The boundary conditions (BCs) for this problem include stress balance, a kinematic condition, an insulated free surface, and axisymmetric flow conditions. The first two conditions apply at the free-surface (liquid-gas interface), while the flow conditions apply along the axis of the jet. The stress balance at the free-surface can be written as

ɵ ɵ n⋅T =−2 Hσ n +∇ s σ , (2.6) where T is the stress tensor in the fluid (we assume that the external gas is stress free),

  1 1 hzz H =21/2 − 23/2  (2.7) 2h (1+ hz ) (1 + h z )  is the mean curvature of the jet surface, and nˆ and tˆ are unit vectors normal and tangential to the free-surface (Figure 2.1),

1 h nˆ= r ˆ − zˆ z , (2.8) 2 2 1+hz 1 + h z

h 1 tˆ = rˆ z + zˆ . (2.9) 2 2 1+hz 1 + h z

The surface gradient ∇s is given by

h ∂1 ∂ ∇ =rˆ z + zˆ . (2.10) s 2 2 ()1+hz∂z() 1 + h z ∂ z

In these expressions h( z , t ) defines the radial height of the free-surface, and

∂h ∂ 2 h h=, h = . z∂z zz ∂ z 2

Figure 2.1. Slender jet geometry and reference frame.

Equation (2.6) can be decomposed into normal and tangential components:

(nˆ⋅T ) ⋅ n ˆ =− 2 H σ , (normal stress) (2.11)

ˆ ˆ (T⋅nˆ ) ⋅ t = t ⋅∇ sσ , (tangential stress) (2.12) where

ɵ 1 ∂σ t ⋅∇σ = . (2.13) s 2 ∂z 1+ hz

The gradient of surface tension, ∇sσ , produces a Marangoni flow towards regions of higher surface tension, which deforms the free-surface and ultimately causes breakup (Furlani, 2005a).

The second (kinematic) boundary condition implies that fluid does not cross the free- surface,

D ()r− hzt(,) = 0 , (r = h ). (2.14) Dt

The flow conditions along the axis of the jet (r = 0) are

vr = 0 , (2.15)

and

∂v z = 0 , (2.16) ∂r

An order of magnitude analysis revealed that any heat transfer due to convection was very small in comparison to the diffusion of heat within the jet; therefore, convection is ignored. So, the thermal boundary condition consists of zero heat transfer to the outside environment:

∇T ⋅ n ˆ = 0 , (r=h) (2.17)

SOLUTION METHOD

The solution method is based on a perturbation analysis described by Eggers and Dupont

(1994) in which the jet variables are expanded in power series of the radial variable r . First, we

write all equations and boundary conditions in component form. For axisymmetric flow, the

Navier-Stokes equation (2.3) reduces to,

  2  ∂∂∂vvvr r r  ∂p ∂ 1 ∂ (rv r ) ∂ v r ρ++vr v z  =−+ µ   + 2  , (2.18) ∂∂∂t r z  ∂∂∂ r rrr  ∂ z 

and

2  ∂∂∂vvvz z z ∂p 1 ∂  ∂∂ vvz z ρ++vr v z  =−+ µ   r +  . (2.19) ∂∂t r ∂ z  ∂ zrrrz ∂∂∂  2 

The energy equation becomes

∂TTT ∂ ∂ 1 ∂∂∂  TT2  ++=vr v z  α  r  + 2  ∂∂t r ∂ z  rrrz ∂∂∂    . (2.20)

where the thermal diffusivity is given by α= k ρ c p .

The continuity condition (2.5) becomes,

1 ∂(rv ) ∂ v r+ z = 0 . (2.21) r∂ r ∂ z

The normal and tangential stress boundary conditions (2.11) and (2.12) can be written as

2µ  ∂∂vv  ∂ v ∂ v  p+ hzr +−− r h 2 z = ... 2 z  z  ()1+ hz ∂∂rz  ∂ r ∂ z    , (r = h) (2.22) 1 h −σ  − zz  21/2 2 3/2  h()1+ hz() 1 + h z 

and

  µ ∂∂vvrz 2  ∂∂ vv rz  2hz−  +−() 1 h z  +   = ... 2     ()1+ hz ∂∂rz ∂∂ zr  , (r = h) . (2.23) 1 ∂σ 2 ∂z 1+ hz

Similarly, the kinematic condition (2.14) gives

∂h +v h = v , (r = h). (2.24) ∂t z z r

We seek a solution to Eqs. (2.18) - (2.24). To this end, we expand vz ( rzt , ,) and pr( , zt ,)

in powers of r (Eggers and Dupont, 1994),

2 vrztz (,,)= vzt0 (,) + vztr 2 (,) + ..., (2.25)

2 przt(,,)= pzt0 (,) + pztr 2 (,) + ..., . (2.26)

From the continuity condition (2.5) and the expansion (2.25) we obtain

∂v( z , t ) r∂v( z , t ) r 3 v(,,) rzt =−0 −2 + ... (2.27) r ∂z2 ∂ z 4

Notice that these expansions are compatible with the boundary conditions (2.15) and (2.16).

Using expansions (2.25) through (2.27), we find that the equation of motion (2.18) for vr is

identically satisfied to lowest order. However, Eq. (2.19) for vz gives

2  ∂∂vv001 ∂ p 0µ ∂ v 0 +v0 =− +4 v 2 +  . (2.28) ∂∂tzρ ∂ z ρ  ∂ z 2 

To solve for v0 we need to eliminate the second-order term v2 from (2.28). To this end, we evaluate the tangential stress condition (2.23) at r = h , collecting lowest-order terms to obtain

1∂σ 3 ∂ h ∂v 1 ∂ 2 v v = +0 + 0 . (2.29) 2 2µhz∂ 2 hzz ∂∂ 4 ∂ z 2

Furthermore, from the normal stress boundary condition (2.22) we find that

∂v p= −µ0 + 2 σ H . (2.30) 0 ∂z

A similar analysis is performed on the energy equation (2.20), which reduces to

2  ∂T0 ∂ T 0 ∂ T 0 +v0 =α 4 T 2 +  . (2.31) ∂t ∂ z ∂ z 2 

From the thermal boundary condition, (2.17), we have

1 ∂h ∂T T = 0 . (2.32) 2 2h∂ z ∂ z

We substitute (2.29) and (2.30) into (2.28) and drop subscripts to obtain

∂∂∂v v 1 2 ∂σ 3 µ ∂∂v  =−−v()2σ H + +  h 2  . (2.33) ∂∂∂t zzρ ρ hzhzz ∂ ρ 2 ∂∂ 

For the energy equation, we substitute (2.32) into (2.31) to get

∂T ∂ T2 ∂∂∂ hTT2  =−v +α  +  . (2.34) ∂t ∂ z hzz ∂∂∂ z 2 

Finally, the kinematic condition (2.24) can be rewritten as

∂(h2) ∂ ( h 2 v ) = − , (2.35) ∂t ∂ z which is also statement of the conservation of volume applied to a cylindrical slice of the jet.

We solve the slender jet equations, (2.33) through (2.35), subject to appropriate boundary conditions using the MOL (Schiesser, 1991). Specifically, we define a uniform staggered grid along the jet and write the spatial derivatives as finite differences with respect to this grid. The partial time derivatives of the variables v , T and h become ordinary time derivatives of the

respective nodal values. We evaluate h , T and p on one set of nodes, and v on a different set

of interlaced nodes as shown in Figure 2.2 (there is one less node point for the interlaced grid).

Thus, for example, Eq. (2.35) reduces to a system of N ODEs of the form

∂h h2 vh − 2 v i=− i+1/2 ii − 1/2 i − 1 () 1 ≤≤i N , (2.36) ∂t2 hi ∆ z

1 1 where N is the number of nodes, and h=() h + h , h=() h + h . A similar system of i+1/22 i i + 1 i−1/22 i i − 1

ODEs is obtained for Eqs. (2.33) and (2.34), and therefore a total of 3N-1 ODEs need to be solved for each simulation. For numerical stability, we apply a numerical upwind differencing scheme for the advection term in Eq. (2.33). Forward or central differencing is used for the remaining terms.

Figure 2.2. The staggered computational grid employed in the model is shown for (a) an infinite cylinder at pinch-off with periodic boundary conditions, and (b) a nozzle driven jet.

We have implemented the MOL in MATLAB® using the ODE solver routines for our

numerical studies. MATLAB® provides several different ODE solvers; they differ in order of

accuracy and robustness to stiff equations. We investigated the performance of various solvers

and ultimately chose the ode23t solver, which employs the trapezoidal integration rule using a

"free" interpolant. This solver provided the fastest solution times with adequate accuracy. We developed models to study both infinite threads of fluid with arbitrary modulation of the free- surface or surface tension along the jet, and nozzle-driven jets, wherein the modulation is applied in a time-wise fashion at the orifice, and then convected downstream.

RESULTS

In this section we demonstrate the model via application to both an infinite microthread of

Newtonian fluid as well as a continuous nozzle driven jet. We validate the model using

established data from the literature as well as VOF CFD simulations. In order to consistently

compare our results, we scale the equations of motion using characteristic values for velocity,

length and time. The velocity is scaled by the capillary velocity, which is given by

σ vc = . (2.37) ρr0

The most appropriate length scale is the initial jet radius, r0 . Hence, the time scale is

determined by dividing the length scale by the velocity scale,

ρr3 t = 0 . (2.38) c σ

* In our analysis below, we present our results using a scaled time t= t/ t c , where t is the

physical time. Since the temporal analysis presented essentially models a stationary microthread

of fluid, the Reynolds number based on velocity has no immediate meaning. Hence, we

substitute the capillary velocity into the familiar Reynolds number equation to achieve a temporal

Reynolds number that is also the inverse of the Ohnesorge number,

ρv r ρσ r Re =c 0 = 0 . (2.39) µ µ

Thus, for example, given the density, surface tension, and initial jet radius, the Reynolds number precisely determines the viscosity. The nominal fluid properties used in our analyses are those of water at standard temperature and pressure, ρ = 998 kg/m 3 , σ = 0.073 N/m , and

µ =0.001 N ⋅ s/m 2 .

We begin our study with an analysis of an infinite micro-thread of Newtonian fluid. While we could easily impose an arbitrary initial perturbation to the free-surface, velocity or surface tension

along the thread, for simplicity we choose a sinusoidal perturbation of the free-surface,

2π z   h(,0) z= r  1 + ε cos    , (2.40) 0 λ  

where λ is the wavelength. The initial perturbation has a peak amplitude equal to 5% of

unperturbed jet radius, i.e. ε = 0.05 . For our test cases we use data published by Ashgriz and

Mashayek (1995), who used a Galerkin finite-element method to study the instability of an axisymmetric, incompressible Newtonian liquid cylinder. They employed a moving mesh, which required time-dependent shape functions to capture the surface deformation.

We set the jet radius to r0 =100 µ m and model the following parameters: Re = 200, k = 0.7;

Re = 200, k = 0.45; Re = 0.1, k = 0.45; and Re = 0.1, k = 0.7, where k= 2π r 0 / λ is the

wavenumber scaled by the initial jet radius. Hereafter, we make reference specifically to this

scaled wavenumber. Some typical free-surface plots at pinch-off are shown in Figure 2.3. The

number of grid points for this analysis ranged from 135 for k = 0.7 to 210 for k = 0.45, and the

corresponding computational time ranged from 4 to 12 seconds using a single processor

* workstation. Our predictions of the scaled time, t= t/ t c , are compared to those obtained by

Ashgriz and Mashayek in Table 2.1. Note that our 1-D model provides excellent agreement with

the FEA-based predictions for all but one case: Re = 200, k = 0.9. This is presumably due to the

fact that our analysis is limited to low order (parabolic) radial variation in the fluid variables,

whereas the FEA takes all higher order terms into account. However, the 1-D is much easier to

implement and requires less time to run.

Figure 2.3. The shape of the free-surface at pinch-off (clock- wise from top left): Re = 200, k = 0.7; Re = 200, k = 0.45; Re = 0.1, k = 0.45; Re = 0.1, k = 0.7.

Table 2.1. This table compares the scaled break-up times from the 1-D model (bold font) with the model of Ashgriz and Mashayek (1995) (regular font).

Next, we perform similar calculations as above, but this time we impose a sinusoidal variation of surface tension along the length of the microthread. This gives rise to Marangoni

instability, wherein a shear stress develops that acts in the tangential direction at the free-surface.

This must be balanced by viscous forces, which give rise to Marangoni flow at the interface with

fluid moving from regions of lower surface tension towards regions of higher surface tension.

This flow causes the free-surface to deform, thereby creating surface curvature and a

corresponding pressure gradient within the jet that perpetuates and amplifies the curvature and

flow. This ultimately leads to breakup (Furlani, 2005a; Eggers and Dupont, 1994).

The surface tension at a liquid-gas interface may be perturbed by various factors including

variations in temperature or surfactant along the interface. Here, we do not specify how the

surface tension is modulated; rather we simply impose a fixed sinusoidal variation in the surface

tension of the form

2π z    σ()z = σ0  1 − β  1cos −     , (2.41) λ   

where σ 0 is the surface tension at ambient temperature. We choose a peak-to-peak variation in

σ of approximately 1%, i.e. β = 0.0048 .

Figure 2.4. The shape of the free-surface as a function of scaled time (t* = t /t c) for Re = 100: Wavenumbers (from left): k = 0.45, k = 0.7 and k = 0.9.

Figure 2.5. The shape of the free-surface as a function of scaled time (t* = t /t c) for Re = 1: Wavenumbers (from left): k = 0.45, k = 0.7 and k = 0.9.

We perform analysis for three different values of the Reynolds number, Re = 1, 10 and 100,

and for each value we track the behavior of the free-surface to pinch-off for three different

wavenumbers: k = 0.45, 0.7 and 0.9. The free-surface shapes for several of these cases at different

times are shown in Figures 2.4 and 2.5. The computational grids for the three values of k

consisted of 205, 135 and 105 nodes, respectively, and the analysis took 4 seconds for the highest

wavenumber and 16 seconds for the lowest wavenumber. These grid densities were established

by successively increasing the number of nodes until the break-up times reached a steady state; it

should be noted that for grid densities half of those previously mentioned, the break-up times only

differed by 0.5%. In order to validate the model we repeat the analysis above using axisymmetric

VOF CFD, which is performed using a commercial program, FLOW-3D™ (www.flow3d.com).

In the CFD models the jet radius is as above, r0 = 100 µm, and we use a uniform computational

mesh with 2 µm cell spacing in both the r and z directions.

No significant changes were observed with a finer mesh. We compare the free-surface plots at pinch-off for the Re = 100 cases in Figure 2.6. The computational time for these cases ranged from 12,000-15,000 seconds, four orders of magnitude longer than the 1-D analysis. The CFD free-surface profiles are somewhat different than the 1-D analysis, especially for the shortest wavelength case, k = 0.9 . Presumably this is due to radial dependencies that we ignore.

Nevertheless, we obtain excellent agreement with the scaled time to pinch-off for both cases, as shown in Table 2.2. Also, as noted above, while it is easy to impose initial perturbations to a microthread of fluid in our model, it can be very difficult to impose similar perturbations, especially infinitesimal free-surface displacements, using commercial VOF CFD programs.

Table 2.3 gives the drop and satellite volumes in nanoLiters for the cases presented in Figure 2.6.

To determine the drop volumes for the 1-D model, we numerically integrated the free surface using the disk method. The CFD code reported the satellite drop volumes directly. Note that the

1-D model predictions are more accurate at longer wavelengths, but less so at shorter wavelengths. This is presumably due to limitations of the low order radial variation that is assumed for the fluid behavior in the slender jet analysis.

Table 2.2. A comparison of the scaled time to pinch-off from our model (bold-face) with an axisymmetric VOF CFD simulation (regular font) is presented.

Table 2.3. A comparison of main drop and satellite volumes from our model (bold-face) with an axisymmetric CFD model is presented for the example presented in Figure 2.5.

The various linear theories of jet break-up predict that the break-up time has a logarithmic

dependence on the initial perturbation amplitude. The 1-D model with surface tension

perturbation possesses that same dependence as shown in Figure 2.7 (Furlani, 2005a). The

breakup time versus wavenumber for several different Reynolds numbers is presented in Figure

2.8. The minimum breakup time shifts to lower wavenumbers as the Reynolds number decreases

(increasing viscosity). Also in Figure 2.8, we plot the wavenumber at the maximum growth rate

( kMGR ) versus Reynolds number. At low Reynolds number this plot reveals a near-logartihmic

dependence. However, beyond a Reynolds number of about 20, kMGR levels out.

Figure 2.7. Scaled break-up time vs. peak-to-peak surface tension variation for Re = 100 and k = 0.7.

Figure 2.8. A plot of the break-up time versus wavenumber for Re= 50, 20, 10, 5, 3, 1, 0.5. The inset shows the wavenumber at

the maximum growth rate, kMGR , versus Reynolds number.

Lastly, we apply our model to a continuous nozzle driven jet. We model a continuous jet of

water with a radius r0 = 5 µ m and a velocity of 10 m/s. For our 1-D analysis, we define the

computational domain from z = 0 (the orifice) to z = 315 µm and use 1050 uniformly spaced grid

points. An inflow boundary condition is imposed at the orifice (z = 0), and an outflow boundary

condition is imposed downstream, at the opposite end of the computational domain. The jet

velocity is held constant at the orifice, and the surface-tension σ is varied in a time-wise step

fashion at this boundary, from a high of 0.073 N/m to a low of 0.0696 N/m, as shown in Figure

2.9. The former value occurs at an ambient temperature of 20 °C, while the latter value can be

achieved by heating the fluid to 40 °C. The time-dependent surface tension boundary condition

mimics the time-dependent thermal stimulation used in the novel micro-fluidic drop generator

discussed earlier and shown in Figure 1.1.

We use the 1-D model to track the free-surface of the jet to pinch-off, and we compare this analysis with an axisymmetric VOF CFD simulation of the same system. In the CFD analysis, the computational domain spanned 0 ≤ z ≤ 250 µm, and we used a uniform mesh with a 0.2 µm cell

spacing in both the r and z directions. The jet profiles at pinch-off for the two models are

compared in Figure 2.10. We found that the jet pinched-off in regions of lower surface tension,

which is consistent with a linear theory of Marangoni instability developed by Furlani (2005).

The time to pinch-off is predicted to be 28.0 µs using the 1-D model and 28.4 µs using CFD, a

difference of 1.4%. Also, the 1-D analysis took approximately 5 minutes to complete, whereas the

CFD simulation took 188 minutes on a comparable workstation, an increase in 380% in

computational time. Furthermore, it is important to note that while it is easy to impose any

arbitrary time-dependent modulation of the free-surface, velocity or surface tension at the orifice

in our model, it can be very difficult to impose similar conditions in a commercial VOF CFD

program.

Figure 2.9. The surface tension at the orifice varies with time for the nozzle driven jet.

Figure 2.10. Predicted jet profiles at pinch-off are shown for the nozzle driven jet: (a) 1-D analysis, (b) VOF CFD.

CHAPTER III: BEYOND BREAK-UP: DROP TRACKING AND MERGING

INTRODUCTION

As our ability to control the jet perturbation process increases in resolution and precision, we find ourselves more interested in the downstream behavior of the drops as they break off from the main filament. As mentioned early, we can create drops of different sizes by modulating the waveform of thermal pulses in real time. These larger drops do not instantly form; they are often the formed from smaller drops that have different momentums which causes them to collide. Of course, we desire instantaneous formation of the large drops. Failing that, it is imperative to know when these large drops form. It is this reason that a model beyond break up is needed.

This type of analysis is missing from current literature. While commercial CFD or full 3-D research codes can model the physics robustly, they are computationally very expensive, often taking days of iteration. One rather impressive 3-D simulation of the merging of drops is given by Quan et al. (2009).

Thus, we have little choice but to extend the slender jet model presented in Chapter II to include drop merging. In what follows, we may use the terms “drop”, “ligament”, and “filament” to describe the mass of fluid we are tracking. When a mass first breaks up, it appears as a ligament with an undulating and rapidly changing shape; only after it has evolved over time does it emerge as a spherical drop. The main part of the jet that stays attached to the nozzle is often called the filament.

MATHEMATICAL FORMULATION

Moving Mesh

In order to accurately model the motion of free drops and ligaments, that is, those without

fixed boundary conditions (such as a wall) we need to modify the numerical method to include a

moving mesh. As the drop or ligament evolves in space and time, its left and right boundaries

change dynamically. This effectively expands and contracts the domain over which the

governing equations are applied. This gives rise to grid advection that must be incorporated in

the numerical method.

In order to formulate the grid advection term, we map the boundaries of the drop from the

dynamically changing physical space to a fixed computational space (Brenner, 2006). The details

of this method can be found in Huang et al. (1994) and Mulholland et al. (1997). Namely, let B

be the distance from the nozzle (fixed in space) to the beginning of the drop and L be the length

of the drop, both given in the physical z coordinate (see Figure 3.1). We then transform the z

variable to the computational space variable x ∈[0,1] using the relation

z− B x= or zxLBzBBLx =+∈+ , [ , ], ∈ [0,1] . (3.1) L

So, z = z(x,t), L = L(t), B = B(t ). Now, using the chain rule, we calculate the total time derivative in the computational space. To fix the point, let us operate on a scalar field given by

φ(,)zt= φ ((,),) zxtt , namely,

dφ∂ φ ∂ φ dz = + . (3.2) dt∂ t ∂ zdt

The dz dt term in (3.2) is determined by taking the time derivative of (3.1) which gives us

dφ∂ φ ∂ φ = +()xLɺ + B ɺ . (3.3) dtt∂ ∂ z

We can now substitute the Eulerian form of the governing equations derived in Chapter II into the

RHS of (3.3). The derivative of the first boundary point, Bɺ , is given by the value of the fluid ɺ velocity calculated there, v1 . Likewise, the value of L is given by the difference between the

right end velocity and the left end velocity, namely, vend − v 1 . For example, the PDE governing

jet velocity, equation (2.33), becomes the following when substituted into (3.3),

dvɺ ɺ ∂∂ v 1 2 ∂σ 3 µ ∂∂2 v  =−−v xL− B −()2σ H + +  h  . (3.4) dt() ∂∂ zzρ ρ hzhzz ∂ ρ 2 ∂∂ 

As evident in the first term of the RHS of (3.4), the moving mesh introduces an extra advection

term similar to the material derivative advection.

Thus for each grid point in a ligament, we need its value in x ∈[0,1] , the physical grid

spacing dz, which changes every time step, and the values of Bɺ and Lɺ . Since the values of B and

L change with time, they too are passed to the ODE solver hence updated every time step. This

process is repeated for the other governing equations and for every ligament.

Figure 3.1. A fluid ligament is tracked by its beginning position and length, both a function of time.

Curvature

The pressure inside a cylindrical jet is a direct function of the surface curvature given by the

Young-Laplace equation, repeated here from equation (2.1),

  1 1 1 hzz p=+=+=σ() k1 k 2 σ  σ 21/2 − 23/2  (3.5) rr1 2 hh(1+z ) (1 + h z ) 

The second curvature term in (3.5) is the in-plane curvature, that is, the standard curvature

formula given in calculus. The first term is the orthogonal (out-of-plane) curvature that results

from revolving the surface profile of the jet around the z axis (hence axisymmetric); it is derived

from trigonometry (see Figure 3.2). The negative coefficient of the second term arises from the

physical observation that a negative in-plane curvature actually increases the pressure in the jet.

Stated differently, the pressure is higher on the concave side of the fluid interface.

Several problems arise when attempting to calculate the curvature based on (3.5) at the ends

of a ligament or drop. As we approach the end of the drop, the height h approaches zero and the

slope of the interface hz diverges. Both cause numerical instabilities. Furthermore, the terms fail to represent the true physics at the endpoints; we recognize since this is an axisymmetric system that both curvatures must match at the end of the drop.

To alleviate these concerns the model uses the following strategy: 1) for the in-plane curvature, we directly calculate the curvature by algebraically fitting an osculating circle to the jet surface for every set of three adjacent grid points, 2) calculate the orthogonal curvature using the formula above except when we approach the endpoints within a given threshold (where it would diverge), and 3) blend the out-of-plane curvature linearly so it exactly matches the in-plane curvature at the endpoints. This method greatly increased numerical stability and confidence that the model accurately portrays the physics at the end points. Finding the osculating circle tangent

to the free surface at a point, hi, amounts to algebraically calculating the center-point that is equidistant from the three given points ( hi-1, h i, and hi+1 ); the radius is a natural result of this

calculation.

Drop Break-up

The code must robustly break a ligament or filament into two separate domains when the

minimum jet radius reaches a prescribed threshold. However, just calculating the minimum point

will give false breaks at near the endpoints as we cannot guarantee that the free surface height at

grid points there are greater than the breaking height threshold. So, we must only test the minima

of the free surface. We do this by calculating straightforward spatial derivatives to determine all

minima along the jet; they are monitored at the end of every successful ODE solver time step.

Once the threshold is satisfied, the code segments the filament into two separate computational domains (see Figure 3.3). Each pass of the ODE solver evolves each ligament separately before proceeding to the next time-step. As a consequence of the way the jet curvature is calculated, the jet may be broken by simply setting the new endpoint heights to zero without regard for smoothing or blending. In essence, robustness is achieved more during the curvature calculation than trying to resolve small-scale surface features at the break-up point.

Because of the uniform grid used to discretize the model, every break will remove grid points from the main filament. Thus, it is imperative to add grid points to both sides of the break using linear interpolation. The best numerical method found was piecewise cubic interpolation. Unlike spline interploation, the cubic method does not match second-order derivatives; however, splines were found to overshoot the original data creating negative drop volumes and thus numerical failure. Also, splines are more oscillatory and could add or remove extra mass.

Figure 3.3. When the smallest minimum of the filament reaches a given threshold, it is broken into two separate computational domains.

Drop Merging

A substantially harder problem is modeling the merging of two separate drops or ligaments.

The conditions for a merge are as follows: 1) the right endpoint of the left ligament and the left endpoint of the right ligament have to approach each other within a given threshold distance, and

2) the velocity of the left ligament must exceed the right one. The second condition ensures that a freshly broken segment does not immediately re-merge into its parent ligament. Also, before we can join the drops, we must interpolate the domains of either or both drops to match grid spacing, dx . Diagrammatically, this is shown in Figure 3.4.

Once the conditions above are met, we have to apply an algorithm for merging the two drops seamlessly. Through experimentation it was found that smoothness of the blend between the two ligaments is paramount for numerical stability. Therefore, we use polynomial fits in both the free surface and the fluid velocity. Specifically, for the free surface, both ends of both ligaments are blended to a new minimum height with zero slope using two third-order polynomials; the two polynomials are then joined where they match height and slope (see Figure 3.5). The fluid velocity and temperature are blended in a similar fashion without the intermediate step of blending first to a minimum point. For completeness, let us examine the third-order polynomial,

y= ax3 + bx 2 ++ cx d . (3.6)

The value of this function at the endpoints must match the two heights prescribed; thus, two

conditions are met. The other two conditions concern the slope of the curve at the endpoints

given by

y′ =3 ax2 + 2 bx + c . (3.7)

We now formulate a matrix equation to solve for the unknowns, namely,

3 2 y  x x x 1  a  1 1 1 1   3 2    y2 x2 x 2 x 2 1  b    =   . (3.8) 3x2 2 x 10  c y ′ 1 1   1  2    3x2 2 x 2 10  d  ′ y2 

The matrix is inverted and the coefficients are found. The polynomial is now complete and is used to generate the new free surface at the grid points between the two drops.

While this procedure is well behaved, it does add a small amount of mass to the system.

Likewise, for breaking drops, a small amount of mass is removed from the system. We examined several instances of both cases (breaking and merging) and found the extra mass gained or lost rarely exceed 3% of the total mass of the two drops in question. This was accomplished by numerically integrating the profiles to find their respective volumes before and after a break/merge.

Figure 3.4. When two drops or ligaments approach each other within a given threshold, their computational domains are joined by a smooth liquid neck.

ALGORITHM

Figure 3.6 shows the decision-making process used in the flow of the numerical code. For each mass of fluid, there exists a separate computational domain and unique grid spacing. For each ligament and time-step, we track the following: surface height y, velocity u, temperature T, number of grid points N, left endpoint position B, and ligament length L. These are passed to the

ODE solver as a large row vector. The function called by the ODE solver applies the boundary conditions, calculates the time-derivatives based on the equations of motion, then increments the row vector for one time step. Then, the program checks for a break or merge. If one is found, the

ODE solver is halted and the main program partitions or merges the data streams accordingly.

Flag variables are used to communicate back to the main program which ligament is breaking or merging. Then, the new data stream is passed to the ODE solver to re-start the sequence. All

44 data is saved at every output time step of the ODE solver. This way, we have access to the entire time history of the sequence and can plot or animate any event of our choosing.

Figure 3.6. Program flow chart is shown.

The model was initially driven with a simple sinusoidal input of temperature at the nozzle.

More interesting, however, is to exploit the device’s ability to have arbitrary thermal inputs; therefore, the model allows for a binary train of pulses such as was shown in Figure 1.4.

Unfortunately, the presence of the sharp transitions of the binary waveform causes numerical instabilities and increased run times due to the adaptive time stepping of the ODE solver. To combat this, the input pulse train is convolved with a Gaussian shape to produce a smooth input.

A typical input is shown in Figure 3.7. This is, in fact, physically more valid than a sharp-edged

profile; the heat pulse from the micro-heater does diffuse considerably before reaching the jet

surface (Sasmal, 2006).

Once the temperature versus time profile is established, the boundary condition is applied inside the ODE solver loop by interpolating against the current time; this is required because of the automatic time-stepping employed by the solver.

RESULTS

A typical, graphical output of the model is shown in Figure 3.8. This represents one jet modeled over a long enough time period to show multiple drop breaks and merges. The number of drops carried through the computation can be user-set so drops that are no longer of interest can be discarded. Here we see seven separate ligaments/drops being modeled. All of the data is saved to the workspace so a user may explore the results at will. In this figure, time advances from bottom to top and the motion of the jet is from left to right. The number of jet snap-shots in time can be selected thus all figures hereafter may have different scales.

A detail view of several satellite ligaments breaking and merging into fundamental drops is shown in Figure 3.9. In Figure 3.10 we see a detail of the formation and oscillation of a satellite drop. The satellite starts as a ligament and then coalesces into a drop with a decaying sinusoidal

46 oscillation. A plot of a satellite filament length versus time is shown in Figure 3.11. The period of oscillation for this 5.1 micrometer diameter satellite is approximately 0.9 microseconds. This agrees quite well with the calculated period of largest oscillation determined by the equation

(Rayleigh, 1879),

2σ 1 f =, τ = , (3.9) π2 ρ r 3 f for which we get 1.06 microseconds.

Using realistic values for the magnitude of the thermal pulse, we have not yet found a satellite-free regime of drop generation. This is in stark contrast to the more tradition drop generation that uses a vibrating body driven by piezoelectric crystals to perturb the jets. In fact, inkjet printing systems that rely on electro-statically charged drops can not operate in the presence of satellites. We are left to conclude, given the presence of satellites and the long break- up lengths of the filaments, that the thermal perturbation of the jet is less energetic than the traditional vibrating body method (Fagerquist, 2006).

A larger drop engulfing a smaller drop is shown in Figure 3.12a. The smaller drop is overtaken due to the small relative velocity of the two. Figure 3.12b shows detail of the merge point between the two drops where the artificial liquid bridge can be seen evolving. Although, as mentioned above, a satellite-free region has not been found, all satellites do eventually merge either forward or rearward. The distance downstream at which this happens is extremely important to the implementation of inkjet printing systems as described in Chapter I. The introduction of the deflection air stream must only occur after satellites have merged, lest they cause system contamination. However, the current model does not include air drag effects which may increase or decrease the distance before the satellites merge.

Shown in Figure 3.13 is the formation of a large drop that is twice the volume of the fundamental drop. The larger drop is created by a temporary absence of one thermal pulse period

of the nozzle boundary condition. Notice how a large drop does not spontaneously form at the

end of the filament; it first breaks into smaller pieces which coalesce downstream because of their

relative velocities. In this case, there are also two satellites that merge into the large drop. A

detail view of the two fundamental drops merging is shown in Figure 3.14. Figures 3.15 and 3.16

show the creation of a 3x and a 4x large drop, respectively.

A plot of the main filament length versus time for a typical case is illustrated in Figure 3.17.

The break-up length was pseudo-steady at 240 micrometers until a large drop was commanded;

then the break-up length fell to about 165 micrometers.

It is helpful to compare the break-up length of the present model to those predicted by theory.

Figure 3.18 shows the comparison between various theories and experimental data. The

theoretical curves were generated using the theory of Rayleigh (1878), Eggers and Dupont

(1994), and Furlani (2005). The equations usually describe the disturbance growth rate rather

than break-up length versus wavenumber; however, the latter description is usually more

informative. A simple relation exists between the growth rate and the break-up length, namely,

ωt the radius of the jet versus time is rt( ) = R0 − ε e ; setting this equal to zero and solving for time reveals the break-up time. The break-up length is the break-up time multiplied by the jet velocity.

The model appears to over-estimate the wavenumber of maximum instability compared to the linear theory of Rayleigh, which gives 0.70. The model of Eggers and Dupont is one-dimensional in nature and includes viscosity which many authors have shown decreases the value of the most unstable wavenumber, their model predicts 0.67 for our jet parameters. However, the theory given by Furlani (2005), which shares the same governing equations as this model, predicts the most unstable wavenumber to be 0.74; this model gives 0.74, an error of less than 1%.

For a wavenumber of 0.6, the model predicts a fundamental drop volume of 3.47 picoLiters.

This ligament initially breaks into a parent drop and satellite drop; the satellite drops merges

within three wavelengths to form the fundamental drop. The velocity of the ligament/drop as it

breaks off is found by summing the momentum of each numerical slice and divided by the total

mass. The velocity for this 3.47 picoLiter drop is 18.9 meters per second. This agrees well with

the theory of Schneider et al. (1967) who predicted the drop velocity based on a mass and

momentum balance, namely,

v 2  σ v= v1 −c  , v = , (3.10) d j2  c vj  ρ a

where vc is the capillary velocity based on the radius of the jet, a. For the model’s parameters,

equation (3.10) predicts a drop velocity of 19.2 meters per second, an error of 2% from the

model. For a wavenumber of 0.7, the fundamental drop volume is 2.94 picoLiters.

The break-up length exhibits a logarithmic dependence on the perturbation amplitude. Figure

3.19 illustrates this dependence. From images of the jet break-up, we can see a transition from forward-merging satellite drops to rearward-merging satellite drops. This is shown in figure 3.20.

Thus, a region should exist that exhibits “infinite” satellites, that is, where satellites do not merge into a neighboring drop.

Most of the theory pertaining to the break up of liquid jets starts with the assumption of a small, sinusoidal radial perturbation of the free surface of an infinite cylinder of fluid. Therefore, it could be enlightening to compare the temperature perturbation we have applied to that of an equivalent radial perturbation at the jet’s surface. We can do this by examining the plot of the free surface at the point of break-up (Figure 3.21). We can fit an exponentially increasing sinusoid curve to this data of the form

2π  rzRae( )= +bz cos  z + φ  . (3.11) 0 λ 

The exponent can be found be examining the logarithmic decrement, that is, the ratio of

successive peaks of the curve. To that end, divide (3.11) evaluated at one peak by itself evaluated

at the next peak one wavelength downstream to arrive at

1 r  b = ln 2  . (3.12) λ r1 

To calculate the initial radial disturbance, a, we recognize that peaks occur at integer multiples of the wavelength and rewrite (3.11) as

−nb λ a=( rn − Re0 ) . (3.13)

Using these two equations, we can fit the exponentially increasing sinusoid also shown in Figure

3.21. We calculate the phase shift by subtracting a peak of the curve fit with its corresponding peak of the raw data; the curve fit is calculated again with the phase shift. The curve fit deviates as the jet radius starts to pinch. This is due to higher order effects near the point of break up.

However, the radial curve and the jet surface match very well. It is expected that any perturbation method used will causes a free surface deformation that drives the break-up process due to the capillary instability. This may be why Rayleigh’s result still bears fruit to this day.

For a variety of perturbation amplitudes, the equivalent radial perturbation is calculated.

Figure 3.22 shows the relationship between the two; a linear fit is also shown.

We end the results section with comparisons between the model and experimental data.

Figure 3.23 shows the generation of fundamental drops. The perturbation amplitude of the model was varied until the break-up length was matched to the experimental setup, approximately 0.55 millimeters. While the shape of filament at the point of break-up is well matched, the behavior of the satellite drops is markedly different from experiment. This may be explained by the lack of radial terms in the model; at or near break-up there could be large radial gradients of velocity in the experimental jet. This disparity carries over into large drop formation as well.

In Figure 3.24, we see the formation of a large drop four times the volume of the fundamental for two different waveform pulse trains. The exact time of fission of the ligament and its constituent drops does not match as well as hoped. However, the minutia of the break-up region is less interesting to researchers than determining the point downstream at which the large drops coalesce. The model predicts 3x drop formation at 1.2 millimeters from the orifice plate.

Experimental values for the 3x drop formation range from 1.2 to 1.5 millimeters (Fagerquist,

2006).

Figure 3.8. A long sequence of drop generation from a single jet is shown. The nozzle is at the left end. Time is increasing from bottom to top.

Figure 3.9. This figure shows a detail view of the satellite formation and subsequent merging shown in the previous figure.

Figure 3.10. Detail of the recoil of a fluid ligament into a satellite drop is shown. Time is increasing from bottom to top. The bulk motion of the ligament is from left to right.

(a) (b)

Figure 3.14. This figure shows the detail of two main drops merging to create a 2x drop. The top of the first column continues at the bottom of the second. Time is increasing from bottom to top.

Figure 3.17. Illustrated is a plot of the main filament length versus time for the program run given in Figure 3.12. The break- up length was steady at 240 micrometers until a large drop was commanded. Then the break-up length fell to about 165 micrometers before increasing to the previous value.

Model Rayleigh, 1878 Eggers and Dupont, 1994 Furlani, 2005 650

600

550

500

450

Break-up Length (micrometers) 400

350 0.4 0.5 0.6 0.7 0.8 0.9 Dimensionless Wavenumber

600

550

500

450

400

350

300

Break-up Length (micrometers) Break-up . Length (micrometers) 250

200 1 10 100 Temperature Perturbation Range (ºC)

Equivalent Radial Perturbation for a Given Temperature Perturbation 100 y = 2.2941x 90 R2 = 0.906 80

20 Radial PerturbationRadial Amplitude (nm) . 10

0 0 10 20 30 40 Temperature Perturbation Range (ºC)

Figure 3.22. The equivalent radial perturbation shows a linear dependence on the actual temperature perturbation.

Figure 3.23. Shown is a comparison of the model to micrographs of physical hardware during fundamental drop generation. While the filament shape is well-matched, the satellite behavior differs.

1 2 3 4

CHAPTER IV: CONCLUSIONS

A numerical model was developed for predicting the instability and breakup of infinite microthreads and continuous nozzle driven microjets of Newtonian fluid. The model is based on a slender jet approximation, and has been implemented using the method of lines numerical technique. The model uses explicit time stepping and a uniform spatial grid, which are relatively easy to implement as compared to other numerical techniques. The model takes into account arbitrary free-surface and velocity perturbations as well as Marangoni instability via variations in surface tension. The model was validated using established data from the literature as well as

VOF CFD simulations. The model enables jet instability predictions that are orders of magnitude faster than axisymmetric VOF CFD simulations. Furthermore, while it is easy to impose arbitrary free-surface, velocity and/or surface tension perturbations (modulation) in our model, similar conditions can be very difficult to implement in commercial CFD software. The model is well suited for parametric analysis of jet breakup and satellite formation as a function of jet dimensions, modulation parameters and fluid rheology. It should be useful for the development and optimization of novel micro-fluidic droplet generators.

Also developed were methods for modeling beyond break-up, specifically, the break-up and merge of ligaments and drops downstream. The model allows for the input of a sinusoidal or binary pulse train for the input temperature boundary condition. It was shown that a larger drop does not spontaneously form at the end of the filament; it first breaks into smaller pieces which coalesce downstream because of their relative velocities. This distance downstream of satellite or main drop merging is important to developers of inkjet systems that use this type of perturbation.

The 1-D model presented here shows great promise to researchers studying the dynamics of the formation and merging of drops. It relative ease of use and speed make it ideal for rapid development of candidate waveforms. It is a realizable goal to make this program available to every liquid jet researcher.

FUTURE WORK

The goal of this research is to produce a useful model that predicts drop formation for a variety of thermal modulation waveforms. Therefore, the human interface needs improving to the point where a user can input the modulation waveform by point-and-click methods, set the program in motion, and be presented later with a table of merge distances from the nozzle, both satellite-to-main drop and main drops-to-large drops.

In order to improve usability, the run time could be reduced by coding certain aspects of the

program in a compiled language which is then called by MATLAB®. Furthermore, the entire

program could be re-written in a compiled language. However, MATLAB® offers a rich set of

plotting and computational tools that are ideal for post-processing the data; it remains the

language of choice for the applied sciences.

The percentage contribution of the thermal variation of surface tension, density, and viscosity

to the break-up process is unknown. For this reason, the model will be extended to include these

variations. The governing equations become more difficult as previously constant physical

properties now must remain inside the spatial derivative terms in the equations of motion.

However, it should not be much effort to include these variations in the model. It seems that no

matter what the perturbation, as soon as the free surface becomes deformed the capillary

instability becomes the driving force behind break-up.

A more ambitious and potentially fruitful project would be to extend the model to include non-Newtonian fluids. The constitutive equations for such fluids contain time derivatives of the stress tensor; these could be calculated in the ODE solver at the same time as the other variables.

It would be interesting to examine the post break-up behavior of these types of jets as they tend to display “beads-on-a-string” phenomena due to the viscoelastic properties of the fluid (Clasen et al., 2006).

APPENDIX A: AXISYMMETRIC EQUATIONS OF MOTION

Here is short summary of the derivation of the axisymmetric Navier-Stokes equations governing incompressible fluids.

VECTOR CALCULUS

First, it is helpful to review some basic formulas from vector calculus and continuum mechanics. The differential operator, “del,” in cylindrical coordinates is given by

∂1 ∂ ∂ ∇=eˆ + e ˆ + e ˆ . (A.1) r∂rθ r ∂θ z ∂ z

The gradient of a scalar field is a vector field, namely

∂φ1 ∂ φ ∂ φ ∇=φ eˆ + e ˆ + e ˆ . (A.2) r∂rθ r ∂θ z ∂ z

The gradient of a vector field is a second-order tensor field. Here is the velocity gradient in cylindrical coordinates.

  ∂vr1  ∂ v r  ∂ v r − vθ   ∂r r ∂θ  ∂ z    ∂vθ1  ∂ v θ  ∂ v θ ∇v = + v r   (A.3) ∂r r ∂θ  ∂ z  ∂v1 ∂ v ∂ v  z z z  ∂r r ∂θ ∂ z 

The divergence of a vector field is a scalar field; it is the trace of the vector gradient.

∂v1 ∂v  ∂ v ∇=·vr +θ ++ v  z (A.4) ∂r r ∂θ r  ∂ z

An important operation in the derivation of the Navier-Stokes equation is the divergence of a tensor field. While this has an intuitive and easy to remember form in Cartesian coordinates, the cylindrical representation is non-trivial. More detail can be found in Bower (2008), namely

∂Trr T rr1 ∂Tθr ∂ T zr T θθ  ++ + −  ∂rrr ∂θ ∂ zr  1 ∂T ∂ TTT ∂ T  ∇=·T θθ +r θ +++ r θ θ r z θ  . (A.5) r∂θ ∂ rrrz ∂  ∂T ∂ T T 1 ∂T  zz+ rz + rz + θ z  ∂z ∂ r rr ∂ θ 

Axisymmetric Coordinates

Let's repeat the equations written above for the axisymmetric case. The derivatives with respect to θ are zero; also, the component of velocity in the θ direction is zero. For now,

however, we cannot assume that any of the tensor terms with a θ component are zero. Later we

will show that several of these are in fact equal to zero when we relate the stress in a fluid to the

rate of strain through a constitutive relation.

The differential operator becomes

∂ ∂ ∇ =eˆ + e ˆ . (A.6) r∂r z ∂ z

Hence, the gradient of a scalar field is

∂φ ∂ φ ∇φ =eˆ + e ˆ . (A.7) r∂r z ∂ z

Here is the velocity gradient in axisymmetric coordinates,

∂vr ∂ v r  0  ∂r ∂ z  v  ∇v = 0r 0 . (A.8) r    ∂v ∂ v z0 z  ∂r ∂ z 

The divergence of the velocity field is then

∂v v ∂ v ∇=·v r + r + z . (A.9) ∂r r ∂ z

Finally, the divergence of a tensor field in axisymmetric coordinates is given by the following. Again, the components of the tensor cannot be assumed to vanish at this point.

∂Trr T rr ∂ T zr Tθθ  + + −  ∂r r ∂ z r  ∂T T T ∂ T  ∇=·T rθ +++ r θθ r z θ  (A.10) ∂r r r ∂ z  ∂T ∂ T T  zz+ rz + rz  ∂z ∂ r r 

FLUID KINEMATICS

1 The rate of deformation tensor is given by D=( ∇ v +∇ ( v ))T . Its form in cylindrical 2

coordinates is given by

    ∂vr1 1∂ v r∂vθ 1 ∂ vv rz ∂   −vθ +   +    ∂r2 r ∂∂∂∂θ rzr  2    11 ∂v∂v   1 ∂ v 11  ∂ v ∂v   D=+− θ r v   θ + v  θ + z   . (A.11) 2 ∂∂rrθθ   r ∂ θr 2  ∂∂ zr θ     1∂∂vv  1 1 ∂ v∂v  ∂ v  zr+   z + θ  z   2∂∂rz  2  rz ∂∂θ  ∂ z 

In the axisymmetric flow equations all velocity components and derivatives with respect to θ

vanish. Here is the rate of deformation tensor in axisymmetric cylindrical coordinates, where all

off-diagonal terms are zero,

 ∂v1  ∂ v ∂ v   r0  r+ z   ∂r2  ∂ z ∂ r   v  D = 0r 0 . (A.12) axi   r  1 ∂vz ∂ v r  ∂ v z  +  0  2 ∂r ∂ z  ∂ z 

According to the assumptions of an incompressible, isotropic medium, as well as a linear relationship between stress and strain rate, the stress in a fluid can be represented by the following constitutive equation (Lai et al., 2005),

T= − pδ + 2 µ D , (A.13) where p is the hydrostatic pressure that exists in the absence of strain, and µ is the dynamic

viscosity. Here no distinction is made between the hydrostatic pressure and the thermodynamic

pressure. Therefore, the stress tensor for the axisymmetric fluid is

 ∂v ∂ v ∂ v    −p +2µr 0 µ  r + z    ∂r ∂ z ∂ r    v  T=0 − p + 2µ r 0 . (A.14) axi    r   ∂vz ∂ v r  ∂ v z  µ+  0 −p + 2 µ   ∂r ∂ z  ∂ z 

CONTINUITY EQUATION

The continuity equation for an incompressible, axisymmetric fluid is

∂v v ∂ v ∇=·v r ++ r z = 0 . (A.15) ∂r r ∂ z

MOMENTUM EQUATION

Newton's second law in invariant form is Dv ρ= ρ g + ∇ · T , (A.16) Dt

D where is the material derivative given by Dt Dv∂ v = +( ∇ v ) v . (A.17) Dt∂ t

Equation (A.16) is valid for all continua. Now we can use equations (A.10) and (A.14) to formulate the divergence of the axisymmetric stress tensor.

    ∂p ∂∂vrµ ∂ v r ∂ ∂ vv zr ∂  v r −+2µ ++ 2 µ  +−   2 µ 2   ∂∂∂rrr rrz ∂∂  ∂∂ rz   r    ∇·T =  0  . (A.18)   ∂p ∂ ∂v∂  ∂ vv ∂  µ ∂ vv ∂   −+2 µz + µ  rz +++   rz    ∂∂∂∂zzzr  ∂∂ zr   rzr ∂∂  

Furthermore, if constant properties are assumed, (A.18) becomes

 2 2 2   ∂p ∂vrµ ∂ v r ∂∂ vv zr v r −+2µ2 + 2 + µ +−2  2 µ 2   ∂r ∂ r rr ∂ ∂∂∂ zrz  r    ∇·T =  0  . (A.19)   2 2 2   ∂p ∂vz ∂∂ vv rzµ  ∂∂ vv rz   −+2µ2 + µ  ++ 2   +   ∂z ∂ z ∂∂∂ rzr  rzr ∂∂  

Equation (A.19) can be further simplified by using partial derivatives of the continuity

equation (A.15) with respect to r and separately with respect to z . This results in the following

equation.

2 2   ∂p ∂vr1 ∂ v r ∂ vv rr −+µ  + +−2 2   ∂r ∂ rrr ∂∂ zr     ∇·T =  0  . (A.20)   2 2  ∂p ∂vz1 ∂ v z ∂ v z  −+µ 2 + + 2  ∂z ∂ rrr ∂∂ z  

Finally, referring to (A.16), we arrive at the axisymmetric Navier-Stokes equations:

2 2  ∂∂∂vvvrrr1∂p ∂∂∂ vvvv rrrr 1 +vr + v z =− +ν  + +−  ∂∂t r ∂ zρ ∂ r ∂ rrrzr ∂∂ 2 2  , (A.21) 2 2  ∂∂∂vvvzzz1∂p ∂∂∂ vvv zzz 1 +vr + v z =− +ν  + +  ∂∂t r ∂ zρ ∂ z ∂ rrrz2 ∂∂ 2 

µ where ν = is the kinematic viscosity. The right-hand side of (A.21) can be simplified by ρ

recognizing several application of the product rule, which results in

2  ∂∂∂vvvr r r 1∂p ∂  1 ∂∂( rvvr )  r ++=−+vr v z ν   +  ∂∂∂t r zρ ∂∂∂ r rrr  ∂ z 2  . (A.22) 2  ∂∂∂vvvz z z 1∂p 1 ∂  ∂∂ vvz  z ++=−+vr v z ν  r  +  ∂∂∂t r zρ ∂ zrrrz ∂∂∂  2 

ENERGY EQUATION

The energy equation is given by

DT ρc= k ∇2 T +Φ , (A.23) p Dt

76 where Φ is the viscous dissipation function (Tannehill et al., 1997) presented here in axisymmetric form,

 ∂v222 v ∂ v  ∂∂ vv  2  Φ=µ 2 r ++ r z  ++ rz   . (A.24)  ∂r r ∂ z  ∂∂ zr  

Expanding (A.23) and using (A.24) gives us the desired form of the axisymmetric energy equation, namely,

∂TTT ∂ ∂ 1 ∂∂∂  TT2  ρcp++= vv r z  k r  ++2  ... ∂∂t r ∂ z  rrrz ∂∂∂   . (A.25)  ∂v222 v ∂ v  ∂∂ vv  2  µ 2 r+ r + z  ++ rz    ∂r r ∂ z  ∂∂ zr  

APPENDIX B: LINEAR THEORY OF JET BREAKUP: RAYLEIGH ANALYSIS

What follows is a detailed discussion of the linear stability analysis of an axisymmetric, inviscid liquid jet. While Rayleigh (1878) approached this problem from the standpoint of kinetic and potential energy, we shall formulate his results in terms of the Navier-Stokes equations.

EQUATIONS OF MOTION

Let us start with the equations of motion for a cylindrical, axisymmetric liquid jet. Denoting

z as the axial direction and r as the radial direction, the continuity equation is given by

∂u 1 ∂ z +(ru ) = 0. (B.1) ∂z r ∂ r r

For the momentum equations we have

2 2  ∂∂∂uuurrr1∂p ∂∂∂ uuuu rrrr 1 +uz + u r =−+ν  ++ −  ∂∂∂t z rρ ∂∂∂ r zrrrr2 2 ∂ 2  . (B.2) 2 2  ∂∂∂uuuzzz1∂p ∂∂∂ uuu zzz 1 +uz + u r =− +ν  ++  ∂∂∂t z rρ ∂∂∂ z zrrr2 2 ∂ 

In our discussion we shall use the inviscid version of (B.2) where ν = 0 , hence

∂u ∂ u ∂ u 1 ∂p r+u r + u r =− ∂tz ∂ z r ∂ rρ ∂ r . (B.3) ∂u ∂ u ∂ u 1 ∂p z+u z + u z =− ∂tz ∂ z r ∂ rρ ∂ z

LINEAR STABILITY ANALYSIS

Let ur= u r + u r′ , uz = uz + u ′z , and p= p + p ′ where the barred and primed quantities are mean and perturbed components, respectively. Substituting these into the first equation of (B.3), setting the mean components equal to zero, and ignoring products of primed components yields

∂u 1 ∂p′ r′ = − . (B.4) ∂tρ ∂ r

Likewise, the second equation of (B.3) becomes

∂u 1 ∂p′ z′ = − , (B.5) ∂tρ ∂ z and the continuity equation becomes

∂u ∂ u u z′+ r ′ + r ′ = 0. (B.6) ∂z ∂ r r

Now we assume that each relevant quantity in the above equations is periodic in nature and

ωt+ ikz ωt+ ikz grows exponentially over time. To that end, we set ur′ = U r e , uz′ = U z e , and

ωt+ ikz p′ = Pe where k = 2π / λ is the wavenumber. Note that Ur , U z , and P are all functions

of r . Substituting these relations into equations (B.4) through (B.6) and canceling the exponential terms produces

1 ∂P ωU = − (B.7) r ρ ∂r

ik ωU= − P (B.8) z ρ

∂U U ikU +r + r = 0 (B.9) z ∂r r

To eliminate U z and P from the above equations, we take the partial derivatives of (B.8)

and (B.9)with respect to r and eliminate ∂Uz / ∂ r between the two to get

k2 ∂ P ∂2u1 ∂ U U +r′ + r −= r 0. (B.10) ρω ∂r ∂ r2 rr ∂ r 2

Now substitute (B.7) into (B.10) to obtain the second order differential equation

∂2U ∂ U r2r+ r r −+(1 krU 2 2 ) = 0. (B.11) ∂r2 ∂ r r

The solution to (B.11) is given in terms of the modified Bessel functions, namely

Urr ( ) = cIkr11 ( ) + cKkr 21 ( ) .

However, to force a bounded solution as r approaches zero, we set c2 = 0 and c1 = c to yield

Ur ( r) = c I1 ( kr ) . (B.12)

We can also determine the solution of P by substituting (B.12) into (B.7) and integrating with respect to r ; this yields

ρω c P( r )= − I ( kr ). (B.13) k 0

BOUNDARY CONDITIONS

Let the free surface of the liquid jet be perturbed such that

εωt+ ikz ε ≪ Rzt(,)= R0 + e , 1. (B.14)

The time derivative of the free surface must equal the radial velocity at the surface. Thus our first

boundary condition is

∂R( z , t ) = u ′ | . (B.15) ∂t r r= R 0

We use this boundary condition to solve for the constant c. This is accomplished by substituting

ωt+ ikz (B.12) and (B.14) into (B.15) (recall that ur′ = U r e ):

ε ωt+ ikz ω t + ikz ω e= cI1( kR 0 ) e , ωε (B.16) ⇒ c = . I1( kR 0 )

Now we are left with the task of determining the growth rate of the disturbance, ω . To this end, we introduce another boundary condition; the pressure in the jet is governed by the local curvature of the surface and the surface tension σ . Thus,

1 1  p+ p ′| =σ  +  (B.17) 0 r= R 0 R1 R 2 

where the unperturbed pressure is given by p0= σ / R 0 . Since the perturbations are small, we

can approximate the radii of curvature by the following relations:

2 1 ∂ εωt+ ikz ε 2 ω t + ikz ≈− 2 ()Re0 + ≈ ke (B.18) R2 ∂ z and

1 1 1 εeωt+ ikz ≈ ≈ − . (B.19) ε ωt+ ikz 2 R10 Re+ RR 00

To obtain the expression in (B.19) , a two-term Taylor expansion was used about eωt+ ikz . Now

substitute (B.18) and (B.19) into the boundary condition (B.17) with p0= σ / R 0 :

ε ωt+ ikz  1 e ε 2 ωt+ ikz pp0 +′ |r= R =σ  − + ke  0 R R 2  0 0 (B.20) σε p′ |= − (1 − kRe2 2 ) ωt+ ikz r= R 0 2 0 R0

The left-hand side of (B.20) is determined by using (B.13) and (B.16) with p′ = Pe ωt+ ikz and

evaluating at r= R 0 , viz.,

2ε ε ω ρI0( kR 0 ) ωt+ ikz σ 2 2 ω t + ikz −e =−−2 (1 kRe0 ) (B.21) k I1( kR 0 ) R 0

After rearranging (B.21), we get the dispersion equation that relates the growth rate of the disturbance as a function of the wavenumber:

2σ 2 2 I1( kR 0 ) ω =3 kR0(1 − k R 0 ) . (B.22) ρR0 I 0( kR 0 )

Disturbances grow only when ω is real; this is satisfied when kR 0 < 1 or λ/ D > π . The

dispersion relation is plotted in Figure B.1. The maximum growth rate occurs at kR 0 = 0.697 (or

2 3 λ /D = 4.507 ) where the value of ω ρR0 / σ = 0.118 .

Figure B.1. The dispersion equation relates growth rate to the dimensionless wavenumber.

We can exploit this relation to find both the time of break-up and the length of the liquid

filament. If the jet is perturbed equally across all wavenumbers, the disturbance corresponding to

kR 0 = 0.697 is physically preferred to cause break-up. Recall that the radial perturbation is given by (B.14). At break-up, R = 0 and (B.14) can be arranged as

R −0 = eωtbo ε (B.23) 1 R ⇒ t = ln0 . bo ω ε

Now substitute the maximum value of the dispersion equation into (B.23) to yield

ρR3 R t = 2.910 ln 0 . (B.24) bo σ ε

ε The ratio R0 / can be estimated or calculated given experimental data. Of course, if the jet is subject to a periodic excitation, it is that wavenumber that ultimately drives the break-up.

Finally, the break-up length is simply the jet velocity times the break-up time.

BIBLIOGRAPHY

Ambravaneswaran B, Wilkes ED, and Basaran OA. Drop formation from a capillary tube: Comparison of one-dimensional and two-dimensional analyses and occurrence of satellite drops. Phys. Fluids 2002; 14 (8):2606 −2621.

Anagnostopoulos CN, Chwalek JM, Delametter CN, Hawkins G, Jeanmaire DL, Lebens JA, Lopez A, and Trauernicht DP. Micro-Jet Nozzle Array for Precise Droplet Metering and Steering Having Increased Droplet Deflection, Proceedings of the 12th International Conference on Solid State Sensors, Actuators and Microsystems, sponsored by IEEE , Boston, June 8-12, 2003; 368-71.

Anderson DM, McFadden GB, and Wheeler AA. Diffuse interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 1998; 30: 139–165.

Ashgriz N and Mashayek F. Temporal analysis of capillary jet breakup. J. Fluid Mech. 1995; 291: 163.

Ashley CT. Emerging applications for inkjet technology. Int. Conf. Digital Printing Tech. 2002; 18 : 540 −542.

Basaran OA. Small-scale free surface flows with breakup: Drop formation and emerging applications. AIChE J. 2002; 48 (9):842 −1848.

Bower AF. Applied Mechanics of Solids , http://solidmechanics.org/, 2008.

Brenner MP. Private communication, 2006.

Brenner MP, Eggers J, Joseph K, Nagel SR, and Shi XD. Breakdown of scaling in droplet fission at high Reynolds number. Phys. Fluids 1997; 9:1573.

Calvert P. Inkjet printing for materials and devices. Chem. Mater. 2001; 13 :3299 −3305.

Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability , Oxford University Press, 1961.

Chwalek JM, Trauernicht DP, Delametter CN, Sharma R, Jeanmaire DL, Anagnostopoulos CN, Hawkins GA, Ambravaneswaran B, Panditaratne JC, and Basaran OA. A new method for deflecting liquid microjets. Phys. Fluids 2002; 14 6:37 −40.

Clasen C, Eggers J, Fontelos MA, Li J, and McKinley GH. The Beads-on-String Structure of Viscoelastic Threads. J. Fluid Mech. 2006; vol. 556, pp. 283–308.

Cline HE and Anthony TR. The effects of harmonics on the capillary instability of liquid jets. J. Appl. Phys. 1978; 49 :3203-3208.

Delametter CN, Trauernicht DP, Chwalek JM. Novel Microfluidic Jet Deflection: Significant Modeling Challenge with Great Application Potential, Technical Proceedings of the 2002 International Conference on Modeling and Simulation of Microsystems sponsored by NSTI, San Juan, Puerto Rico , April 21-25, 2002; 44-47

Eggers, J. Nonlinear dynamics and breakup of free-surface flows, Reviews of Modern Physics 1997; 69 :865-929.

Eggers J and Dupont TF. Drop formation in a one-dimensional approximation of the Navier– Stokes equation. J. Fluid Mech. 1994; 262 :205.

Eggers J and Villermaux E. Physics of liquid jets. Rep. Prog. Phys. 2008; 71: 036601.

Fagerquist, R. Private communication, Eastman Kodak Company, 2006.

Furlani EP. Temporal instability of viscous liquid microjets with spatially varying surface tension. J. Phys. A: Math. Gen. 2005; 38: 263 −276.

Furlani EP. Thermal modulation and instability of viscous microjets. Proc. NSTI Nanotechnology Conference 2005.

Furlani EP, Price BG, Hawkins G, and Lopez AG. Thermally induced Marangoni instability of liquid microjets with application to continuous inkjet printing. Proc. NSTI Nanotechnology Conference 2006.

Gao Z. Instability of non-Newtonian jets with surface tension gradient. J. Phys. A: Math. Theor. 2008; 42 :065501.

Green AE. On the nonlinear behavior of fluid jets. Intl. J. Engng Sci, 1976; 14 :49-63.

Huang W, Ren Y, and Russel RD. Moving Mesh Methods Based on Moving Mesh Partial Differential Equations. J. Comp. Phys. 1994; 113 :279-290.

Hilbing JH and Heister SD. Droplet size control in liquid jet breakup. Phys. Fluids 1996; 8:1574- 1581.

Hirt CW and Nichols BD. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys . 1981; 39 :201.

Lai WM, Rubin D, and Krempl E. Introduction to Continuum Mechanics, Third Edition . McGraw-Hill Science/Engineering/Math, 2005.

Lee HC. Drop formation in a liquid jet. IBM J. Res. Develop. 1974; 18 :364-369.

Lundgren TS and Mansour NN. Oscillations of drops in zero gravity with weak viscous effect. J. Fluid Mech. 1988; 194 :479-510.

Maronnier V, Picasso M, and Rappaz J. Numerical simulation of free surface flows. J. Comp. Phys. 1999; 155 :439–455.

Mullholland LS, Qiu Y, and Sloan DM. Solution of Evolutionary Partial Differential Equations Using Adaptive Finite Differences with Pseudospectral Post-processing. J. Comp. Phys. 1997; 131 :280-298.

Pimbley WT. Drop formation from a liquid jet: a linear one-dimensional analysis considered as a Boundary Value Problem. IBM J. Res. Develop. 1976; 20 :148.

Pimbley WT and Lee HC. Satellite drop formation in a liquid jet. IBM J. Res. Develop . 1997; 21 :21-30.

Plateau JAF. Acad. Sci. Brux. Mem. 1849; 23 :5.

Quan S, Lou J, and Schmidt DP. Modeling merging and breakup in the moving mesh interface tracking method for multiphase flow simulations. J. Comp. Phys. 2009; 228 :2660–2675

Rayleigh, Lord. On the instability of jets. Proc. London Math. Soc. 1878; X.

Rayleigh, Lord. On the capillary phenomena of jets. Proc. London Math Soc. 1879; 10 :4 −13.

Roth EA, Xu T, Das M, Gregory C, Hickman JJ, and Boland T. Inkjet printing for high- throughput cell patterning. Biomaterials 2004; 25: 3707–3715.

Sasmal G. Private communication, Eastman Kodak Company, 2006.

Savart F. Mémoire sur la constitution des veines liquides lancées par des orifices circulaires en mince paroi. Ann. Chim. 1833; 53 :337 −386.

Scardovelli R and Zaleski S. Direct numerical simulation of free-surface and interfacial flow. Annu. Rev. Fluid Mech. 1999; 31 :567–603.

Schiesser WE. The Numerical Method of Lines Integration of Partial Differential Equations , Academic Press, San Diego 1991.

Schneider JM, Lindblad NR, Hendricks Jr. CD, and Crowley JM. Stability of an Electrified Liquid Jet. Journal of Applied Physics, 1967; 38 :2599-2605

Sellens RW. A One-Dimensional Numerical Model of the Capillary Instability. Atomization and Sprays 1992; 2(4):239-251.

Sethian JA. Level Set Methods , Cambridge Monographs on Applied and Computational Mathematics (Cambridge University Press, Cambridge, UK) 1996.

Setiawan ER and Heister SD. Nonlinear modeling of an inﬁnite electriﬁed jet. J. Electrostat. 1997; 42 :243–257.

Sirringhaus H and Shimoda T. Inkjet printing of functional materials. MRS Bull. 2003; 28 (11):802 −803.

Tannehill JC, Anderson DA, and Pletcher RH. Computational Fluid Mechanics and Heat Transfer, 2 nd Edition , Taylor & Francis, 1997.

Tekin E, Smith PJ, and Schubert US. Inkjet printing as a deposition and patterning tool for polymers and inorganic particles. Soft Matter 2008; 4(4):703 −713.