Preface xi

List of Symbols xv

**1 Introduction 1**

References 10

**2 Single Degree of Freedom System 13**

2.1 Introduction 13

2.2 Governing Equations and Solution for Square Pulse Driving 15

2.2.1 Entrance and Exit Effects (Entrance Pressure Drop, Exit Loss) 22

2.2.2 Corrected Speed of Sound 34

2.2.3 Effect of Surface Tension on Resonance Frequency 36

2.2.4 Rayleigh’s Method for Calculating the Resonance Frequency 37

2.2.5 Logarithmic Decrement Method to Estimate Damping 38

2.2.6 Bulk Viscosity 40

2.2.7 First Estimate on the Frequency Dependence of Damping 41

2.3 Solution for Ramped Pulse Driving 42

2.4 Solution for Exponential Pulse Driving 47

2.5 Solution for Harmonic Driving and Fourier Analysis 50

2.5.1 Frequency-dependent Damping (Full Solution) 56

2.6 Non-linear Effects Associated with Non-complete Filling of the Nozzle 61

References 71

**3 Two Degrees of Freedom System 75**

3.1 Introduction 75

3.1.1 Rayleigh’s Method to Determine Approximately the Resonance Frequencies of a Two Degrees of Freedom System for the Case with Surface Tension 79

3.1.2 Calculation of the Damping of Two Degrees of Freedom System with Low Viscosity Using the Logarithmic Decrement Method 84

3.1.3 FlowThrough a Conical Nozzle 87

3.1.4 FlowThrough a Bell-mouth-shaped Nozzle 91

3.2 Governing Equations and Solutions for Square Pulse Driving 98

3.2.1 Special Cases 101

3.2.2 Solutions for the Low Viscosity Inks to Square Pulse Driving 105

3.2.3 Solutions for Inks with a Moderate Viscosity to Square Pulse Driving 111

3.2.4 Solutions for a High Viscosity Ink to Square Pulse Driving 115

3.3 Solutions for Ramped Pulse Driving 119

3.3.1 Solutions for Low Viscosity Inks to Ramp Actuation 121

3.3.2 Solutions for Moderate Viscosity Inks to Ramp Actuation 122

3.3.3 Solution for Large Viscosity Inks to Ramp Actuation 122

3.3.4 Solution to Ramped Pulse Driving 123

3.4 Solutions for Exponential Pulse Driving 128

3.4.1 Solution for Low Viscosity Inks to Exponential Ramp Driving 130

3.4.2 Solution for Moderate Viscosity Inks to Exponential Ramp Driving 131

3.4.3 Solution for Large Viscosity Inks to Exponential Ramp Actuation 131

3.4.4 Solutions to Exponential Pulse Driving (Pulse Consisting of Two Exponential Ramps) 132

3.5 Solution for Harmonic Driving and Fourier Analysis 134

3.5.1 Frequency Dependent Damping (Full Solution) 144

3.6 Non-linear Analysis 148

3.6.1 Capillary Pressure and Force in Conical Nozzle 157

3.6.2 Capillary Pressure and Force in Bell-mouth-shaped Nozzle 161

References 163

**4 Multi-cavity Helmholtz Resonator Theory 167**

4.1 Introduction 167

4.2 Governing Equations 169

4.2.1 Speed of Sound in Main Supply Channel 172

4.3 Solutions for Ramped Pulse Driving for Low Viscosity Inks 174

4.4 Solution for Harmonic Driving and Fourier Analysis 183

References 192

**5 Waveguide Theory of Single-nozzle Print Head 193**

5.1 Introduction 193

5.2 Long Waveguide Theory 197

5.2.1 Characteristics of a Closed End/Closed Pump of the Waveguide Type Without Connecting Ducts 202

5.2.2 Characteristics of an Open End/Closed End Pump of the Waveguide Type Without Connecting Ducts 204

5.2.3 Viscous Drag in Non-circular Channels 206

5.3 Solutions for Ramped Pulse Driving of the Waveguide-type Inkjet Pump 207

5.3.1 The Closed End/Closed End Case 207

5.3.2 Damping of the Closed End/Closed End Print Head 216

5.3.3 Open End/Closed End Case 219

5.4 Solutions for Harmonic Driving and Fourier Analysis Including the Effect of Damping 221

5.4.1 Solution of Wave Equation with Poiseuille Damping in Nozzle and Throttle 224

5.4.2 Sample Calculation and Results for Closed End/Closed End Print Head Channel Arrangement 227

5.4.3 Sample Calculation and Results for Open End/Closed End Print Head Channel Arrangement 230

5.4.4 Full Solution of Wave Equation Including Frequency-dependent Damping 233

5.4.5 Closed End/Closed End Case 238

5.4.6 Open End/Closed End Case 240

5.5 Non-linear Analysis of the Waveguide Type of Print Head Including Inertia, Viscous, and Surface Tension Effects in the Nozzle 243

5.5.1 Results for the Closed End/Closed End Arrangement 245

5.5.2 Results for the Open End/Closed End Type of Waveguide Pump 246

5.5.3 High Frequency Pulsing, Start-up, and Nozzle Front Flooding 249

5.5.4 Effect of an Air Bubble on the Internal Acoustics of a Print Head 252

5.5.5 Higher Order Meniscus Oscillations 254

5.6 Means and Methods to Enhance Fluid Velocity in Nozzle 258

References 259

**6 Multi-cavity Waveguide Theory 263**

6.1 Introduction to Multi-cavity Acoustics 263

6.2 Analysis of Cross-talk in an Open End/Closed End Linear Array Print Head with Alternately Activated and Non-activated Pumps 266

6.3 Analysis of Cross-talk in an Open End/Closed End Linear Array Print Head with Alternately One Pump Activated and Two Pumps Idling 277

6.4 Analysis of Cross-talk in an Open End/Closed End Linear Array Print Head with Alternately One Pump Activated and Three Pumps Idling 285

6.5 Analysis of Cross-talk in an Open End/Closed End Linear Array-shared Wall Shear-mode Print Head with Alternately One Pump Activated and Two Pumps Non-activated 297

6.6 Analysis of Cross-talk in a Closed End/Closed End Linear Array Print Head with Alternately Activated and Non-activated Pumps 302

References 307

**7 Droplet Formation 309**

7.1 Introduction 309

7.2 Analysis of Droplet Formation (Positive Pulse) 312

7.2.1 Force (Impulse) Consideration 313

7.2.2 Energy Consideration 316

7.2.3 Droplet Formation Criterion from a Retracted Meniscus 319

7.3 Analysis of Droplet Formation (Negative Pulse) 320

7.3.1 Force Consideration 321

7.3.2 Energy Consideration 324

7.4 Deceleration Due to Elongational and Surface Tension Effects Prior to Pinching Off 326

7.5 Non-linear Two Degrees of Freedom Analysis Including the Effects of Droplet Formation 332

7.6 Non-linear Waveguide Theory Including the Effects of Droplet Formation 335

7.6.1 Results for the Closed End/Closed End Arrangement 336

7.6.2 Results for the Open End/Closed End Type ofWaveguide Pump 340

References 344

**8 Droplet Flight, Evaporation, Impact, Spreading, Permeation, and Drying 347**

8.1 Introduction 347

8.2 Evaporation of a Free-flying Droplet Exposed to Still Air 348

8.3 Cooling of a Free-flying Droplet During Flight Through Still Air 353

8.4 Deceleration of a Free-flying Droplet due to Air Friction 355

8.5 Spreading 357

8.5.1 Static Spreading 359

8.5.2 Surface-tension-driven Spreading 362

8.5.3 Inertia-controlled Spreading 366

8.6 Permeation into Porous Substrates 389

8.7 Evaporation of Dome-shaped Blobs of Fluid 391

References 393

Appendix A: Solving Algebraic Equations 399

A.1 Second-order Algebraic Equation 399

A.2 Third-order Algebraic Equation 399

A.3 Fourth-order Algebraic Equation 402

References 404

Appendix B: Fourier Decomposition of a Pulse 407

B.1 Pulse with Two Ramps 407

B.2 Exponential Pulse 409

B.3 Pulse with Three Ramps and Two Stationary Levels 413

References 416

Appendix C: Toroidal Co-ordinate System 417

C.1 Introduction 417

C.2 Definition with Respect to Rectangular Co-ordinate System 417

C.3 Scale Factors 417

C.4 Elementary Line Element 418

C.5 Unit Vectors 418

C.6 Nabla Operator ∇ 419

C.7 Gradient of Scalar 419

C.8 Divergence of a Vector Field 419

C.9 Dyadic Product ∇v 420

C.10 Laplacian of Vector Field ∇. ∇v (∇2v) 421

C.11 Indefinite Integrals Involving Hyperbolic Functions 422

References 422

Index 423