This book is an introduction to fundamentals of fluid dynamics and is intended for undergraduate students and also low-level graduate students. The amount of materials in the book is reasonable to cover in one semester. An instructor can choose to skip some parts of the book according to her/his class schedules and the level of students taking this course. Many figures and illustrations are included to facilitate firmer grasp of the physical concepts and the mathematical details addressed. Each chapter contains a number of solved problems. The enclosed solutions are very detailed, in general, so that students understand how each problem is approached. More problems with solutions will be added to the website of this book for instructors.

### Kwang-Yul Kim

Prof. Kwang-Yul Kim received a Ph.D. in physical oceanography from Texas A&M University in 1986.

He taught meteorology and oceanography at Texas A&M University and Florida State University until

he came back to Seoul National University in 2009 as a professor in the School of Earth and Environmental

Sciences. His notable research career includes more than 100 publications in major SCI journals

with over 1,500 citations and H index of 23. He is a member of Phi Kappa Phi Honor Society.

e-mail: kwang56@snu.ac.kr###

He taught meteorology and oceanography at Texas A&M University and Florida State University until

he came back to Seoul National University in 2009 as a professor in the School of Earth and Environmental

Sciences. His notable research career includes more than 100 publications in major SCI journals

with over 1,500 citations and H index of 23. He is a member of Phi Kappa Phi Honor Society.

e-mail: kwang56@snu.ac.kr

Preface v

Chapter 1. Vector Calculus 1

1.1. Review / 2

1.2. The Del (∇)Operator / 3

1.3. Indicial Notation / 4

1.4. The ε–δ Theorem / 7

1.5. Vector Calculus / 8

1.6. Curvilinear Coordinate System / 10

1.7. Integral Theorems / 15

Exercise Problems for Chapter 1 / 20

Solved Problems for Chapter 1 / 22

Chapter 2. Thermodynamic Properties of Fluids 35

2.1. Equation of State / 36

2.2. The First Law of Thermodynamics / 37

2.3. Isentropic Processes / 43

2.4. Compressibility / 48

Exercise Problems for Chapter 2 / 49

Solved Problems for Chapter 2 / 51

Chapter 3. Fluid Statics 63

3.1. Pressure / 64

3.2. Equation of Fluid Statics / 65

3.3. Examples of Hydrostatics / 67

3.4. Hypsometric Equation / 70

3.5. Stability of the Atmospheric Column / 73

Exercise Problems for Chapter 3 / 78

Solved Problems for Chapter 3 / 81

Chapter 4. Fluid Kinematics 97

4.1. Pictorial Method of Description / 98

4.2. Eulerian Representation / 98

4.3. Particle Derivative of the Velocity Vector / 103

4.4. Physical Meaning of Gradient, Divergence and Curl / 106

4.5. Linear and Shear Strain Rates / 110

Exercise Problems for Chapter 4 / 113

Solved Problems for Chapter 4 / 116

Chapter 5. Conservation Laws 131

5.1. Equation of Continuity / 132

5.2. Mass Conservation for Material Volume / 134

5.3. Conservation of Momentum / 135

5.4. Conservation of Vorticity / 137

Exercise Problems for Chapter 5 / 139

Solved Problems for Chapter 5 / 142

Chapter 6. Frictionless Flow 151

6.1. Force / 152

6.2. Equation of Motion / 153

6.3. Euler’s Equation / 155

6.4. Bernoulli Equation / 158

6.5. Equation of Motion in Cylindrical Coordinates / 162

Exercise Problems for Chapter 6 / 166

Solved Problems for Chapter 6 / 169

Chapter 7. Motion with Rotation 181

7.1. Vorticity / 182

7.2. Vortex Motions / 182

7.3. Circulation Theorem / 185

7.4. Vorticity Equation / 187

7.5. Vorticity Equation in Cylindrical Coordinates / 190

Exercise Problems for Chapter 7 / 194

Solved Problems for Chapter 7 / 197

Chapter 8. Velocity Potential and Stream Function 215

8.1. Velocity Potential / 216

8.2. Stream Function / 219

8.3. Vortex Filament and Vortex Tube / 224

8.4. Superposition of Flows / 226

8.5. Complex Potential / 231

Exercise Problems for Chapter 8 / 236

Solved Problems for Chapter 8 / 239

Chapter 9. Viscous Flow 253

9.1. Physical Meaning of Viscous Stress / 254

9.2. Symmetry of Viscous Stress / 255

9.3. Constitutive Equation for a Newtonian Fluid / 257

9.4. Navier-Stokes Equation / 261

9.5. Simple Examples of Viscous Flow / 262

9.6. The Reynolds Number / 269

Exercise Problems for Chapter 9 / 271

Solved Problems for Chapter 9 / 274

Chapter 10. The Effect of the Earth’s Rotation 295

10.1. Motion in a Rotating Frame of Reference / 296

10.2. Fictitious Acceleration in a Rotating Frame / 297

10.3. Equation of Motion in a Rotating Frame / 299

10.4. Vorticity Equation in a Rotating Frame / 300

10.5. The Geostrophic Equation / 304

10.6. Vorticity Conservation on the β-Plane / 306

10.7. The Rossby Number / 309

10.8. The Ekman Number / 310

Exercise Problems for Chapter 10 / 312

Solved Problems for Chapter 10 / 316

Chapter 11. Energy Conservation Principles 333

11.1. Conservation of Mechanical Energy / 334

11.2. Conservation of Internal Energy / 336

11.3. General Relations among Physical Properties / 338

11.4. Entropy Equation and the Second Law / 339

Exercise Problems for Chapter 11 / 342

Solved Problems for Chapter 11 / 345

Appendix 355

A.1. Useful Constants and Parameters / 356

A.2. MKS Units of Physical Quantities / 358

A.3. Equation of Motion in Spherical Coordinates / 359

A.4. Vorticity Equation in Spherical Coordinates / 360

Index 363

Chapter 1. Vector Calculus 1

1.1. Review / 2

1.2. The Del (∇)Operator / 3

1.3. Indicial Notation / 4

1.4. The ε–δ Theorem / 7

1.5. Vector Calculus / 8

1.6. Curvilinear Coordinate System / 10

1.7. Integral Theorems / 15

Exercise Problems for Chapter 1 / 20

Solved Problems for Chapter 1 / 22

Chapter 2. Thermodynamic Properties of Fluids 35

2.1. Equation of State / 36

2.2. The First Law of Thermodynamics / 37

2.3. Isentropic Processes / 43

2.4. Compressibility / 48

Exercise Problems for Chapter 2 / 49

Solved Problems for Chapter 2 / 51

Chapter 3. Fluid Statics 63

3.1. Pressure / 64

3.2. Equation of Fluid Statics / 65

3.3. Examples of Hydrostatics / 67

3.4. Hypsometric Equation / 70

3.5. Stability of the Atmospheric Column / 73

Exercise Problems for Chapter 3 / 78

Solved Problems for Chapter 3 / 81

Chapter 4. Fluid Kinematics 97

4.1. Pictorial Method of Description / 98

4.2. Eulerian Representation / 98

4.3. Particle Derivative of the Velocity Vector / 103

4.4. Physical Meaning of Gradient, Divergence and Curl / 106

4.5. Linear and Shear Strain Rates / 110

Exercise Problems for Chapter 4 / 113

Solved Problems for Chapter 4 / 116

Chapter 5. Conservation Laws 131

5.1. Equation of Continuity / 132

5.2. Mass Conservation for Material Volume / 134

5.3. Conservation of Momentum / 135

5.4. Conservation of Vorticity / 137

Exercise Problems for Chapter 5 / 139

Solved Problems for Chapter 5 / 142

Chapter 6. Frictionless Flow 151

6.1. Force / 152

6.2. Equation of Motion / 153

6.3. Euler’s Equation / 155

6.4. Bernoulli Equation / 158

6.5. Equation of Motion in Cylindrical Coordinates / 162

Exercise Problems for Chapter 6 / 166

Solved Problems for Chapter 6 / 169

Chapter 7. Motion with Rotation 181

7.1. Vorticity / 182

7.2. Vortex Motions / 182

7.3. Circulation Theorem / 185

7.4. Vorticity Equation / 187

7.5. Vorticity Equation in Cylindrical Coordinates / 190

Exercise Problems for Chapter 7 / 194

Solved Problems for Chapter 7 / 197

Chapter 8. Velocity Potential and Stream Function 215

8.1. Velocity Potential / 216

8.2. Stream Function / 219

8.3. Vortex Filament and Vortex Tube / 224

8.4. Superposition of Flows / 226

8.5. Complex Potential / 231

Exercise Problems for Chapter 8 / 236

Solved Problems for Chapter 8 / 239

Chapter 9. Viscous Flow 253

9.1. Physical Meaning of Viscous Stress / 254

9.2. Symmetry of Viscous Stress / 255

9.3. Constitutive Equation for a Newtonian Fluid / 257

9.4. Navier-Stokes Equation / 261

9.5. Simple Examples of Viscous Flow / 262

9.6. The Reynolds Number / 269

Exercise Problems for Chapter 9 / 271

Solved Problems for Chapter 9 / 274

Chapter 10. The Effect of the Earth’s Rotation 295

10.1. Motion in a Rotating Frame of Reference / 296

10.2. Fictitious Acceleration in a Rotating Frame / 297

10.3. Equation of Motion in a Rotating Frame / 299

10.4. Vorticity Equation in a Rotating Frame / 300

10.5. The Geostrophic Equation / 304

10.6. Vorticity Conservation on the β-Plane / 306

10.7. The Rossby Number / 309

10.8. The Ekman Number / 310

Exercise Problems for Chapter 10 / 312

Solved Problems for Chapter 10 / 316

Chapter 11. Energy Conservation Principles 333

11.1. Conservation of Mechanical Energy / 334

11.2. Conservation of Internal Energy / 336

11.3. General Relations among Physical Properties / 338

11.4. Entropy Equation and the Second Law / 339

Exercise Problems for Chapter 11 / 342

Solved Problems for Chapter 11 / 345

Appendix 355

A.1. Useful Constants and Parameters / 356

A.2. MKS Units of Physical Quantities / 358

A.3. Equation of Motion in Spherical Coordinates / 359

A.4. Vorticity Equation in Spherical Coordinates / 360

Index 363