Fluid Mechanics Theory and Applications

Prepared by

Dr. Hossam S.S. AbdelMeguid F

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Fluid Mechanics 2 Theory and Applications

Prepared by

Dr. Hossam S.S. AbdelMeguid

2014

Fluid Mechanic-2

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ISBN:

DEPOSIT NUMBER:

All rights reserved.

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Preface

This book evolved from lecture notes developed for teaching this material, in

Fluid Mechanics. The course is taken by graduate students, along with post

graduate students.

Exercises and assignments are an important aspect of any such course and

many have been developed in conjunction with this book. Rather than

lengthening the text, they are available on the book's webpage:

https://sites.google.com/site/hssaleh/

The book is organized into four main parts. Part 1 deals with Introduction in

fluid dynamics, Physical laws in fluid mechanics field, Reynolds transport

theorem, and Analysis of some engineering applications by control volume

method. Part 2 deals with Deducing the Navier-Stocks equations, and the use

of Navier-Stocks equations in solving some engineering application. Part 3

concerns boundary layer theory, its exact solution (Blasuis solution) and its

solution using Von-Karmen equations. Part 4 concerns Potential flow theory,

and its applications.

The emphasis is on building an understanding of the essential ideas that

underlie the development, analysis, and practical use of fluid mechanics

concepts for solving mechanical engineering application. I have tried to give an

indication of some of the more practical aspects. My goal is to form a

foundation from which students can approach the vast literature on more

advanced topics and further explore the theory and/or use of fluid mechanics

theories and equations for engineering application according to their interests

and needs.

I have also been influenced by other books covering these same topics, and

many excellent ones exist at all levels (See References section). Advanced

books go into more detail on countless subjects only briefly discussed here.

There are also a number of general introductory books that may be useful as a

complement to the presentation found here, See the References list at the last

page of this book.

Dr. Hossam S.S. AbdelMeguid,

Jan, 2014

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Contents

PREFACE ................................................................................................................................. 3

1 INTEGRAL FORM FOR A CONTROL VOLUME ANALYSIS ................................................ 11

1.1 INTRODUCTION ............................................................................................................ 11

1.2 BASIC LAWS FOR A SYSTEM ............................................................................................. 12

1.2.1 Conservation of Mass ..................................................................................... 13

1.2.2 Newtons Second Law ..................................................................................... 13

1.2.3 The Angular-Momentum Principle .................................................................. 13

1.2.4 The First Law of Thermodynamics ................................................................... 14

1.2.5 The Second Law of Thermodynamics ............................................................... 15

1.3 CONTROL VOLUME ....................................................................................................... 15

1.4 REYNOLDS TRANSPORT THEOREM .................................................................................... 16

1.4.1 Derivation ...................................................................................................... 18

1.4.2 Physical Interpretation .................................................................................... 21

1.5 CONSERVATION OF MASS ............................................................................................... 23

1.5.1 Non Deformable Control Volume .................................................................... 26

1.5.2 Incompressible Fluids ...................................................................................... 26

1.5.3 Steady Compressible Fluids ............................................................................. 30

1.5.4 Problems (Conservation of Mass) .................................................................... 38

1.6 MOMENTUM EQUATION FOR CONTROL VOLUME ................................................................. 43

1.6.1 Introduction ................................................................................................... 43

1.6.2 External Forces ............................................................................................... 43

1.6.3 Momentum Governing Equation ..................................................................... 45

1.6.4 Differential Control Volume Analysis ............................................................... 60

1.6.5 Moving Control Volume with Constant Velocity ............................................... 67

1.6.6 Linearly Accelerating Control Volume .............................................................. 70

1.6.7 Arbitrary Accelerating Control Volume ............................................................ 81

1.6.8 The Angular-Momentum Principle .................................................................. 87

1.6.9 Problems (Conservation of Momentum) ........................................................ 101

1.7 THE FIRST LAW OF THERMODYNAMICS ............................................................................ 120

1.7.1 Rate of Work Done by a Control Volume ....................................................... 121

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1.7.2 Control Volume Equation .............................................................................. 124

1.7.3 Problems (The 1st Law of Thermodynamics)................................................... 130

1.8 THE SECOND LAW OF THERMODYNAMICS......................................................................... 132

2 DIFFERENTIAL ANALYSIS OF FLUID MOTION .............................................................. 135

2.1 MASS CONSERVATION ................................................................................................. 135

2.1.1 Steady Compressible Flow ............................................................................. 140

2.1.2 Incompressible Flow ..................................................................................... 140

2.1.3 Cylindrical Coordinate System ....................................................................... 144

2.1.4 Problem (Mass Conservation) ....................................................................... 149

2.2 MOTION OF A FLUID PARTICLE....................................................................................... 154

2.2.1 The Acceleration Field of a Fluid .................................................................... 155

2.2.2 Fluid Rotation ............................................................................................... 160

2.2.3 Fluid Deformation ......................................................................................... 164

2.2.4 Problems (Motion of a Fluid Particle) ............................................................ 166

2.3 MOMENTUM EQUATION IN CARTESIAN COORDINATES ........................................................ 171

2.3.1 Newtonian Fluid: Navier-Stokes Equations .................................................... 182

2.3.2 Inviscid Flow: Eulers Equation ...................................................................... 183

2.4 MOMENTUM EQUATION IN CYLINDRICAL COORDINATES....................................................... 185

2.4.1 Newtonian Fluid: Navier-Stokes Equations .................................................... 194

2.4.2 Inviscid Flow: Eulers Equation ...................................................................... 196

2.5 CLASSICAL EXACT SOLUTIONS TO NAVIER-STOKES EQUATIONS ............................................... 197

2.5.1 Couette Flow ................................................................................................ 197

2.5.2 Poiseuille Flow .............................................................................................. 211

2.5.3 Flow between Rotating Cylinders .................................................................. 216

2.5.4 Problems (Momentum Equation) .................................................................. 226

3 BOUNDARY LAYER THEORY ....................................................................................... 237

3.1 THE BOUNDARY-LAYER CONCEPT ................................................................................... 238

3.2 BOUNDARY-LAYER THICKNESSES .................................................................................... 240

3.2.1 The Disturbance Thickness ......................................................................... 240

3.2.2 The Displacement Thickness, * .................................................................... 241

3.2.3 The Momentum Thickness, ........................................................................ 244

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3.3 DERIVATION FOR TWO-DIMENSIONAL FLOW ..................................................................... 246

3.4 LAMINAR FLAT-PLATE BOUNDARY LAYER: EXACT SOLUTION.................................................. 248

3.5 MOMENTUM INTEGRAL EQUATION................................................................................. 255

3.5.1 Momentum Integral Equation for Flow with Zero Pressure Gradient .............. 262

3.5.2 Boundary Layers with Pressure Gradient ....................................................... 279

3.6 PROBLEMS (BOUNDARY LAYER) ..................................................................................... 283

4 POTENTIAL FLOW ...................................................................................................... 293

4.1 INTRODUCTION .......................................................................................................... 293

4.2 IRROTATIONAL FLOW .................................................................................................. 294

4.3 CIRCULATION AND VORTICITY ........................................................................................ 295

4.4 STREAM LINES, STREAM TUBE, PATH, STREAK AND TIME LINES ............................................. 296

4.5 STREAM LINE ............................................................................................................ 298

4.6 STREAM FUNCTION ..................................................................................................... 299

4.7 POTENTIAL FUNCTION ................................................................................................. 301

4.8 ELEMENTARY PLANE-FLOW SOLUTIONS ........................................................................... 304

4.8.1 Rectilinear Flow Field (Uniform Stream) ........................................................ 304

4.8.2 Source Flow .................................................................................................. 306

4.8.3 Sink Flow ...................................................................................................... 308

4.8.4 Irrotational Vortex ........................................................................................ 309

4.8.5 Doublet ........................................................................................................ 311

4.9 SUPERPOSITION OF ELEMENTARY PLANE FLOWS................................................................. 313

4.9.1 Flow Past a Half Body ................................................................................... 314

4.9.2 Source and Sink of Equal Strength ................................................................. 318

4.9.3 Doublet (dipole flow) .................................................................................. 320

4.9.4 Bathtub Vortex (Sink and Vortex) .................................................................. 320

4.9.5 Flow Past a Rankine Body ............................................................................. 322

4.9.6 Flow Around a Circular Cylinder .................................................................... 325

4.9.7 Flow Past a Circular Cylinder with Circulation ................................................ 329

4.10 PROBLEMS ........................................................................................................... 334

5 REFERENCES .............................................................................................................. 347

6 APPENDICES .............................................................................................................. 348

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