fluidMechanics

Basics of Fluid Mechanics

Genick Bar–Meir, Ph. D.
7449 North Washtenaw Ave
Chicago, IL 60645
email:genick at potto.org

Copyright © 2013, 2011, 2010, 2009, 2008, 2007, and 2006 by Genick Bar-Meir
See the file copying.fdl or copyright.tex for copying conditions.
Version (0.3.4.0

July 25, 2013)

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How to cite this book:
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‘We are like dwarfs sitting on the shoulders of giants”

from The Metalogicon by John in 1159

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CONTENTS

Nomenclature
GNU Free Documentation License . . . . . . . . . . . . . . . .
1. APPLICABILITY AND DEFINITIONS . . . . . . . . .
2. VERBATIM COPYING . . . . . . . . . . . . . . . . . .
3. COPYING IN QUANTITY . . . . . . . . . . . . . . . .
4. MODIFICATIONS . . . . . . . . . . . . . . . . . . . .
5. COMBINING DOCUMENTS . . . . . . . . . . . . . .
6. COLLECTIONS OF DOCUMENTS . . . . . . . . . . .
7. AGGREGATION WITH INDEPENDENT WORKS . . .
8. TRANSLATION . . . . . . . . . . . . . . . . . . . . .
9. TERMINATION . . . . . . . . . . . . . . . . . . . . .
10. FUTURE REVISIONS OF THIS LICENSE . . . . . . .
ADDENDUM: How to use this License for your documents
How to contribute to this book . . . . . . . . . . . . . . . . .

Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Steven from artofproblemsolving.com . . . . . . . . . . .
Dan H. Olson . . . . . . . . . . . . . . . . . . . . . . . .
Richard Hackbarth . . . . . . . . . . . . . . . . . . . . . .
John Herbolenes . . . . . . . . . . . . . . . . . . . . . . .
Eliezer Bar-Meir . . . . . . . . . . . . . . . . . . . . . .
Henry Schoumertate . . . . . . . . . . . . . . . . . . . . .
Your name here . . . . . . . . . . . . . . . . . . . . . . .
Typo corrections and other ”minor” contributions . . . . .
Version 0.3.2.0 March 18, 2013 . . . . . . . . . . . . . . . . . .
pages 617 size 4.8M . . . . . . . . . . . . . . . . . . . . .
Version 0.3.0.5 March 1, 2011 . . . . . . . . . . . . . . . . . .

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vi

CONTENTS
pages 400 size 3.5M . .
Version 0.1.8 August 6, 2008
pages 189 size 2.6M . .
Version 0.1 April 22, 2008 . .
pages 151 size 1.3M . .
Properties . . . . . . .
Open Channel Flow . .

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liii
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lxi

1 Introduction to Fluid Mechanics
1.1 What is Fluid Mechanics? . . . . .
1.2 Brief History . . . . . . . . . . . .
1.3 Kinds of Fluids . . . . . . . . . . .
1.4 Shear Stress . . . . . . . . . . . .
1.5 ViscosityViscosity . . . . . . . . .
1.5.1 General . . . . . . . . . .
1.5.2 Non–Newtonian Fluids . .
1.5.3 Kinematic Viscosity . . . .
1.5.4 Estimation of The Viscosity
1.6 Fluid Properties . . . . . . . . . .
1.6.1 Fluid Density . . . . . . .
1.6.2 Bulk Modulus . . . . . . .
1.7 Surface Tension . . . . . . . . . .
1.7.1 Wetting of Surfaces . . . .

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1
1
3
5
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9
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24
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35

2 Review of Thermodynamics
2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45
45

3 Review of Mechanics
3.1 Kinematics of of Point Body . . . . .
3.2 Center of Mass . . . . . . . . . . . .
3.2.1 Actual Center of Mass . . . .
3.2.2 Aproximate Center of Area . .
3.3 Moment of Inertia . . . . . . . . . . .
3.3.1 Moment of Inertia for Mass . .
3.3.2 Moment of Inertia for Area . .
3.3.3 Examples of Moment of Inertia
3.3.4 Product of Inertia . . . . . . .
3.3.5 Principal Axes of Inertia . . . .
3.4 Newton’s Laws of Motion . . . . . . .
3.5 Angular Momentum and Torque . . .
3.5.1 Tables of geometries . . . . .

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CONTENTS

vii

4 Fluids Statics
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 The Hydrostatic Equation . . . . . . . . . . . . . . . . . . .
4.3 Pressure and Density in a Gravitational Field . . . . . . . . .
4.3.1 Constant Density in Gravitational Field . . . . . . . .
4.3.2 Pressure Measurement . . . . . . . . . . . . . . . .
4.3.3 Varying Density in a Gravity Field . . . . . . . . . .
4.3.4 The Pressure Effects Due To Temperature Variations
4.3.5 Gravity Variations Effects on Pressure and Density .
4.3.6 Liquid Phase . . . . . . . . . . . . . . . . . . . . . .
4.4 Fluid in a Accelerated System . . . . . . . . . . . . . . . . .
4.4.1 Fluid in a Linearly Accelerated System . . . . . . . .
4.4.2 Angular Acceleration Systems: Constant Density . .
4.4.3 Fluid Statics in Geological System . . . . . . . . . .
4.5 Fluid Forces on Surfaces . . . . . . . . . . . . . . . . . . . .
4.5.1 Fluid Forces on Straight Surfaces . . . . . . . . . . .
4.5.2 Forces on Curved Surfaces . . . . . . . . . . . . . .
4.6 Buoyancy and Stability . . . . . . . . . . . . . . . . . . . .
4.6.1 Stability . . . . . . . . . . . . . . . . . . . . . . . .
4.6.2 Surface Tension . . . . . . . . . . . . . . . . . . . .
4.7 Rayleigh–Taylor Instability . . . . . . . . . . . . . . . . . . .
4.8 Qualitative questions . . . . . . . . . . . . . . . . . . . . .

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Integral Analysis

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126
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139
143

145

5 Mass Conservation
5.1 Introduction . . . . . . . . . . . . . . . . . . . .
5.2 Control Volume . . . . . . . . . . . . . . . . . .
5.3 Continuity Equation . . . . . . . . . . . . . . . .
5.3.1 Non Deformable Control Volume . . . . .
5.3.2 Constant Density Fluids . . . . . . . . . .
5.4 Reynolds Transport Theorem . . . . . . . . . . .
5.5 Examples For Mass Conservation . . . . . . . . .
5.6 The Details Picture – Velocity Area Relationship
5.7 More Examples for Mass Conservation . . . . . .

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147
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151
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169

6 Momentum Conservation
6.1 Momentum Governing Equation . . . . . . . . . . . .
6.1.1 Introduction to Continuous . . . . . . . . . . .
6.1.2 External Forces . . . . . . . . . . . . . . . . .
6.1.3 Momentum Governing Equation . . . . . . . .
6.1.4 Momentum Equation in Acceleration System .
6.1.5 Momentum For Steady State and Uniform Flow
6.2 Momentum Equation Application . . . . . . . . . . . .

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173
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180

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CONTENTS
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183
183
190
192
194

7 Energy Conservation
7.1 The First Law of Thermodynamics . . . . . . . . . . . . . . .
7.2 Limitation of Integral Approach . . . . . . . . . . . . . . . . .
7.3 Approximation of Energy Equation . . . . . . . . . . . . . . .
7.3.1 Energy Equation in Steady State . . . . . . . . . . . .
7.3.2 Energy Equation in Frictionless Flow and Steady State
7.4 Energy Equation in Accelerated System . . . . . . . . . . . .
7.4.1 Energy in Linear Acceleration Coordinate . . . . . . .
7.4.2 Linear Accelerated System . . . . . . . . . . . . . . .
7.4.3 Energy Equation in Rotating Coordinate System . . . .
7.4.4 Simplified Energy Equation in Accelerated Coordinate .
7.4.5 Energy Losses in Incompressible Flow . . . . . . . . .
7.5 Examples of Integral Energy Conservation . . . . . . . . . . .

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6.3
6.4

II

6.2.1 Momentum for Unsteady State and Uniform
6.2.2 Momentum Application to Unsteady State .
Conservation Moment Of Momentum . . . . . . .
More Examples on Momentum Conservation . . . .
6.4.1 Qualitative Questions . . . . . . . . . . . .

Flow

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Differential Analysis

225

8 Differential Analysis
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.1 Mass Conservation Examples . . . . . . . . . . . . . . .
8.2.2 Simplified Continuity Equation . . . . . . . . . . . . . .
8.3 Conservation of General Quantity . . . . . . . . . . . . . . . . .
8.3.1 Generalization of Mathematical Approach for Derivations
8.3.2 Examples of Several Quantities . . . . . . . . . . . . . .
8.4 Momentum Conservation . . . . . . . . . . . . . . . . . . . . .

8.5 Derivations of the Momentum Equation . . . . . . . . . . . . .
8.6 Boundary Conditions and Driving Forces . . . . . . . . . . . . .
8.6.1 Boundary Conditions Categories . . . . . . . . . . . . .
8.7 Examples for Differential Equation (Navier-Stokes) . . . . . . .
8.7.1 Interfacial Instability . . . . . . . . . . . . . . . . . . . .
9 Dimensional Analysis
9.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . .
9.1.1 Brief History . . . . . . . . . . . . . . . . . . . . . .
9.1.2 Theory Behind Dimensional Analysis . . . . . . . . .
9.1.3 Dimensional Parameters Application for Experimental
9.1.4 The Pendulum Class Problem . . . . . . . . . . . . .
9.2 Buckingham–π–Theorem . . . . . . . . . . . . . . . . . . .

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227
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CONTENTS

9.3
9.4

9.5
9.6

ix

9.2.1 Construction of the Dimensionless Parameters . . . . . . . . .
9.2.2 Basic Units Blocks . . . . . . . . . . . . . . . . . . . . . . .
9.2.3 Implementation of Construction of Dimensionless Parameters .
9.2.4 Similarity and Similitude . . . . . . . . . . . . . . . . . . . .
Nusselt’s Technique . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary of Dimensionless Numbers . . . . . . . . . . . . . . . . . .
9.4.1 The Significance of these Dimensionless Numbers . . . . . . .
9.4.2 Relationship Between Dimensionless Numbers . . . . . . . . .
9.4.3 Examples for Dimensional Analysis . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix summary of Dimensionless Form of Navier–Stokes Equations

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281
282
285
294
298
308
312
315
316
319
319

10 Potential Flow
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1.1 Inviscid Momentum Equations . . . . . . . . . . . . . . . . . .
10.2 Potential Flow Function . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.1 Streamline and Stream function . . . . . . . . . . . . . . . . .
10.2.2 Compressible Flow Stream Function . . . . . . . . . . . . . . .
10.2.3 The Connection Between the Stream Function and the Potential
10.3 Potential Flow Functions Inventory . . . . . . . . . . . . . . . . . . . .
10.3.1 Flow Around a Circular Cylinder . . . . . . . . . . . . . . . . .
10.4 Conforming Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.1 Complex Potential and Complex Velocity . . . . . . . . . . . .
10.5 Unsteady State Bernoulli in Accelerated Coordinates . . . . . . . . . .
10.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

325
325
326
332

333
336
Function338
342
357
369
369
373
373

11 Compressible Flow One Dimensional
11.1 What is Compressible Flow? . . . . . . . . . . . . . . . . . .
11.2 Why Compressible Flow is Important? . . . . . . . . . . . . .
11.3 Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
11.3.2 Speed of Sound in Ideal and Perfect Gases . . . . . . .
11.3.3 Speed of Sound in Almost Incompressible Liquid . . . .
11.3.4 Speed of Sound in Solids . . . . . . . . . . . . . . . .
11.3.5 The Dimensional Effect of the Speed of Sound . . . .
11.4 Isentropic Flow . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.1 Stagnation State for Ideal Gas Model . . . . . . . . .
11.4.2 Isentropic Converging-Diverging Flow in Cross Section
11.4.3 The Properties in the Adiabatic Nozzle . . . . . . . . .
11.4.4 Isentropic Flow Examples . . . . . . . . . . . . . . . .
11.4.5 Mass Flow Rate (Number) . . . . . . . . . . . . . . .
11.4.6 Isentropic Tables . . . . . . . . . . . . . . . . . . . .
11.4.7 The Impulse Function . . . . . . . . . . . . . . . . . .
11.5 Normal Shock . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5.1 Solution of the Governing Equations . . . . . . . . . .

377
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377
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378
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381
382
382
384
384
386
387
391
394
401
403
406
408

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x

CONTENTS
11.5.2 Prandtl’s Condition . . . . . . . . . . . . . . . . . . . .
11.5.3 Operating Equations and Analysis . . . . . . . . . . . .
11.5.4 The Moving Shocks . . . . . . . . . . . . . . . . . . . .
11.5.5 Shock or Wave Drag Result from a Moving Shock . . . .
11.5.6 Tables of Normal Shocks, k = 1.4 Ideal Gas . . . . . . .
11.6 Isothermal Flow . . . . . . . . . . . . . . . . . . . . . . . . . .
11.6.1 The Control Volume Analysis/Governing equations . . .
11.6.2 Dimensionless Representation . . . . . . . . . . . . . .
11.6.3 The Entrance Limitation of Supersonic Branch . . . . .
11.6.4 Supersonic Branch . . . . . . . . . . . . . . . . . . . .
11.6.5 Figures and Tables . . . . . . . . . . . . . . . . . . . .
11.6.6 Isothermal Flow Examples . . . . . . . . . . . . . . . . .
11.7 Fanno Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
11.7.2 Non–Dimensionalization of the Equations . . . . . . . .
11.7.3 The Mechanics and Why the Flow is Choked? . . . . . .
11.7.4 The Working Equations . . . . . . . . . . . . . . . . . .
11.7.5 Examples of Fanno Flow . . . . . . . . . . . . . . . . .
11.7.6 Working Conditions . . . . . . . . . . . . . . . . . . . .
11.7.7 The Pressure Ratio, P2 / P1 , effects . . . . . . . . . . .
11.7.8 Practical Examples for Subsonic Flow . . . . . . . . . .
fL
11.7.9 Subsonic Fanno Flow for Given 4 D
and Pressure Ratio
11.7.10 Subsonic Fanno Flow for a Given M1 and Pressure Ratio
11.7.11 More Examples of Fanno Flow . . . . . . . . . . . . . .
11.8 The Table for Fanno Flow . . . . . . . . . . . . . . . . . . . .
11.9 Rayleigh Flow . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.10Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11.10.1 Governing Equations . . . . . . . . . . . . . . . . . . .
11.10.2 Rayleigh Flow Tables and Figures . . . . . . . . . . . . .
11.10.3 Examples For Rayleigh Flow . . . . . . . . . . . . . . .

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411
413
414
416
418
421
421
422
426
428
429
430
436
436
438
441
442
445
451
456
463
463

466
468
469
471
471
472
475
478

12 Compressible Flow 2–Dimensional
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.1 Preface to Oblique Shock . . . . . . . . . . . . . . . . . .
12.2 Oblique Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.1 Solution of Mach Angle . . . . . . . . . . . . . . . . . . .
12.2.2 When No Oblique Shock Exist or the case of D > 0 . . .
12.2.3 Application of Oblique Shock . . . . . . . . . . . . . . . .
12.3 Prandtl-Meyer Function . . . . . . . . . . . . . . . . . . . . . . .
12.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
12.3.2 Geometrical Explanation . . . . . . . . . . . . . . . . . .
12.3.3 Alternative Approach to Governing Equations . . . . . . .
12.3.4 Comparison And Limitations between the Two Approaches
12.4 The Maximum Turning Angle . . . . . . . . . . . . . . . . . . . .

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485
485
485
487
489
492
508
520
520
521
522
525
526

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CONTENTS
12.5
12.6
12.7
12.8
12.9

xi

The Working Equations for the Prandtl-Meyer Function . . .
d’Alembert’s Paradox . . . . . . . . . . . . . . . . . . . . .

Flat Body with an Angle of Attack . . . . . . . . . . . . . .
Examples For Prandtl–Meyer Function . . . . . . . . . . . .
Combination of the Oblique Shock and Isentropic Expansion

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526
526
527
527
530

13 Multi–Phase Flow
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3 What to Expect From This Chapter . . . . . . . . . . . . .
13.4 Kind of Multi-Phase Flow . . . . . . . . . . . . . . . . . . .
13.5 Classification of Liquid-Liquid Flow Regimes . . . . . . . . .
13.5.1 Co–Current Flow . . . . . . . . . . . . . . . . . . .
13.6 Multi–Phase Flow Variables Definitions . . . . . . . . . . . .
13.6.1 Multi–Phase Averaged Variables Definitions . . . . .
13.7 Homogeneous Models . . . . . . . . . . . . . . . . . . . . .
13.7.1 Pressure Loss Components . . . . . . . . . . . . . .
13.7.2 Lockhart Martinelli Model . . . . . . . . . . . . . .
13.8 Solid–Liquid Flow . . . . . . . . . . . . . . . . . . . . . . .
13.8.1 Solid Particles with Heavier Density ρS > ρL . . . .

13.8.2 Solid With Lighter Density ρS < ρ and With Gravity
13.9 Counter–Current Flow . . . . . . . . . . . . . . . . . . . . .
13.9.1 Horizontal Counter–Current Flow . . . . . . . . . . .
13.9.2 Flooding and Reversal Flow . . . . . . . . . . . . . .
13.10Multi–Phase Conclusion . . . . . . . . . . . . . . . . . . . .

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535
535
535
536
537
538
539
543
544
547
548
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552
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555
557
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567
567
568
570
572
578
578
579
581
584
586
589
591
593
593
594
595

A Mathematics For Fluid Mechanics
A.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1.1 Vector Algebra . . . . . . . . . . . . . . . . . .
A.1.2 Differential Operators of Vectors . . . . . . . .
A.1.3 Differentiation of the Vector Operations . . . .

A.2 Ordinary Differential Equations (ODE) . . . . . . . . .
A.2.1 First Order Differential Equations . . . . . . . .
A.2.2 Variables Separation or Segregation . . . . . .
A.2.3 Non–Linear Equations . . . . . . . . . . . . . .
A.2.4 Second Order Differential Equations . . . . . .
A.2.5 Non–Linear Second Order Equations . . . . . .
A.2.6 Third Order Differential Equation . . . . . . .
A.2.7 Forth and Higher Order ODE . . . . . . . . . .
A.2.8 A general Form of the Homogeneous Equation
A.3 Partial Differential Equations . . . . . . . . . . . . . .
A.3.1 First-order equations . . . . . . . . . . . . . .
A.4 Trigonometry . . . . . . . . . . . . . . . . . . . . . .

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xii

CONTENTS

Index
597
Subjects Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
Authors Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

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LIST OF FIGURES

1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14

1.15
1.16
1.17
1.18
1.19
1.20
1.21
1.22
1.23
1.24
1.25

Diagram to explain fluid mechanics branches . . . . . . .
Density as a function of the size of sample . . . . . . . .
Schematics to describe the shear stress in fluid mechanics
The deformation of fluid due to shear stress . . . . . . .
The difference of power fluids . . . . . . . . . . . . . . .
Nitrogen and Argon viscosity. . . . . . . . . . . . . . .
The shear stress as a function of the shear rate . . . . .
Air viscosity as a function of the temperature . . . . . .
Water viscosity as a function temperature. . . . . . . . .
Liquid metals viscosity as a function of the temperature .
Reduced viscosity as function of the reduced temperature
Reduced viscosity as function of the reduced temperature
Concentrating cylinders with the rotating inner cylinder .
Rotating disc in a steady state . . . . . . . . . . . . . .
Water density as a function of temperature . . . . . . .
Two liquid layers under pressure . . . . . . . . . . . . .
Surface tension control volume analysis . . . . . . . . .
Surface tension erroneous explanation . . . . . . . . . .

Glass tube inserted into mercury . . . . . . . . . . . . .
Capillary rise between two plates . . . . . . . . . . . . .
Forces in Contact angle . . . . . . . . . . . . . . . . . .
Description of wetting and non–wetting fluids . . . . . .
Description of the liquid surface . . . . . . . . . . . . .
The raising height as a function of the radii . . . . . . .
The raising height as a function of the radius . . . . . .

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2
6
6
7
9
10
10
11

12
13
17
18
20
21
22
27
30
31
32
34
35
35
37
40
40

3.1

Description of the extinguish nozzle . . . . . . . . . . . . . . . . . . .

54

xiii

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xiv

LIST OF FIGURES
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13

Description of how the center of mass is calculated . . . . . . . .

Thin body center of mass/area schematic. . . . . . . . . . . . . .
The schematic that explains the summation of moment of inertia.
The schematic to explain the summation of moment of inertia. . .
Cylinder with an element for calculation moment of inertia . . . .
Description of rectangular in x–y plane. . . . . . . . . . . . . . .
A square element for the calculations of inertia. . . . . . . . . . .
The ratio of the moment of inertia 2D to 3D. . . . . . . . . . . .
Moment of inertia for rectangular . . . . . . . . . . . . . . . . . .
Description of parabola - moment of inertia and center of area . .
Triangle for example 3.7 . . . . . . . . . . . . . . . . . . . . . . .
Product of inertia for triangle . . . . . . . . . . . . . . . . . . . .

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55
56
57
58
59
59
60
60
61

61
62
64

4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19
4.20
4.21
4.22
4.23
4.24
4.25
4.26

4.27
4.28
4.29
4.30
4.31

Description of a fluid element in accelerated system. . . . . .
Pressure lines in a static constant density fluid . . . . . . . . .
A schematic to explain the atmospheric pressure measurement
The effective gravity is for accelerated cart . . . . . . . . . . .
Tank and the effects different liquids . . . . . . . . . . . . .
Schematic of gas measurement utilizing the “U” tube . . . . .
Schematic of sensitive measurement device . . . . . . . . . . .
Inclined manometer . . . . . . . . . . . . . . . . . . . . . . .
Inverted manometer . . . . . . . . . . . . . . . . . . . . . .
Hydrostatic pressure under a compressible liquid phase . . . .
Two adjoin layers for stability analysis . . . . . . . . . . . . .
The varying gravity effects on density and pressure . . . . . .
The effective gravity is for accelerated cart . . . . . . . . . . .
A cart slide on inclined plane . . . . . . . . . . . . . . . . . .
Forces diagram of cart sliding on inclined plane . . . . . . . .
Schematic to explain the angular angle . . . . . . . . . . . . .
Schematic angular angle to explain example 4.11 . . . . . . .
Earth layers not to scale . . . . . . . . . . . . . . . . . . . . .
Illustration of the effects of the different radii . . . . . . . . .
Rectangular area under pressure . . . . . . . . . . . . . . . .
Schematic of submerged area . . . . . . . . . . . . . . . . . .
The general forces acting on submerged area . . . . . . . . . .
The general forces acting on non symmetrical straight area . .
The general forces acting on a non symmetrical straight area .

The effects of multi layers density on static forces . . . . . . .
The forces on curved area . . . . . . . . . . . . . . . . . . . .
Schematic of Net Force on floating body . . . . . . . . . . . .
Circular shape Dam . . . . . . . . . . . . . . . . . . . . . . .
Area above the dam arc subtract triangle . . . . . . . . . . .
Area above the dam arc calculation for the center . . . . . . .
Moment on arc element around Point “O” . . . . . . . . . . .

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69
71
72
73
74
76
77
78
79
82
88
90
93
94
95
95

96
97
98
100
101
102
104
105
108
109
110
111
112
113
113

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LIST OF FIGURES

xv

4.32
4.33
4.34

4.35
4.36
4.37
4.38
4.39
4.40
4.41
4.42
4.43
4.44
4.45
4.46
4.47
4.48
4.49
4.50
4.51

Polynomial shape dam description . . . . . . . . . . . . . . . . .
The difference between the slop and the direction angle . . . . . .
Schematic of Immersed Cylinder . . . . . . . . . . . . . . . . . .
The floating forces on Immersed Cylinder . . . . . . . . . . . . .
Schematic of a thin wall floating body . . . . . . . . . . . . . . .
Schematic of floating bodies . . . . . . . . . . . . . . . . . . . .
Schematic of floating cubic . . . . . . . . . . . . . . . . . . . . .
Stability analysis of floating body . . . . . . . . . . . . . . . . . .
Cubic body dimensions for stability analysis . . . . . . . . . . . .
Stability of cubic body infinity long . . . . . . . . . . . . . . . . .
The maximum height reverse as a function of density ratio . . . .
Stability of two triangles put tougher . . . . . . . . . . . . . . . .

The effects of liquid movement on the GM . . . . . . . . . . . .
Measurement of GM of floating body . . . . . . . . . . . . . . . .
Calculations of GM for abrupt shape body . . . . . . . . . . . . .
A heavy needle is floating on a liquid. . . . . . . . . . . . . . . .
Description of depression to explain the Rayleigh–Taylor instability
Description of depression to explain the instability . . . . . . . . .
The cross section of the interface for max liquid. . . . . . . . . .
Three liquids layers under rotation . . . . . . . . . . . . . . . . .

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115
115
117
118
118
126
127
127
130
131
131
132
134
135
136
138
139
141

142
143

5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12

Control volume and system in motion . . . . . . . . . .
Piston control volume . . . . . . . . . . . . . . . . . . .
Schematics of velocities at the interface . . . . . . . . .
Schematics of flow in a pipe with varying density . . . .
Filling of the bucket and choices of the control volumes .
Height of the liquid for example 5.4 . . . . . . . . . . .
Boundary Layer control mass . . . . . . . . . . . . . . .
Control volume usage to calculate local averaged velocity
Control volume and system in the motion . . . . . . . .
Circular cross section for finding Ux . . . . . . . . . . .
Velocity for a circular shape . . . . . . . . . . . . . . . .
Boat for example 5.14 . . . . . . . . . . . . . . . . . .

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147
148
149
150
153
156
161
166
167
168
169
169

6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11

The explanation for the direction relative to surface . . . . . . .

Schematics of area impinged by a jet . . . . . . . . . . . . . . .
Nozzle schematic for forces calculations . . . . . . . . . . . . .
Propeller schematic to explain the change of momentum . . . .
Toy Sled pushed by the liquid jet . . . . . . . . . . . . . . . . .
A rocket with a moving control volume . . . . . . . . . . . . . .
Schematic of a tank seating on wheels . . . . . . . . . . . . . .
A new control volume to find the velocity in discharge tank . . .
The impeller of the centrifugal pump and the velocities diagram
Nozzle schematics water rocket . . . . . . . . . . . . . . . . . .
Flow out of un symmetrical tank . . . . . . . . . . . . . . . . .

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174

177
179
181
182
183
185
186
191
192
195

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xvi

LIST OF FIGURES
6.12 The explanation for the direction relative to surface . . . . . . . . . . . 196
7.1
7.2
7.3

7.4

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198
200
202
210
210
210
210
210
218
220
221

8.12
8.13
8.14
8.15
8.16
8.17
8.18
8.19
8.20
8.21

The mass balance on the infinitesimal control volume . . . . . . . .
The mass conservation in cylindrical coordinates . . . . . . . . . . .
Mass flow due to temperature difference . . . . . . . . . . . . . . .
Mass flow in coating process . . . . . . . . . . . . . . . . . . . . .
Stress diagram on a tetrahedron shape . . . . . . . . . . . . . . . .
Diagram to analysis the shear stress tensor . . . . . . . . . . . . . .
The shear stress creating torque . . . . . . . . . . . . . . . . . . .
The shear stress at different surfaces . . . . . . . . . . . . . . . . .
Control volume at t and t + dt under continuous angle deformation
Shear stress at two coordinates in 45◦ orientations . . . . . . . . . .
Different rectangles deformations . . . . . . . . . . . . . . . . . . .
(a) Deformations of the isosceles triangular . . . . . . . . . . . .
(b) Deformations of the straight angle triangle . . . . . . . . . .
Linear strain of the element . . . . . . . . . . . . . . . . . . . . . .
1–Dimensional free surface . . . . . . . . . . . . . . . . . . . . . .
Flow driven by surface tension . . . . . . . . . . . . . . . . . . . .
Flow in kendle with a surfece tension gradient . . . . . . . . . . . .
Flow between two plates when the top moving . . . . . . . . . . . .
One dimensional flow with shear between plates . . . . . . . . . . .
The control volume of liquid element in “short cut” . . . . . . . . .
Flow of Liquid between concentric cylinders . . . . . . . . . . . . .
Mass flow due to temperature difference . . . . . . . . . . . . . . .
Liquid flow due to gravity . . . . . . . . . . . . . . . . . . . . . . .

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228
230
232
234
241
243
243
245
247
248
249
249
249
251
256
259
259

260
261
262
264
267
269

9.1
9.2
9.3
9.4

Fitting rod into a hole . . . . . . . . .
Pendulum for dimensional analysis . .
Resistance of infinite cylinder . . . . .
Oscillating Von Karman Vortex Street

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278
279
285

312

7.5
7.6
7.7
7.8
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
8.11

The work on the control volume . . . . . . . . . .
Discharge from a Large Container . . . . . . . . .
Kinetic Energy and Averaged Velocity . . . . . . .
Typical resistance for selected outlet configuration .
(a) Projecting pipe K= 1 . . . . . . . . . . . . .
(b) Sharp edge pipe connection K=0.5 . . . . . .
(c) Rounded inlet pipe K=0.04 . . . . . . . . . .
Flow in an oscillating manometer . . . . . . . . . .
A long pipe exposed to a sudden pressure difference
Liquid exiting a large tank trough a long tube . . .
Tank control volume for Example 7.2 . . . . . . .

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10.1 Streamlines to explain stream function . . . . . . . . . . . . . . . . . . 334
10.2 Streamlines with different different element direction to explain stream function335

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LIST OF FIGURES

xvii

(a) Streamlines with element in X direction to explain stream function 335
(b) Streamlines with element in the Y direction to explain stream function335
10.3 Constant Stream lines and Constant Potential lines . . . . . . . . . . . 339
10.4 Stream lines and potential lines are drawn as drawn for two dimensional flow.340
10.5 Stream lines and potential lines for Example 10.3 . . . . . . . . . . . . 341
10.6 Uniform Flow Streamlines and Potential Lines . . . . . . . . . . . . . . 343
10.7 Streamlines and Potential lines due to Source or sink . . . . . . . . . . 344
10.8 Vortex free flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
10.9 Circulation path to illustrate varies calculations . . . . . . . . . . . . . 347
10.10Combination of the Source and Sink . . . . . . . . . . . . . . . . . . . 350
10.11Stream and Potential line for a source and sink . . . . . . . . . . . . . 352
10.12Stream and potential lines for doublet . . . . . . . . . . . . . . . . . . 358
10.13Stream function of uniform flow plus doublet . . . . . . . . . . . . . . 360
10.14Source in the Uniform Flow . . . . . . . . . . . . . . . . . . . . . . . . 361
10.15Velocity field around a doublet in uniform velocity . . . . . . . . . . . . 362
10.16Doublet in a uniform flow with Vortex in various conditions. . . . . . . 366
(a) Streamlines of doublet in uniform field with Vortex . . . . . . . . 366
(b) Boundary case for streamlines of doublet in uniform field with Vortex366
10.17Schematic to explain Magnus’s effect . . . . . . . . . . . . . . . . . . . 368
10.18Wing in a typical uniform flow . . . . . . . . . . . . . . . . . . . . . . 368
11.1 A very slow moving piston in a still gas . . . . . . . . . . . . . . .
11.2 Stationary sound wave and gas moves relative to the pulse . . . .
11.3 Moving object at three relative velocities . . . . . . . . . . . . .
(a) Object travels at 0.005 of the speed of sound . . . . . . . .

(b) Object travels at 0.05 of the speed of sound . . . . . . . . .
(c) Object travels at 0.15 of the speed of sound . . . . . . . . .
11.4 Flow through a converging diverging nozzle . . . . . . . . . . . .
11.5 Perfect gas flows through a tube . . . . . . . . . . . . . . . . . .
11.7 Control volume inside a converging-diverging nozzle. . . . . . . .
11.6 Station properties as f (M ) . . . . . . . . . . . . . . . . . . . . .
11.8 The relationship between the cross section and the Mach number
11.9 Schematic to explain the significances of the Impulse function . .
11.10Schematic of a flow through a nozzle example (??) . . . . . . . .
11.11A shock wave inside a tube . . . . . . . . . . . . . . . . . . . . .
11.12The Mexit and P0 as a function Mupstream . . . . . . . . . . . .
11.13The ratios of the static properties of the two sides of the shock. .
11.14Stationary and moving coordinates for the moving shock . . . . .
(a) Stationary coordinates . . . . . . . . . . . . . . . . . . . .
(b) Moving coordinates . . . . . . . . . . . . . . . . . . . . . .
11.15The shock drag diagram for moving shock . . . . . . . . . . . . .
11.16The diagram for the common explanation for shock drag . . . . .
11.17Control volume for isothermal flow . . . . . . . . . . . . . . . . .
11.18Working relationships for isothermal flow . . . . . . . . . . . . . .

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378
378
383
383
383
383
384
385
386
387
391
403
405
406
412
413
415
415
415
416
417
421
427


xviii

LIST OF FIGURES

11.19Control volume of the gas flow in a constant cross section for Fanno
11.20Various parameters in fanno flow . . . . . . . . . . . . . . . . . . .
11.21Schematic of Example 11.18 . . . . . . . . . . . . . . . . . . . . .
11.22The schematic of Example (11.19) . . . . . . . . . . . . . . . . . .
fL
11.23The effects of increase of 4 D
on the Fanno line . . . . . . . . . .
4f L
11.24The effects of the increase of D on the Fanno Line . . . . . . . .
11.25Min and m
˙ as a function of the 4fDL . . . . . . . . . . . . . . . . .
11.26M1 as a function M2 for various 4fDL . . . . . . . . . . . . . . . . .
11.27 M1 as a function M2 . . . . . . . . . . . . . . . . . . . . . . . . .
fL
11.28 The pressure distribution as a function of 4 D
. . . . . . . . . . .
4f L
11.29Pressure as a function of long D . . . . . . . . . . . . . . . . . .
11.30 The effects of pressure variations on Mach number profile . . . . .
fL
fL
11.31 Pressure ratios as a function of 4 D
when the total 4 D
= 0.3 . .
11.32 The maximum entrance Mach number as a function of 4fDL . . .
11.33 Unchoked flow showing the hypothetical “full” tube . . . . . . . .
fL

11.34Pressure ratio obtained for fix 4 D
for k=1.4 . . . . . . . . . . . .
4f L
11.35Conversion of solution for given
= 0.5 and pressure ratio . . .
D
11.36 The results of the algorithm showing the conversion rate . . . . . .
11.37The control volume of Rayleigh Flow . . . . . . . . . . . . . . . . .
11.38The temperature entropy diagram for Rayleigh line . . . . . . . . .
11.39The basic functions of Rayleigh Flow (k=1.4) . . . . . . . . . . . .
11.40Schematic of the combustion chamber . . . . . . . . . . . . . . . .

Flow436
. . 445
. . 445
. . 447
. . 451
. . 452
. . 452
. . 454
. . 455
. . 456
. . 457
. . 458
. . 459
. . 460
. . 463
. . 464
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465
467
471
473
478
483

12.1 A view of a normal shock as a limited case for oblique shock
12.2 The oblique shock or Prandtl–Meyer function regions . . . .
12.3 A typical oblique shock schematic . . . . . . . . . . . . . .
12.4 Flow around spherically blunted 30◦ cone-cylinder . . . . . .
12.5 The different views of a large inclination angle . . . . . . . .
12.6 The three different Mach numbers . . . . . . . . . . . . . .
12.7 The “imaginary” Mach waves at zero inclination . . . . . . .
12.8 The possible range of solutions . . . . . . . . . . . . . . .
12.9 Two dimensional wedge . . . . . . . . . . . . . . . . . . . .
12.10 A local and a far view of the oblique shock. . . . . . . . .
12.11 Oblique shock around a cone . . . . . . . . . . . . . . . .

12.12 Maximum values of the properties in an oblique shock . . .
12.13 Two variations of inlet suction for supersonic flow . . . . .
12.14Schematic for Example (12.5) . . . . . . . . . . . . . . . . .
12.15Schematic for Example (12.6) . . . . . . . . . . . . . . . . .
12.16 Schematic of two angles turn with two weak shocks . . . .
12.17Schematic for Example (12.11) . . . . . . . . . . . . . . . .
12.18Illustration for Example (12.14) . . . . . . . . . . . . . . . .
12.19 Revisiting of shock drag diagram for the oblique shock. . .
12.21Definition of the angle for the Prandtl–Meyer function . . .

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485
486
486
492
493
495
499
501

503
504
506
507
508
508
510
510
514
517
519
520

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LIST OF FIGURES

xix

12.22The angles of the Mach line triangle . . . . . . . . . . . . . .
12.23The schematic of the turning flow . . . . . . . . . . . . . . .
12.24The mathematical coordinate description . . . . . . . . . . . .
12.25Prandtl-Meyer function after the maximum angle . . . . . . .
12.27Diamond shape for supersonic d’Alembert’s Paradox . . . . .
12.28The definition of attack angle for the Prandtl–Meyer function
12.29Schematic for Example (12.5) . . . . . . . . . . . . . . . . . .
12.30 Schematic for the reversed question of Example 12.17 . . . .
12.20Oblique δ − θ − M relationship figure . . . . . . . . . . . . .
12.26The angle as a function of the Mach number . . . . . . . . .
12.31 Schematic of the nozzle and Prandtl–Meyer expansion. . . .

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520
521
522
526
527
527
528
529
533
534

534

13.1 Different fields of multi phase flow. . . . . . . . . . . . . . . . . . . .
13.2 Stratified flow in horizontal tubes when the liquids flow is very slow. .
13.3 Kind of Stratified flow in horizontal tubes. . . . . . . . . . . . . . . .
13.4 Plug flow in horizontal tubes with the liquids flow is faster. . . . . . .
13.5 Modified Mandhane map for flow regime in horizontal tubes. . . . . .
13.6 Gas and liquid in Flow in verstical tube against the gravity. . . . . . .
13.7 A dimensional vertical flow map low gravity against gravity. . . . . . .
13.8 The terminal velocity that left the solid particles. . . . . . . . . . . .
13.9 The flow patterns in solid-liquid flow. . . . . . . . . . . . . . . . . . .
13.10Counter–flow in vertical tubes map. . . . . . . . . . . . . . . . . . .
13.11Counter–current flow in a can. . . . . . . . . . . . . . . . . . . . . .
13.12Image of counter-current flow in liquid–gas/solid–gas configurations. .
13.13Flood in vertical pipe. . . . . . . . . . . . . . . . . . . . . . . . . . .
13.14A flow map to explain the horizontal counter–current flow. . . . . . .
13.15A diagram to explain the flood in a two dimension geometry. . . . . .
13.16General forces diagram to calculated the in a two dimension geometry.

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537
539
540
540
541
542
543
553
554
555
555
556
557
557
558
563

A.1
A.2
A.3
A.4
A.5
A.6
A.7

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567
568
574
575
576
577
595

Vector in Cartesian coordinates system . . . . . . . .
The right hand rule . . . . . . . . . . . . . . . . . .
Cylindrical Coordinate System . . . . . . . . . . . .
Spherical Coordinate System . . . . . . . . . . . . .
The general Orthogonal with unit vectors . . . . . .
Parabolic coordinates by user WillowW using Blender
The tringle angles sides . . . . . . . . . . . . . . . .

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xx

LIST OF FIGURES

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LIST OF TABLES

1

Books Under Potto Project . . . . . . . . . . . . . . . . . . . . . . . .

1.3

1.3
1.1
1.2
1.4
1.5
1.5
1.6
1.7
1.7

Viscosity of selected liquids . . . . . . . . . . . . . . .
continue . . . . . . . . . . . . . . . . . . . . . . . . .
Sutherland’s equation coefficients . . . . . . . . . . . .
Viscosity of selected gases . . . . . . . . . . . . . . . .
Properties at the critical stage . . . . . . . . . . . . .
Bulk modulus for selected materials . . . . . . . . . .
continue . . . . . . . . . . . . . . . . . . . . . . . . .
The contact angle for air/water with selected materials.
The surface tension for selected materials. . . . . . . .
continue . . . . . . . . . . . . . . . . . . . . . . . . .

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12
13
14
14
15
24
25
36
42
43

2.1

Properties of Various Ideal Gases [300K] . . . . . . . . . . . . . . . . .

50

3.1
3.2

Moments of Inertia full shape. . . . . . . . . . . . . . . . . . . . . . .
Moment of inertia for various plane surfaces . . . . . . . . . . . . . . .

67
68

9.1
9.1

9.2
9.3
9.3
9.3
9.4
9.5
9.6
9.7

Basic Units of Two Common Systems . .
continue . . . . . . . . . . . . . . . . . .
Units of the Pendulum . . . . . . . . . .
Physical Units for Two Common Systems
continue . . . . . . . . . . . . . . . . . .
continue . . . . . . . . . . . . . . . . . .
Dimensional matrix . . . . . . . . . . . .
Units of the Pendulum . . . . . . . . . .
gold grain dimensional matrix . . . . . . .
Units of the Pendulum . . . . . . . . . .

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xxi

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l

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276
279
283
284
285
287
293
294
298

xxii

LIST OF TABLES
9.8
9.8
9.8

Common Dimensionless Parameters of Thermo–Fluid in the Field . . . . 309
continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

10.1 Simple Solution to Laplaces’ Equation . . . . . . . . . . . . . . . . . . 374
10.2 Axisymetrical 3–D Flow . . . . . . . . . . . . . . . . . . . . . . . . . 374
10.2 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
11.1
11.1
11.1
11.1
11.1
11.2
11.2
11.3
11.3
11.3
11.3
11.4
11.4
11.5

11.6
11.6
11.6
11.7
11.7
11.7
11.7

Fliegner’s number a function of Mach number . .
continue . . . . . . . . . . . . . . . . . . . . . .
continue . . . . . . . . . . . . . . . . . . . . . .
continue . . . . . . . . . . . . . . . . . . . . . .
continue . . . . . . . . . . . . . . . . . . . . . .
Isentropic Table k = 1.4 . . . . . . . . . . . . .
continue . . . . . . . . . . . . . . . . . . . . . .
The shock wave table for k = 1.4 . . . . . . . .
continue . . . . . . . . . . . . . . . . . . . . . .
continue . . . . . . . . . . . . . . . . . . . . . .
continue . . . . . . . . . . . . . . . . . . . . . .
The Isothermal Flow basic parameters . . . . .
The Isothermal Flow basic parameters (continue)
The flow parameters for unchoked flow . . . . .
Fanno Flow Standard basic Table k=1.4 . . . . .
continue . . . . . . . . . . . . . . . . . . . . . .
continue . . . . . . . . . . . . . . . . . . . . . .
Rayleigh Flow k=1.4 . . . . . . . . . . . . . . .
continue . . . . . . . . . . . . . . . . . . . . . .
continue . . . . . . . . . . . . . . . . . . . . . .
continue . . . . . . . . . . . . . . . . . . . . . .

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397
398
399
400
401
402

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418
419
420
421
429
430
436
469
470
471
475
476
477
478

12.1 Table of maximum values of the oblique Shock k=1.4 . . . . . . . . . 504
12.1 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
A.1 Orthogonal coordinates systems (under construction please ignore)

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. . 578


NOMENCLATURE

¯
R

Universal gas constant, see equation (2.26), page 49

τ

The shear stress Tenser, see equation (6.7), page 174
Units length., see equation (2.1), page 45

ˆ
n

unit vector normal to surface of constant property, see equation (10.17), page 329

λ

bulk viscosity, see equation (8.101), page 253

M

Angular Momentum, see equation (6.38), page 190

µ

viscosity at input temperature T, see equation (1.17), page 12

µ0

reference viscosity at reference temperature, Ti0 , see equation (1.17), page 12

F ext

External forces by non–fluids means, see equation (6.11), page 175

U

The velocity taken with the direction, see equation (6.1), page 173

ρ

Density of the fluid, see equation (11.1), page 379

Ξ

Martinelli parameter, see equation (13.43), page 551

A

The area of surface, see equation (4.139), page 110

a

The acceleration of object or system, see equation (4.0), page 69

Bf

Body force, see equation (2.9), page 47

BT

bulk modulus, see equation (11.16), page 382

c

Speed of sound, see equation (11.1), page 379

xxiii

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xxiv

LIST OF TABLES

c.v.

subscribe for control volume, see equation (5.0), page 148

Cp

Specific pressure heat, see equation (2.23), page 49

Cv

Specific volume heat, see equation (2.22), page 49

E

Young’s modulus, see equation (11.17), page 382

EU

Internal energy, see equation (2.3), page 46

Eu

Internal Energy per unit mass, see equation (2.6), page 46

Ei

System energy at state i, see equation (2.2), page 46

G

The gravitation constant, see equation (4.69), page 91

gG

general Body force, see equation (4.0), page 69

H

Enthalpy, see equation (2.18), page 48

h

Specific enthalpy, see equation (2.18), page 48

k

the ratio of the specific heats, see equation (2.24), page 49

kT

Fluid thermal conductivity, see equation (7.3), page 198

L

Angular momentum, see equation (3.40), page 65

M

Mach number, see equation (11.24), page 385

P

Pressure, see equation (11.3), page 379

Patmos Atmospheric Pressure, see equation (4.107), page 102
q

Energy per unit mass, see equation (2.6), page 46

Q12

The energy transfered to the system between state 1 and state 2, see equation (2.2), page 46

R

Specific gas constant, see equation (2.27), page 50

S

Entropy of the system, see equation (2.13), page 48

Suth

Suth is Sutherland’s constant and it is presented in the Table 1.1, see equation (1.17), page 12

Tτ

Torque, see equation (3.42), page 66

Ti0

reference temperature in degrees Kelvin, see equation (1.17), page 12

Tin

input temperature in degrees Kelvin, see equation (1.17), page 12

U

velocity , see equation (2.4), page 46

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LIST OF TABLES

xxv

w

Work per unit mass, see equation (2.6), page 46

W12

The work done by the system between state 1 and state 2, see equation (2.2),
page 46

z

the coordinate in z direction, see equation (4.14), page 72

says

Subscribe says, see equation (5.0), page 148

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