Fluid Mechanics Through Problems

R.J. Garde Pro Vice-chancellor

Indira Gandhi National Open University New Delhi, India

(Formerly, Professor of Hydraulic Engineering University of Roorkee

Roorkee, India)

JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore

Contents

Preface v

I. Properties of Fluids 1 1.1 Introduction / 1.2 Mass Density, Specific Weight, Specific Volume,

Relative Density, Pressure 2 1.3 Viscosity 2 1.4 Bulk Modulus of Elasticity 4 1.5 Gas Laws 4 1.6 Surface Tension and Capillarity 5 1.7 Flow Regimes 6 Illustrative Examples 7 Problems 75 Descriptive Questions 18

II. Kinematics of Fluid Flow 21 2.1 Velocity 21 2.2 Types of Flow 22 2.3 Stream Lines, Path Lines, Streak Lines 22 2.4 Continuity Equation 22 2.5 Acceleration 23 2.6 Rotation, Vorticity, Circulation 24 2.7 Velocity Potential and Stream Function 26 2.8 Flow-Net 27 2.9 Relaxation Method for Drawing Flow-net 28 Illustrative Examples 29 Problems 46 Descriptive Questions 53

III. Equations of Motion and Energy Theorem 57 3.1 Equations of Motion 57

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3.2 Integration of Euler's Equation of Motion Along a Stream Line 58

3.3 Physical Meaning of Terms in Eq. (3.4. a) and Applications 59

3.4 Energy Equation with Heat Transfer 60 Illustrative Examples 60 Problems 78 Descriptive Questions 82

IV. Fluid Statics 85 4.1 Fluid Pressure 85 4.2 Pressure Variation in Static Compressible Fluid 85 4.3 Manometers 86 4.4 Forces on Immersed Plane Surfaces 87 4.5 Total Force on Immersed Curved Surfaces 88 4.6 Floating Bodies 89 4.7 Stability of Floating and Submerged Bodies 89 4.8 Fluid Mass Subjected to Uniform Acceleration 91 4.9 Stresses in Pipes Due to Internal Pressure 92 Illustrative Examples 92 Problems 118 Descriptive Questions 129

V. Applications of Bernoulli's Equation 131 5.1 Introduction 131 5.2 Pressure Distribution in Irrotational Flow 131 5.3 Hydraulic Grade Line and Total Energy Line 132 5.4 Cavitation 132 5.5 Flow Through Small and Large Orifices 132 5.6 Flow Through Mouthpieces 134 5.7 Two and Three Dimensional Orifices and Flow

Measuring Devices in Pipes 755 5.8 Weirs 137 5.9 Flow Under a Sluice Gate 141 Illustrative Examples 142 Problems 161 Descriptive Questions 167

VI. Momentum Equation and Its Applications 169 6.1 Linear Momentum Equation 169 6.2 Momentum Correction Factor ß 170 6.3 Applications 171 6.4 Angular Momentum 171 6.5 Applications of Energy and Momentum

Equations 172

Contents ix

Illustrative Examples 172 Problems 200 Descriptive Questions 210

VII. Dimensional Analysis and Similitude 212 7.1 Dimensional Analysis 212 7.2 Buckingham's « Theorem 212 7.3 Variables in Fluid Mechanics 212 7.4 Procedures of Dimensional Analysis 214 7.5 Similitude 214 7.6 Important Dimensionless Parameters 215 7.7 Model Scales 216 Illustrative Examples 217 Problems 239 Descriptive Questions 245

VIII. Laminar Flow 249 8.1 Introduction 249 8.2 Navier-Stokes' Equations 249 8.3 Hagen-Poiseuille's Equation for Laminar Flow in

Pipes 251 8.4 Stokes' Law for Fall Velocity 252 8.5 Darcy'sLaw 254 8.6 Fluidization 254 8.7 Viscometers 255 8.8 Lubrication Mechanics 255 8.9 Transition from Laminar to Turbulent Flow 255 Illustrative Examples 256 Problems 282 Descriptive Questions 288

IX. Boundary Layer Theory 291 9.1 Introduction 291 9.2 Boundary Layer Definitions and Characteristics 292 9.3 Laminar Boundary Layer 293 9.4 Turbulent Boundary Layer 294 9.5 Hydrodynamically Smooth and Rough Surfaces 296 9.6 Application of Momentum Equation 296 9.7 Establishment of Flow in Pipes 297 9.8 Separation and Its Control 298 Illustrative Examples 299 Problems 318 Descriptive Questions 321

X. Turbulent Flow 10.1 Characteristics of Turbulent Flow 324

324

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10.2 Turbulent Shear 326 10.3 Velocity Distribution in Turbulent Flow 327 10.4 Friction Factor Variation in Smooth and Rough

Pipes 328 10.5 Friction Factors for Commercial Pipes 330 10.6 Noncircular Conduits 332 10.7 Aging of Pipes 332 Illustrative Examples 332 Problems 350 Descriptive Questions 353

XI. Problems in Pipe Flow 355 11.1 Introduction 355 11.2 Sudden Expansions and Diffusers 355 11.3 Contractions and Contracting Transitions 357 11.4 Flow in Pipe Bends 358 11.5 Loss Coefficients for Pipe Fittings 360 11.6 Equivalent Length of Pipe 361 11.7 Siphons 361 11.8 Pipe Flow Problems 362 11.9 Parallel Pipes 363 11.10 Branching Pipes 364 11.11 Pipe Net-Works 366 Illustrative Examples 367 Problems 394 Descriptive Questions 403

XII. Forces on Immersed Bodies 407 12.1 Introduction 407 12.2 Deformation Drag, Friction Drag, Form Drag 408 12.3 Variation of CD with Re 409 12.4 Effects of Free Surface and Compressibility on

Drag 415 12.5 Lift 417 Illustrative Examples 420 Problems 439 Descriptive Questions 444

XIII. Open Channel Flow 447 13.1 Some Definitions 447 13.2 Steady Uniform Flow 447 13.3 Most Efficient Channel Section 450 13.4 Specific Energy and Specific Force 450 13.5 Brink Depth, Broad Crested Weirs, Contracting

Transitions 451

Contents xi

13.6 Hydraulic Jump 455 13.7 Flow in a Bend 456 13.8 Gradually Varied Flow 457 13.9 Surges and Gravity Waves 459 Illustrative Examples 461 Problems 507 Descriptive Questions 5i7

XIV. Compressible Flow 520 14.1 Effects of Compressibility 520 14.2 Perfect Gas Relationships 520 14.3 Internal Energy, Entropy and Enthalpy 521 14.4 Propagation of Elastic Wave and Flow Classification 522 14.5 Equations of Motion for One Dimensional Steady

Compressible Flows 523 14.6 Normal Shock Waves 524 14.7 Stagnation Pressure in Compressible Fluids 525 14.8 Flow through Nozzles, Orifices, Venturimeters 525 Illustrative Examples 527 Problems 541 Descriptive Questions 544

XV. Pumps 547 15.1 Introduction 547 15.2 Pump Classification 547 15.3 Reciprocating Pumps 548 15.4 Air Vessels 550 15.5 Rotodynamic Pumps 551 15.6 Velocity Diagram 554 15.7 Specified Speed 555 15.8 Pump Intakes and Sumps 556 15-9 Miscellaneous 556 Illustr-attve-Examples 557 Problems 575 Descriptive Questions 579

XVI. Turbines 581 16.1 Introduction 581 16.2 Impulse Turbines: Pelton Wheel 581 16.3 Reaction Turbines 584 16.4 Francis Turbine 586 16.5 Kaplan Turbine 588 16.6 Selection of Turbines 590 Illustrative Examples 590 Problems 606 Descriptive Questions 610

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XVII. Unsteady Flows 612 17.1 Types of Unsteady Flows 612 17.2 Equation of Motion 612 17.2 Propagation of Pressure Wave, Water Hammer 613 17.4 Surges in Open Channels 616 17.5 Oscillations in U-Tube and Surge Tanks 676 17.6 Non-Periodic Flows 618 Illustrative Examples 619 Problems 635 Descriptive Questions 638

APPENDIX APPENDIX A. Properties of Some Common Fluids at 20°C and

Atmospheric Pressure 640 APPBNDIX B. Properties of Water at Different Temperatures 641 APPENDIX C. Properties of Air at Different Temperatures and at

Atmospheric Pressure 643 APPENDIX D. Properties of Common Gases at 273.16 K. and at

Atmospheric Pressure 644 APPENDIX E. Properties of Standard Atmosphere 645 APPENDIX F. Properties of Areas 646

Notations 649 Greek Notations 654

Index 655