**Fluid Mechanics and Machinery - Kothandaraman**

**1 Physical Properties of Fluids .................................................................... 1**
1.0 Introduction .............................................................................................................. 1

1.1 Three Phases of Matter............................................................................................ 2

1.2 Compressible and Incompressible Fluids ............................................................... 2

1.3 Dimensions and Units.............................................................................................. 3

1.4 Continuum ................................................................................................................ 4

1.5 Definition of Some Common Terminology ............................................................. 4

1.6 Vapour and Gas ........................................................................................................ 5

1.7 Characteristic Equation for Gases .......................................................................... 6

1.8 Viscosity .................................................................................................................... 7

1.8.1 Newtonian and Non Newtonian Fluids................................................ 10

1.8.2 Viscosity and Momentum Transfer ...................................................... 11

1.8.3 Effect of Temperature on Viscosity ...................................................... 11

1.8.4 Significance of Kinematic Viscosity...................................................... 11

1.8.5 Measurement of Viscosity of Fluids ..................................................... 12

1.9 Application of Viscosity Concept .......................................................................... 13

1.9.1 Viscous Torque and Power—Rotating Shafts ...................................... 13

1.9.2 Viscous Torque—Disk Rotating Over a Parallel Plate ....................... 14

1.9.3 Viscous Torque—Cone in a Conical Support ....................................... 16

1.10 Surface Tension ...................................................................................................... 17

1.10.1 Surface Tension Effect on Solid-Liquid Interface ............................... 17

1.10.2 Capillary Rise or Depression ................................................................ 18

1.10.3 Pressure Difference Caused by Surface Tension on a Doubly

Curved Surface....................................................................................... 19

1.10.4 Pressure Inside a Droplet and a Free Jet ............................................ 20

1.11 Compressibility and Bulk Modulus ...................................................................... 21

1.11.1 Expressions for the Compressibility of Gases ..................................... 22

1.12 Vapour Pressure ..................................................................................................... 23

1.12.1 Partial Pressure ..................................................................................... 23

Solved Problems ..................................................................................................... 24

Objective Questions ................................................................................................ 33

Review Questions .................................................................................................... 38

Exercise Problems ................................................................................................... 39

**2 Pressure Distribution in Fluids ............................................................... 42**
2.0 Introduction ............................................................................................................ 42

2.1 Pressure .................................................................................................................. 42

2.2 Pressure Measurement .......................................................................................... 43

2.3 Pascal’s Law........................................................................................................... 45

2.4 Pressure Variation in Static Fluid (Hydrostatic Law) ........................................ 46

2.4.1 Pressure Variation in Fluid with Constant Density ........................... 47

2.4.2 Pressure Variation in Fluid with Varying Density ............................. 48

2.5 Manometers ............................................................................................................ 49

2.5.1 Micromanometer .................................................................................... 51

2.6 Distribution of Pressure in Static Fluids Subjected to Acceleration, as .......... 53

2.6.1 Free Surface of Accelerating Fluid ....................................................... 54

2.6.2 Pressure Distribution in Accelerating Fluids along Horizontal

Direction ................................................................................................. 55

2.7 Forced Vortex ......................................................................................................... 58

Solved Problems ..................................................................................................... 60

Review Questions .................................................................................................... 71

Objective Questions ................................................................................................ 71

Exercise Problems ................................................................................................... 74

**3 Forces on Surfaces Immersed in Fluids ................................................ 80**
3.0 Introduction ............................................................................................................ 80

3.1 Centroid and Moment of Inertia of Areas ............................................................ 81

3.2 Force on an Arbitrarily Shaped Plate Immersed in a Liquid ............................. 83

3.3 Centre of Pressure for an Immersed Inclined Plane ........................................... 84

3.3.1 Centre of Pressure for Immersed Vertical Planes .............................. 86

3.4 Component of Forces on Immersed Inclined Rectangles .................................... 87

3.5 Forces on Curved Surfaces .................................................................................... 89

3.6 Hydrostatic Forces in Layered Fluids .................................................................. 92

Solved Problems ..................................................................................................... 93

Review Questions .................................................................................................. 111

Objective Questions .............................................................................................. 112

Exercise Problems ................................................................................................. 115

**4 Buoyancy Forces and Stability of Floating Bodies ............................. 119**
4.0 Archimedes Principle ........................................................................................... 119

4.1 Buoyancy Force .................................................................................................... 119

4.2 Stability of Submerged and Floating Bodies ..................................................... 121

4.3 Conditions for the Stability of Floating Bodies .................................................. 123

4.4 Metacentric Height .............................................................................................. 124

4.4.1 Experimental Method for the Determination of Metacentric

Height ................................................................................................... 125

Solved Problems ................................................................................................... 125

Review Questions .................................................................................................. 136

Objective Questions .............................................................................................. 137

Exercise Problems ................................................................................................. 139

**5 Fluid Flow—Basic Concepts—Hydrodynamics .................................. 142**
5.0 Introduction .......................................................................................................... 142

5.1 Lagrangian and Eularian Methods of Study of Fluid Flow .............................. 143

5.2 Basic Scientific Laws Used in the Analysis of Fluid Flow ................................ 143

5.3 Flow of Ideal / Inviscid and Real Fluids ............................................................. 143

5.4 Steady and Unsteady Flow .................................................................................. 144

5.5 Compressible and Incompressible Flow ............................................................. 144

5.6 Laminar and Turbulent Flow .............................................................................. 144

5.7 Concepts of Uniform Flow, Reversible Flow and Three

Dimensional Flow................................................................................................. 145

5.8 Velocity and Acceleration Components .............................................................. 145

5.9 Continuity Equation for Flow—Cartesian Co-ordinates .................................. 146

5.10 Irrotational Flow and Condition for Such Flows ............................................... 148

5.11 Concepts of Circulation and Vorticity ................................................................ 148

5.12 Stream Lines, Stream Tube, Path Lines, Streak Lines and Time Lines ........ 149

5.13 Concept of Stream Line ....................................................................................... 150

5.14 Concept of Stream Function ................................................................................ 151

5.15 Potential Function ................................................................................................ 153

5.16 Stream Function for Rectilinear Flow Field (Positive X Direction) ................. 154

5.17 Two Dimensional Flows—Types of Flow ............................................................ 154

5.17.1 Source Flow .......................................................................................... 155

5.17.2 Sink Flow .............................................................................................. 155

5.17.3 Irrotational Vortex of Strength K....................................................... 155

5.17.4 Doublet of Strength Λ .......................................................................... 156

5.18 Principle of Superposing of Flows (or Combining of Flows) ............................. 157

5.18.1 Source and Uniform Flow (Flow Past a Half Body) .......................... 157

5.18.2 Source and Sink of Equal Strength with Separation of 2a

Along x-Axis .......................................................................................... 157

5.18.3 Source and Sink Displaced at 2a and Uniform Flow

(Flow Past a Rankine Body) ................................................................ 158

5.18.4 Vortex (Clockwise) and Uniform Flow ............................................... 158

5.18.5 Doublet and Uniform Flow (Flow Past a Cylinder) .......................... 158

5.18.6 Doublet, Vortex (Clockwise) and Uniform Flow ................................ 158

(xi)

5.18.7 Source and Vortex (Spiral Vortex Counterclockwise)....................... 159

5.18.8 Sink and Vortex (Spiral Vortex Counterclockwise) .......................... 159

5.18.9 Vortex Pair (Equal Strength, Opposite Rotation,

Separation by 2a) ................................................................................. 159

5.19 Concept of Flow Net ............................................................................................. 159

Solved Problems ................................................................................................... 160

Objective Questions .............................................................................................. 174

Exercise Problems ................................................................................................. 178

**6 Bernoulli Equation and Applications.................................................... 180**
6.0 Introduction .......................................................................................................... 180

6.1 Forms of Energy Encountered in Fluid Flow..................................................... 180

6.1.1 Kinetic Energy ..................................................................................... 181

6.1.2 Potential Energy .................................................................................. 181

6.1.3 Pressure Energy (Also Equals Flow Energy) .................................... 182

6.1.4 Internal Energy.................................................................................... 182

6.1.5 Electrical and Magnetic Energy ......................................................... 183

6.2 Variation in the Relative Values of Various Forms of Energy

During Flow .......................................................................................................... 183

6.3 Euler’s Equation of Motion for Flow Along a Stream Line .............................. 183

6.4 Bernoulli Equation for Fluid Flow ...................................................................... 184

6.5 Energy Line and Hydraulic Gradient Line ........................................................ 187

6.6 Volume Flow Through a Venturimeter .............................................................. 188

6.7 Euler and Bernoulli Equation for Flow with Friction ....................................... 191

6.8 Concept and Measurement of Dynamic, Static and Total Head ..................... 192

6.8.1 Pitot Tube ............................................................................................. 193

Solved Problems ................................................................................................... 194

Objective Questions .............................................................................................. 213

Exercise Problems ................................................................................................. 215

**7 Flow in Closed Conduits (Pipes)........................................................... 219**
7.0 Parameters Involved in the Study of Flow Through Closed Conduits ............ 219

7.1 Boundary Layer Concept in the Study of Fluid Flow ....................................... 220

7.2 Boundary Layer Development Over A Flat Plate ............................................. 220

7.3 Development of Boundary Layer in Closed Conduits (Pipes) .......................... 221

7.4 Features of Laminar and Turbulent Flows ........................................................ 222

7.5 Hydraulically “Rough” and “Smooth” Pipes ....................................................... 223

7.6 Concept of “Hydraulic Diameter”: (Dh) .............................................................. 223

7.7 Velocity Variation with Radius for Fully Developed Laminar

Flow in Pipes ........................................................................................................ 224

7.8 Darcy–Weisbach Equation for Calculating Pressure Drop .............................. 226

7.9 Hagen–Poiseuille Equation for Friction Drop ................................................... 228

7.10 Significance of Reynolds Number in Pipe Flow ................................................. 229

7.11 Velocity Distribution and Friction Factor for Turbulent Flow in Pipes .......... 230

7.12 Minor Losses in Pipe Flow ................................................................................... 231

7.13 Expression for the Loss of Head at Sudden Expansion in Pipe Flow ............ 232

7.14 Losses in Elbows, Bends and Other Pipe Fittings ............................................. 234

7.15 Energy Line and Hydraulic Grade Line in Conduit Flow ................................ 234

7.16 Concept of Equivalent Length............................................................................. 235

7.17 Concept of Equivalent Pipe or Equivalent Length ............................................ 235

7.18 Fluid Power Transmission Through Pipes ......................................................... 238

7.18.1 Condition for Maximum Power Transmission ................................... 238

7.19 Network of Pipes .................................................................................................. 239

7.19.1 Pipes in Series—Electrical Analogy ................................................... 240

7.19.2 Pipes in Parallel ................................................................................... 241

7.19.3 Branching Pipes ................................................................................... 243

7.19.4 Pipe Network........................................................................................ 245

Solved Problems ................................................................................................... 245

Objective Questions .............................................................................................. 256

Exercise Problems ................................................................................................. 259

**8 Dimensional Analysis............................................................................. 263**
8.0 Introduction .......................................................................................................... 263

8.1 Methods of Determination of Dimensionless Groups........................................ 264

8.2 The Principle of Dimensional Homogeneity ...................................................... 265

8.3 Buckingham Pi Theorem ..................................................................................... 265

8.3.1 Determination of π Groups.................................................................. 265

8.4 Important Dimensionless Parameters ............................................................... 270

8.5 Correlation of Experimental Data ...................................................................... 270

8.5.1 Problems with One Pi Term ................................................................ 271

8.5.2 Problems with Two Pi Terms .............................................................. 271

8.5.3 Problems with Three Dimensionless Parameters ............................. 273

Solved Problems ................................................................................................... 273

Objective Questions .............................................................................................. 291

Exercise Problems ................................................................................................. 293

**9 Similitude and Model Testing ................................................................ 296**
9.0 Introduction .......................................................................................................... 296

9.1 Model and Prototype ............................................................................................ 296

9.2 Conditions for Similarity Between Models and Prototype ............................... 297

9.2.1 Geometric Similarity ........................................................................... 297

9.2.2 Dynamic Similarity.............................................................................. 297

9.2.3 Kinematic Similarity ........................................................................... 298

9.3 Types of Model Studies ........................................................................................ 298

9.3.1 Flow Through Closed Conduits .......................................................... 298

9.3.2 Flow Around Immersed Bodies........................................................... 299

9.3.3 Flow with Free Surface ....................................................................... 300

9.3.4 Models for Turbomachinery ................................................................ 301

9.4 Nondimensionalising Governing Differential Equations.................................. 302

9.5 Conclusion ............................................................................................................. 303

Solved Problems ................................................................................................... 303

Objective Questions .............................................................................................. 315

Exercise Problems ................................................................................................. 317

**10 Boundary Layer Theory and Flow Over Surfaces ............................... 321**
10.0 Introduction .......................................................................................................... 321

10.1 Boundary Layer Thickness .................................................................................. 321

10.1.1 Flow Over Flat Plate ........................................................................... 322

10.1.2 Continuity Equation ............................................................................ 322

10.1.3 Momentum Equation ........................................................................... 324

10.1.4 Solution for Velocity Profile ................................................................ 325

10.1.5 Integral Method ................................................................................... 327

10.1.6 Displacement Thickness ...................................................................... 330

10.1.7 Momentum Thickness ......................................................................... 331

10.2 Turbulent Flow ..................................................................................................... 332

10.3 Flow Separation in Boundary Layers ................................................................. 334

10.3.1 Flow Around Immersed Bodies – Drag and Lift ............................... 334

10.3.2 Drag Force and Coefficient of Drag .................................................... 335

10.3.3 Pressure Drag ...................................................................................... 336

10.3.4 Flow Over Spheres and Cylinders ...................................................... 337

10.3.5 Lift and Coefficient of Lift ................................................................... 338

10.3.6 Rotating Sphere and Cylinder ............................................................ 339

Solved Problems ................................................................................................... 341

Objective Questions .............................................................................................. 353

Exercise Problems ................................................................................................. 356

**11 Flow Measurements ............................................................................... 359**
11.1 Introduction .......................................................................................................... 359

11.2 Velocity Measurements........................................................................................ 359

11.2.1 Pitot Tube ............................................................................................. 360

11.2.2 Vane Anemometer and Currentmeter ............................................... 362

11.2.3 Hot Wire Anemometer......................................................................... 362

11.2.4 Laser Doppler Anemometer ................................................................ 363

11.3 Volume Flow Rate Measurement ........................................................................ 364

11.3.1 Rotameter (Float Meter) ..................................................................... 364

11.3.2 Turbine Type Flowmeter ..................................................................... 364

11.3.3 Venturi, Nozzle and Orifice Meters .................................................... 365

11.3.4 Elbow Meter ......................................................................................... 367

11.4 Flow Measurement Using Orifices, Notches and Weirs ................................... 367

11.4.1 Discharge Measurement Using Orifices ............................................ 367

11.4.2 Flow Measurements in Open Channels ............................................. 368

Solved Problems ................................................................................................... 371

Review Questions .................................................................................................. 379

Objective Questions .............................................................................................. 380

Exercise Problems ................................................................................................. 381

**12 Flow in Open Channels .......................................................................... 383**
12.0 Introduction .......................................................................................................... 383

12.1.1 Characteristics of Open Channels ...................................................... 383

12.1.2 Classification of Open Channel Flow ................................................. 384

12.2 Uniform Flow: (Also Called Flow at Normal Depth) ......................................... 384

12.3 Chezy’s Equation for Discharge .......................................................................... 385

12.4 Determination of Chezy’s Constant .................................................................... 386

12.4.1 Bazin’s Equation for Chezy’s Constant .............................................. 386

12.4.2 Kutter’s Equation for Chezy’s Constant C......................................... 387

12.4.3 Manning’s Equation for C ................................................................... 388

12.5 Economical Cross-Section for Open Channels ................................................... 390

12.6 Flow with Varying Slopes and Areas.................................................................. 395

12.6.1 Velocity of Wave Propagation in Open Surface Flow ....................... 395

12.6.2 Froude Number .................................................................................... 397

12.6.3 Energy Equation for Steady Flow and Specific Energy.................... 397

12.6.4 Non Dimensional Representation of Specific Energy Curve............ 400

12.7 Effect of Area Change .......................................................................................... 404

12.7.1 Flow Over a Bump ............................................................................... 404

12.7.2 Flow Through Sluice Gate, from Stagnant Condition ...................... 406

12.7.3 Flow Under a Sluice Gate in a Channel............................................. 407

12.8 Flow with Gradually Varying Depth .................................................................. 409

12.8.1 Classification of Surface Variations ................................................... 410

12.9 The Hydraulic Jump (Rapidly Varied Flow) ...................................................... 411

12.10 Flow Over Broad Crested Weir ........................................................................... 414

12.11 Effect of Lateral Contraction............................................................................... 415

Solved Problems ................................................................................................... 416

Review Questions .................................................................................................. 430

Objective Questions .............................................................................................. 430

Exercise Problems ................................................................................................. 432

**13 Dynamics of Fluid Flow.......................................................................... 435**
13.0 Introduction .......................................................................................................... 435

13.1 Impulse Momentum Principle ............................................................................. 435

13.1.1 Forces Exerted on Pressure Conduits ................................................ 436

13.1.2 Force Exerted on a Stationary Vane or Blade ................................... 438

13.2 Absolute and Relative Velocity Relations .......................................................... 439

13.3 Force on a Moving Vane or Blade ....................................................................... 439

13.4 Torque on Rotating Wheel ................................................................................... 443

Solved Problems ................................................................................................... 445

Exercise Questions ................................................................................................ 450

**14 Hydraulic Turbines.................................................................................. 452**
14.0 Introduction .......................................................................................................... 452

14.1 Hydraulic Power Plant......................................................................................... 452

14.2 Classification of Turbines .................................................................................... 453

14.3 Similitude and Model Testing ............................................................................. 453

14.3.1 Model and Prototype............................................................................ 457

14.3.2 Unit Quantities .................................................................................... 459

14.4 Turbine Efficiencies ............................................................................................. 460

14.5 Euler Turbine Equation ....................................................................................... 461

14.5.1 Components of Power Produced ......................................................... 462

14.6 Pelton Turbine ...................................................................................................... 464

14.6.1 Power Development ............................................................................. 466

14.6.2 Torque and Power and Efficiency Variation with Speed Ratio........ 470

14.7 Reaction Turbines ................................................................................................ 472

14.7.1 Francis Turbines .................................................................................. 473

14.8 Axial Flow Turbines ............................................................................................. 480

14.9 Cavitation in Hydraulic Machines ...................................................................... 482

14.9 Governing of Hydraulic Turbines ....................................................................... 484

Worked Examples ................................................................................................. 486

Review Questions .................................................................................................. 513

Objective Questions .............................................................................................. 514

Exercise Problems ................................................................................................. 515

**15 Rotodynamic Pumps .............................................................................. 519**
15.0 Introduction .......................................................................................................... 519

15.1 Centrifugal Pumps ............................................................................................... 519

15.1.1 Impeller ................................................................................................ 521

15.1.2 Classification ........................................................................................ 521

15.2 Pressure Developed by the Impeller ................................................................... 522

15.2.1 Manometric Head ................................................................................ 523

15.3 Energy Transfer by Impeller ............................................................................... 523

15.3.1 Slip and Slip Factor ............................................................................. 525

15.3.3 Losses in Centrifugal Pumps .............................................................. 525

15.3.4 Effect of Outlet Blade Angle ............................................................... 526

15.4 Pump Characteristics........................................................................................... 527

15.5 Operation of Pumps in Series and Parallel ........................................................ 529

15.6 Specific Speed and Significance .......................................................................... 531

15.7 Cavitation ............................................................................................................. 532

15.8 Axial Flow Pump .................................................................................................. 533

15.9 Power Transmitting Systems .............................................................................. 535

15.9.1 Fluid Coupling...................................................................................... 535

15.9.2 Torque Converter ................................................................................. 536

Solved Examples................................................................................................... 538

Revierw Questions ................................................................................................ 556

Objective Questions .............................................................................................. 556

Exercise Problems ................................................................................................. 557

**16 Reciprocating Pumps ............................................................................. 560**
16.0 Introduction .......................................................................................................... 560

16.1 Comparison ........................................................................................................... 560

16.2 Description and Working ..................................................................................... 560

16.3 Flow Rate and Power .......................................................................................... 562

16.3.1 Slip ........................................................................................................ 563

16.4 Indicator Diagram ................................................................................................ 564

16.4.1 Acceleration Head ................................................................................ 565

16.4.2 Minimum Speed of Rotation of Crank................................................ 569

16.4.3 Friction Head ....................................................................................... 570

16.5 Air Vessels ............................................................................................................ 572

16.5.1 Flow into and out of Air Vessel ........................................................... 575

16.6 Rotary Positive Displacement Pumps ................................................................ 576

16.6.1 Gear Pump............................................................................................ 577

16.6.2 Lobe Pump ............................................................................................ 577

16.6.3 Vane Pump ........................................................................................... 577

Solved Problems ................................................................................................... 578

Review Questions .................................................................................................. 587

Objective Questions .............................................................................................. 587

Exercise Problems ................................................................................................. 587

Appendix ............................................................................................................. 590

Index .................................................................................................................... 595

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