Fluid Mechanics

544 Pages·2016·12.894 MB·English

by  Bireswar Majumdar

Fluid Mechanics
with Laboratory Manual
SECOND EDITION
Bireswar Majumdar
Professor
Department of Power Engineering
Jadavpur University
Kolkata
PHI Learning 11)7°M@
Delhi-110092
2016
FLUID MECHANICS WITH LABORATORY MANUAL, Second Edition
Bireswar Majumdar
© 2016 by PHI Learning Private Limited, Delhi. All rights reserved. No part of this book may
be reproduced in any form, by mimeograph or any other means, without permission in writing
from the publisher.
ISBN-978-81-203-5180-6
The export rights of this book are vested solely with the publisher.
Second Printing (Second Edition) November, 2015
Published by Asoke K. Ghosh, PHI Learning Private Limited, Rimjhim House, 111, Patparganj
Industrial Estate, Delhi-110092 and Printed by Rajkamal Electric Press, Plot No. 2, Phase IV,
HSIDC, Kundli-131028, Sonepat, Haryana.
To
My Daughters
Basupurna and Basumita

CONTENTS
Preface
Preface to the First Edition
1. Introduction
1.1 Definition of Fluid
1.2 Fluid Continuum 3
1.3 Properties of Fluid 3
1.4 Units and Dimensions 8
1.5 Governing Equations of Fluid Flows 9
1.6 Forces on Fluid Elements 10
1.6.1 Internal Forces Acting on a Fluid Element 11
1.6.2 Internal Forces Acting on a Fluid Element in Presence of
Shear Stresses 12
2. Fluid Static 15-53
2.1 Pressure 15
2.2 Euler's Equation of Motion 16
2.3 Absolute and Gauge Pressure 20
2.4 Pressure Measuring Devices 21
2.4.1 Tube Gauges 21
2.4.2 Mechanical Pressure Gauges (Bourdon Gauge) 25
2.4.3 Other Types of Pressure Gauges 25
2.5 Hydrostatic Force on Submerged Surfaces 26
2.5.1 Force on Submerged Plane Surface 26
2.5.2 Force on Submerged Curved Surface 28
2.6 Buoyancy and Stability of Submerged and Floating Bodies 29
2.6.1 Determination of Metacentric Height 30
2.7 Equilibrium of Moving Fluids 32
2.7.1 Static Fluid Subjected to Uniform Rotation 32
2.7.2 Static Fluid Subjected to Uniform Acceleration 34
Solved Examples 37
Problems 52
Vi Contents
3. Fluid Kinematics 54-77
3.1 Types of Fluid Flow 54
3.2 Deformation, Translation and Rotation 58
3.3 Circulation and Vorticity 58
3.4 Streamline, Pathline and Streak Line 60
3.4.1 Stream Filament 62
3.4.2 Stream Tube 63
3.5 Vortex Line 63
3.5.1 Vortex Tube 63
3.5.2 Vortex Filament 63
3.5.3 Vortex Sheet 64
3.6 Fluid Acceleration 64
Solved Examples 66
Problems 77
4. Inviscid Fluid Flow 78-156
4.1 Law of Conservation of Mass 78
4.2 Euler's Equation of Motion 79
4.3 Bernoulli's Equation of Energy 82
4.3.1 Physical Significance of Bernoulli's Equation 83
4.4 Application of Bernoulli's Equation 83
4.4.1 Flow of Liquid from a Tank Through a Small Orifice 84
4.4.2 Flow Through Obstruction Type Flow Meter 85
4.4.3 Flow over Notches/Weir 88
4.4.4 Pitot Tube 90
4.4.5 Application in Fluid Machines 91
4.5 Equation of State 92
4.5.1 Bernoulli's Equation for Isothermal Flow 92
4.5.2 Bernoulli's Equation for Adiabatic Flow 92
4.6 Potential Flow 93
4.6.1 Consequences of the Definition of 0 94
4.6.2 Euler's Equation for Potential Flow 95
4.6.3 Special Cases 98
4.7 Unsteady Potential Flow (Surface Waves) 100
4.7.1 Straight, Moving Waves 100
4.7.2 Particle Path 105
4.7.3 Effect of Surface Tension 106
4.7.4 Standing Waves and Group Velocity 107
4.8 Stream Function (v') 108
4.8.1 Consequences of the Definition of Stream Function yr 109
4.9 Different Type of Inviscid Flows 111
4.9.1 Uniform Rectilinear Flow 112
4.9.2 Source Flow 112
4.9.3 Sink Flow 114
4.9.4 Irrotational Vortex 114
4.10 Principle of Superimposition 115
4.10.1 Doublet 115
Contents Vii
4.11 Axi-symmetric Flow 127
4.11.1 Properties of yi 128
4.11.2 Uniform Flow Parallel to x-Axis 131
4.11.3 3-D Source Flow 132
4.11.4 Superimposition of Uniform Flow and a Source Flow 133
4.11.5 Uniform Flow with a Source and a Sink Flow 136
4.11.6 3-D Doublet 137
4.11.7 Uniform Rectilinear Flow with a Doublet at Origin 138
4.11.8 Line Source or Line Sink 140
4.11.9 Flow over a Streamlined Body 141
Solved Examples 142
Problems 155
5. Integral Analysis of Fluid Flow 157-186
5.1 Relation between Extensive Property N and Intensive Property ti 159
5.2 Reynolds Transport Theorem (RTT) 159
5.2.1 RTT Based on Inertial/Fixed Frame of Reference 159
5.3 Conservation of Mass 161
5.4 Conservation of Linear Momentum (Inertial/Non-accelerating Frame of
Reference) 164
5.5 Principle of Angular Momentum (Inertial Frame of Reference) 170
5.6 Conservation of Linear Momentum (Non-inertial Frame
of Reference) 173
5.7 Principle of Conservation of Energy 175
Solved Examples 177
Problems 185
6. Differential Analysis of Fluid Flow 187-214
6.1 Conservation of Mass 187
6.2 Newton's Second Law of Motion 188
6.2.1 Stresses on a Fluid Element and Their Nature 189
6.2.2 Properties of Stress Tensor 190
6.3 Equation of Motion 192
6.4 Volumetric Deformation 193
6.5 Angular Deformation (Shear Deformation/Volumetric Strain) 194
6.6 Rotation of the Fluid Element 196
6.7 Vorticity () 197
6.8 Circulation (F) 197
6.9 Constitutive Equation Relating ru, cif, co;, (Navier—Stokes' Equation) 199
6.10 Energy Equation 203
6.11 Fundamental Equations in Cylindrical and Spherical Co-ordinates 207
Solved Examples 208
Problems 214
Viii Contents
7. Exact Solutions of Newtonian Fluid Flow 215-230
7.1 Parallel Flows 215
7.2 Flow between Two Parallel Plates 215
7.3 Flow Through Circular Pipe 218
7.4 Flow Through Annulus Pipe 223
7.5 Flow between Concentric Rotating Cylinders 224
7.6 Suddenly Accelerating Plate (Rayleigh's First Problem)
(Unsteady Flow) 226
7.7 Flow with Thermal Parameters 228
8. Low Reynolds Number Flow 231-242
8.1 Flow Past a Sphere 233
8.2 Drag on the Sphere 235
8.3 Ossen Approximation (1910) 236
8.4 Hydrodynamic Theory of Lubrication 237
8.4.1 Pressure Distribution 241
9. Large Reynolds Number Flow 243-291
9.1 Governing Equation for Boundary Layer and Order of Magnitude 245
9.2 Properties of Boundary Layer Equation 248
9.3 Solution of Laminar Boundary Layer Equation
(Blausius Solution, 1908) 250
9.3.1 Flow over a Flat Plate 250
9.3.2 Characteristics of Blasius Solution 255
9.3.3 Blasius Solution is Exact but Boundary Layer Equations are
not Exact 259
9.3.4 Drag on the Plate 260
9.3.5 Flow over Surfaces for dp/dx # 0 261
9.4 Integral Approach for Boundary Layer Flows 263
9.4.1 Procedure for Solving Momentum Integral Equation 268
9.4.2 Solution for Boundary Layer Equation for a Body for
Arbitrary Shape 272
9.5 Solution Procedure 274
9.5.1 Marching Technique 274
9.5.2 Energy Integral Method 276
Solved Examples 279
10. Transition and Turbulent Flows 292-321
10.1 Transition Phenomenon in Various Practical Flow Situations 293
10.1.1 Flow Through a Circular Pipe (Reynolds Experiment) 293
10.1.2 Transition Process in Boundary Layer 293
10.1.3 Flow over Bluff Bodies 294
10.1.4 Why Transition Takes Place? 295
10.1.5 How to Represent Disturbances Mathematically? 295
10.2 Fully Developed Turbulent Flows 298
10.2.1 Governing Equation of Motion for Turbulent Flows 301
Contents ix
10.3 Phenomenological Theories of Turbulence 304
10.3.1 Boussinesq Hypothesis 304
10.3.2 Prandtl's Mixing Length Hypothesis 305
10.4 Example 308
10.4.1 Channel Flow 308
10.4.2 No-slip Condition at the Edge of Laminar Sub-layer 308
10.4.3 Turbulent Flows in Absence of the Wall 310
10.5 Examples of Turbulent Flows in Practical Situation 311
10.5.1 Couette Flow 311
10.5.2 Turbulent Flow through a Circular Pipe 312
10.6 When a Pipe Surface is Considered as Smooth? 316
10.7 Correlation for 1-,„ 318
10.8 Criteria for Treating a Surface as Smooth 320
10.9 Comparison of Laminar and Turbulent Boundary Layers 321
11. Fundamentals of Compressible Fluid Flow 322-373
11.1 Review of Basic Thermodynamic Principle 322
11.2 Elastic Waves in a Compressible Fluid 326
11.3 Mach Number and Its Effect 329
11.3.1 Incompressible Medium 329
11.3.2 Compressible Medium 329
11.4 Isentropic Stagnation State 331
11.4.1 Stagnation Enthalpy 331
11.4.2 Stagnation Temperature 332
11.4.3 Isentropic Temperature Ratio for a Perfect Gas (To/T) 332
11.4.4 Isentropic Pressure Ratio for a Perfect Gas (po/p) 333
11.4.5 Compressibility Correction Factor (CCF) 334
11.5 Steady 1-D Isentropic Flow of a Perfect Gas Through a Variable
Area Duct 335
11.5.1 Flow Through Nozzle 339
11.5.2 Flow Through Convergent—Divergent Nozzle (De-Laval nozzle) 344
11.6 Shock Wave 345
11.6.1 Normal Shock Wave 346
11.6.2 Oblique Shock Wave 350
11.7 Compressible Flow in Pipe of Constant Cross-Section 353
11.7.1 Adiabatic Flow with Friction 353
11.7.2 Steady Isothermal Flow in a Constant Area Duct with Friction 361
11.7.3 Comparison of the Results Obtained for Adiabatic Flow
and Isothermal Flow 364
11.8 Steady One-dimensional Flow with Heat Transfer in a Constant
Area Duct 364
11.8.1 Physical Interpretation of Rayleigh Line 366
11.8.2 Intersection of a Fanno Line and a Rayleigh Line 367
Solved Examples 368
Problems 371