1. Conversion Factors from BG to SI Units To convert from To
Multiply by Acceleration ft/s2 m/s2 0.3048 Area ft2 m2 9.2903 E 2
mi2 m2 2.5900 E 6 acres m2 4.0469 E 3 Density slug/ft3 kg/m3 5.1538
E 2 lbm/ft3 kg/m3 1.6019 E 1 Energy ft-lbf J 1.3558 Btu J 1.0551 E
3 cal J 4.1868 Force lbf N 4.4482 kgf N 9.8067 Length ft m 0.3048
in m 2.5400 E 2 mi (statute) m 1.6093 E 3 nmi (nautical) m 1.8520 E
3 Mass slug kg 1.4594 E 1 lbm kg 4.5359 E 1 Mass ow slug/s kg/s
1.4594 E 1 lbm/s kg/s 4.5359 E 1 Power ftlbf/s W 1.3558 hp W 7.4570
E 2
2. Conversion Factors from BG to SI Units (Continued) To
convert from To Multiply by Pressure lbf/ft2 Pa 4.7880 E 1 lbf/in2
Pa 6.8948 E 3 atm Pa 1.0133 E 5 mm Hg Pa 1.3332 E 2 Specic weight
lbf/ft3 N/m3 1.5709 E 2 Specic heat ft2 /(s2 R) m2 /(s2 K) 1.6723 E
1 Surface tension lbf/ft N/m 1.4594 E 1 Temperature F C tC (tF 32)
R K 0.5556 Velocity ft/s m/s 0.3048 mi/h m/s 4.4704 E 1 knot m/s
5.1444 E 1 Viscosity lbfs/ft2 Ns/m2 4.7880 E 1 g/(cms) Ns/m2 0.1
Volume ft3 m3 2.8317 E 2 L m3 0.001 gal (U.S.) m3 3.7854 E 3 uid
ounce (U.S.) m3 2.9574 E 5 Volume ow ft3 /s m3 /s 2.8317 E 2
gal/min m3 /s 6.3090 E 5 5 9
3. EQUATION SHEET Ideal-gas law: p RT, Rair 287 J/kg-K
Hydrostatics, constant density: p2 p1 (z2 z1), g Buoyant force:
(displaced volume) CV momentum: Steady ow energy: (p/V2 /2gz)out
hfriction hpump hturbine Incompressible continuity: Incompressible
stream function Bernoulli unsteady irrotational ow: Pipe head loss:
where f Moody chart friction factor Laminar flat plate ow: , ,
Isentropic ow: , , Prandtl-Meyer expansion: K (k1)/(k1), K1/2 tan1
[(Ma2 1)/K]1/2 tan1 (Ma2 1)1/2 Gradually varied channel ow: , Fr
V/Vcritdy/dx (S0 S)/(1 Fr2 ) p0/p (T0/T)k(k1) 0/ (T0/T)1/(k1) T0
/T15(k1)/26Ma2 CD Drag/11 2V2 A2; CL Lift/11 2V2 A2 CD 1.328/ReL
1/2 cf 0.664/Rex 1/2 /x 5.0/Rex 1/2 hf f(L/d)V2 /(2g) /t dp/ V2 /2
gz Const u /y; v /x (x,y): V 0 (p/V2 /2gz)in g 3(AV)V4out g 3(AV
)V4in gF d/dt1 CV Vd2 FB fluid Surface tension: Hydrostatic panel
force: , yCPIxxsin/(hCGA), xCPIxysin/(hCGA) CV mass: CV angular
momentum: Acceleration: Navier-Stokes: Velocity potential :
Turbulent friction factor: Orice, nozzle, venturi ow: , Turbulent
at plate ow: , 2-D potential ow: One-dimensional isentropic area
change: A/A*(1/Ma)[1{(k1)/2}Ma2 ](1/2)(k1)/(k1) Uniform ow,
Mannings n, SI units: Euler turbine formula: Power Q(u2Vt2 u1Vt1),
u r V0(m/s) (1.0/n)3Rh(m)42/3 S0 1/2 2 2 0 cf 0.027/Rex 1/7 , CD
0.031/ReL 1/7 /x 0.16/Rex 1/7 d/DQCdAthroat 32p/5(14 )641/2 2.0
log10 3/(3.7d) 2.51/1Red 1f)4 1/1f u /x; v /y; w /z (x,y,z)
(dV/dt)gp2 V u(V/x) v(V/y)w(V/z) dV/dt V/t gAV(r0V)outgAV(r0V)in
gM0 d/dt(CV (r0V)d) g(AV)in 0 d/dt(CV d) g(AV)out F hCGA p Y(R1 1
R2 1 )
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6. McGraw-Hill Series in Mechanical Engineering
Alciatore/Histand Introduction to Mechatronics and Measurement
Systems Anderson Computational Fluid Dynamics: The Basics with
Applications Anderson Fundamentals of Aerodynamics Anderson
Introduction to Flight Anderson Modern Compressible Flow
Beer/Johnston Vector Mechanics for Engineers: Statics and Dynamics
Beer/Johnston Mechanics of Materials Budynas Advanced Strength and
Applied Stress Analysis Budynas/Nisbett Shigleys Mechanical
Engineering Design engel Heat and Mass Transfer: A Practical
Approach engel Introduction to Thermodynamics & Heat Transfer
engel/Boles Thermodynamics: An Engineering Approach engel/Cimbala
Fluid Mechanics: Fundamentals and Applications engel/Turner
Fundamentals of Thermal-Fluid Sciences Dieter Engineering Design: A
Materials & Processing Approach Dieter Mechanical Metallurgy
Dorf/Byers Technology Ventures: From Idea to Enterprise
Finnemore/Franzini Fluid Mechanics with Engineering Applications
Hamrock/Schmid/Jacobson Fundamentals of Machine Elements Heywood
Internal Combustion Engine Fundamentals Holman Experimental Methods
for Engineers Holman Heat Transfer Kays/Crawford/Weigand Convective
Heat and Mass Transfer Meirovitch Fundamentals of Vibrations Norton
Design of Machinery Palm System Dynamics Reddy An Introduction to
Finite Element Method Schey Introduction to Manufacturing Processes
Smith/Hashemi Foundations of Materials Science and Engineering
Turns An Introduction to Combustion: Concepts and Applications
Ugural Mechanical Design: An Integrated Approach Ullman The
Mechanical Design Process White Fluid Mechanics White Viscous Fluid
Flow whi29346_fm_i-xvi.qxd 12/14/09 7:09PM Page ii ntt
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7. Seventh Edition Frank M. White University of Rhode Island
Fluid Mechanics whi29346_fm_i-xvi.qxd 12/14/09 7:09PM Page iii ntt
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8. FLUID MECHANICS, SEVENTH EDITION Published by McGraw-Hill, a
business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of
the Americas, New York, NY 10020. Copyright 2011 by The McGraw-Hill
Companies, Inc. All rights reserved. Previous editions 2008, 2003,
and 1999. No part of this publication may be reproduced or
distributed in any form or by any means, or stored in a database or
retrieval system, without the prior written consent of The
McGraw-Hill Companies, Inc., including, but not limited to, in any
network or other electronic storage or transmission, or broadcast
for distance learning. Some ancillaries, including electronic and
print components, may not be available to customers outside the
United States. This book is printed on acid-free paper. 1 2 3 4 5 6
7 8 9 0 DOC/DOC 1 0 9 8 7 6 5 4 3 2 1 0 www.mhhe.com ISBN
978-0-07-352934-9 MHID 0-07-352934-6 Vice President &
Editor-in-Chief: Marty Lange Vice President, EDP/Central Publishing
Services: Kimberly Meriwether-David Global Publisher: Raghothaman
Srinivasan Senior Sponsoring Editor: Bill Stenquist Director of
Development: Kristine Tibbetts Developmental Editor: Lora Neyens
Senior Marketing Manager: Curt Reynolds Senior Project Manager:
Lisa A. Bruodt Production Supervisor: Nicole Baumgartner Design
Coordinator: Brenda A. Rolwes Cover Designer: Studio Montage, St.
Louis, Missouri (USE) Cover Image: Copyright SkySails Senior Photo
Research Coordinator: John C. Leland Photo Research: Emily
Tietz/Editorial Image, LLC Compositor: Aptara, Inc. Typeface: 10/12
Times Roman Printer: R. R. Donnelley All credits appearing on page
or at the end of the book are considered to be an extension of the
copyright page. Library of Congress Cataloging-in-Publication Data
White, Frank M. Fluid mechanics / Frank M. White. 7th ed. p. cm.
(Mcgraw-Hill series in mechanical engineering) Includes
bibliographical references and index. ISBN 9780073529349 (alk.
paper) 1. Fluid mechanics. I. Title. TA357.W48 2009 620.106dc22
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9. Frank M. White is Professor Emeritus of Mechanical and Ocean
Engineering at the University of Rhode Island. He studied at
Georgia Tech and M.I.T. In 1966 he helped found, at URI, the rst
department of ocean engineering in the country. Known primarily as
a teacher and writer, he has received eight teaching awards and has
written four textbooks on uid mechanics and heat transfer. From
1979 to 1990 he was editor-in-chief of the ASME Journal of Fluids
Engineering and then served from 1991 to 1997 as chairman of the
ASME Board of Editors and of the Publications Committee. He is a
Fellow of ASME and in 1991 received the ASME Fluids Engineering
Award. He lives with his wife, Jeanne, in Narragansett, Rhode
Island. v About the Author whi29346_fm_i-xvi.qxd 12/14/09 7:09PM
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11. Preface xi Chapter 1 Introduction 3 1.1 Preliminary Remarks
3 1.2 History and Scope of Fluid Mechanics 4 1.3 Problem-Solving
Techniques 6 1.4 The Concept of a Fluid 6 1.5 The Fluid as a
Continuum 8 1.6 Dimensions and Units 9 1.7 Properties of the
Velocity Field 17 1.8 Thermodynamic Properties of a Fluid 18 1.9
Viscosity and Other Secondary Properties 25 1.10 Basic Flow
Analysis Techniques 40 1.11 Flow Patterns: Streamlines,
Streaklines, and Pathlines 41 1.12 The Engineering Equation Solver
46 1.13 Uncertainty in Experimental Data 46 1.14 The Fundamentals
of Engineering (FE) Examination 48 Problems 49 Fundamentals of
Engineering Exam Problems 57 Comprehensive Problems 58 References
61 Chapter 2 Pressure Distribution in a Fluid 65 2.1 Pressure and
Pressure Gradient 65 2.2 Equilibrium of a Fluid Element 67 2.3
Hydrostatic Pressure Distributions 68 2.4 Application to Manometry
75 2.5 Hydrostatic Forces on Plane Surfaces 78 2.6 Hydrostatic
Forces on Curved Surfaces 86 2.7 Hydrostatic Forces in Layered
Fluids 89 2.8 Buoyancy and Stability 91 2.9 Pressure Distribution
in Rigid-Body Motion 97 2.10 Pressure Measurement 105 Summary 109
Problems 109 Word Problems 132 Fundamentals of Engineering Exam
Problems 133 Comprehensive Problems 134 Design Projects 135
References 136 Chapter 3 Integral Relations for a Control Volume
139 3.1 Basic Physical Laws of Fluid Mechanics 139 3.2 The Reynolds
Transport Theorem 143 3.3 Conservation of Mass 150 3.4 The Linear
Momentum Equation 155 3.5 Frictionless Flow: The Bernoulli Equation
169 3.6 The Angular Momentum Theorem 178 3.7 The Energy Equation
184 Summary 195 Problems 195 Word Problems 224 Fundamentals of
Engineering Exam Problems 224 Comprehensive Problems 226 Design
Project 227 References 227 Contents vii whi29346_fm_i-xvi.qxd
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12. Chapter 4 Differential Relations for Fluid Flow 229 4.1 The
Acceleration Field of a Fluid 230 4.2 The Differential Equation of
Mass Conservation 232 4.3 The Differential Equation of Linear
Momentum 238 4.4 The Differential Equation of Angular Momentum 244
4.5 The Differential Equation of Energy 246 4.6 Boundary Conditions
for the Basic Equations 249 4.7 The Stream Function 253 4.8
Vorticity and Irrotationality 261 4.9 Frictionless Irrotational
Flows 263 4.10 Some Illustrative Incompressible Viscous Flows 268
Summary 276 Problems 277 Word Problems 288 Fundamentals of
Engineering Exam Problems 288 Comprehensive Problems 289 References
290 Chapter 5 Dimensional Analysis and Similarity 293 5.1
Introduction 298 5.2 The Principle of Dimensional Homogeneity 296
5.3 The Pi Theorem 302 5.4 Nondimensionalization of the Basic
Equations 312 5.5 Modeling and Its Pitfalls 321 Summary 333
Problems 333 Word Problems 342 Fundamentals of Engineering Exam
Problems 342 Comprehensive Problems 343 Design Projects 344
References 344 Chapter 6 Viscous Flow in Ducts 347 6.1 Reynolds
Number Regimes 347 6.2 Internal versus External Viscous Flow 352
6.3 Head LossThe Friction Factor 355 6.4 Laminar Fully Developed
Pipe Flow 357 6.5 Turbulence Modeling 359 6.6 Turbulent Pipe Flow
365 6.7 Four Types of Pipe Flow Problems 373 6.8 Flow in
Noncircular Ducts 379 6.9 Minor or Local Losses in Pipe Systems 388
6.10 Multiple-Pipe Systems 397 6.11 Experimental Duct Flows:
Diffuser Performance 403 6.12 Fluid Meters 408 Summary 429 Problems
430 Word Problems 448 Fundamentals of Engineering Exam Problems 449
Comprehensive Problems 450 Design Projects 452 References 453
Chapter 7 Flow Past Immersed Bodies 457 7.1 Reynolds Number and
Geometry Effects 457 7.2 Momentum Integral Estimates 461 7.3 The
Boundary Layer Equations 464 7.4 The Flat-Plate Boundary Layer 467
7.5 Boundary Layers with Pressure Gradient 476 7.6 Experimental
External Flows 482 Summary 509 Problems 510 Word Problems 523
Fundamentals of Engineering Exam Problems 524 Comprehensive
Problems 524 Design Project 525 References 526 Chapter 8 Potential
Flow and Computational Fluid Dynamics 529 8.1 Introduction and
Review 529 8.2 Elementary Plane Flow Solutions 532 8.3
Superposition of Plane Flow Solutions 539 8.4 Plane Flow Past
Closed-Body Shapes 545 8.5 Other Plane Potential Flows 555 8.6
Images 559 8.7 Airfoil Theory 562 8.8 Axisymmetric Potential Flow
574 8.9 Numerical Analysis 579 viii Contents whi29346_fm_i-xvi.qxd
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13. Summary 593 Problems 594 Word Problems 604 Comprehensive
Problems 605 Design Projects 606 References 606 Chapter 9
Compressible Flow 609 9.1 Introduction: Review of Thermodynamics
609 9.2 The Speed of Sound 614 9.3 Adiabatic and Isentropic Steady
Flow 616 9.4 Isentropic Flow with Area Changes 622 9.5 The Normal
Shock Wave 629 9.6 Operation of Converging and Diverging Nozzles
637 9.7 Compressible Duct Flow with Friction 642 9.8 Frictionless
Duct Flow with Heat Transfer 654 9.9 Two-Dimensional Supersonic
Flow 659 9.10 Prandtl-Meyer Expansion Waves 669 Summary 681
Problems 682 Word Problems 695 Fundamentals of Engineering Exam
Problems 696 Comprehensive Problems 696 Design Projects 698
References 698 Chapter 10 Open-Channel Flow 701 10.1 Introduction
701 10.2 Uniform Flow: The Chzy Formula 707 10.3 Efcient
Uniform-Flow Channels 712 10.4 Specic Energy: Critical Depth 714
10.5 The Hydraulic Jump 722 10.6 Gradually Varied Flow 726 10.7
Flow Measurement and Control by Weirs 734 Summary 741 Problems 741
Word Problems 754 Fundamentals of Engineering Exam Problems 754
Comprehensive Problems 754 Design Projects 756 References 756
Chapter 11 Turbomachinery 759 11.1 Introduction and Classication
759 11.2 The Centrifugal Pump 762 11.3 Pump Performance Curves and
Similarity Rules 768 11.4 Mixed- and Axial-Flow Pumps: The Specic
Speed 778 11.5 Matching Pumps to System Characteristics 785 11.6
Turbines 793 Summary 807 Problems 807 Word Problems 820
Comprehensive Problems 820 Design Project 822 References 822
Appendix A Physical Properties of Fluids 824 Appendix B
Compressible Flow Tables 829 Appendix C Conversion Factors 836
Appendix D Equations of Motion in Cylindrical Coordinates 838
Answers to Selected Problems 840 Index 847 Contents ix
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14. This page intentionally left blank
15. The seventh edition of Fluid Mechanics sees some additions
and deletions but no philosophical change. The basic outline of
eleven chapters, plus appendices, remains the same. The triad of
integral, differential, and experimental approaches is retained.
Many problem exercises, and some fully worked examples, have been
changed. The informal, student-oriented style is retained. A number
of new photographs and gures have been added. Many new references
have been added, for a total of 435. The writer is a rm believer in
further reading, especially in the postgraduate years. The total
number of problem exercises continues to increase, from 1089 in the
rst edition, to 1675 in this seventh edition. There are
approximately 20 new problems added to each chapter. Most of these
are basic end-of-chapter problems, classied according to topic.
There are also Word Problems, multiple-choice Fundamentals of
Engineering Problems, Comprehensive Problems, and Design Projects.
The appendix lists approximately 700 Answers to Selected Problems.
The example problems are structured in the text to follow the
sequence of recom- mended steps outlined in Sect. 1.3,
Problem-Solving Techniques. The Engineering Equation Solver (EES)
is available with the text and continues its role as an attractive
tool for uid mechanics and, indeed, other engineering prob- lems.
Not only is it an excellent solver, but it also contains
thermophysical proper- ties, publication-quality plotting, units
checking, and many mathematical functions, including numerical
integration. The author is indebted to Sanford Klein and William
Beckman, of the University of Wisconsin, for invaluable and
continuous help in preparing and updating EES for use in this text.
For newcomers to EES, a brief guide to its use is found on this
books website. There are some revisions in each chapter. Chapter 1
has added material on the history of late 20th century uid
mechanics, notably the development of Computational Fluid Dynamics.
A very brief introduction to the acceleration eld has been added.
Boundary conditions for slip ow have been added. There is more
discussion of the speed of sound in liquids. The treatment of
thermal conductivity has been moved to Chapter 4. Content Changes
Learning Tools General Approach xi Preface whi29346_fm_i-xvi.qxd
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16. Chapter 2 introduces a photo, discussion, and new problems
for the deep ocean submersible vehicle, ALVIN. The density
distribution in the troposphere is now given explicitly. There are
brief remarks on the great Greek mathematician, Archimedes. Chapter
3 has been substantially revised. Reviewers wanted Bernoullis
equation moved ahead of angular velocity and energy, to follow
linear momentum. I did this and followed their specic improvements,
but truly extensive renumbering and rear- ranging was necessary.
Pressure and velocity conditions at a tank surface have an improved
discussion. A brief history of the control volume has been added.
There is a better treatment of the relation between Bernoullis
equation and the energy equa- tion. There is a new discussion of
stagnation, static and dynamic pressures, and boundary conditions
at a jet exit. Chapter 4 has a great new opener: CFD for ow past a
spinning soccer ball. The total time derivative of velocity is now
written out in full. Fouriers Law, and its appli- cation to the
differential energy equation, have been moved here from Chapter 1.
There are 21 new problems, including several slip-ow analyses. The
Chapter 5 introduction expands on the effects of Mach number and
Froude number, instead of concentrating only on the Reynolds
number. Ipsens method, which the writer admires, is retained as an
alternative to the pi theorem. The new opener, a giant
disk-band-gap parachute, allows for several new dimensional
analysis problems. Chapter 6 has a new formula for entrance length
in turbulent duct ow, sent to me by two different researchers.
There is a new problem describing the ow in a fuel cell. The new
opener, the Trans-Alaska Pipeline, allows for several innovative
prob- lems, including a related one on the proposed Alaska-Canada
natural gas pipeline. Chapter 7 has an improved description of
turbulent ow past a at plate, plus recent reviews of progress in
turbulence modeling with CFD. Two new aerodynamic advances are
reported: the Finaish-Witherspoon redesign of the Kline-Fogelman
air- foil and the increase in stall angle achieved by tubercles
modeled after a humpback whale. The new Transition ying car, which
had a successful maiden ight in 2009, leads to a number of good
problem assignments. Two other photos, Rocket Man over the Alps,
and a cargo ship propelled by a kite, also lead to interesting new
problems. Chapter 8 is essentially unchanged, except for a bit more
discussion of modern CFD software. The Transition autocar, featured
in Chapter 7, is attacked here by aerodynamic theory, including
induced drag. Chapter 9 beneted from reviewer improvement. Figure
9.7, with its 30-year-old curve-ts for the area ratio, has been
replaced with ne-gridded curves for the area- change properties.
The curve-ts are gone, and Mach numbers follow nicely from Fig. 9.7
and either Excel or EES. New Trends in Aeronautics presents the
X-43 Scram- jet airplane, which generates several new problem
assignments. Data for the proposed Alaska-to-Canada natural gas
pipeline provides a different look at frictional choking. Chapter
10 is basically the same, except for new photos of both plane and
circu- lar hydraulic jumps, plus a tidal bore, with their
associated problem assignments. Chapter 11 has added a section on
the performance of gas turbines, with applica- tion to turbofan
aircraft engines. The section on wind turbines has been updated,
with new data and photos. A wind-turbine-driven vehicle, which can
easily move directly into the wind, has inspired new problem
assignments. xii Preface whi29346_fm_i-xvi.qxd 12/14/09 7:09PM Page
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17. Appendix A has new data on the bulk modulus of various
liquids. Appendix B, Compressible Flow Tables, has been shortened
by using coarser increments (0.1) in Mach number. Tables with much
smaller increments are now on the bookswebsite. Appendix E,
Introduction to EES, has been deleted and moved to the website, on
the theory that most students are now quite familiar with EES. A
number of supplements are available to students and/or instructors
at the text website www.mhhe.com/white7e. Students have access to a
Student Study Guide developed by Jerry Dunn of Texas A&M
University. They are also able to utilize Engineering Equation
Solver (EES), uid mechanics videos developed by Gary Set- tles of
Pennsylvania State University, and CFD images and animations
prepared by Fluent Inc. Also available to students are Fundamentals
of Engineering (FE) Exam quizzes, prepared by Edward Anderson of
Texas Tech University. Instructors may obtain a series of
PowerPoint slides and images, plus the full Solu- tions Manual, in
PDF format. The Solutions Manual provides complete and detailed
solutions, including problem statements and artwork, to the
end-of-chapter problems. It may be photocopied for posting or
preparing transparencies for the classroom. Instruc- tors can also
obtain access to C.O.S.M.O.S. for the seventh edition. C.O.S.M.O.S.
is a Complete Online Solutions Manual Organization System
instructors can use to create exams and assignments, create custom
content, and edit supplied problems and solutions. Ebooks are an
innovative way for students to save money and create a greener
environment at the same time. An ebook can save students about half
the cost of a traditional textbook and offers unique features like
a powerful search engine, high- lighting, and the ability to share
notes with classmates using ebooks. McGraw-Hill offers this text as
an ebook. To talk about the ebook options, con- tact your
McGraw-Hill sales rep or visit the site www.coursesmart.com to
learn more. Online Supplements Preface xiii Electronic Textbook
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18. As usual, so many people have helped me that I may fail to
list them all. Sheldon Green of the University of British Columbia,
Gordon Holloway of the University of New Brunswick, Sukanta K. Dash
of The Indian Institute of Technology at Kharagpur, and Pezhman
Shirvanian of Ford Motor Co. gave many helpful suggestions. Hubert
Chanson of the University of Queensland, Frank J. Cunha of
Pratt&Whitney, Samuel Schweighart of Terrafugia Inc., Mark
Spear of the Woods Hole Oceanographic Institution, Keith Hanna of
ANSYS Inc., Elena Mejia of the Jet Propulsion Laboratory, Anne
Staack of SkySails, Inc., and Ellen Emerson White provided great
new images. Samuel S. Sih of Walla Walla College, Timothy Singler
of SUNY Binghamton, Saeed Moaveni of Minnesota State University,
and John Borg of Marquette University were especially helpful with
the solutions manual. The following prereviewers gave many
excellent suggestions for improving the manuscript: Rolando Bravo
of Southern Illinois University; Joshua B. Kollat of Penn State
University; Daniel Maynes of Brigham Young University; Joseph
Schaefer of Iowa State University; and Xiangchun Xuan of Clemson
University. In preparation, the writer got stuck on Chapter 3 but
was rescued by the following reviewers: Serhiy Yarusevych of the
University of Waterloo; H. Pirouz Kavehpour and Jeff Eldredge of
the University of California, Los Angeles; Rayhaneh Akhavan of the
University of Michigan; Soyoung Steve Cha of the University of
Illinois, Chicago; Georgia Richardson of the University of Alabama;
Krishan Bhatia of Rowan Univer- sity; Hugh Coleman of the
University of Alabama-Huntsville; D.W. Ostendorf of the University
of Massachusetts; and Donna Meyer of the University of Rhode
Island. The writer continues to be indebted to many others who have
reviewed this book over the various years and editions. Many other
reviewers and correspondents gave good suggestions, encouragement,
corrections, and materials: Elizabeth J. Kenyon of MathWorks; Juan
R. Cruz of NASA Langley Research Center; LiKai Li of University of
Science and Technology of China; Tom Robbins of National
Instruments; Tapan K. Sengupta of the Indian Institute of
Technology at Kanpur; Paulo Vatavuk of Unicamp University; Andris
Skattebo of Scand- power A/S; Jeffrey S. Allen of Michigan
Technological University; Peter R. Spedding of Queens University,
Belfast, Northern Ireland; Iskender Sahin of Western Michigan
University; Michael K. Dailey of General Motors; Cristina L. Archer
of Stanford University; Paul F. Jacobs of Technology Development
Associates; Rebecca Cullion- Webb of the University of Colorado at
Colorado Springs; Debendra K. Das of the Uni- versity of Alaska
Fairbanks; Kevin OSullivan and Matthew Lutts of the Associated
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19. Press; Lennart Luttig and Nina Koliha of REpower Systems
AG, Hamburg, Germany; Chi-Yang Cheng of ANSYS Inc.; Debabrata
Dasgupta of The Indian Institute of Tech- nology at Kharagpur;
Fabian Anselmet of the Institut de Recherche sur les Phenomenes
Hors Equilibre, Marseilles; David Chelidze, Richard Lessmann, Donna
Meyer, Arun Shukla, Peter Larsen, and Malcolm Spaulding of the
University of Rhode Island; Craig Swanson of Applied Science
Associates, Inc.; Jim Smay of Oklahoma State University; Deborah
Pence of Oregon State University; Eric Braschoss of Philadelphia,
PA.; and Dale Hart of Louisiana Tech University. The McGraw-Hill
staff was, as usual, enormously helpful. Many thanks are due to
Bill Stenquist, Lora Kalb-Neyens, Curt Reynolds, John Leland, Jane
Mohr, Brenda Rolwes, as well as all those who worked on previous
editions. Finally, the continuing support and encouragement of my
wife and family are, as always, much appreciated. Special thanks
are due to our dog, Sadie, and our cats, Cole and Kerry.
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22. Hurricane Rita in the Gulf of Mexico, Sept. 22, 2005. Rita
made landfall at the Texas-Louisiana border and caused billions of
dollars in wind and ooding damage. Though more dramatic than
typical applications in this text, Rita is a true uid ow, strongly
inuenced by the earths rota- tion and the ocean temperature. (Photo
courtesy of NASA.) 2 whi29346_ch01_002-063.qxd 10/14/09 19:57 Page
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23. 3 Fluid mechanics is the study of uids either in motion
(uid dynamics) or at rest (uid statics). Both gases and liquids are
classied as uids, and the number of uid engi- neering applications
is enormous: breathing, blood ow, swimming, pumps, fans, tur-
bines, airplanes, ships, rivers, windmills, pipes, missiles,
icebergs, engines, lters, jets, and sprinklers, to name a few. When
you think about it, almost everything on this planet either is a
uid or moves within or near a uid. The essence of the subject of
uid ow is a judicious compromise between theory and experiment.
Since uid ow is a branch of mechanics, it satises a set of well-
documented basic laws, and thus a great deal of theoretical
treatment is available. How- ever, the theory is often frustrating
because it applies mainly to idealized situations, which may be
invalid in practical problems. The two chief obstacles to a
workable the- ory are geometry and viscosity. The basic equations
of uid motion (Chap. 4) are too difcult to enable the analyst to
attack arbitrary geometric congurations. Thus most textbooks
concentrate on at plates, circular pipes, and other easy
geometries. It is pos- sible to apply numerical computer techniques
to complex geometries, and specialized textbooks are now available
to explain the new computational uid dynamics (CFD) approximations
and methods [14].1 This book will present many theoretical results
while keeping their limitations in mind. The second obstacle to a
workable theory is the action of viscosity, which can be neglected
only in certain idealized ows (Chap. 8). First, viscosity increases
the dif- culty of the basic equations, although the boundary-layer
approximation found by Ludwig Prandtl in 1904 (Chap. 7) has greatly
simplied viscous-ow analyses. Sec- ond, viscosity has a
destabilizing effect on all uids, giving rise, at frustratingly
small velocities, to a disorderly, random phenomenon called
turbulence. The theory of tur- bulent ow is crude and heavily
backed up by experiment (Chap. 6), yet it can be quite serviceable
as an engineering estimate. This textbook only introduces the
standard experimental correlations for turbulent time-mean ow.
Meanwhile, there are advanced texts on both time-mean turbulence
and turbulence modeling [5, 6] and on the newer, computer-intensive
direct numerical simulation (DNS) of uctuating turbulence [7, 8].
1.1 Preliminary Remarks Chapter 1 Introduction 1 Numbered
references appear at the end of each chapter.
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24. Thus there is theory available for uid ow problems, but in
all cases it should be backed up by experiment. Often the
experimental data provide the main source of information about
specic ows, such as the drag and lift of immersed bodies (Chap. 7).
Fortunately, uid mechanics is a highly visual subject, with good
instru- mentation [911], and the use of dimensional analysis and
modeling concepts (Chap. 5) is widespread. Thus experimentation
provides a natural and easy comple- ment to the theory. You should
keep in mind that theory and experiment should go hand in hand in
all studies of uid mechanics. Like most scientic disciplines, uid
mechanics has a history of erratically occurring early
achievements, then an intermediate era of steady fundamental
discoveries in the eighteenth and nineteenth centuries, leading to
the twenty-rst-century era of modern practice, as we
self-centeredly term our limited but up-to-date knowledge. Ancient
civilizations had enough knowledge to solve certain ow problems.
Sailing ships with oars and irrigation systems were both known in
prehistoric times. The Greeks pro- duced quantitative information.
Archimedes and Hero of Alexandria both postulated the parallelogram
law for addition of vectors in the third century B.C. Archimedes
(285212 B.C.) formulated the laws of buoyancy and applied them to
oating and sub- merged bodies, actually deriving a form of the
differential calculus as part of the analysis. The Romans built
extensive aqueduct systems in the fourth century B.C. but left no
records showing any quantitative knowledge of design principles.
From the birth of Christ to the Renaissance there was a steady
improvement in the design of such ow systems as ships and canals
and water conduits but no recorded evidence of fundamental
improvements in ow analysis. Then Leonardo da Vinci (14521519)
stated the equation of conservation of mass in one-dimensional
steady ow. Leonardo was an excellent experimentalist, and his notes
contain accurate descriptions of waves, jets, hydraulic jumps, eddy
formation, and both low-drag (streamlined) and high-drag
(parachute) designs. A Frenchman, Edme Mariotte (16201684), built
the rst wind tunnel and tested models in it. Problems involving the
momentum of uids could nally be analyzed after Isaac Newton
(16421727) postulated his laws of motion and the law of viscosity
of the lin- ear uids now called newtonian. The theory rst yielded
to the assumption of a per- fect or frictionless uid, and
eighteenth-century mathematicians (Daniel Bernoulli, Leonhard
Euler, Jean dAlembert, Joseph-Louis Lagrange, and Pierre-Simon
Laplace) produced many beautiful solutions of frictionless-ow
problems. Euler, Fig. 1.1, devel- oped both the differential
equations of motion and their integrated form, now called the
Bernoulli equation. DAlembert used them to show his famous paradox:
that a body immersed in a frictionless uid has zero drag. These
beautiful results amounted to overkill, since perfect-uid
assumptions have very limited application in practice and most
engineering ows are dominated by the effects of viscosity.
Engineers began to reject what they regarded as a totally
unrealistic theory and developed the science of hydraulics, relying
almost entirely on experiment. Such experimentalists as Chzy,
Pitot, Borda, Weber, Francis, Hagen, Poiseuille, Darcy, Manning,
Bazin, and Weisbach produced data on a variety of ows such as open
channels, ship resistance, pipe ows, waves, and turbines. All too
often the data were used in raw form without regard to the
fundamental physics of ow. 1.2 History and Scope of Fluid Mechanics
4 Chapter 1 Introduction Fig. 1.1 Leonhard Euler (1707 1783) was
the greatest mathemati- cian of the eighteenth century and used
Newtons calculus to develop and solve the equations of motion of
inviscid ow. He published over 800 books and papers. [Courtesy of
the School of Mathematics and Statistics, University of St Andrew,
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25. At the end of the nineteenth century, unication between
experimental hydraulics and theoretical hydrodynamics nally began.
William Froude (18101879) and his son Robert (18461924) developed
laws of model testing; Lord Rayleigh (18421919) proposed the
technique of dimensional analysis; and Osborne Reynolds (18421912)
published the classic pipe experiment in 1883, which showed the
importance of the dimensionless Reynolds number named after him.
Meanwhile, viscous-ow theory was available but unexploited, since
Navier (17851836) and Stokes (18191903) had successfully added
newtonian viscous terms to the equations of motion. The result- ing
Navier-Stokes equations were too difcult to analyze for arbitrary
ows. Then, in 1904, a German engineer, Ludwig Prandtl (18751953),
Fig. 1.2, published perhaps the most important paper ever written
on uid mechanics. Prandtl pointed out that uid ows with small
viscosity, such as water ows and airows, can be divided into a thin
vis- cous layer, or boundary layer, near solid surfaces and
interfaces, patched onto a nearly inviscid outer layer, where the
Euler and Bernoulli equations apply. Boundary-layer theory has
proved to be a very important tool in modern ow analysis. The
twentieth- century foundations for the present state of the art in
uid mechanics were laid in a series of broad-based experiments and
theories by Prandtl and his two chief friendly competi- tors,
Theodore von Krmn (18811963) and Sir Geoffrey I. Taylor (18861975).
Many of the results sketched here from a historical point of view
will, of course, be discussed in this textbook. More historical
details can be found in Refs. 12 to 14. The second half of the
twentieth century introduced a new tool: Computational Fluid
Dynamics (CFD). The earliest paper on the subject known to this
writer was by A. Thom in 1933 [47], a laborious, but accurate, hand
calculation of ow past a cylinder at low Reynolds numbers.
Commercial digital computers became available in the 1950s, and
personal computers in the 1970s, bringing CFD into adulthood. A
legendary rst textbook was by Patankar [3]. Presently, with
increases in computer speed and memory, almost any laminar ow can
be modeled accurately. Turbulent ow is still calculated with
empirical models, but Direct Numerical Simulation [7, 8] is
possible for low Reynolds numbers. Another ve orders of magnitude
in computer speed are needed before general turbulent ows can be
calculated. That may not be possible, due to size limits of nano-
and pico-elements. But, if general DNS devel- ops, Gad-el-Hak [14]
raises the prospect of a shocking future: all of uid mechanics
reduced to a black box, with no real need for teachers,
researchers, writers, or uids engineers. Since the earth is 75
percent covered with water and 100 percent covered with air, the
scope of uid mechanics is vast and touches nearly every human
endeavor. The sciences of meteorology, physical oceanography, and
hydrology are concerned with naturally occurring uid ows, as are
medical studies of breathing and blood circu- lation. All
transportation problems involve uid motion, with well-developed
spe- cialties in aerodynamics of aircraft and rockets and in naval
hydrodynamics of ships and submarines. Almost all our electric
energy is developed either from water ow or from steam ow through
turbine generators. All combustion problems involve uid motion as
do the more classic problems of irrigation, ood control, water
supply, sewage disposal, projectile motion, and oil and gas
pipelines. The aim of this book is to present enough fundamental
concepts and practical applications in uid mechan- ics to prepare
you to move smoothly into any of these specialized elds of the sci-
ence of owand then be prepared to move out again as new
technologies develop. 1.2 History and Scope of Fluid Mechanics 5
Fig. 1.2 Ludwig Prandtl (1875 1953), often called the father of
modern uid mechanics [15], developed boundary layer theory and many
other innovative analy- ses. He and his students were pioneers in
ow visualization techniques. [Aufnahme von Fr. Struckmeyer,
Gottingen, courtesy AIP Emilio Segre Visual Archives, Lande
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26. Fluid ow analysis is packed with problems to be solved. The
present text has more than 1700 problem assignments. Solving a
large number of these is a key to learning the subject. One must
deal with equations, data, tables, assumptions, unit systems, and
solution schemes. The degree of difculty will vary, and we urge you
to sample the whole spectrum of assignments, with or without the
Answers in the Appendix. Here are the recommended steps for problem
solution: 1. Read the problem and restate it with your summary of
the results desired. 2. From tables or charts, gather the needed
property data: density, viscosity, etc. 3. Make sure you understand
what is asked. Students are apt to answer the wrong questionfor
example, pressure instead of pressure gradient, lift force instead
of drag force, or mass ow instead of volume ow. Read the problem
carefully. 4. Make a detailed, labeled sketch of the system or
control volume needed. 5. Think carefully and list your
assumptions. You must decide if the ow is steady or unsteady,
compressible or incompressible, viscous or inviscid, and whether a
control volume or partial differential equations are needed. 6.
Find an algebraic solution if possible. Then, if a numerical value
is needed, use either the SI or BG unit systems, to be reviewed in
Sec. 1.6. 7. Report your solution, labeled, with the proper units
and the proper number of signicant gures (usually two or three)
that the data uncertainty allows. We shall follow these steps,
where appropriate, in our example problems. From the point of view
of uid mechanics, all matter consists of only two states, uid and
solid. The difference between the two is perfectly obvious to the
layperson, and it is an interesting exercise to ask a layperson to
put this difference into words. The tech- nical distinction lies
with the reaction of the two to an applied shear or tangential
stress. A solid can resist a shear stress by a static deection; a
uid cannot . Any shear stress applied to a uid, no matter how
small, will result in motion of that uid. The uid moves and deforms
continuously as long as the shear stress is applied. As a
corollary, we can say that a uid at rest must be in a state of zero
shear stress, a state often called the hydrostatic stress condition
in structural analysis. In this condition, Mohrs circle for stress
reduces to a point, and there is no shear stress on any plane cut
through the element under stress. Given this denition of a uid,
every layperson also knows that there are two classes of uids,
liquids and gases. Again the distinction is a technical one
concern- ing the effect of cohesive forces. A liquid, being
composed of relatively close-packed molecules with strong cohesive
forces, tends to retain its volume and will form a free surface in
a gravitational eld if unconned from above. Free-surface ows are
dom- inated by gravitational effects and are studied in Chaps. 5
and 10. Since gas mole- cules are widely spaced with negligible
cohesive forces, a gas is free to expand until it encounters
conning walls. A gas has no denite volume, and when left to itself
without connement, a gas forms an atmosphere that is essentially
hydrostatic. The hydrostatic behavior of liquids and gases is taken
up in Chap. 2. Gases cannot form a free surface, and thus gas ows
are rarely concerned with gravitational effects other than
buoyancy. 1.4 The Concept of a Fluid 1.3 Problem-Solving Techniques
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27. Figure 1.3 illustrates a solid block resting on a rigid
plane and stressed by its own weight. The solid sags into a static
deection, shown as a highly exaggerated dashed line, resisting
shear without ow. A free-body diagram of element A on the side of
the block shows that there is shear in the block along a plane cut
at an angle through A. Since the block sides are unsupported,
element A has zero stress on the left and right sides and
compression stress p on the top and bottom. Mohrs circle does not
reduce to a point, and there is nonzero shear stress in the block.
By contrast, the liquid and gas at rest in Fig. 1.3 require the
supporting walls in order to eliminate shear stress. The walls
exert a compression stress of p and reduce Mohrs circle to a point
with zero shear everywherethat is, the hydrostatic condi- tion. The
liquid retains its volume and forms a free surface in the
container. If the walls are removed, shear develops in the liquid
and a big splash results. If the container is tilted, shear again
develops, waves form, and the free surface seeks a horizontal con-
guration, pouring out over the lip if necessary. Meanwhile, the gas
is unrestrained and expands out of the container, lling all
available space. Element A in the gas is also hydrostatic and
exerts a compression stress p on the walls. 1.4 The Concept of a
Fluid 7 Static deflection Free surface Hydrostatic condition
LiquidSolid A A A (a) (c) (b) (d) 0 0 A A Gas (1) p p p p p = 0 2 1
= p = p 1 Fig. 1.3 A solid at rest can resist shear. (a) Static
deection of the solid; (b) equilibrium and Mohrs circle for solid
element A. A uid cannot resist shear. (c) Containing walls are
needed; (d) equilibrium and Mohrs circle for uid element A.
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28. In the previous discussion, clear decisions could be made
about solids, liquids, and gases. Most engineering uid mechanics
problems deal with these clear casesthat is, the common liquids,
such as water, oil, mercury, gasoline, and alcohol, and the com-
mon gases, such as air, helium, hydrogen, and steam, in their
common temperature and pressure ranges. There are many borderline
cases, however, of which you should be aware. Some apparently solid
substances such as asphalt and lead resist shear stress for short
periods but actually deform slowly and exhibit denite uid behavior
over long periods. Other substances, notably colloid and slurry
mixtures, resist small shear stresses but yield at large stress and
begin to ow as uids do. Specialized textbooks are devoted to this
study of more general deformation and flow, a field called rheology
[16]. Also, liquids and gases can coexist in two-phase mixtures,
such as steamwater mixtures or water with entrapped air bubbles.
Specialized textbooks pres- ent the analysis of such multiphase ows
[17]. Finally, in some situations the distinc- tion between a
liquid and a gas blurs. This is the case at temperatures and
pressures above the so-called critical point of a substance, where
only a single phase exists, pri- marily resembling a gas. As
pressure increases far above the critical point, the gaslike
substance becomes so dense that there is some resemblance to a
liquid and the usual thermodynamic approximations like the
perfect-gas law become inaccurate. The criti- cal temperature and
pressure of water are Tc 647 K and pc 219 atm (atmosphere2 ) so
that typical problems involving water and steam are below the
critical point. Air, being a mixture of gases, has no distinct
critical point, but its principal component, nitrogen, has Tc 126 K
and pc 34 atm. Thus typical problems involving air are in the range
of high temperature and low pressure where air is distinctly and
denitely a gas. This text will be concerned solely with clearly
identiable liquids and gases, and the borderline cases just
discussed will be beyond our scope. We have already used technical
terms such as uid pressure and density without a rig- orous
discussion of their denition. As far as we know, uids are
aggregations of mol- ecules, widely spaced for a gas, closely
spaced for a liquid. The distance between mol- ecules is very large
compared with the molecular diameter. The molecules are not xed in
a lattice but move about freely relative to each other. Thus uid
density, or mass per unit volume, has no precise meaning because
the number of molecules occupying a given volume continually
changes. This effect becomes unimportant if the unit volume is
large compared with, say, the cube of the molecular spacing, when
the number of molecules within the volume will remain nearly
constant in spite of the enormous interchange of particles across
the boundaries. If, however, the chosen unit volume is too large,
there could be a noticeable variation in the bulk aggregation of
the particles. This situation is illustrated in Fig. 1.4, where the
density as calculated from molecular mass m within a given volume
is plotted versus the size of the unit volume. There is a limiting
vol- ume * below which molecular variations may be important and
above which aggre- gate variations may be important. The density of
a uid is best dened as (1.1) lim S* m 1.5 The Fluid as a Continuum
8 Chapter 1 Introduction 2 One atmosphere equals 2116 lbf/ft2
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29. The limiting volume * is about 109 mm3 for all liquids and
for gases at atmo- spheric pressure. For example, 109 mm3 of air at
standard conditions contains approx- imately 3 107 molecules, which
is sufcient to dene a nearly constant density according to Eq.
(1.1). Most engineering problems are concerned with physical dimen-
sions much larger than this limiting volume, so that density is
essentially a point func- tion and uid properties can be thought of
as varying continually in space, as sketched in Fig. 1.4a. Such a
uid is called a continuum, which simply means that its varia- tion
in properties is so smooth that differential calculus can be used
to analyze the substance. We shall assume that continuum calculus
is valid for all the analyses in this book. Again there are
borderline cases for gases at such low pressures that molec- ular
spacing and mean free path3 are comparable to, or larger than, the
physical size of the system. This requires that the continuum
approximation be dropped in favor of a molecular theory of rareed
gas ow [18]. In principle, all uid mechanics problems can be
attacked from the molecular viewpoint, but no such attempt will be
made here. Note that the use of continuum calculus does not
preclude the possibility of discon- tinuous jumps in uid properties
across a free surface or uid interface or across a shock wave in a
compressible uid (Chap. 9). Our calculus in analyzing uid ow must
be exible enough to handle discontinuous boundary conditions. A
dimension is the measure by which a physical variable is expressed
quantitatively. A unit is a particular way of attaching a number to
the quantitative dimension. Thus length is a dimension associated
with such variables as distance, displacement, width, deection, and
height, while centimeters and inches are both numerical units for
expressing length. Dimension is a powerful concept about which a
splendid tool called dimensional analysis has been developed (Chap.
5), while units are the numerical quantity that the customer wants
as the nal answer. In 1872 an international meeting in France
proposed a treaty called the Metric Con- vention, which was signed
in 1875 by 17 countries including the United States. It was an
improvement over British systems because its use of base 10 is the
foundation of our number system, learned from childhood by all.
Problems still remained because 1.6 Dimensions and Units 1.6
Dimensions and Units 9 Microscopic uncertainty Macroscopic
uncertainty 0 1200 * 10-9 mm3 Elemental volume Region containing
fluid = 1000 kg/m3 = 1100 = 1200 = 1300 (a) (b) Fig. 1.4 The limit
denition of continuum uid density: (a) an elemental volume in a uid
region of variable continuum density; (b) calculated density versus
size of the elemental volume. 3 The mean distance traveled by
molecules between collisions (see Prob. P1.5).
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30. even the metric countries differed in their use of
kiloponds instead of dynes or newtons, kilograms instead of grams,
or calories instead of joules. To standardize the metric system, a
General Conference of Weights and Measures, attended in 1960 by 40
countries, proposed the International System of Units (SI). We are
now undergo- ing a painful period of transition to SI, an
adjustment that may take many more years to complete. The
professional societies have led the way. Since July 1, 1974, SI
units have been required by all papers published by the American
Society of Mechanical Engineers, and there is a textbook explaining
the SI [19]. The present text will use SI units together with
British gravitational (BG) units. In uid mechanics there are only
four primary dimensions from which all other dimen- sions can be
derived: mass, length, time, and temperature.4 These dimensions and
their units in both systems are given in Table 1.1. Note that the
kelvin unit uses no degree symbol. The braces around a symbol like
{M} mean the dimension of mass. All other variables in uid
mechanics can be expressed in terms of {M}, {L}, {T}, and {}. For
example, acceleration has the dimensions {LT2 }. The most crucial
of these secondary dimensions is force, which is directly related
to mass, length, and time by Newtons second law. Force equals the
time rate of change of momentum or, for constant mass, F ma (1.2)
From this we see that, dimensionally, {F} {MLT 2 }. The use of a
constant of proportionality in Newtons law, Eq. (1.2), is avoided
by dening the force unit exactly in terms of the other basic units.
In the SI system, the basic units are newtons {F}, kilograms {M},
meters {L}, and seconds {T}. We dene 1 newton of force 1 N 1 kg 1
m/s2 The newton is a relatively small force, about the weight of an
apple (0.225 lbf). In addi- tion, the basic unit of temperature { }
in the SI system is the degree Kelvin, K. Use of these SI units (N,
kg, m, s, K) will require no conversion factors in our equations.
In the BG system also, a constant of proportionality in Eq. (1.2)
is avoided by den- ing the force unit exactly in terms of the other
basic units. In the BG system, the basic units are pound-force {F},
slugs {M}, feet {L}, and seconds {T}. We dene 1 pound of force 1
lbf 1 slug 1 ft/s2# The British Gravitational (BG) System # The
International System (SI) Primary Dimensions 10 Chapter 1
Introduction Table 1.1 Primary Dimensions in SI and BG Systems
Primary dimension SI unit BG unit Conversion factor Mass {M}
Kilogram (kg) Slug 1 slug 14.5939 kg Length {L} Meter (m) Foot (ft)
1 ft 0.3048 m Time {T} Second (s) Second (s) 1 s 1 s Temperature {}
Kelvin (K) Rankine (R) 1 K 1.8R 4 If electromagnetic effects are
important, a fth primary dimension must be included, electric
current {I}, whose SI unit is the ampere (A).
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31. One lbf 4.4482 N and approximates the weight of four
apples. We will use the abbreviation lbf for pound-force and lbm
for pound-mass. The slug is a rather hefty mass, equal to 32.174
lbm. The basic unit of temperature {} in the BG system is the
degree Rankine, R. Recall that a temperature difference 1 K 1.8R.
Use of these BG units (lbf, slug, ft, s, R) will require no
conversion factors in our equations. There are other unit systems
still in use. At least one needs no proportionality constant: the
CGS system (dyne, gram, cm, s, K). However, CGS units are too small
for most applications (1 dyne 105 N) and will not be used here. In
the USA, some still use the English Engineering system, (lbf, lbm,
ft, s, R), where the basic mass unit is the pound of mass. Newtons
law (1.2) must be rewritten: (1.3) The constant of proportionality,
gc, has both dimensions and a numerical value not equal to 1.0. The
present text uses only the SI and BG systems and will not solve
prob- lems or examples in the English Engineering system. Because
Americans still use them, a few problems in the text will be stated
in truly awkward units: acres, gallons, ounces, or miles. Your
assignment will be to convert these and solve in the SI or BG
systems. In engineering and science, all equations must be
dimensionally homogeneous, that is, each additive term in an
equation must have the same dimensions. For example, take
Bernoullis incompressible equation, to be studied and used
throughout this text: Each and every term in this equation must
have dimensions of pressure {ML1 T2 }. We will examine the
dimensional homogeneity of this equation in detail in Ex. 1.3. A
list of some important secondary variables in uid mechanics, with
dimensions derived as combinations of the four primary dimensions,
is given in Table 1.2. A more complete list of conversion factors
is given in App. C. p 1 2 V2 gZ constant The Principle of
Dimensional Homogeneity F ma gc , where gc 32.174 ft # lbm lbf # s2
Other Unit Systems 1.6 Dimensions and Units 11 Table 1.2 Secondary
Dimensions in Fluid Mechanics Secondary dimension SI unit BG unit
Conversion factor Area {L2 } m2 ft2 1 m2 10.764 ft2 Volume {L3 } m3
ft3 1 m3 35.315 ft3 Velocity {LT 1 } m/s ft/s 1 ft/s 0.3048 m/s
Acceleration {LT2 } m/s2 ft/s2 1 ft/s2 0.3048 m/s2 Pressure or
stress {ML1 T2 } Pa N/m2 lbf/ft2 1 lbf/ft2 47.88 Pa Angular
velocity {T1 } s1 s1 1 s1 1 s1 Energy, heat, work {ML2 T2 } J N m
ft lbf 1 ft lbf 1.3558 J Power {ML2 T3 } W J/s ft lbf/s 1 ft lbf/s
1.3558 W Density {ML3 } kg/m3 slugs/ft3 1 slug/ft3 515.4 kg/m3
Viscosity {ML1 T1 } kg/(m s) slugs/(ft s) 1 slug/(ft s) 47.88 kg/(m
s) Specic heat {L2 T2 1 } m2 /(s2 K) ft2 /(s2 R) 1 m2 /(s2 K) 5.980
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32. EXAMPLE 1.1 A body weighs 1000 lbf when exposed to a
standard earth gravity g 32.174 ft/s2 . (a) What is its mass in kg?
(b) What will the weight of this body be in N if it is exposed to
the moons standard acceleration gmoon 1.62 m/s2 ? (c) How fast will
the body accelerate if a net force of 400 lbf is applied to it on
the moon or on the earth? Solution We need to nd the (a) mass; (b)
weight on the moon; and (c) acceleration of this body. This is a
fairly simple example of conversion factors for differing unit
systems. No prop- erty data is needed. The example is too low-level
for a sketch. Newtons law (1.2) holds with known weight and
gravitational acceleration. Solve for m: Convert this to kilograms:
m 31.08 slugs (31.08 slugs)(14.5939 kg/slug) 454 kg Ans. (a) The
mass of the body remains 454 kg regardless of its location.
Equation (1.2) applies with a new gravitational acceleration and
hence a new weight: F Wmoon mgmoon (454 kg)(1.62 m/s2 ) 735 N Ans.
(b) This part does not involve weight or gravity or location. It is
simply an application of Newtons law with a known mass and known
force: F 400 lbf ma (31.08 slugs) a Solve for Ans. (c) Comment (c):
This acceleration would be the same on the earth or moon or
anywhere. Many data in the literature are reported in inconvenient
or arcane units suitable only to some industry or specialty or
country. The engineer should convert these data to the SI or BG
system before using them. This requires the systematic application
of conversion factors, as in the following example. EXAMPLE 1.2
Industries involved in viscosity measurement [27, 36] continue to
use the CGS system of units, since centimeters and grams yield
convenient numbers for many uids. The absolute viscosity () unit is
the poise, named after J. L. M. Poiseuille, a French physician who
in 1840 performed pioneering experiments on water ow in pipes; 1
poise 1 g/(cm-s). The kinematic viscosity () unit is the stokes,
named after G. G. Stokes, a British physicist who a 400 lbf 31.08
slugs 12.87 ft s2 a0.3048 m ft b 3.92 m s2 Part (c) Part (b) F W
1000 lbf mg (m)(32.174 ft/s2 ), or m 1000 lbf 32.174 ft/s2 31.08
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33. in 1845 helped develop the basic partial differential
equations of uid momentum; 1 stokes 1 cm2 /s. Water at 20C has 0.01
poise and also 0.01 stokes. Express these results in (a) SI and (b)
BG units. Solution Approach: Systematically change grams to kg or
slugs and change centimeters to meters or feet. Property values:
Given 0.01 g/(cm-s) and 0.01 cm2 /s. Solution steps: (a) For
conversion to SI units, Ans. (a) For conversion to BG units Ans.
(b) Comments: This was a laborious conversion that could have been
shortened by using the direct viscosity conversion factors in App.
C. For example, BG SI/47.88. We repeat our advice: Faced with data
in unusual units, convert them immediately to either SI or BG units
because (1) it is more professional and (2) theoretical equa- tions
in uid mechanics are dimensionally consistent and require no
further conver- sion factors when these two fundamental unit
systems are used, as the following example shows. EXAMPLE 1.3 A
useful theoretical equation for computing the relation between
pressure, velocity, and alti- tude in a steady ow of a nearly
inviscid, nearly incompressible uid with negligible heat transfer
and shaft work5 is the Bernoulli relation, named after Daniel
Bernoulli, who pub- lished a hydrodynamics textbook in 1738: (1)
where p0 stagnation pressure p pressure in moving uid V velocity
density Z altitude g gravitational acceleration p0 p 1 2 V2 gZ 0.01
cm2 s 0.01 cm2 (0.01 m/cm)2 (1 ft/0.3048 m)2 s 0.0000108 ft2 s 0.01
g cm # s 0.01 g(1 kg/1000 g)(1 slug/14.5939 kg) (0.01 m/cm)(1
ft/0.3048 m)s 0.0000209 slug ft # s Part (b) 0.01 cm2 s 0.01 cm2
(0.01 m/cm)2 s 0.000001 m2 s 0.01 g cm # s 0.01 g(1 kg/1000g)
cm(0.01 m/cm)s 0.001 kg m # s Part (a) 1.6 Dimensions and Units 13
5 Thats an awful lot of assumptions, which need further study in
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34. (a) Show that Eq. (1) satises the principle of dimensional
homogeneity, which states that all additive terms in a physical
equation must have the same dimensions. (b) Show that consis- tent
units result without additional conversion factors in SI units. (c)
Repeat (b) for BG units. Solution We can express Eq. (1)
dimensionally, using braces, by entering the dimensions of each
term from Table 1.2: {ML1 T2 } {ML1 T2 } {ML3 }{L2 T2 } {ML3 }{LT2
}{L} {ML1 T2 } for all terms Ans. (a) Enter the SI units for each
quantity from Table 1.2: {N/m2 } {N/m2 } {kg/m3 }{m2 /s2 } {kg/m3
}{m/s2 }{m} {N/m2 } {kg/(m s2 )} The right-hand side looks bad
until we remember from Eq. (1.3) that 1 kg 1 N s2 /m. Ans. (b) Thus
all terms in Bernoullis equation will have units of pascals, or
newtons per square meter, when SI units are used. No conversion
factors are needed, which is true of all theo- retical equations in
uid mechanics. Introducing BG units for each term, we have {lbf/ft2
} {lbf/ft2 } {slugs/ft3 }{ft2 /s2 } {slugs/ft3 }{ft/s2 }{ft}
{lbf/ft2 } {slugs/(ft s2 )} But, from Eq. (1.3), 1 slug 1 lbf s2
/ft, so that Ans. (c) All terms have the unit of pounds-force per
square foot. No conversion factors are needed in the BG system
either. There is still a tendency in English-speaking countries to
use pound-force per square inch as a pressure unit because the
numbers are more manageable. For example, stan- dard atmospheric
pressure is 14.7 lbf/in2 2116 lbf/ft2 101,300 Pa. The pascal is a
small unit because the newton is less than lbf and a square meter
is a very large area. Note that not only must all (uid) mechanics
equations be dimensionally homogeneous, one must also use
consistent units; that is, each additive term must have the same
units. There is no trouble doing this with the SI and BG systems,
as in Example 1.3, but woe unto those who try to mix colloquial
English units. For example, in Chap. 9, we often use the assumption
of steady adiabatic compressible gas ow: h 1 2V2 constant
Consistent Units 1 4 5slugs/(ft # s2 )6 5lbf # s2 /ft6 5ft # s2 6
5lbf/ft2 6 Part (c) 5kg/(m # s2 )6 5N # s2 /m6 5m # s2 6 5N/m2 6
Part (b) Part (a) 14 Chapter 1 Introduction
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35. where h is the uid enthalpy and V2 /2 is its kinetic energy
per unit mass. Colloquial thermodynamic tables might list h in
units of British thermal units per pound mass (Btu/lb), whereas V
is likely used in ft/s. It is completely erroneous to add Btu/lb to
ft2 /s2 . The proper unit for h in this case is ft lbf/slug, which
is identical to ft2 /s2 . The conversion factor is 1 Btu/lb 25,040
ft2 /s2 25,040 ft lbf/slug. All theoretical equations in mechanics
(and in other physical sciences) are dimension- ally homogeneous;
that is, each additive term in the equation has the same
dimensions. However, the reader should be warned that many
empirical formulas in the engi- neering literature, arising
primarily from correlations of data, are dimensionally incon-
sistent. Their units cannot be reconciled simply, and some terms
may contain hidden variables. An example is the formula that pipe
valve manufacturers cite for liquid vol- ume ow rate Q (m3 /s)
through a partially open valve: where p is the pressure drop across
the valve and SG is the specic gravity of the liquid (the ratio of
its density to that of water). The quantity CV is the valve ow
coefcient, which manufacturers tabulate in their valve brochures.
Since SG is dimen- sionless {1}, we see that this formula is
totally inconsistent, with one side being a ow rate {L3 /T} and the
other being the square root of a pressure drop {M1/2 /L1/2 T}. It
follows that CV must have dimensions, and rather odd ones at that:
{L7/2 /M1/2 }. Nor is the resolution of this discrepancy clear,
although one hint is that the values of CV in the literature
increase nearly as the square of the size of the valve. The pres-
entation of experimental data in homogeneous form is the subject of
dimensional analysis (Chap. 5). There we shall learn that a
homogeneous form for the valve ow relation is where is the liquid
density and A the area of the valve opening. The discharge coef-
cient C d is dimensionless and changes only moderately with valve
size. Please believeuntil we establish the fact in Chap. 5that this
latter is a much better for- mulation of the data. Meanwhile, we
conclude that dimensionally inconsistent equations, though they
occur in engineering practice, are misleading and vague and even
dangerous, in the sense that they are often misused outside their
range of applicability. Engineering results often are too small or
too large for the common units, with too many zeros one way or the
other. For example, to write p 114,000,000 Pa is long and awkward.
Using the prex M to mean 106 , we convert this to a concise p 114
MPa (megapascals). Similarly, t 0.000000003 s is a proofreaders
night- mare compared to the equivalent t 3 ns (nanoseconds). Such
prexes are com- mon and convenient, in both the SI and BG systems.
A complete list is given in Table 1.3. Convenient Prexes in Powers
of 10 Q Cd Aopeninga p b 1/2 Q CV a p SG b 1/2 Homogeneous versus
Dimensionally Inconsistent Equations 1.6 Dimensions and Units 15
Table 1.3 Convenient Prexes for Engineering Units Multiplicative
factor Prex Symbol 1012 tera T 109 giga G 106 mega M 103 kilo k 102
hecto h 10 deka da 101 deci d 102 centi c 103 milli m 106 micro 109
nano n 1012 pico p 1015 femto f 1018 atto a
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36. EXAMPLE 1.4 In 1890 Robert Manning, an Irish engineer,
proposed the following empirical formula for the average velocity V
in uniform ow due to gravity down an open channel (BG units): (1)
where R hydraulic radius of channel (Chaps. 6 and 10) S channel
slope (tangent of angle that bottom makes with horizontal) n
Mannings roughness factor (Chap. 10) and n is a constant for a
given surface condition for the walls and bottom of the channel.
(a) Is Mannings formula dimensionally consistent? (b) Equation (1)
is commonly taken to be valid in BG units with n taken as
dimensionless. Rewrite it in SI form. Solution Assumption: The
channel slope S is the tangent of an angle and is thus a
dimensionless ratio with the dimensional notation {1}that is, not
containing M, L, or T. Approach (a): Rewrite the dimensions of each
term in Mannings equation, using brackets {}: This formula is
incompatible unless {1.49/n} {L1/3 /T}. If n is dimensionless (and
it is never listed with units in textbooks), the number 1.49 must
carry the dimensions of {L1/3 /T}. Ans. (a) Comment (a): Formulas
whose numerical coefcients have units can be disastrous for
engineers working in a different system or another uid. Mannings
formula, though pop- ular, is inconsistent both dimensionally and
physically and is valid only for water ow with certain wall
roughnesses. The effects of water viscosity and density are hidden
in the numerical value 1.49. Approach (b): Part (a) showed that
1.49 has dimensions. If the formula is valid in BG units, then it
must equal 1.49 ft1/3 /s. By using the SI conversion for length, we
obtain (1.49 ft1/3 /s)(0.3048 m/ft)1/3 1.00 m1/3 /s Therefore
Mannings inconsistent formula changes form when converted to the SI
system: Ans. (b) with R in meters and V in meters per second.
Comment (b): Actually, we misled you: This is the way Manning, a
metric user, rst proposed the formula. It was later converted to BG
units. Such dimensionally inconsis- tent formulas are dangerous and
should either be reanalyzed or treated as having very limited
application. SI units: V 1.0 n R2/3 S1/2 5V6 e 1.49 n f 5R2/3 6
5S1/2 6 or e L T f e 1.49 n f 5L2/3 6 516 V 1.49 n R2/3 S1/2 16
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37. In a given ow situation, the determination, by experiment
or theory, of the proper- ties of the uid as a function of position
and time is considered to be the solution to the problem. In almost
all cases, the emphasis is on the spacetime distribution of the uid
properties. One rarely keeps track of the actual fate of the specic
uid parti- cles.6 This treatment of properties as continuum-eld
functions distinguishes uid mechanics from solid mechanics, where
we are more likely to be interested in the tra- jectories of
individual particles or systems. There are two different points of
view in analyzing problems in mechanics. The rst view, appropriate
to uid mechanics, is concerned with the eld of ow and is called the
eulerian method of description. In the eulerian method we compute
the pressure eld p(x, y, z, t) of the ow pattern, not the pressure
changes p(t) that a particle expe- riences as it moves through the
eld. The second method, which follows an individual particle moving
through the ow, is called the lagrangian description. The
lagrangian approach, which is more appro- priate to solid
mechanics, will not be treated in this book. However, certain
numeri- cal analyses of sharply bounded uid ows, such as the motion
of isolated uid droplets, are very conveniently computed in
lagrangian coordinates [1]. Fluid dynamic measurements are also
suited to the eulerian system. For example, when a pressure probe
is introduced into a laboratory ow, it is xed at a specic position
(x, y, z). Its output thus contributes to the description of the
eulerian pres- sure eld p(x, y, z, t). To simulate a lagrangian
measurement, the probe would have to move downstream at the uid
particle speeds; this is sometimes done in oceano- graphic
measurements, where owmeters drift along with the prevailing
currents. The two different descriptions can be contrasted in the
analysis of trafc ow along a freeway. A certain length of freeway
may be selected for study and called the eld of ow. Obviously, as
time passes, various cars will enter and leave the eld, and the
iden- tity of the specic cars within the eld will constantly be
changing. The trafc engi- neer ignores specic cars and concentrates
on their average velocity as a function of time and position within
the eld, plus the ow rate or number of cars per hour pass- ing a
given section of the freeway. This engineer is using an eulerian
description of the trafc ow. Other investigators, such as the
police or social scientists, may be interested in the path or speed
or destination of specic cars in the eld. By following a specic car
as a function of time, they are using a lagrangian description of
the ow. Foremost among the properties of a ow is the velocity eld
V(x, y, z, t). In fact, determining the velocity is often
tantamount to solving a ow problem, since other properties follow
directly from the velocity eld. Chapter 2 is devoted to the calcu-
lation of the pressure eld once the velocity eld is known. Books on
heat transfer (for example, Ref. 20) are largely devoted to nding
the temperature eld from known velocity elds. The Velocity Field
Eulerian and Lagrangian Descriptions 1.7 Properties of the Velocity
Field 1.7 Properties of the Velocity Field 17 6 One example where
uid particle paths are important is in water quality analysis of
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38. In general, velocity is a vector function of position and
time and thus has three components u, , and w, each a scalar eld in
itself: (1.4) The use of u, , and w instead of the more logical
component notation Vx, Vy, and Vz is the result of an almost
unbreakable custom in uid mechanics. Much of this text- book,
especially Chaps. 4, 7, 8, and 9, is concerned with nding the
distribution of the velocity vector V for a variety of practical
ows. The acceleration vector, a dV/dt, occurs in Newtons law for a
uid and thus is very important. In order to follow a particle in
the Eulerian frame of reference, the nal result for acceleration is
nonlinear and quite complicated. Here we only give the formula:
(1.5) where (u, v, w) are the velocity components from Eq. (1.4).
We shall study this formula in detail in Chap. 4. The last three
terms in Eq. (1.5) are nonlinear products and greatly complicate
the analysis of general fluid motions, especially viscous ows.
While the velocity eld V is the most important uid property, it
interacts closely with the thermodynamic properties of the uid. We
have already introduced into the discussion the three most common
such properties: 1. Pressure p 2. Density 3. Temperature T These
three are constant companions of the velocity vector in ow
analyses. Four other intensive thermodynamic properties become
important when work, heat, and energy balances are treated (Chaps.
3 and 4): 4. Internal energy 5. Enthalpy h p/ 6. Entropy s 7.
Specic heats cp and cv In addition, friction and heat conduction
effects are governed by the two so-called transport properties: 8.
Coefcient of viscosity 9. Thermal conductivity k All nine of these
quantities are true thermodynamic properties that are determined by
the thermodynamic condition or state of the uid. For example, for a
single-phase 1.8 Thermodynamic Properties of a Fluid a dV dt V t u
V x v V y w V z The Acceleration Field V(x, y, z, t) iu(x, y, z, t)
jv(x, y, z, t) kw(x, y, z, t) 18 Chapter 1 Introduction
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39. substance such as water or oxygen, two basic properties
such as pressure and temperature are sufcient to x the value of all
the others: (p, T) h h(p, T) (p, T) and so on for every quantity in
the list. Note that the specic volume, so important in
thermodynamic analyses, is omitted here in favor of its inverse,
the density . Recall that thermodynamic properties describe the
state of a systemthat is, a collection of matter of xed identity
that interacts with its surroundings. In most cases here the system
will be a small uid element, and all properties will be assumed to
be continuum properties of the ow eld: (x, y, z, t), and so on.
Recall also that thermodynamics is normally concerned with static
systems, whereas uids are usually in variable motion with
constantly changing properties. Do the properties retain their
meaning in a uid ow that is technically not in equilibrium? The
answer is yes, from a statistical argument. In gases at normal
pres- sure (and even more so for liquids), an enormous number of
molecular collisions occur over a very short distance of the order
of 1 m, so that a uid subjected to sudden changes rapidly adjusts
itself toward equilibrium. We therefore assume that all the
thermodynamic properties just listed exist as point functions in a
owing uid and follow all the laws and state relations of ordinary
equilibrium thermodynamics. There are, of course, important
nonequilibrium effects such as chemical and nuclear reactions in
owing uids, which are not treated in this text. Pressure is the
(compression) stress at a point in a static uid (Fig. 1.3). Next to
velocity, the pressure p is the most dynamic variable in uid
mechanics. Differences or gradients in pressure often drive a uid
ow, especially in ducts. In low-speed ows, the actual magnitude of
the pressure is often not important, unless it drops so low as to
cause vapor bubbles to form in a liquid. For convenience, we set
many such problem assignments at the level of 1 atm 2116 lbf/ft2
101,300 Pa. High-speed (compressible) gas ows (Chap. 9), however,
are indeed sensitive to the magnitude of pressure. Temperature T is
related to the internal energy level of a uid. It may vary
consider- ably during high-speed ow of a gas (Chap. 9). Although
engineers often use Celsius or Fahrenheit scales for convenience,
many applications in this text require absolute (Kelvin or Rankine)
temperature scales: If temperature differences are strong, heat
transfer may be important [20], but our concern here is mainly with
dynamic effects. The density of a uid, denoted by (lowercase Greek
rho), is its mass per unit volume. Density is highly variable in
gases and increases nearly proportionally to the pressure level.
Density in liquids is nearly constant; the density of water (about
1000 kg/m3 ) Density K C 273.16 R F 459.69 Temperature Pressure 1.8
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40. increases only 1 percent if the pressure is increased by a
factor of 220. Thus most liq- uid ows are treated analytically as
nearly incompressible. In general, liquids are about three orders
of magnitude more dense than gases at atmospheric pressure. The
heaviest common liquid is mercury, and the lightest gas is
hydrogen. Compare their densities at 20C and 1 atm: They differ by
a factor of 162,000! Thus the physical parameters in various liquid
and gas ows might vary considerably. The differences are often
resolved by the use of dimensional analysis (Chap. 5). Other uid
densities are listed in Tables A.3 and A.4 (in App. A) and in Ref.
21. The specic weight of a uid, denoted by (lowercase Greek gamma),
is its weight per unit volume. Just as a mass has a weight W mg,
density and specic weight are simply related by gravity: (1.6) The
units of are weight per unit volume, in lbf/ft3 or N/m3 . In
standard earth grav- ity, g 32.174 ft/s2 9.807 m/s2 . Thus, for
example, the specic weights of air and water at 20C and 1 atm are
approximately Specic weight is very useful in the hydrostatic
pressure applications of Chap. 2. Specic weights of other uids are
given in Tables A.3 and A.4. Specic gravity, denoted by SG, is the
ratio of a uid density to a standard reference uid, usually water
at 4C (for liquids) and air (for gases): (1.7) For example, the
specic gravity of mercury (Hg) is SGHg 13,580/1000 13.6. Engineers
nd these dimensionless ratios easier to remember than the actual
numer- ical values of density of a variety of uids. In
thermostatics the only energy in a substance is that stored in a
system by molec- ular activity and molecular bonding forces. This
is commonly denoted as internal energy . A commonly accepted
adjustment to this static situation for uid ow is to add two more
energy terms that arise from newtonian mechanics: potential energy
and kinetic energy. Potential and Kinetic Energies SGliquid liquid
water liquid 1000 kg/m3 SGgas gas air gas 1.205 kg/m3 Specic
Gravity water (998 kg/m3 )(9.807 m/s2 ) 9790 N/m3 62.4 lbf/ft3 air
(1.205 kg/m3 )(9.807 m/s2 ) 11.8 N/m3 0.0752 lbf/ft3 g Specic
Weight Mercury: 13,580 kg/m3 Hydrogen: 0.0838 kg/m3 20 Chapter 1
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41. The potential energy equals the work required to move the
system of mass m from the origin to a position vector r ix jy kz
against a gravity eld g. Its value is mg r, or g r per unit mass.
The kinetic energy equals the work required to change the speed of
the mass from zero to velocity V. Its value is mV2 or V2 per unit
mass. Then by common convention the total stored energy e per unit
mass in uid mechanics is the sum of three terms: e V2 (g r) (1.8)
Also, throughout this book we shall dene z as upward, so that g gk
and g r gz. Then Eq. (1.8) becomes e V2 gz (1.9) The molecular
internal energy is a function of T and p for the single-phase pure
substance, whereas the potential and kinetic energies are kinematic
quantities. Thermodynamic properties are found both theoretically
and experimentally to be related to each other by state relations
that differ for each substance. As mentioned, we shall conne
ourselves here to single-phase pure substances, such as water in
its liquid phase. The second most common uid, air, is a mixture of
gases, but since the mixture ratios remain nearly constant between
160 and 2200 K, in this temperature range air can be considered to
be a pure substance. All gases at high temperatures and low
pressures (relative to their critical point) are in good agreement
with the perfect-gas law (1.10) where the specic heats cp and cv
are dened in Eqs. (1.14) and (1.15). Since Eq. (1.10) is
dimensionally consistent, R has the same dimensions as spe- cic
heat, {L2 T2 1 }, or velocity squared per temperature unit (kelvin
or degree Rankine). Each gas has its own constant R, equal to a
universal constant divided by the molecular weight (1.11) where
49,700 ft-lbf/(slugmol R) 8314 J/(kmol K). Most applications in
this book are for air, whose molecular weight is M 28.97/mol:
(1.12) Standard atmospheric pressure is 2116 lbf/ft2 2116 slug/(ft
s2 ), and standard temperature is 60F 520R. Thus standard air
density is (1.13) This is a nominal value suitable for problems.
For other gases, see Table A.4. air 2116 slug/(ft # s2 ) 31716 ft2
/(s2 # R)4(520R) 0.00237 slug/ft3 1.22 kg/m3 Rair 49,700 ft #
lbf/(slugmol # R) 28.97/mol 1716 ft # lbf slug # R 1716 ft2 s2 R
287 m2 s2 # K ## Rgas Mgas p RT R cp cv gas constant State
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42. One proves in thermodynamics that Eq. (1.10) requires that
the internal molecular energy of a perfect gas vary only with
temperature: (T). Therefore the spe- cic heat cv also varies only
with temperature: or (1.14) In like manner h and cp of a perfect
gas also vary only with temperature: (1.15) The ratio of specic
heats of a perfect gas is an important dimensionless parameter in
compressible ow analysis (Chap. 9) (1.16) As a rst approximation in
airow analysis we commonly take cp, cv, and k to be constant:
(1.17) Actually, for all gases, cp and cv increase gradually with
temperature, and k decreases gradually. Experimental values of the
specic-heat ratio for eight common gases are shown in Fig. 1.5.
Many ow problems involve steam. Typical steam operating conditions
are relatively close to the critical point, so that the perfect-gas
approximation is inaccurate. Since no simple formulas apply
accurately, steam properties are available both in EES (see Sec.
1.12) and on a CD-ROM [23] and even on the Internet, as a MathPad
Corp. applet [24]. Meanwhile, the error of using the perfect-gas
law can be moderate, as the following example shows. EXAMPLE 1.5
Estimate and cp of steam at 100 lbf/in2 and 400F, in English units,
(a) by the perfect-gas approximation and (b) by the ASME Steam
Tables [23] or by EES. Solution Approach (a)the perfect-gas law:
Although steam is not an ideal gas, we can estimate these
properties with moderate accuracy from Eqs. (1.10) and (1.17).
First convert pressure cp kR k 1 6010 ft2 /(s2 # R) 1005 m2 /(s2 #
K) cv R k 1 4293 ft2 /(s2 # R) 718 m2 /(s2 # K) kair 1.4 k cp cv
k(T) 1 dh cp(T)dT cp a h T b p dh dT cp(T) h p RT h(T) d cv(T)dT cv
a T b d dT cv(T) 22 Chapter 1 Introduction
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43. from 100 lbf/in2 to 14,400 lbf/ft2 , and use absolute
temperature, (400F 460) 860R. Then we need the gas constant for
steam, in English units. From Table A.4, the molecular weight of
H2O is 18.02, whence Then the density estimate follows from the
perfect-gas law, Eq. (1.10): Ans. (a) At 860R, from Fig. 1.5,
ksteam cp /cv 1.30. Then, from Eq. (1.17), Ans. (a) Approach
(b)tables or software: One can either read the steam tables or
program a few lines in EES. In either case, the English units (psi,
Btu, lbm) are awkward when applied to uid mechanics formulas. Even
so, when using EES, make sure that the cp kR k 1 (1.3)(2758 ft #
lbf/(slug R)) (1.3 1) 12,000 ft # lbf slug R p RT 14,400 lbf/ft2
32758 ft # lbf/(slug # R)4(860 R) 0.00607 slug ft3 Rsteam English
MH2O 49,700 ft # lbf/(slugmol R) 18.02/mol 2758 ft # lbf slug R 1.8
Thermodynamic Properties of a Fluid 23 Ar Atmospheric pressure H2
CO Air and N2 O2 Steam CO2 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 10000
3000 4000 5000 =k cp c 2000 Temperature, R Fig. 1.5 Specic-heat
ratio of eight common gases as a function of temperature. (Data
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44. Variable Information menu species English units: psia and
F. EES statements for evaluating density and specic heat of steam
are, for these conditions, Rho DENSITY(steam, P 100,T 400) Cp
SPECHEAT(steam, P 100,T 400) Note that the software is set up for
psia and F, without converting. EES returns the curve- t values Rho
0.2027 lbm/ft3 ; Cp 0.5289 Btu/(lbm-F) As just stated, Btu and lbm
are extremely unwieldy when applied to mass, momentum, and energy
problems in uid mechanics. Therefore, either convert to ft-lbf and
slugs using your own resources, or use the Convert function in EES,
placing the old and new units in single quote marks: Rho2
Rho*CONVERT(lbm/ft^3,slug/ft^3) Cp2
Cp*CONVERT(Btu/lbm-F,ft^2/s^2-R) Note that (1) you multiply the old
Rho and Cp by the CONVERT function; and (2) units to the right of
the division sign / in CONVERT are assumed to be in the
denominator. EES returns these results: Rho2 0.00630 slug/ft3 Cp2
13,200 ft2 /(s2 -R) Ans. (b) Comments: The steam tables would yield
results quite close to EES. The perfect-gas estimate of is 4
percent low, and the estimate of cp is 9 percent low. The chief
reason for the discrepancy is that this temperature and pressure
are rather close to the critical point and saturation line of
steam. At higher temperatures and lower pressures, say, 800F and 50
lbf/in2 , the perfect-gas law yields properties with an accuracy of
about 1 percent. Once again let us warn that English units (psia,
lbm Btu) are awkward and must be converted in most uid mechanics
formulas. EES handles SI units nicely, with no con- version factors
needed. The writer knows of no perfect-liquid law comparable to
that for gases. Liquids are nearly incompressible and have a
single, reasonably constant specic heat. Thus an idealized state
relation for a liquid is const cp cv const dh cp dT (1.18) Most of
the ow problems in this book can be attacked with these simple
assump- tions. Water is normally taken to have a density of 998
kg/m3 and a specic heat cp 4210 m2 /(s2 K). The steam tables may be
used if more accuracy is required. The density of a liquid usually
decreases slightly with temperature and increases moderately with
pressure. If we neglect the temperature effect, an empirical
pressure density relation for a liquid is (1.19) where B and n are
dimensionless parameters that vary slightly with temperature and pa
and a are standard atmospheric values. Water can be tted
approximately to the values B 3000 and n 7. p pa (B 1)a a b n B
State Relations for Liquids 24 Chapter 1 Introduction
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45. Seawater is a variable mixture of water and salt and thus
requires three thermo- dynamic properties to dene its state. These
are normally taken as pressure, temper- ature, and the salinity ,
dened as the weight of the dissolved salt divided by the weight of
the mixture. The average salinity of seawater is 0.035, usually
written as 35 parts per 1000, or 35 . The average density of
seawater is 2.00 slugs/ft3 1030 kg/m3 . Strictly speaking, seawater
has three specic heats, all approximately equal to the value for
pure water of 25,200 ft2 /(s2 R) 4210 m2 /(s2 K). EXAMPLE 1.6 The
pressure at the deepest part of the ocean is approximately 1100
atm. Estimate the den- sity of seawater in slug/ft3 at this
pressure. Solution Equation (1.19) holds for either water or
seawater. The ratio p/pa is given as 1100: or Assuming an average
surface seawater density a 2.00 slugs/ft3 , we compute Ans. Even at
these immense pressures, the density increase is less than 5
percent, which justies the treatment of a liquid ow as essentially
incompressible. The quantities such as pressure, temperature, and
density discussed in the previous section are primary thermodynamic
variables characteristic of any system. Certain secondary variables
also characterize specic uid mechanical behavior. The most
important of these is viscosity, which relates the local stresses
in a moving uid to the strain rate of the uid element. Viscosity is
a quantitative measure of a uids resistance to ow. More specically,
it determines the uid strain rate that is generated by a given
applied shear stress. We can easily move through air, which has
very low viscosity. Movement is more dif- cult in water, which has
50 times higher viscosity. Still more resistance is found in SAE 30
oil, which is 300 times more viscous than water. Try to slide your
hand through glycerin, which is ve times more viscous than SAE 30
oil, or blackstrap molasses, another factor of ve higher than
glyce