1. Conversion Factors from BG to SI UnitsTo convert from To Multiply byAcceleration ft/s2 m/s2 0.3048Area ft2 m2 9.2903 E2mi2 m2 2.5900 E6acres m2 4.0469 E3Density slug/ft3 kg/m3 5.1538 E2lbm/ft3 kg/m3 1.6019 E1Energy ft-lbf J 1.3558Btu J 1.0551 E3cal J 4.1868Force lbf N 4.4482kgf N 9.8067Length ft m 0.3048in m 2.5400 E2mi (statute) m 1.6093 E3nmi (nautical) m 1.8520 E3Mass slug kg 1.4594 E1lbm kg 4.5359 E1Mass flow slug/s kg/s 1.4594 E1lbm/s kg/s 4.5359 E1Power ftlbf/s W 1.3558hp W 7.4570 E2

2. Conversion Factors from BG to SI Units (Continued)To convert from To Multiply byPressure lbf/ft2 Pa 4.7880 E1lbf/in2 Pa 6.8948 E3atm Pa 1.0133 E5mm Hg Pa 1.3332 E2Specific weight lbf/ft3 N/m3 1.5709 E2Specific heat ft2/(s2R) m2/(s2K) 1.6723 E1Surface tension lbf/ft N/m 1.4594 E1Temperature F C tC(tF32)59R K 0.5556Velocity ft/s m/s 0.3048mi/h m/s 4.4704 E1knot m/s 5.1444 E1Viscosity lbfs/ft2 Ns/m2 4.7880 E1g/(cms) Ns/m2 0.1Volume ft3 m3 2.8317 E2L m3 0.001gal (U.S.) m3 3.7854 E3fluid ounce (U.S.) m3 2.9574 E5Volume flow ft3/s m3/s 2.8317 E2gal/min m3/s 6.3090 E5 3. EQUATION SHEETIdeal-gas law: pRT, Rair287 J/kg-KHydrostatics, constant density:p2p1(z2z1), gBuoyant force:(displaced volume)FBfluidCV momentum:d/dt1 CVVd2 g 3 (AV)V4 outg 3 (AV)V4 ingF(p/V2/2gz)in Steady flow energy:(p/V2/2gz)outhfrictionhpumphturbineIncompressible continuity:V0Incompressible stream function(x,y):u/y; v/xBernoulli unsteady irrotational flow:/t dp/V2/2 gzConstPipe head loss:hff(L/d)V2/(2g)where fMoody chart friction factor1/2/x5.0/RexLaminar flat plate flow: ,c0.664/Rex1/2f ,CD1.328/ReL1/2 V2A2; CLLift/ 112V2A2T0 /T15(k1)/26Ma2CDDrag/112Isentropic flow: ,0/(T0/T)1/(k1),p0/p(T0/T)k(k1) Prandtl-Meyer expansion: K(k1)/(k1),K1/2tan1[(Ma21)/K]1/2tan1(Ma21)1/2Gradually varied channel flow:dy/dx(S0S)/(1Fr2), FrV/Vcrit1R2pY(R11)Surface tension:Hydrostatic panel force: ,yCPIxxsin /(hCGA), xCPIxysin /(hCGA)CV mass:d/dt(CVFhCGAd) g(AV)out g(AV)in0CV angular momentum:d/dt(CV (r0V)d)gAV(r0V)outgAV(r0V)in gM0Acceleration:dV/dtV/tu(V/x)v(V/y)w(V/z)(dV/dt)gp 2VNavier-Stokes:Velocity potential :(x, y, z)u/x; v/y; w/zTurbulent friction factor:1/1f 2.0 log10 3 4. /(3.7d)2.51/1Red1f) 4Orifice, nozzle, venturi flow:QCdAthroat 32p/5(1 4)6 4 1/2,d/D 1/7/x0.16/RexTurbulent flat plate flow: ,1/7, CD0.031/ReL220cf0.027/Rex1/72-D potential flow:One-dimensional isentropic area change:A/A*(1/Ma)[1{(k1)/2}Ma2](1/2)(k1)/(k1)Uniform flow, Mannings n, SI units:V0(m/s)(1.0/n) 3Rh(m) 4 2/3S0Euler turbine formula:1/2PowerQ(u2Vt2u1Vt1), ur 5. This page intentionally left blank 6. Fluid Mechanics 7. McGraw-Hill Series in Mechanical EngineeringAlciatore/HistandIntroduction to Mechatronics and Measurement SystemsAndersonComputational Fluid Dynamics: The Basics with ApplicationsAndersonFundamentals of AerodynamicsAndersonIntroduction to FlightAndersonModern Compressible FlowBeer/JohnstonVector Mechanics for Engineers: Statics and DynamicsBeer/JohnstonMechanics of MaterialsBudynasAdvanced Strength and Applied Stress AnalysisBudynas/NisbettShigley s Mechanical Engineering DesignengelHeat and Mass Transfer: A Practical ApproachengelIntroduction to ThermodynamicsHeat Transferengel/BolesThermodynamics: An Engineering Approachengel/CimbalaFluid Mechanics: Fundamentals and Applicationsengel/TurnerFundamentals of Thermal-Fluid SciencesDieterEngineering Design: A MaterialsProcessing ApproachDieterMechanical MetallurgyDorf/ByersTechnology Ventures: From Idea to EnterpriseFinnemore/FranziniFluid Mechanics with Engineering ApplicationsHamrock/Schmid/JacobsonFundamentals of Machine ElementsHeywoodInternal Combustion Engine FundamentalsHolmanExperimental Methods for EngineersHolmanHeat TransferKays/Crawford/WeigandConvective Heat and Mass TransferMeirovitchFundamentals of VibrationsNortonDesign of MachineryPalmSystem DynamicsReddyAn Introduction to Finite Element MethodScheyIntroduction to Manufacturing ProcessesSmith/HashemiFoundations of Materials Science and EngineeringTurnsAn Introduction to Combustion: Concepts and ApplicationsUguralMechanical Design: An Integrated ApproachUllmanThe Mechanical Design ProcessWhiteFluid MechanicsWhiteViscous Fluid Flow 8. Fluid MechanicsSeventh EditionFrank M. WhiteUniversity of Rhode Island 9. FLUID MECHANICS, SEVENTH EDITIONPublished by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of theAmericas, New York, NY 10020. Copyright 2011 by The McGraw-Hill Companies, Inc. All rightsreserved. Previous editions 2008, 2003, and 1999. No part of this publication may be reproduced ordistributed in any form or by any means, or stored in a database or retrieval system, without the priorwritten consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or otherelectronic storage or transmission, or broadcast for distance learning.Some ancillaries, including electronic and print components, may not be available to customers outside theUnited 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 0ISBN 978-0-07-352934-9MHID 0-07-352934-6Vice PresidentEditor-in-Chief: Marty LangeVice President, EDP/Central Publishing Services: Kimberly Meriwether-DavidGlobal Publisher: Raghothaman SrinivasanSenior Sponsoring Editor: Bill StenquistDirector of Development: Kristine TibbettsDevelopmental Editor: Lora NeyensSenior Marketing Manager: Curt ReynoldsSenior Project Manager: Lisa A. BruflodtProduction Supervisor: Nicole BaumgartnerDesign Coordinator: Brenda A. RolwesCover Designer: Studio Montage, St. Louis, Missouri(USE) Cover Image: Copyright SkySailsSenior Photo Research Coordinator: John C. LelandPhoto Research: Emily Tietz/Editorial Image, LLCCompositor: Aptara, Inc.Typeface: 10/12 Times RomanPrinter: R. R. DonnelleyAll credits appearing on page or at the end of the book are considered to be an extension of the copyright page.www.mhhe.comLibrary of Congress Cataloging-in-Publication DataWhite, 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 2009620.106dc222009047498 10. Frank M. White is Professor Emeritus of Mechanical and Ocean Engineering at theUniversity of Rhode Island. He studied at Georgia Tech and M.I.T. In 1966 he helpedfound, at URI, the first department of ocean engineering in the country. Knownprimarily as a teacher and writer, he has received eight teaching awards and has writtenfour textbooks on fluid mechanics and heat transfer.From 1979 to 1990 he was editor-in-chief of the ASME Journal of FluidsEngineering and then served from 1991 to 1997 as chairman of the ASME Boardof Editors and of the Publications Committee. He is a Fellow of ASME and in 1991received the ASME Fluids Engineering Award. He lives with his wife, Jeanne, inNarragansett, Rhode Island.vAbout the Author 11. To Jeanne 12. Preface xiChapter 1Introduction 31.1 Preliminary Remarks 31.2 History and Scope of Fluid Mechanics 41.3 Problem-Solving Techniques 61.4 The Concept of a Fluid 61.5 The Fluid as a Continuum 81.6 Dimensions and Units 91.7 Properties of the Velocity Field 171.8 Thermodynamic Properties of a Fluid 181.9 Viscosity and Other Secondary Properties 251.10 Basic Flow Analysis Techniques 401.11 Flow Patterns: Streamlines, Streaklines, andPathlines 411.12 The Engineering Equation Solver 461.13 Uncertainty in Experimental Data 461.14 The Fundamentals of Engineering (FE)ContentsExamination 48Problems 49Fundamentals of Engineering Exam Problems 57Comprehensive Problems 58References 61Chapter 2Pressure Distribution in a Fluid 652.1 Pressure and Pressure Gradient 652.2 Equilibrium of a Fluid Element 672.3 Hydrostatic Pressure Distributions 682.4 Application to Manometry 752.5 Hydrostatic Forces on Plane Surfaces 782.6 Hydrostatic Forces on Curved Surfaces 862.7 Hydrostatic Forces in Layered Fluids 892.8 Buoyancy and Stability 912.9 Pressure Distribution in Rigid-Body Motion 972.10 Pressure Measurement 105Summary 109Problems 109Word Problems 132Fundamentals of Engineering ExamProblems 133Comprehensive Problems 134Design Projects 135References 136Chapter 3Integral Relations for a Control Volume 1393.1 Basic Physical Laws of Fluid Mechanics 1393.2 The Reynolds Transport Theorem 1433.3 Conservation of Mass 1503.4 The Linear Momentum Equation 1553.5 Frictionless Flow: The Bernoulli Equation 1693.6 The Angular Momentum Theorem 1783.7 The Energy Equation 184Summary 195Problems 195Word Problems 224Fundamentals of Engineering Exam Problems 224Comprehensive Problems 226Design Project 227References 227vii 13. Chapter 4Differential Relations for Fluid Flow 2294.1 The Acceleration Field of a Fluid 2304.2 The Differential Equation of Mass Conservation 2324.3 The Differential Equation of Linear Momentum 2384.4 The Differential Equation of Angular Momentum 2444.5 The Differential Equation of Energy 2464.6 Boundary Conditions for the Basic Equations 2494.7 The Stream Function 2534.8 Vorticity and Irrotationality 2614.9 Frictionless Irrotational Flows 2634.10 Some Illustrative Incompressible Viscous Flows 268Summary 276Problems 277Word Problems 288Fundamentals of Engineering Exam Problems 288Comprehensive Problems 289References 290Chapter 5Dimensional Analysis and Similarity 2935.1 Introduction 2985.2 The Principle of Dimensional Homogeneity 2965.3 The Pi Theorem 3025.4 Nondimensionalization of the Basic Equations 3125.5 Modeling and Its Pitfalls 321Summary 333Problems 333Word Problems 342Fundamentals of Engineering Exam Problems 342Comprehensive Problems 343Design Projects 344References 344Chapter 6Viscous Flow in Ducts 3476.1 Reynolds Number Regimes 3476.2 Internal versus External Viscous Flow 3526.3 Head LossThe Friction Factor 3556.4 Laminar Fully Developed Pipe Flow 3576.5 Turbulence Modeling 3596.6 Turbulent Pipe Flow 3656.7 Four Types of Pipe Flow Problems 3736.8 Flow in Noncircular Ducts 3796.9 Minor or Local Losses in Pipe Systems 3886.10 Multiple-Pipe Systems 3976.11 Experimental Duct Flows: Diffuser Performance 4036.12 Fluid Meters 408Summary 429Problems 430Word Problems 448Fundamentals of Engineering Exam Problems 449Comprehensive Problems 450Design Projects 452References 453Chapter 7Flow Past Immersed Bodies 4577.1 Reynolds Number and Geometry Effects 4577.2 Momentum Integral Estimates 4617.3 The Boundary Layer Equations 4647.4 The Flat-Plate Boundary Layer 4677.5 Boundary Layers with Pressure Gradient 4767.6 Experimental External Flows 482Summary 509Problems 510Word Problems 523Fundamentals of Engineering Exam Problems 524Comprehensive Problems 524Design Project 525References 526Chapter 8Potential Flow and Computational Fluid Dynamics 5298.1 Introduction and Review 5298.2 Elementary Plane Flow Solutions 5328.3 Superposition of Plane Flow Solutions 5398.4 Plane Flow Past Closed-Body Shapes 5458.5 Other Plane Potential Flows 5558.6 Images 5598.7 Airfoil Theory 5628.8 Axisymmetric Potential Flow 5748.9 Numerical Analysis 579viii Contents 14. Summary 593Problems 594Word Problems 604Comprehensive Problems 605Design Projects 606References 606Chapter 9Compressible Flow 6099.1 Introduction: Review of Thermodynamics 6099.2 The Speed of Sound 6149.3 Adiabatic and Isentropic Steady Flow 6169.4 Isentropic Flow with Area Changes 6229.5 The Normal Shock Wave 6299.6 Operation of Converging and Diverging Nozzles 6379.7 Compressible Duct Flow with Friction 6429.8 Frictionless Duct Flow with Heat Transfer 6549.9 Two-Dimensional Supersonic Flow 6599.10 Prandtl-Meyer Expansion Waves 669Summary 681Problems 682Word Problems 695Fundamentals of Engineering Exam Problems 696Comprehensive Problems 696Design Projects 698References 698Chapter 10Open-Channel Flow 70110.1 Introduction 70110.2 Uniform Flow: The Chzy Formula 70710.3 Efficient Uniform-Flow Channels 71210.4 Specific Energy: Critical Depth 71410.5 The Hydraulic Jump 72210.6 Gradually Varied Flow 726Contents ix10.7 Flow Measurement and Control by Weirs 734Summary 741Problems 741Word Problems 754Fundamentals of Engineering Exam Problems 754Comprehensive Problems 754Design Projects 756References 756Chapter 11Turbomachinery 75911.1 Introduction and Classification 75911.2 The Centrifugal Pump 76211.3 Pump Performance Curves and Similarity Rules 76811.4 Mixed- and Axial-Flow Pumps: The Specific Speed 77811.5 Matching Pumps to System Characteristics 78511.6 Turbines 793Summary 807Problems 807Word Problems 820Comprehensive Problems 820Design Project 822References 822Appendix A Physical Properties of Fluids 824Appendix B Compressible Flow Tables 829Appendix C Conversion Factors 836Appendix D Equations of Motion in Cylindrical Coordinates 838Answers to Selected Problems 840Index 847 15. This page intentionally left blank 16. The seventh edition of Fluid Mechanics sees some additions and deletions but nophilosophical change. The basic outline of eleven chapters, plus appendices, remainsthe same. The triad of integral, differential, and experimental approaches is retained.Many problem exercises, and some fully worked examples, have been changed. Theinformal, student-oriented style is retained. A number of new photographs and figureshave been added. Many new references have been added, for a total of 435. The writeris a firm believer in further reading, especially in the postgraduate years.The total number of problem exercises continues to increase, from 1089 in the firstedition, to 1675 in this seventh edition. There are approximately 20 new problemsadded to each chapter. Most of these are basic end-of-chapter problems, classifiedaccording to topic. There are also Word Problems, multiple-choice Fundamentals ofEngineering Problems, Comprehensive Problems, and Design Projects. The appendixlists approximately 700 Answers to Selected Problems.The example problems are structured in the text to follow the sequence of recom-mendedsteps outlined in Sect. 1.3, Problem-Solving Techniques.The Engineering Equation Solver (EES) is available with the text and continuesits role as an attractive tool for fluid 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 WilliamBeckman, of the University of Wisconsin, for invaluable and continuous help inpreparing and updating EES for use in this text. For newcomers to EES, a brief guideto 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 fluid mechanics,notably the development of Computational Fluid Dynamics. A very brief introductionto the acceleration field has been added. Boundary conditions for slip flow have beenadded. There is more discussion of the speed of sound in liquids. The treatment ofthermal conductivity has been moved to Chapter 4.General ApproachLearning ToolsContent ChangesxiPreface 17. Chapter 2 introduces a photo, discussion, and new problems for the deep oceansubmersible vehicle, ALVIN. The density distribution in the troposphere is now givenexplicitly. There are brief remarks on the great Greek mathematician, Archimedes.Chapter 3 has been substantially revised. Reviewers wanted Bernoullis equationmoved ahead of angular velocity and energy, to follow linear momentum. I did thisand followed their specific improvements, but truly extensive renumbering and rear-rangingwas necessary. Pressure and velocity conditions at a tank surface have animproved discussion. A brief history of the control volume has been added. There isa better treatment of the relation between Bernoullis equation and the energy equa-tion.There is a new discussion of stagnation, static and dynamic pressures, andboundary conditions at a jet exit.Chapter 4 has a great new opener: CFD for flow past a spinning soccer ball. Thetotal time derivative of velocity is now written out in full. Fouriers Law, and its appli-cationto the differential energy equation, have been moved here from Chapter 1.There are 21 new problems, including several slip-flow analyses.The Chapter 5 introduction expands on the effects of Mach number and Froudenumber, instead of concentrating only on the Reynolds number. Ipsens method, whichthe writer admires, is retained as an alternative to the pi theorem. The new opener, agiant disk-band-gap parachute, allows for several new dimensional analysis problems.Chapter 6 has a new formula for entrance length in turbulent duct flow, sent to meby two different researchers. There is a new problem describing the flow in a fuelcell. 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 flow past a flat plate, plus recentreviews of progress in turbulence modeling with CFD. Two new aerodynamicadvances are reported: the Finaish-Witherspoon redesign of the Kline-Fogelman air-foiland the increase in stall angle achieved by tubercles modeled after a humpbackwhale. The new Transition flying car, which had a successful maiden flight in 2009,leads to a number of good problem assignments. Two other photos, Rocket Man overthe 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 modernCFD software. The Transition autocar, featured in Chapter 7, is attacked here byaerodynamic theory, including induced drag.Chapter 9 benefited from reviewer improvement. Figure 9.7, with its 30-year-oldcurve-fits for the area ratio, has been replaced with fine-gridded curves for the area-changeproperties. The curve-fits are gone, and Mach numbers follow nicely fromFig. 9.7 and either Excel or EES. New Trends in Aeronautics presents the X-43 Scram-jetairplane, which generates several new problem assignments. Data for the proposedAlaska-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-larhydraulic jumps, plus a tidal bore, with their associated problem assignments.Chapter 11 has added a section on the performance of gas turbines, with applica-tionto turbofan aircraft engines. The section on wind turbines has been updated, withnew data and photos. A wind-turbine-driven vehicle, which can easily move directlyinto the wind, has inspired new problem assignments.xii Preface 18. 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) inMach 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 thetheory that most students are now quite familiar with EES.A number of supplements are available to students and/or instructors at the textwebsite www.mhhe.com/white7e. Students have access to a Student Study Guidedeveloped by Jerry Dunn of Texas AM University. They are also able to utilizeEngineering Equation Solver (EES), fluid mechanics videos developed by Gary Set-tlesof Pennsylvania State University, and CFD images and animations prepared byFluent Inc. Also available to students are Fundamentals of Engineering (FE) Examquizzes, prepared by Edward Anderson of Texas Tech University.Instructors may obtain a series of PowerPoint slides and images, plus the full Solu-tionsManual, in PDF format. The Solutions Manual provides complete and detailedsolutions, including problem statements and artwork, to the end-of-chapter problems. Itmay be photocopied for posting or preparing transparencies for the classroom. Instruc-torscan also obtain access to C.O.S.M.O.S. for the seventh edition. C.O.S.M.O.S. is aComplete Online Solutions Manual Organization System instructors can use to createexams and assignments, create custom content, and edit supplied problems andsolutions.Ebooks are an innovative way for students to save money and create a greenerenvironment at the same time. An ebook can save students about half the cost of atraditional 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-tactyour McGraw-Hill sales rep or visit the site www.coursesmart.com to learn more.Online SupplementsPreface xiiiElectronic Textbook Options 19. AcknowledgmentsAs usual, so many people have helped me that I may fail to list them all. Sheldon Greenof the University of British Columbia, Gordon Holloway of the University of NewBrunswick, Sukanta K. Dash of The Indian Institute of Technology at Kharagpur, andPezhman Shirvanian of Ford Motor Co. gave many helpful suggestions. Hubert Chansonof the University of Queensland, Frank J. Cunha of PrattWhitney, Samuel Schweighartof Terrafugia Inc., Mark Spear of the Woods Hole Oceanographic Institution, KeithHanna of ANSYS Inc., Elena Mejia of the Jet Propulsion Laboratory, Anne Staack ofSkySails, Inc., and Ellen Emerson White provided great new images. Samuel S. Sihof Walla Walla College, Timothy Singler of SUNY Binghamton, Saeed Moaveni ofMinnesota State University, and John Borg of Marquette University were especiallyhelpful with the solutions manual.The following prereviewers gave many excellent suggestions for improving themanuscript: Rolando Bravo of Southern Illinois University; Joshua B. Kollat of PennState University; Daniel Maynes of Brigham Young University; Joseph Schaefer ofIowa State University; and Xiangchun Xuan of Clemson University.In preparation, the writer got stuck on Chapter 3 but was rescued by the followingreviewers: Serhiy Yarusevych of the University of Waterloo; H. Pirouz Kavehpour andJeff Eldredge of the University of California, Los Angeles; Rayhaneh Akhavan of theUniversity 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 ofthe 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 bookover 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 NASALangley Research Center; LiKai Li of University of Science and Technology of China;Tom Robbins of National Instruments; Tapan K. Sengupta of the Indian Institute ofTechnology at Kanpur; Paulo Vatavuk of Unicamp University; Andris Skattebo of Scand-powerA/S; Jeffrey S. Allen of Michigan Technological University; Peter R. Speddingof Queens University, Belfast, Northern Ireland; Iskender Sahin of Western MichiganUniversity; Michael K. Dailey of General Motors; Cristina L. Archer of StanfordUniversity; Paul F. Jacobs of Technology Development Associates; Rebecca Cullion-Webb of the University of Colorado at Colorado Springs; Debendra K. Das of the Uni-versityof Alaska Fairbanks; Kevin OSullivan and Matthew Lutts of the Associatedxiv 20. Acknowledgments xvPress; 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-nologyat Kharagpur; Fabian Anselmet of the Institut de Recherche sur les PhenomenesHors Equilibre, Marseilles; David Chelidze, Richard Lessmann, Donna Meyer, ArunShukla, Peter Larsen, and Malcolm Spaulding of the University of Rhode Island; CraigSwanson of Applied Science Associates, Inc.; Jim Smay of Oklahoma State University;Deborah Pence of Oregon State University; Eric Braschoss of Philadelphia, PA.; andDale Hart of Louisiana Tech University.The McGraw-Hill staff was, as usual, enormously helpful. Many thanks are due toBill Stenquist, Lora Kalb-Neyens, Curt Reynolds, John Leland, Jane Mohr, BrendaRolwes, as well as all those who worked on previous editions.Finally, the continuing support and encouragement of my wife and family are, asalways, much appreciated. Special thanks are due to our dog, Sadie, and our cats, Coleand Kerry. 21. This page intentionally left blank 22. Fluid Mechanics 23. Hurricane Rita in the Gulf of Mexico, Sept. 22, 2005. Rita made landfall at the Texas-Louisianaborder and caused billions of dollars in wind and flooding damage. Though more dramatic thantypical applications in this text, Rita is a true fluid flow, strongly influenced by the earths rota-tionand the ocean temperature. (Photo courtesy of NASA.)2 24. Fluid mechanics is the study of fluids either in motion (fluid dynamics) or at rest (fluidstatics). Both gases and liquids are classified as fluids, and the number of fluid engi-neeringapplications is enormous: breathing, blood flow, swimming, pumps, fans, tur-bines,airplanes, ships, rivers, windmills, pipes, missiles, icebergs, engines, filters, jets,and sprinklers, to name a few. When you think about it, almost everything on thisplanet either is a fluid or moves within or near a fluid.The essence of the subject of fluid flow is a judicious compromise between theoryand experiment. Since fluid flow is a branch of mechanics, it satisfies a set of well-documentedbasic 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-oryare geometry and viscosity. The basic equations of fluid motion (Chap. 4) are toodifficult to enable the analyst to attack arbitrary geometric configurations. Thus mosttextbooks concentrate on flat plates, circular pipes, and other easy geometries. It is pos-sibleto apply numerical computer techniques to complex geometries, and specializedtextbooks are now available to explain the new computational fluid dynamics (CFD)approximations and methods [14].1 This book will present many theoretical resultswhile keeping their limitations in mind.The second obstacle to a workable theory is the action of viscosity, which can beneglected only in certain idealized flows (Chap. 8). First, viscosity increases the diffi-cultyof the basic equations, although the boundary-layer approximation found byLudwig Prandtl in 1904 (Chap. 7) has greatly simplified viscous-flow analyses. Sec-ond,viscosity has a destabilizing effect on all fluids, giving rise, at frustratingly smallvelocities, to a disorderly, random phenomenon called turbulence. The theory of tur-bulentflow is crude and heavily backed up by experiment (Chap. 6), yet it can be quiteserviceable as an engineering estimate. This textbook only introduces the standardexperimental correlations for turbulent time-mean flow. Meanwhile, there are advancedtexts on both time-mean turbulence and turbulence modeling [5, 6] and on the newer,computer-intensive direct numerical simulation (DNS) of fluctuating turbulence [7, 8].31.1 Preliminary RemarksChapter 1Introduction1Numbered references appear at the end of each chapter. 25. Thus there is theory available for fluid flow problems, but in all cases it shouldbe backed up by experiment. Often the experimental data provide the main sourceof information about specific flows, such as the drag and lift of immersed bodies(Chap. 7). Fortunately, fluid 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-mentto the theory. You should keep in mind that theory and experiment should gohand in hand in all studies of fluid mechanics.Like most scientific disciplines, fluid mechanics has a history of erratically occurringearly achievements, then an intermediate era of steady fundamental discoveries in theeighteenth and nineteenth centuries, leading to the twenty-first-century era of modernpractice, as we self-centeredly term our limited but up-to-date knowledge. Ancientcivilizations had enough knowledge to solve certain flow problems. Sailing ships withoars and irrigation systems were both known in prehistoric times. The Greeks pro-ducedquantitative information. Archimedes and Hero of Alexandria both postulatedthe 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 floating and sub-mergedbodies, actually deriving a form of the differential calculus as part of theanalysis. The Romans built extensive aqueduct systems in the fourth century B.C. butleft no records showing any quantitative knowledge of design principles.From the birth of Christ to the Renaissance there was a steady improvement in thedesign of such flow systems as ships and canals and water conduits but no recordedevidence of fundamental improvements in flow analysis. Then Leonardo da Vinci(14521519) stated the equation of conservation of mass in one-dimensional steadyflow. Leonardo was an excellent experimentalist, and his notes contain accuratedescriptions of waves, jets, hydraulic jumps, eddy formation, and both low-drag(streamlined) and high-drag (parachute) designs. A Frenchman, Edme Mariotte(16201684), built the first wind tunnel and tested models in it.Problems involving the momentum of fluids could finally be analyzed after IsaacNewton (16421727) postulated his laws of motion and the law of viscosity of the lin-earfluids now called newtonian. The theory first yielded to the assumption of a per-fector frictionless fluid, and eighteenth-century mathematicians (Daniel Bernoulli,Leonhard Euler, Jean dAlembert, Joseph-Louis Lagrange, and Pierre-Simon Laplace)produced many beautiful solutions of frictionless-flow problems. Euler, Fig. 1.1, devel-opedboth the differential equations of motion and their integrated form, now calledthe Bernoulli equation. DAlembert used them to show his famous paradox: that a bodyimmersed in a frictionless fluid has zero drag. These beautiful results amounted tooverkill, since perfect-fluid assumptions have very limited application in practice andmost engineering flows are dominated by the effects of viscosity. Engineers began toreject what they regarded as a totally unrealistic theory and developed the science ofhydraulics, relying almost entirely on experiment. Such experimentalists as Chzy,Pitot, Borda, Weber, Francis, Hagen, Poiseuille, Darcy, Manning, Bazin, and Weisbachproduced data on a variety of flows such as open channels, ship resistance, pipe flows,waves, and turbines. All too often the data were used in raw form without regard tothe fundamental physics of flow.4 Chapter 1 Introduction1.2 History and Scope ofFluid MechanicsFig. 1.1 Leonhard Euler (17071783) was the greatest mathemati-cianof the eighteenth century andused Newtons calculus to developand solve the equations of motionof inviscid flow. He published over800 books and papers. [Courtesyof the School of Mathematics andStatistics, University of St Andrew,Scotland.] 26. 1.2 History and Scope of Fluid Mechanics 5At the end of the nineteenth century, unification between experimental hydraulicsand theoretical hydrodynamics finally began. William Froude (18101879) and his sonRobert (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 thedimensionless Reynolds number named after him. Meanwhile, viscous-flow theorywas available but unexploited, since Navier (17851836) and Stokes (18191903) hadsuccessfully added newtonian viscous terms to the equations of motion. The result-ingNavier-Stokes equations were too difficult to analyze for arbitrary flows. Then, in1904, a German engineer, Ludwig Prandtl (18751953), Fig. 1.2, published perhaps themost important paper ever written on fluid mechanics. Prandtl pointed out that fluid flowswith small viscosity, such as water flows and airflows, can be divided into a thin vis-couslayer, or boundary layer, near solid surfaces and interfaces, patched onto a nearlyinviscid outer layer, where the Euler and Bernoulli equations apply. Boundary-layertheory has proved to be a very important tool in modern flow analysis. The twentieth-centuryfoundations for the present state of the art in fluid mechanics were laid in a seriesof broad-based experiments and theories by Prandtl and his two chief friendly competi-tors,Theodore von Krmn (18811963) and Sir Geoffrey I. Taylor (18861975). Manyof the results sketched here from a historical point of view will, of course, be discussedin this textbook. More historical details can be found in Refs. 12 to 14.The second half of the twentieth century introduced a new tool: ComputationalFluid Dynamics (CFD). The earliest paper on the subject known to this writer wasby A. Thom in 1933 [47], a laborious, but accurate, hand calculation of flow past acylinder at low Reynolds numbers. Commercial digital computers became availablein the 1950s, and personal computers in the 1970s, bringing CFD into adulthood. Alegendary first textbook was by Patankar [3]. Presently, with increases in computerspeed and memory, almost any laminar flow can be modeled accurately. Turbulentflow is still calculated with empirical models, but Direct Numerical Simulation [7, 8]is possible for low Reynolds numbers. Another five orders of magnitude in computerspeed are needed before general turbulent flows can be calculated. That may not bepossible, 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 fluid mechanicsreduced to a black box, with no real need for teachers, researchers, writers, or fluidsengineers.Since the earth is 75 percent covered with water and 100 percent covered with air,the scope of fluid mechanics is vast and touches nearly every human endeavor. Thesciences of meteorology, physical oceanography, and hydrology are concerned withnaturally occurring fluid flows, as are medical studies of breathing and blood circu-lation.All transportation problems involve fluid motion, with well-developed spe-cialtiesin aerodynamics of aircraft and rockets and in naval hydrodynamics of shipsand submarines. Almost all our electric energy is developed either from water flowor from steam flow through turbine generators. All combustion problems involve fluidmotion as do the more classic problems of irrigation, flood control, water supply,sewage disposal, projectile motion, and oil and gas pipelines. The aim of this bookis to present enough fundamental concepts and practical applications in fluid mechan-icsto prepare you to move smoothly into any of these specialized fields of the sci-enceof flowand then be prepared to move out again as new technologies develop.Fig. 1.2 Ludwig Prandtl (18751953), often called the father ofmodern fluid mechanics [15],developed boundary layer theoryand many other innovative analy-ses.He and his students werepioneers in flow visualizationtechniques. [Aufnahme von Fr.Struckmeyer, Gottingen, courtesyAIP Emilio Segre Visual Archives,Lande Collection.] 27. Fluid flow analysis is packed with problems to be solved. The present text has morethan 1700 problem assignments. Solving a large number of these is a key to learningthe subject. One must deal with equations, data, tables, assumptions, unit systems,and solution schemes. The degree of difficulty will vary, and we urge you to samplethe 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 wrongquestionfor example, pressure instead of pressure gradient, lift force insteadof drag force, or mass flow instead of volume flow. 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 flow is steadyor unsteady, compressible or incompressible, viscous or inviscid, and whethera control volume or partial differential equations are needed.6. Find an algebraic solution if possible. Then, if a numerical value is needed, useeither 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 ofsignificant figures (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 fluid mechanics, all matter consists of only two states, fluidand solid. The difference between the two is perfectly obvious to the layperson, and itis an interesting exercise to ask a layperson to put this difference into words. The tech-nicaldistinction lies with the reaction of the two to an applied shear or tangential stress.A solid can resist a shear stress by a static deflection; a fluid cannot . Any shear stressapplied to a fluid, no matter how small, will result in motion of that fluid. The fluidmoves and deforms continuously as long as the shear stress is applied. As a corollary,we can say that a fluid at rest must be in a state of zero shear stress, a state often calledthe hydrostatic stress condition in structural analysis. In this condition, Mohrs circlefor stress reduces to a point, and there is no shear stress on any plane cut through theelement under stress.Given this definition of a fluid, every layperson also knows that there are twoclasses of fluids, liquids and gases. Again the distinction is a technical one concern-ingthe effect of cohesive forces. A liquid, being composed of relatively close-packedmolecules with strong cohesive forces, tends to retain its volume and will form a freesurface in a gravitational field if unconfined from above. Free-surface flows are dom-inatedby gravitational effects and are studied in Chaps. 5 and 10. Since gas mole-culesare widely spaced with negligible cohesive forces, a gas is free to expand untilit encounters confining walls. A gas has no definite volume, and when left to itselfwithout confinement, a gas forms an atmosphere that is essentially hydrostatic. Thehydrostatic behavior of liquids and gases is taken up in Chap. 2. Gases cannot forma free surface, and thus gas flows are rarely concerned with gravitational effects otherthan buoyancy.6 Chapter 1 Introduction1.3 Problem-Solving Techniques1.4 The Concept of a Fluid 28. 1.4 The Concept of a Fluid 7StaticdeflectionFreesurfaceA A ASolid Liquid(a) (c)1 p1A A = p = pHydrostaticcondition02 p p(b) (d)Gas(1)pp= 0Figure 1.3 illustrates a solid block resting on a rigid plane and stressed by its own0weight. The solid sags into a static deflection, shown as a highly exaggerated dashedline, resisting shear without flow. A free-body diagram of element A on the side ofthe block shows that there is shear in the block along a plane cut at an anglethroughA. Since the block sides are unsupported, element A has zero stress on the left andright sides and compression stress p on the top and bottom. Mohrs circle doesnot 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 inorder to eliminate shear stress. The walls exert a compression stress of p and reduceMohrs 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 wallsare removed, shear develops in the liquid and a big splash results. If the container istilted, shear again develops, waves form, and the free surface seeks a horizontal con-figuration,pouring out over the lip if necessary. Meanwhile, the gas is unrestrainedand expands out of the container, filling all available space. Element A in the gas isalso hydrostatic and exerts a compression stress p on the walls.Fig. 1.3 A solid at rest can resistshear. (a) Static deflection of thesolid; (b) equilibrium and Mohrscircle for solid element A. A fluidcannot resist shear. (c) Containingwalls are needed; (d ) equilibriumand Mohrs circle for fluidelement A. 29. In the previous discussion, clear decisions could be made about solids, liquids, andgases. Most engineering fluid mechanics problems deal with these clear casesthat is,the common liquids, such as water, oil, mercury, gasoline, and alcohol, and the com-mongases, such as air, helium, hydrogen, and steam, in their common temperature andpressure ranges. There are many borderline cases, however, of which you should beaware. Some apparently solid substances such as asphalt and lead resist shear stressfor short periods but actually deform slowly and exhibit definite fluid behavior overlong periods. Other substances, notably colloid and slurry mixtures, resist small shearstresses but yield at large stress and begin to flow as fluids do. Specialized textbooksare devoted to this study of more general deformation and flow, a field calledrheology [16]. Also, liquids and gases can coexist in two-phase mixtures, such assteamwater mixtures or water with entrapped air bubbles. Specialized textbooks pres-entthe analysis of such multiphase flows [17]. Finally, in some situations the distinc-tionbetween a liquid and a gas blurs. This is the case at temperatures and pressuresabove the so-called critical point of a substance, where only a single phase exists, pri-marilyresembling a gas. As pressure increases far above the critical point, the gaslikesubstance becomes so dense that there is some resemblance to a liquid and the usualthermodynamic approximations like the perfect-gas law become inaccurate. The criti-caltemperature and pressure of water are Tc647 K and pc219 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 Tc126 K and pc34 atm. Thus typical problems involving air arein the range of high temperature and low pressure where air is distinctly and definitelya gas. This text will be concerned solely with clearly identifiable liquids and gases,and the borderline cases just discussed will be beyond our scope.We have already used technical terms such as fluid pressure and density without a rig-orousdiscussion of their definition. As far as we know, fluids are aggregations of mol-ecules,widely spaced for a gas, closely spaced for a liquid. The distance between mol-eculesis very large compared with the molecular diameter. The molecules are not fixedin a lattice but move about freely relative to each other. Thus fluid density, or mass perunit volume, has no precise meaning because the number of molecules occupying a givenvolume continually changes. This effect becomes unimportant if the unit volume is largecompared with, say, the cube of the molecular spacing, when the number of moleculeswithin the volume will remain nearly constant in spite of the enormous interchange ofparticles across the boundaries. If, however, the chosen unit volume is too large, therecould be a noticeable variation in the bulk aggregation of the particles. This situation isillustrated in Fig. 1.4, where the density as calculated from molecular mass m withina given volumeis 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-gatevariations may be important. The densityof a fluid is best defined aslim (1.1)S*m8 Chapter 1 Introduction1.5 The Fluid as a Continuum2One atmosphere equals 2116 lbf/ft2101,300 Pa. 30. 1200 The limiting volume * is about 109 mm3 for all liquids and for gases at atmo-sphericpressure. For example, 109 mm3 of air at standard conditions contains approx-imately3107 molecules, which is sufficient to define a nearly constant densityaccording to Eq. (1.1). Most engineering problems are concerned with physical dimen-sionsmuch larger than this limiting volume, so that density is essentially a point func-tionand fluid properties can be thought of as varying continually in space, as sketchedin Fig. 1.4a. Such a fluid is called a continuum, which simply means that its varia-tionin properties is so smooth that differential calculus can be used to analyze thesubstance. We shall assume that continuum calculus is valid for all the analyses inthis book. Again there are borderline cases for gases at such low pressures that molec-ularspacing and mean free path3 are comparable to, or larger than, the physical sizeof the system. This requires that the continuum approximation be dropped in favor ofa molecular theory of rarefied gas flow [18]. In principle, all fluid mechanics problemscan 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-tinuousjumps in fluid properties across a free surface or fluid interface or across ashock wave in a compressible fluid (Chap. 9). Our calculus in analyzing fluid flowmust be flexible 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. Thuslength is a dimension associated with such variables as distance, displacement, width,deflection, and height, while centimeters and inches are both numerical units forexpressing length. Dimension is a powerful concept about which a splendid tool calleddimensional analysis has been developed (Chap. 5), while units are the numericalquantity that the customer wants as the final 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 wasan improvement over British systems because its use of base 10 is the foundation ofour number system, learned from childhood by all. Problems still remained because1.6 Dimensions and Units1.6 Dimensions and Units 9MicroscopicuncertaintyMacroscopicuncertainty0 * 10-9 mm3ElementalvolumeRegion containing fluid= 1000 kg/m3= 1100= 1200= 1300(a) (b)Fig. 1.4 The limit definition ofcontinuum fluid density: (a) anelemental volume in a fluid regionof variable continuum density;(b) calculated density versus sizeof the elemental volume.3The mean distance traveled by molecules between collisions (see Prob. P1.5). 31. even the metric countries differed in their use of kiloponds instead of dynes ornewtons, kilograms instead of grams, or calories instead of joules. To standardize themetric system, a General Conference of Weights and Measures, attended in 1960 by40 countries, proposed the International System of Units (SI). We are now undergo-inga painful period of transition to SI, an adjustment that may take many more yearsto complete. The professional societies have led the way. Since July 1, 1974, SI unitshave been required by all papers published by the American Society of MechanicalEngineers, and there is a textbook explaining the SI [19]. The present text will use SIunits together with British gravitational (BG) units.In fluid mechanics there are only four primary dimensions from which all other dimen-sionscan be derived: mass, length, time, and temperature.4 These dimensions and theirunits in both systems are given in Table 1.1. Note that the kelvin unit uses no degreesymbol. The braces around a symbol like {M} mean the dimension of mass. All othervariables in fluid mechanics can be expressed in terms of {M}, {L}, {T}, and {}. Forexample, acceleration has the dimensions {LT2}. The most crucial of these secondarydimensions is force, which is directly related to mass, length, and time by Newtonssecond law. Force equals the time rate of change of momentum or, for constant mass,Fma (1.2)From this we see that, dimensionally, {F}{MLT2}.The use of a constant of proportionality in Newtons law, Eq. (1.2), is avoided bydefining the force unit exactly in terms of the other basic units. In the SI system, thebasic units are newtons {F}, kilograms {M}, meters {L}, and seconds {T}. We define1 newton of force1 N1 kg 1 m/s2The 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 ofthese 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 defin-ingthe force unit exactly in terms of the other basic units. In the BG system, the basicunits are pound-force {F}, slugs {M}, feet {L}, and seconds {T}. We define1 pound of force1 lbf1 slug # 1 ft/s210 Chapter 1 IntroductionTable 1.1 Primary Dimensions inSI and BG SystemsPrimary DimensionsThe British Gravitational (BG)System#The International System (SI)Primary dimension SI unit BG unit Conversion factorMass {M} Kilogram (kg) Slug 1 slug14.5939 kgLength {L} Meter (m) Foot (ft) 1 ft0.3048 mTime {T} Second (s) Second (s) 1 s1 sTemperature {} Kelvin (K) Rankine (R) 1 K1.8R4If electromagnetic effects are important, a fifth primary dimension must be included, electric current{I}, whose SI unit is the ampere (A). 32. One lbf 4.4482 N and approximates the weight of four apples. We will use theabbreviation lbf for pound-force and lbm for pound-mass. The slug is a rather heftymass, equal to 32.174 lbm. The basic unit of temperature {} in the BG system is thedegree Rankine, R. Recall that a temperature difference 1 K1.8R. Use of theseBG 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 mostapplications (1 dyne105 N) and will not be used here.In the USA, some still use the English Engineering system, (lbf, lbm, ft, s, R), wherethe 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 notequal to 1.0. The present text uses only the SI and BG systems and will not solve prob-lemsor 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, thatis, 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:p 12 V2gZconstantEach and every term in this equation must have dimensions of pressure {ML1T2}.We will examine the dimensional homogeneity of this equation in detail in Ex. 1.3.A list of some important secondary variables in fluid mechanics, with dimensionsderived as combinations of the four primary dimensions, is given in Table 1.2. A morecomplete list of conversion factors is given in App. C.The Principle ofDimensional HomogeneityF magc, where gc32.174ft # lbmlbf # s2Other Unit Systems1.6 Dimensions and Units 11Table 1.2 Secondary Dimensionsin Fluid Mechanics Secondary dimension SI unit BG unit Conversion factorArea {L2} m2 ft2 1 m210.764 ft2Volume {L3} m3 ft3 1 m335.315 ft3Velocity {LT1} m/s ft/s 1 ft/s0.3048 m/sAcceleration {LT2} m/s2 ft/s2 1 ft/s20.3048 m/s2Pressure or stress {ML1T2} PaN/m2 lbf/ft2 1 lbf/ft247.88 PaAngular velocity {T1} s1 s1 1 s11 s1Energy, heat, work {ML2T2} JNm ftlbf 1 ftlbf1.3558 JPower {ML2T3} W J/s ftlbf/s 1 ftlbf/s1.3558 WDensity {ML3} kg/m3 slugs/ft3 1 slug/ft3515.4 kg/m3Viscosity {ML1T1} kg/(ms) slugs/(fts) 1 slug/(fts)47.88 kg/(ms)Specific heat {L2T21} m2/(s2K) ft2/(s2R) 1 m2/(s2K)5.980 ft2/(s2R) 33. EXAMPLE 1.1A body weighs 1000 lbf when exposed to a standard earth gravity g32.174 ft/s2. (a) Whatis its mass in kg? (b) What will the weight of this body be in N if it is exposed to themoons standard acceleration gmoon1.62 m/s2? (c) How fast will the body accelerate ifa net force of 400 lbf is applied to it on the moon or on the earth?SolutionWe need to find 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-ertydata 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:m31.08 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 witha new gravitational acceleration and hence a new weight:FWmoonmgmoon(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 ofNewtons law with a known mass and known force:F400 lbfma(31.08 slugs) aSolve forAns. (c)a 400 lbf31.08 slugs 12.87fts2 a0.3048mftb3.92ms2Comment (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 suitableonly to some industry or specialty or country. The engineer should convert these datato the SI or BG system before using them. This requires the systematic applicationof conversion factors, as in the following example.EXAMPLE 1.2Industries involved in viscosity measurement [27, 36] continue to use the CGS system ofunits, since centimeters and grams yield convenient numbers for many fluids. The absoluteviscosity () unit is the poise, named after J. L. M. Poiseuille, a French physician who in1840 performed pioneering experiments on water flow in pipes; 1 poise1 g/(cm-s). Thekinematic viscosity () unit is the stokes, named after G. G. Stokes, a British physicist who12 Chapter 1 IntroductionPart (b)Part (c)FW1000 lbfmg(m)(32.174 ft/s2), or m 1000 lbf32.174 ft/s231.08 slugsPart (a) 34. in 1845 helped develop the basic partial differential equations of fluid momentum; 1 stokes 1 cm2/s. Water at 20C has 0.01 poise and also 0.01 stokes. Express theseresults in (a) SI and (b) BG units.Solution Approach: Systematically change grams to kg or slugs and change centimeters to metersor 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 unitsAns. (b)0.01cm2s 0.01cm2(0.01 m/cm)2(1 ft/0.3048 m)2 Comments: This was a laborious conversion that could have been shortened by usingthe direct viscosity conversion factors in App. C. For example, BGSI/47.88.We repeat our advice: Faced with data in unusual units, convert them immediatelyto either SI or BG units because (1) it is more professional and (2) theoretical equa-tionsin fluid mechanics are dimensionally consistent and require no further conver-sionfactors when these two fundamental unit systems are used, as the followingexample shows.EXAMPLE 1.3A useful theoretical equation for computing the relation between pressure, velocity, and alti-tudein a steady flow of a nearly inviscid, nearly incompressible fluid with negligible heattransfer and shaft work5 is the Bernoulli relation, named after Daniel Bernoulli, who pub-lisheda hydrodynamics textbook in 1738:(1)p0p12V2gZwhere p0stagnation pressureppressure in moving fluidVvelocitydensityZaltitudeggravitational accelerations 0.0000108ft2s0.01gcm # s 0.01g(1 kg/1000 g)(1 slug/14.5939 kg)(0.01 m/cm)(1 ft/0.3048 m)s 0.0000209slugft # sPart (b)0.01cm2s 0.01cm2(0.01 m/cm)2s 0.000001m2s0.01gcm # s 0.01g(1 kg/1000g)cm(0.01 m/cm)s 0.001kgm # sPart (a)1.6 Dimensions and Units 135Thats an awful lot of assumptions, which need further study in Chap. 3. 35. (a) Show that Eq. (1) satisfies the principle of dimensional homogeneity, which states that alladditive terms in a physical equation must have the same dimensions. (b) Show that consis-tentunits result without additional conversion factors in SI units. (c) Repeat (b) for BG units.SolutionWe can express Eq. (1) dimensionally, using braces, by entering the dimensions of eachterm from Table 1.2:{ML1T2}{ML1T2}{ML3}{L2T2}{ML3}{LT2}{L} {ML1T2} 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/(ms2)}The right-hand side looks bad until we remember from Eq. (1.3) that 1 kg1 Ns2/m.Ans. (b)Thus all terms in Bernoullis equation will have units of pascals, or newtons per squaremeter, when SI units are used. No conversion factors are needed, which is true of all theo-reticalequations in fluid 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/(fts2)}But, from Eq. (1.3), 1 slug1 lbfs2/ft, so thatAns. (c)All terms have the unit of pounds-force per square foot. No conversion factors are neededin the BG system either.There is still a tendency in English-speaking countries to use pound-force per squareinch as a pressure unit because the numbers are more manageable. For example, stan-dardatmospheric pressure is 14.7 lbf/in22116 lbf/ft2101,300 Pa. The pascal is asmall unit because the newton is less than lbf and a square meter is a very large area.Note that not only must all (fluid) 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, butwoe unto those who try to mix colloquial English units. For example, in Chap. 9, weoften use the assumption of steady adiabatic compressible gas flow:h12V2constant14 Chapter 1 IntroductionConsistent Units145slugs/(ft # s2)6 5lbf # s2/ft65ft # s26 5lbf/ft26Part (a)Part (c)5kg/(m # s2)6 5N # s2/m65m # s26 5N/m26Part (b) 36. where h is the fluid enthalpy and V2/2 is its kinetic energy per unit mass. Colloquialthermodynamic 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 toft2/s2. The proper unit for h in this case is ftlbf/slug, which is identical to ft2/s2.The conversion factor is 1 Btu/lb25,040 ft2/s225,040 ftlbf/slug.All theoretical equations in mechanics (and in other physical sciences) are dimension-allyhomogeneous; 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-neeringliterature, arising primarily from correlations of data, are dimensionally incon-sistent.Their units cannot be reconciled simply, and some terms may contain hiddenvariables. An example is the formula that pipe valve manufacturers cite for liquid vol-umeflow rate Q (m3/s) through a partially open valve:1/2where p is the pressure drop across the valve and SG is the specific gravity of theliquid (the ratio of its density to that of water). The quantity CV is the valve flowcoefficient, 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 aflow rate {L3/T} and the other being the square root of a pressure drop {M1/2/L1/2T}.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 ofCV in the literature increase nearly as the square of the size of the valve. The pres-entationof experimental data in homogeneous form is the subject of dimensionalanalysis (Chap. 5). There we shall learn that a homogeneous form for the valve flowrelation iswhereis the liquid density and A the area of the valve opening. The discharge coef-ficientC d is dimensionless and changes only moderately with valve size. Pleasebelieveuntil we establish the fact in Chap. 5that this latter is a much better for-mulationof the data.Meanwhile, we conclude that dimensionally inconsistent equations, though theyoccur in engineering practice, are misleading and vague and even dangerous, in thesense that they are often misused outside their range of applicability.Engineering results often are too small or too large for the common units, with toomany zeros one way or the other. For example, to write p114,000,000 Pa is longand awkward. Using the prefix M to mean 106, we convert this to a concisep114 MPa (megapascals). Similarly, t0.000000003 s is a proofreaders night-marecompared to the equivalent t3 ns (nanoseconds). Such prefixes are com-monand convenient, in both the SI and BG systems. A complete list is given inTable 1.3.Convenient Prefixes inPowers of 10QCdAopeningapb1/2QCVapSGbHomogeneous versusDimensionally InconsistentEquations1.6 Dimensions and Units 15Table 1.3 Convenient Prefixesfor Engineering UnitsMultiplicativefactor Prefix Symbol1012 tera T109 giga G106 mega M103 kilo k102 hecto h10 deka da101 deci d102 centi c103 milli m106 micro 109 nano n1012 pico p1015 femto f1018 atto a 37. EXAMPLE 1.4In 1890 Robert Manning, an Irish engineer, proposed the following empirical formulafor the average velocity V in uniform flow due to gravity down an open channel (BGunits):(1)where Rhydraulic radius of channel (Chaps. 6 and 10)Schannel slope (tangent of angle that bottom makes with horizontal)nMannings 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 tobe 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 dimensionlessratio with the dimensional notation {1}that is, not containing M, L, or T. Approach (a): Rewrite the dimensions of each term in Mannings equation, usingbrackets {}:This formula is incompatible unless {1.49/n}{L1/3/T}. If n is dimensionless (and it isnever listed with units in textbooks), the number 1.49 must carry the dimensions of{L1/3/T}. Ans. (a) Comment (a): Formulas whose numerical coefficients have units can be disastrous forengineers working in a different system or another fluid. Mannings formula, though pop-ular,is inconsistent both dimensionally and physically and is valid only for water flowwith certain wall roughnesses. The effects of water viscosity and density are hidden inthe numerical value 1.49. Approach (b): Part (a) showed that 1.49 has dimensions. If the formula is valid in BGunits, 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)1/31.00 m1/3/sTherefore 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, firstproposed the formula. It was later converted to BG units. Such dimensionally inconsis-tentformulas are dangerous and should either be reanalyzed or treated as having verylimited application.SI units: V 1.0nR2/3S1/25V6e1.49nf 5R2/36 5S1/26 or eLTfe1.49nf 5L2/36 516V 1.49nR2/3S1/216 Chapter 1 Introduction 38. In a given flow situation, the determination, by experiment or theory, of the proper-tiesof the fluid as a function of position and time is considered to be the solution tothe problem. In almost all cases, the emphasis is on the spacetime distribution of thefluid properties. One rarely keeps track of the actual fate of the specific fluid parti-cles.6 This treatment of properties as continuum-field functions distinguishes fluidmechanics from solid mechanics, where we are more likely to be interested in the tra-jectoriesof individual particles or systems.There are two different points of view in analyzing problems in mechanics. The firstview, appropriate to fluid mechanics, is concerned with the field of flow and is calledthe eulerian method of description. In the eulerian method we compute the pressurefield p(x, y, z, t) of the flow pattern, not the pressure changes p(t) that a particle expe-riencesas it moves through the field.The second method, which follows an individual particle moving through the flow,is called the lagrangian description. The lagrangian approach, which is more appro-priateto solid mechanics, will not be treated in this book. However, certain numeri-calanalyses of sharply bounded fluid flows, such as the motion of isolated fluiddroplets, 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 flow, it is fixed at a specificposition (x, y, z). Its output thus contributes to the description of the eulerian pres-surefield p(x, y, z, t). To simulate a lagrangian measurement, the probe would haveto move downstream at the fluid particle speeds; this is sometimes done in oceano-graphicmeasurements, where flowmeters drift along with the prevailing currents.The two different descriptions can be contrasted in the analysis of traffic flow alonga freeway. A certain length of freeway may be selected for study and called the field offlow. Obviously, as time passes, various cars will enter and leave the field, and the iden-tityof the specific cars within the field will constantly be changing. The traffic engi-neerignores specific cars and concentrates on their average velocity as a function oftime and position within the field, plus the flow rate or number of cars per hour pass-inga given section of the freeway. This engineer is using an eulerian description of thetraffic flow. Other investigators, such as the police or social scientists, may be interestedin the path or speed or destination of specific cars in the field. By following a specificcar as a function of time, they are using a lagrangian description of the flow.Foremost among the properties of a flow is the velocity field V(x, y, z, t). In fact,determining the velocity is often tantamount to solving a flow problem, since otherproperties follow directly from the velocity field. Chapter 2 is devoted to the calcu-lationof the pressure field once the velocity field is known. Books on heat transfer(for example, Ref. 20) are largely devoted to finding the temperature field from knownvelocity fields.1.7 Properties of theVelocity FieldEulerian and LagrangianDescriptionsThe Velocity Field1.7 Properties of the Velocity Field 176One example where fluid particle paths are important is in water quality analysis of the fate of con-taminantdischarges. 39. In general, velocity is a vector function of position and time and thus has threecomponents u, , and w, each a scalar field in itself:(1.4)The use of u, , and w instead of the more logical component notation Vx, Vy, and Vzis the result of an almost unbreakable custom in fluid mechanics. Much of this text-book,especially Chaps. 4, 7, 8, and 9, is concerned with finding the distribution ofthe velocity vector V for a variety of practical flows.The acceleration vector, adV/dt, occurs in Newtons law for a fluid and thus is veryimportant. In order to follow a particle in the Eulerian frame of reference, the final resultfor 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 thisformula in detail in Chap. 4. The last three terms in Eq. (1.5) are nonlinear productsand greatly complicate the analysis of general fluid motions, especially viscousflows.While the velocity field V is the most important fluid property, it interacts closelywith the thermodynamic properties of the fluid. We have already introduced into thediscussion the three most common such properties:1. Pressure p2. Density 3. Temperature TThese three are constant companions of the velocity vector in flow analyses. Fourother intensive thermodynamic properties become important when work, heat, andenergy balances are treated (Chaps. 3 and 4):4. Internal energy 5. Enthalpy hp/6. Entropy s7. Specific heats cp and cvIn addition, friction and heat conduction effects are governed by the two so-calledtransport properties:8. Coefficient of viscosity 9. Thermal conductivity kAll nine of these quantities are true thermodynamic properties that are determined bythe thermodynamic condition or state of the fluid. For example, for a single-phase1.8 Thermodynamic Propertiesof a Fluida dVdt V t u V x v V y w V zThe Acceleration FieldV(x, y, z, t)iu(x, y, z, t)jv(x, y, z, t)kw(x, y, z, t)18 Chapter 1 Introduction 40. substance such as water or oxygen, two basic properties such as pressure andtemperature are sufficient to fix the value of all the others:(p, T) hh(p, T) (p, T)and so on for every quantity in the list. Note that the specific volume, so importantin thermodynamic analyses, is omitted here in favor of its inverse, the density .Recall that thermodynamic properties describe the state of a systemthat is, acollection of matter of fixed identity that interacts with its surroundings. In most caseshere the system will be a small fluid element, and all properties will be assumed tobe continuum properties of the flow field: (x, y, z, t), and so on.Recall also that thermodynamics is normally concerned with static systems,whereas fluids are usually in variable motion with constantly changing properties.Do the properties retain their meaning in a fluid flow that is technically not inequilibrium? 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 collisionsoccur over a very short distance of the order of 1 m, so that a fluid subjected tosudden changes rapidly adjusts itself toward equilibrium. We therefore assume thatall the thermodynamic properties just listed exist as point functions in a flowing fluidand follow all the laws and state relations of ordinary equilibrium thermodynamics.There are, of course, important nonequilibrium effects such as chemical and nuclearreactions in flowing fluids, which are not treated in this text.Pressure is the (compression) stress at a point in a static fluid (Fig. 1.3). Next tovelocity, the pressure p is the most dynamic variable in fluid mechanics. Differencesor gradients in pressure often drive a fluid flow, especially in ducts. In low-speedflows, the actual magnitude of the pressure is often not important, unless it drops solow as to cause vapor bubbles to form in a liquid. For convenience, we set many suchproblem assignments at the level of 1 atm2116 lbf/ft2101,300 Pa. High-speed(compressible) gas flows (Chap. 9), however, are indeed sensitive to the magnitudeof pressure.Temperature T is related to the internal energy level of a fluid. It may vary consider-ablyduring high-speed flow of a gas (Chap. 9). Although engineers often use Celsiusor Fahrenheit scales for convenience, many applications in this text require absolute(Kelvin or Rankine) temperature scales:RF459.69If temperature differences are strong, heat transfer may be important [20], but ourconcern here is mainly with dynamic effects.The density of a fluid, denoted by(lowercase Greek rho), is its mass per unit volume.Density is highly variable in gases and increases nearly proportionally to the pressurelevel. Density in liquids is nearly constant; the density of water (about 1000 kg/m3)PressureDensityKC273.16Temperature1.8 Thermodynamic Properties of a Fluid 19 41. increases only 1 percent if the pressure is increased by a factor of 220. Thus most liq-uidflows are treated analytically as nearly incompressible.In general, liquids are about three orders of magnitude more dense than gases atatmospheric pressure. The heaviest common liquid is mercury, and the lightest gas ishydrogen. Compare their densities at 20C and 1 atm:They differ by a factor of 162,000! Thus the physical parameters in various liquidand gas flows might vary considerably. The differences are often resolved by the useof dimensional analysis (Chap. 5). Other fluid densities are listed in Tables A.3 andA.4 (in App. A) and in Ref. 21.The specific weight of a fluid, denoted by(lowercase Greek gamma), is its weightper unit volume. Just as a mass has a weight Wmg, density and specific weightare simply related by gravity:(1.6)gThe units ofare weight per unit volume, in lbf/ft3 or N/m3. In standard earth grav-ity,g32.174 ft/s29.807 m/s2. Thus, for example, the specific weights of air andwater at 20C and 1 atm are approximatelyair(1.205 kg/m3)(9.807 m/s2)11.8 N/m30.0752 lbf/ft3Specific weight is very useful in the hydrostatic pressure applications of Chap. 2.Specific weights of other fluids are given in Tables A.3 and A.4.Specific gravity, denoted by SG, is the ratio of a fluid density to a standard referencefluid, usually water at 4C (for liquids) and air (for gases):(1.7)For example, the specific gravity of mercury (Hg) is SGHg13,580/100013.6.Engineers find these dimensionless ratios easier to remember than the actual numer-icalvalues of density of a variety of fluids.In thermostatics the only energy in a substance is that stored in a system by molec-ularactivity and molecular bonding forces. This is commonly denoted as internalenergy . A commonly accepted adjustment to this static situation for fluid flow is toadd two more energy terms that arise from newtonian mechanics: potential energyand kinetic energy.Potential and Kinetic EnergiesSGliquid liquidwaterliquid1000 kg/m3SGgas gasairgas1.205 kg/m3Specific Gravitywater(998 kg/m3)(9.807 m/s2)9790 N/m362.4 lbf/ft3Specific WeightMercury: 13,580 kg/m3 Hydrogen: 0.0838 kg/m320 Chapter 1 Introduction 42. 1.8 Thermodynamic Properties of a Fluid 21The potential energy equals the work required to move the system of mass m fromthe origin to a position vector rixjykz against a gravity field g. Its valueis mgr, or gr per unit mass. The kinetic energy equals the work required tochange the speed of the mass from zero to velocity V. Its value is mV2 or V2 perunit mass. Then by common convention the total stored energy e per unit mass influid mechanics is the sum of three terms:eV2(gr) (1.8)Also, throughout this book we shall define z as upward, so that ggk and gr gz. Then Eq. (1.8) becomeseV2gz (1.9)The molecular internal energy is a function of T and p for the single-phase puresubstance, whereas the potential and kinetic energies are kinematic quantities.Thermodynamic properties are found both theoretically and experimentally to berelated to each other by state relations that differ for each substance. As mentioned,we shall confine ourselves here to single-phase pure substances, such as water in itsliquid phase. The second most common fluid, air, is a mixture of gases, but since themixture ratios remain nearly constant between 160 and 2200 K, in this temperaturerange 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)pRT Rcpcvgas constantwhere the specific heats cp and cv are defined in Eqs. (1.14) and (1.15).Since Eq. (1.10) is dimensionally consistent, R has the same dimensions as spe-cificheat, {L2T21}, or velocity squared per temperature unit (kelvin or degreeRankine). Each gas has its own constant R, equal to a universal constantdividedby the molecular weight(1.11)Rgas Mgas# #where 49,700 ft-lbf/(slugmol R)8314 J/(kmol K). Most applications inthis book are for air, whose molecular weight is M28.97/mol:(1.12)Rair 49,700 ft # lbf/(slugmol # R)28.97/mol 1716ft # lbfslug # R 1716ft2s2 R 287m2s2 # KStandard atmospheric pressure is 2116 lbf/ft22116 slug/(fts2), and standardtemperature is 60F520R. Thus standard air density is(1.13)air 2116 slug/(ft # s2)31716 ft2/(s2 # R)4(520R) 0.00237 slug/ft31.22 kg/m3This is a nominal value suitable for problems. For other gases, see Table A.4.State Relations for Gases12121212 43. One proves in thermodynamics that Eq. (1.10) requires that the internal molecularenergy of a perfect gas vary only with temperature: (T). Therefore the spe-cificheat cv also varies only with temperature:cva TbddT cv(T )dcv(T )dTor (1.14)In like manner h and cp of a perfect gas also vary only with temperature:(1.15)h cpap h T RTh(T )bpdhdT cp(T )dhcp(T )dTThe ratio of specific heats of a perfect gas is an important dimensionless parameterin compressible flow analysis (Chap. 9)(1.16)k cpcv k(T ) 44. 1As a first approximation in airflow analysis we commonly take cp, cv, and k to beconstant:(1.17)cv cp Rk1kRk1kair1.4 4293 ft2/(s2 # R)718 m2/(s2 # K) 6010 ft2/(s2 # R)1005 m2/(s2 # K)Actually, for all gases, cp and cv increase gradually with temperature, and k decreasesgradually. Experimental values of the specific-heat ratio for eight common gases areshown in Fig. 1.5.Many flow problems involve steam. Typical steam operating conditions are relativelyclose to the critical point, so that the perfect-gas approximation is inaccurate. Since nosimple formulas apply accurately, steam properties are available both in EES (seeSec. 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 thefollowing example shows.EXAMPLE 1.5Estimateand cp of steam at 100 lbf/in2 and 400F, in English units, (a) by the perfect-gasapproximation 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 estimatethese properties with moderate accuracy from Eqs. (1.10) and (1.17). First convert pressure22 Chapter 1 Introduction 45. 1.8 Thermodynamic Properties of a Fluid 23ArAtmospheric pressureH2COAir andN2O2SteamCO21.71.61.51.41.31.21.11.00 1000 3000 4000 5000cpc2000Temperature, Rfrom 100 lbf/in2 to 14,400 lbf/ft2, and use absolute temperature, (400F460)860R.Then we need the gas constant for steam, in English units. From Table A.4, the molecularweight of H2O is 18.02, whenceRsteam EnglishMH2O49,700 ft # lbf/(slugmol R)18.02/mol 2758ft # lbfslug RThen the density estimate follows from the perfect-gas law, Eq. (1.10):Ans. (a) pRT14,400 lbf/ft232758 ft # lbf/(slug # R)4(860 R) 0.00607slugft3At 860R, from Fig. 1.5, ksteamcp /cv1.30. Then, from Eq. (1.17),Ans. (a)cp kRk1(1.3)(2758 ft # lbf/(slug R))(1.31) 12,000ft # lbfslug Rk = Approach (b)tables or software: One can either read the steam tables or program afew lines in EES. In either case, the English units (psi, Btu, lbm) are awkward whenapplied to fluid mechanics formulas. Even so, when using EES, make sure that theFig. 1.5 Specific-heat ratio of eightcommon gases as a function oftemperature. (Data from Ref. 22.) 46. Variable Information menu specifies English units: psia and F. EES statements forevaluating density and specific heat of steam are, for these conditions,RhoDENSITY(steam, P100,T400)CpSPECHEAT(steam, P100,T400)Note that the software is set up for psia and F, without converting. EES returns the curve-fitvaluesRho0.2027 lbm/ft3 ; Cp0.5289 Btu/(lbm-F)As just stated, Btu and lbm are extremely unwieldy when applied to mass, momentum,and energy problems in fluid mechanics. Therefore, either convert to ft-lbf and slugs usingyour own resources, or use the Convert function in EES, placing the old and new unitsin single quote marks:Rho2Rho*CONVERT(lbm/ft^3,slug/ft^3)Cp2Cp*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) unitsto the right of the division sign / in CONVERT are assumed to be in the denominator.EES returns these results:Rho20.00630 slug/ft3 Cp213,200 ft2/(s2-R) Ans. (b) Comments: The steam tables would yield results quite close to EES. The perfect-gasestimate ofis 4 percent low, and the estimate of cp is 9 percent low. The chief reasonfor the discrepancy is that this temperature and pressure are rather close to the criticalpoint and saturation line of steam. At higher temperatures and lower pressures, say, 800Fand 50 lbf/in2, the perfect-gas law yields properties with an accuracy of about1 percent.Once again let us warn that English units (psia, lbm Btu) are awkward and must beconverted in most fluid mechanics formulas. EES handles SI units nicely, with no con-versionfactors needed.The writer knows of no perfect-liquid law comparable to that for gases. Liquids arenearly incompressible and have a single, reasonably constant specific heat. Thus anidealized state relation for a liquid isconst cpcvconst dhcp dT (1.18)Most of the flow 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 specific heatcp4210 m2/(s2K). The steam tables may be used if more accuracy is required.The density of a liquid usually decreases slightly with temperature and increasesmoderately with pressure. If we neglect the temperature effect, an empirical pressuredensity relation for a liquid is(1.19)ppa (B1) aan Bbwhere B and n are dimensionless parameters that vary slightly with temperature andpa and a are standard atmospheric values. Water can be fitted approximately to thevalues B3000 and n7.24 Chapter 1 IntroductionState Relations for Liquids 47. Seawater is a variable mixture of water and salt and thus requires three thermo-dynamicproperties to define its state. These are normally taken as pressure, temper-ature,and the salinity , defined as the weight of the dissolved salt divided by theweight of the mixture. The average salinity of seawater is 0.035, usually written as 35parts per 1000, or 35 . The average density of seawater is 2.00 slugs/ft31030 kg/m3.Strictly speaking, seawater has three specific heats, all approximately equal to thevalue for pure water of 25,200 ft2/(s2R)4210 m2/(s2K).EXAMPLE 1.6The pressure at the deepest part of the ocean is approximately 1100 atm. Estimate the den-sityof seawater in slug/ft3 at this pressure.SolutionEquation (1.19) holds for either water or seawater. The ratio p/pa is given as 1100:or1100(3001)aa a41003001a1/7b7 3000b 1.046Assuming an average surface seawater density a2.00 slugs/ft3, we computeAns.Even at these immense pressures, the density increase is less than 5 percent, which justifiesthe treatment of a liquid flow as essentially incompressible.The quantities such as pressure, temperature, and density discussed in the previoussection are primary thermodynamic variables characteristic of any system. Certainsecondary variables also characterize specific fluid mechanical behavior. The mostimportant of these is viscosity, which relates the local stresses in a moving fluid tothe strain rate of the fluid element.Viscosity is a quantitative measure of a fluids resistance to flow. More specifically,it determines the fluid strain rate that is generated by a given applied shear stress. Wecan easily move through air, which has very low viscosity. Movement is more diffi-cultin water, which has 50 times higher viscosity. Still more resistance is found inSAE 30 oil, which is 300 times more viscous than water. Try to slide your handthrough glycerin, which is five times more viscous than SAE 30 oil, or blackstrapmolasses, another factor of five higher than glycerin. Fluids may have a vast rangeof viscosities.Consider a fluid element sheared in one plane by a single shear stress , as inFig. 1.6a. The shear strain anglewill continuously grow with time as long asthe stress is maintained, the upper surface moving at speed u larger than the1.9 Viscosity and OtherSecondary PropertiesViscosity1.046(2.00)2.09 slugs/ft3S1.9 Viscosity and Other Secondary Properties 25 48. u t t u = uu = 0Velocityprofilelower. Such common fluids as water, oil, and air show a linear relation betweenapplied shear and resulting strain rate:(1.20) tFrom the geometry of Fig. 1.4a, we see that(1.21)tanu tyIn the limit of infinitesimal changes, this becomes a relation between shear strain rateand velocity gradient:(1.22)ddtdudyFrom Eq. (1.20), then, the applied shear is also proportional to the velocity gradient forthe common linear fluids. The constant of proportionality is the viscosity coefficient :(1.23)ddt dudyEquation (1.23) is dimensionally consistent; thereforehas dimensions of stresstime:{FT/L2} or {M/(LT)}. The BG unit is slugs per foot-second, and the SI unit is kilo-gramsper meter-second. The linear fluids that follow Eq. (1.23) are called newtonianfluids, after Sir Isaac Newton, who first postulated this resistance law in 1687.We do not really care about the strain angle (t) in fluid mechanics, concentratinginstead on the velocity distribution u(y), as in Fig. 1.6b. We shall use Eq. (1.23) inChap. 4 to derive a differential equation for finding the velocity distribution u(y)and, more generally, V(x, y, z, t)in a viscous fluid. Figure 1.6b illustrates a shearlayer, or boundary layer, near a solid wall. The shear stress is proportional to the slopeof the velocity profile and is greatest at the wall. Further, at the wall, the velocity uis zero relative to the wall: This is called the no-slip condition and is characteristicof all viscous fluid flows.26 Chapter 1 Introduction(a) (b)xu(y)y = dudydudyNo slip at wall y0Fig. 1.6 Shear stress causes contin-uousshear deformation in a fluid:(a) a fluid element straining at arate /t; (b) newtonian shear dis-tributionin a shear layer near awall. 49. The viscosity of newtonian fluids is a true thermodynamic property and varies withtemperature and pressure. At a given state (p, T) there is a vast range of values amongthe common fluids. Table 1.4 lists the viscosity of eight fluids at standard pressureand temperature. There is a variation of six orders of magnitude from hydrogen upto glycerin. Thus there will be wide differences between fluids subjected to the sameapplied stresses.Generally speaking, the viscosity of a fluid increases only weakly with pressure.For example, increasing p from 1 to 50 atm will increaseof air only 10 percent.Temperature, however, has a strong effect, withincreasing with T for gases anddecreasing for liquids. Figure A.1 (in App. A) shows this temperature variation forvarious common fluids. It is customary in most engineering work to neglect the pres-surevariation.The variation ( p, T ) for a typical fluid is nicely shown by Fig. 1.7, from Ref. 25,which normalizes the data with the critical-point state (c, pc, Tc). This behavior, calledthe principle of corresponding states, is characteristic of all fluids, but the actualnumerical values are uncertain to20 percent for any given fluid. For example, val-uesof (T) for air at 1atm, from Table A.2, fall about 8 percent low compared to thelow-density limit in Fig. 1.7.Note in Fig. 1.7 that changes with temperature occur very rapidly near the criticalpoint. In general, critical-point measurements are extremely difficult and uncertain.The primary parameter correlating the viscous behavior of all newtonian fluids is thedimensionless Reynolds number:(1.24)Re VLVLwhere V and L are characteristic velocity and length scales of the flow. The secondform of Re illustrates that the ratio oftohas its own name, the kinematic viscosity:(1.25) It is called kinematic because the mass units cancel, leaving only the dimensions{L2/T}.The Reynolds Number1.9 Viscosity and Other Secondary Properties 27Table 1.4 Viscosity and KinematicViscosity of Eight Fluids at 1 atmand 20C, Ratio ,RatioFluid kg/(ms) /(H2) kg/m3 m2/s /(Hg)Hydrogen 9.0 E6 1.0 0.084 1.05 E4 910Air 1.8 E5 2.1 1.20 1.50 E5 130Gasoline 2.9 E4 33 680 4.22 E7 3.7Water 1.0 E3 114 998 1.01 E6 8.7Ethyl alcohol 1.2 E3 135 789 1.52 E6 13Mercury 1.5 E3 170 13,550 1.16 E7 1.0SAE 30 oil 0.29 33,000 891 3.25 E4 2,850Glycerin 1.5 170,000 1,260 1.18 E3 10,3001 kg/(ms)0.0209 slug/(fts); 1 m2/s10.76 ft2/s. 50. 1235Low-density limit10= T1097 86543210.9LiquidTwo-phaseregionCriticalpoint0.5pr = p/pc = 0.2025Dense gasGenerally, the first thing a fluids engineer should do is estimate the Reynolds num-berrange of the flow under study. Very low Re indicates viscous creeping motion,where inertia effects are negligible. Moderate Re implies a smoothly varying laminarflow. High Re probably spells turbulent flow, which is slowly varying in the time-meanbut has superimposed strong random high-frequency fluctuations. Explicitnumerical values for low, moderate, and high Reynolds numbers cannot be stated here.They depend on flow geometry and will be discussed in Chaps. 5 through 7.Table 1.4 also lists values offor the same eight fluids. The pecking order changesconsiderably, and mercury, the heaviest, has the smallest viscosity relative to its ownweight. All gases have highrelative to thin liquids such as gasoline, water, and alco-hol.Oil and glycerin still have the highest , but the ratio is smaller. For given val-uesof V and L in a flow, these fluids exhibit a spread of four orders of magnitude inthe Reynolds number.A classic problem is the flow induced between a fixed lower plate and an upper platemoving steadily at velocity V, as shown in Fig. 1.8. The clearance between plates is28 Chapter 1 IntroductionFlow between Plates0.4=c rTr Tc0.80.70.60.50.40.30.20.6 0.8 1 2 3 4 5 6 7 8 9 10Fig. 1.7 Fluid viscosity nondimen-sionalizedby critical-point proper-ties.This generalized chart ischaracteristic of all fluids but isaccurate only to20 percent.(From Ref. 25.) 51. 1.9 Viscosity and Other Secondary Properties 29yMovingplate:u = VViscousfluidxu = Vh u(y)Vu = 0Fixed plateh, and the fluid is newtonian and