Fundamentalsof

Fluid MechanicsFourth Edition

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Fourth Edition

Fundamentalsof

Fluid Mechanics

BRUCE R. MUNSONDONALD F. YOUNGDepartment of Aerospace Engineering and Engineering Mechanics

THEODORE H. OKIISHIDepartment of Mechanical EngineeringIowa State UniversityAmes, Iowa, USA

John Wiley & Sons, Inc.

New York Chichester Weinheim Brisbane Singapore Toronto

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No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form orby any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted underSections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of thePublisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center,222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470. Requests to the Publisher forpermission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue,New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ@WILEY.COM.

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To Erik and all others who possess the curiosity,patience, and desire to learn

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Bruce R. Munson, Professor of Engineering Mechanics at Iowa State University since 1974,received his B.S. and M.S. degrees from Purdue University and his Ph.D. degree from theAerospace Engineering and Mechanics Department of the University of Minnesota in 1970.

From 1970 to 1974, Dr. Munson was on the mechanical engineering faculty of DukeUniversity. From 1964 to 1966, he worked as an engineer in the jet engine fuel control de-partment of Bendix Aerospace Corporation, South Bend, Indiana.

Dr. Munsons main professional activity has been in the area of fluid mechanics edu-cation and research. He has been responsible for the development of many fluid mechanicscourses for studies in civil engineering, mechanical engineering, engineering science, andagricultural engineering and is the recipient of an Iowa State University Superior Engineer-ing Teacher Award and the Iowa State University Alumni Association Faculty Citation.

He has authored and coauthored many theoretical and experimental technical paperson hydrodynamic stability, low Reynolds number flow, secondary flow, and the applicationsof viscous incompressible flow. He is a member of The American Society of Mechanical En-gineers and The American Physical Society.

Donald F. Young, Anson Marston Distinguished Professor Emeritus in Engineering, is a fac-ulty member in the Department of Aerospace Engineering and Engineering Mechanics atIowa State University. Dr. Young received his B.S. degree in mechanical engineering, hisM.S. and Ph.D. degrees in theoretical and applied mechanics from Iowa State, and has taughtboth undergraduate and graduate courses in fluid mechanics for many years. In addition tobeing named a Distinguished Professor in the College of Engineering, Dr. Young has also re-ceived the Standard Oil Foundation Outstanding Teacher Award and the Iowa State Univer-sity Alumni Association Faculty Citation. He has been engaged in fluid mechanics researchfor more than 35 years, with special interests in similitude and modeling and the interdisci-plinary field of biomedical fluid mechanics. Dr. Young has contributed to many technicalpublications and is the author or coauthor of two textbooks on applied mechanics. He is aFellow of The American Society of Mechanical Engineers.

About the Authors

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Theodore H. Okiishi, Associate Dean of Engineering and past Chair of Mechanical Engi-neering at Iowa State University, has taught fluid mechanics courses there since 1967. Hereceived his undergraduate and graduate degrees at Iowa State.

From 1965 to 1967, Dr. Okiishi served as a U.S. Army officer with duty assignmentsat the National Aeronautics and Space Administration Lewis Research Center, Cleveland,Ohio, where he participated in rocket nozzle heat transfer research, and at the CombinedIntelligence Center, Saigon, Republic of South Vietnam, where he studied seasonal riverflooding problems.

Professor Okiishi is active in research on turbomachinery fluid dynamics. He and hisgraduate students and other colleagues have written a number of journal articles based ontheir studies. Some of these projects have involved significant collaboration with governmentand industrial laboratory researchers with two technical papers winning the ASME MelvilleMedal.

Dr. Okiishi has received several awards for teaching. He has developed undergraduateand graduate courses in classical fluid dynamics as well as the fluid dynamics of turboma-chines.

He is a licensed professional engineer. His technical society activities include havingbeen chair of the board of directors of The American Society of Mechanical Engineers(ASME) International Gas Turbine Institute. He is a Fellow of The American Society ofMechanical Engineers and the editor of the Journal of Turbomachinery.

viii About the Authors

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Preface

This book is intended for junior and senior engineering students who are interested in learn-ing some fundamental aspects of fluid mechanics. This area of mechanics is mature, and acomplete coverage of all aspects of it obviously cannot be accomplished in a single volume.We developed this text to be used as a first course. The principles considered are classicaland have been well-established for many years. However, fluid mechanics education has im-proved with experience in the classroom, and we have brought to bear in this book our ownideas about the teaching of this interesting and important subject. This fourth edition hasbeen prepared after several years of experience by the authors using the previous editions foran introductory course in fluid mechanics. Based on this experience, along with suggestionsfrom reviewers, colleagues, and students, we have made a number of changes in this newedition. Many of these changes are minor and have been made to simply clarify, update andexpand certain ideas and concepts.

The major changes in the fourth edition involve the CD-ROM that accompanies thebook. This E-book CD-ROM contains the entire print component of the book, plus additionalmaterial not in the print version. This approach allows the inclusion of various materials thatwould either cause the print version to be too big or materials that are ideally (and only)suited for the electronic media. Approximately 25 percent of the homework problems in boththe E-book and the print version are new problems.

The E-book contains the following material. (1) There are 80 video segmentsillustrating many interesting and practical applications of real-world fluid phenomena.Each video segment is identified at the appropriate location in the text material by an iconof the type shown in the left margin. In addition, there are approximately 160 homeworkproblems that are tied-in directly with the topics in the videos. The appropriate videos canbe viewed directly from the problems. (2) There are 30 extended, laboratory-type problemsthat involve actual experimental data for simple experiments of the type that are often foundin the laboratory portion of many introductory fluid mechanics courses. The data for theseproblems are provided in an EXCEL format. (3) There is a set of 186 review problems cov-ering most of the main topics in the book. Complete, detailed solutions to these problemsare provided. (4) Chapter 12, Turbomachines, is contained in the E-book only.

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The E-book material on the CD-ROM with all its links is navigated using AdobeAcrobatTM. The links within the E-book include the following types:

1. Links from the Table of Contents to major segments of the E-book (i.e., chapters, ap-pendices, index, videos, lab problems, review problem).

2. Links from the Index to topics within the E-book.3. Links from reference to a figure, table, equation, or section to the actual figure, table,

equation, or section. All figures can be enlarged and printed.

4. Links from end-of-chapter Key Words and Topics to the appropriate location withinthe chapter.

5. Links from a video icon in the margin to that video segment.6. Links from a video homework problem to the appropriate video segment.7. Links from the beginning of the homework problems at the end of a chapter to the re-

view problems for that chapter.

8. Links from a review problem to the complete solution for that problem.9. Links from a brief problem statement for a lab-type homework problem to the com-

plete detailed problem statement.

10. Links from a lab-type problem statement to the EXCEL data page for that problem.11. Links from an even-numbered problem to its answer.

One of our aims is to represent fluid mechanics as it really isan exciting and usefuldiscipline. To this end, we include analyses of numerous everyday examples of fluid-flowphenomena to which students and faculty can easily relate. In the fourth edition 165 examplesare presented that provide detailed solutions to a variety of problems. Also, a generous setof homework problems in each chapter stresses the practical application of principles. Thoseproblems that can be worked best with a programmable calculator or a computer, about 10%of the problems, are so identified. Also included in most chapters are several open-endedproblems. These problems require critical thinking in that in order to work them one mustmake various assumptions and provide the necessary data. Students are thus required to makereasonable estimates or to obtain additional information outside the classroom. Theseopenended problems are clearly identified. Other features are the inclusion of extended,laboratory-type problems in most chapters and problems directly related to the video segmentsprovided.

Since this is an introductory text, we have designed the presentation of material to al-low for the gradual development of student confidence in fluid mechanics problem solving.Each important concept or notion is considered in terms of simple and easy-to-understandcircumstances before more complicated features are introduced. A brief summary (or high-light) sentence has been added to each page of text. These sentences serve to prepare orremind the reader about an important concept discussed on that page. The entire page muststill be read to understand the materialthe summary sentences merely reinforce thecomprehension.

Two systems of units continue to be used throughout the text: the British GravitationalSystem (pounds, slugs, feet, and seconds), and the International System of Units (newtons,kilograms, meters, and seconds). Both systems are widely used, and we believe that studentsneed to be knowledgeable and comfortable with both systems. Approximately one-half ofthe examples and homework problems use the British System; the other half is based on theInternational System.

In the first four chapters, the student is made aware of some fundamental aspects offluid motion, including important fluid properties, regimes of flow, pressure variations in flu-ids at rest and in motion, fluid kinematics, and methods of flow description and analysis.

x Preface

A summary or(highlight) sen-tence is inserted oneach page of text.

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The Bernoulli equation is introduced in Chapter 3 to draw attention, early on, to some of theinteresting effects of fluid motion on the distribution of pressure in a flow field. We believethat this timely consideration of elementary fluid dynamics will increase student enthusiasmfor the more complicated material that follows. In Chapter 4, we convey the essential ele-ments of kinematics, including Eulerian and Lagrangian mathematical descriptions of flowphenomena, and indicate the vital relationship between the two views. For teachers who wishto consider kinematics in detail before the material on elementary fluid dynamics, Chapters3 and 4 can be interchanged without loss of continuity.

Chapters 5, 6, and 7 expand on the basic analysis methods generally used to solve orto begin solving fluid mechanics problems. Emphasis is placed on understanding how flowphenomena are described mathematically and on when and how to use infinitesimal and fi-nite control volumes. Owing to the importance of numerical techniques in fluid mechanics,we have included introductory material on this subject in Chapter 6. The effects of fluid fric-tion on pressure and velocity distributions are also considered in some detail. A formal coursein thermodynamics is not required to understand the various portions of the text that considersome elementary aspects of the thermodynamics of fluid flow. Chapter 7 features theadvantages of using dimensional analysis and similitude for organizing test data and forplanning experiments and the basic techniques involved.

Chapters 8 to 12 offer students opportunities for the further application of the princi-ples learned early in the text. Also, where appropriate, additional important notions such asboundary layers, transition from laminar to turbulent flow, turbulence modeling, chaos, andflow separation are introduced. Practical concerns such as pipe flow, open-channel flow, flowmeasurement, drag and lift, the effects of compressibility, and the fluid mechanics funda-mentals associated with turbomachines are included.

The compressible flow tables found in the previous editions (and in other texts) havebeen replaced by corresponding graphs. It is felt that in the current era of visual learning,these graphs allow a fuller understanding of the characteristics of the compressible flowfunctions.

An Instructors Manual is available to professors who adopt this book for classroomuse. This manual contains complete, detailed solutions to all the problems in the text and isin CD format. It may be obtained by contacting your local Wiley representative who can befound at www.wiley.com/college.

Students who study this text and who solve a representative set of the exercises pro-vided should acquire a useful knowledge of the fundamentals of fluid mechanics. Faculty whouse this text are provided with numerous topics to select from in order to meet the objectivesof their own courses. More material is included than can be reasonably covered in one term.All are reminded of the fine collection of supplementary material. Where appropriate, wehave cited throughout the text the articles and books that are available for enrichment.

We express our thanks to the many colleagues who have helped in the development ofthis text, including Professor Bruce Reichert of Kansas State University for help with Chap-ter 11 and Professor Patrick Kavanagh of Iowa State University for help with Chapter 12.We wish to express our gratitude to the many persons who supplied the photographs usedthroughout the text and to Milton Van Dyke for his help in this effort. Finally, we thank ourfamilies for their continued encouragement during the writing of this fourth edition.

Working with students over the years has taught us much about fluid mechanicseducation. We have tried in earnest to draw from this experience for the benefit of users ofthis book. Obviously we are still learning, and we welcome any suggestions and commentsfrom you.

BRUCE R. MUNSONDONALD F. YOUNG

THEODORE H. OKIISHI

Preface xi

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Contents

1INTRODUCTION 3

1.1 Some Characteristics of Fluids 41.2 Dimensions, Dimensional

Homogeneity, and Units 51.2.1 Systems of Units 7

1.3 Analysis of Fluid Behavior 121.4 Measures of Fluid Mass and

Weight 121.4.1 Density 121.4.2 Specific Weight 131.4.3 Specific Gravity 13

1.5 Ideal Gas Law 141.6 Viscosity 151.7 Compressibility of Fluids 22

1.7.1 Bulk Modulus 221.7.2 Compression and Expansion

of Gases 231.7.3 Speed of Sound 24

1.8 Vapor Pressure 251.9 Surface Tension 261.10 A Brief Look Back in History 28

Key Words and Topics 31References 31Review Problems 31Problems 32

2FLUID STATICS 41

2.1 Pressure at a Point 412.2 Basic Equation for Pressure Field 43

2.3 Pressure Variation in a Fluid at Rest 452.3.1 Incompressible Fluid 452.3.2 Compressible Fluid 48

2.4 Standard Atmosphere 502.5 Measurement of Pressure 512.6 Manometry 53

2.6.1 Piezometer Tube 542.6.2 U-Tube Manometer 542.6.3 Inclined-Tube Manometer 58

2.7 Mechanical and Electronic Pressure Measuring Devices 59

2.8 Hydrostatic Force on a Plane Surface 612.9 Pressure Prism 682.10 Hydrostatic Force on a Curved

Surface 722.11 Buoyancy, Flotation, and Stability 74

2.11.1 Archimedes Principle 742.11.2 Stability 76

2.12 Pressure Variation in a Fluid with Rigid-Body Motion 782.12.1 Linear Motion 782.12.2 Rigid-Body Rotation 81Key Words and Topics 84References 84Review Problems 84Problems 84

3ELEMENTARY FLUID DYNAMICSTHE BERNOULLIEQUATION 101

3.1 Newtons Second Law 1013.2 F ma Along a Streamline 104

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xiv Contents

3.3 F ma Normal to a Streamline 1093.4 Physical Interpretation 1113.5 Static, Stagnation, Dynamic,

and Total Pressure 1153.6 Examples of Use of the Bernoulli Equation 119

3.6.1 Free Jets 1193.6.2 Confined Flows 1213.6.3 Flowrate Measurement 128

3.7 The Energy Line and the Hydraulic Grade Line 134

3.8 Restrictions on Use of the Bernoulli Equation 1373.8.1 Compressibility Effects 1373.8.2 Unsteady Effects 1403.8.3 Rotational Effects 1423.8.4 Other Restrictions 144Key Words and Topics 144References 144Review Problems 145Problems 145

4FLUID KINEMATICS 161

4.1 The Velocity Field 1614.1.1 Eulerian and Lagrangian Flow

Descriptions 1634.1.2 One-, Two-, and Three-

Dimensional Flows 1654.1.3 Steady and Unsteady Flows 1664.1.4 Streamlines, Streaklines,

and Pathlines 1664.2 The Acceleration Field 171

4.2.1 The Material Derivative 1714.2.2 Unsteady Effects 1744.2.3 Convective Effects 1754.2.4 Streamline Coordinates 178

4.3 Control Volume and System Representations 1804.4 The Reynolds Transport Theorem 181

4.4.1 Derivation of the Reynolds Transport Theorem 184

4.4.2 Physical Interpretation 1894.4.3 Relationship to Material Derivative 1904.4.4 Steady Effects 1914.4.5 Unsteady Effects 1914.4.6 Moving Control Volumes 1934.4.7 Selection of a Control Volume 194Key Words and Topics 195References 195Review Problems 196Problems 196

5FINITE CONTROL VOLUME ANALYSIS 205

5.1 Conservation of MassThe Continuity Equation 2065.1.1 Derivation of the Continuity

Equation 2065.1.2 Fixed, Nondeforming Control

Volume 2085.1.3 Moving, Nondeforming

Control Volume 2155.1.4 Deforming Control Volume 218

5.2 Newtons Second LawThe Linear Momentum and Moment-of-Momentum Equations 2215.2.1 Derivation of the Linear

Momentum Equation 2215.2.2 Application of the Linear

Momentum Equation 2235.2.3 Derivation of the Moment-of-

Momentum Equation 2415.2.4 Application of the Moment-of-

Momentum Equation 2435.3 First Law of ThermodynamicsThe

Energy Equation 2515.3.1 Derivation of the Energy

Equation 2515.3.2 Application of the Energy

Equation 2545.3.3 Comparison of the Energy

Equation with the Bernoulli Equation 259

5.3.4 Application of the Energy Equation to Nonuniform Flows 266

5.3.5 Combination of the Energy Equation and the Moment-of-Momentum Equation 270

5.4 Second Law of ThermodynamicsIrreversible Flow 2725.4.1 Semi-infinitesimal Control

Volume Statement of the Energy Equation 272

5.4.2 Semi-infinitesimal Control Volume Statement of the Second Law of Thermodynamics 273

5.4.3 Combination of the Equations of the First and Second Laws of Thermodynamics 274

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Contents xv

5.4.4 Application of the Loss Form of the Energy Equation 275

Key Words and Topics 277References 277Review Problems 277Problems 277

6DIFFERENTIAL ANALYSIS OF FLUID FLOW 299

6.1 Fluid Element Kinematics 3006.1.1 Velocity and Acceleration

Fields Revisited 3006.1.2 Linear Motion and Deformation 3016.1.3 Angular Motion and

Deformation 3036.2 Conservation of Mass 306

6.2.1 Differential Form of Continuity Equation 306

6.2.2 Cylindrical Polar Coordinates 3096.2.3 The Stream Function 310

6.3 Conservation of Linear Momentum 3136.3.1 Description of Forces Acting

on the Differential Element 3146.3.2 Equations of Motion 316

6.4 Inviscid Flow 3176.4.1 Eulers Equations of Motion 3176.4.2 The Bernoulli Equation 3186.4.3 Irrotational Flow 3206.4.4 The Bernoulli Equation for

Irrotational Flow 3226.4.5 The Velocity Potential 322

6.5 Some Basic, Plane Potential Flows 3266.5.1 Uniform Flow 3286.5.2 Source and Sink 3296.5.3 Vortex 3316.5.4 Doublet 334

6.6 Superposition of Basic, Plane Potential Flows 3366.6.1 Source in a Uniform

StreamHalf-Body 3376.6.2 Rankine Ovals 3406.6.3 Flow Around a Circular Cylinder 342

6.7 Other Aspects of Potential Flow Analysis 348

6.8 Viscous Flow 3496.8.1 Stress-Deformation Relationships 3496.8.2 The NaiverStokes Equations 350

6.9 Some Simple Solutions for Viscous,Incompressible Fluids 352

6.9.1 Steady, Laminar Flow Between Fixed Parallel Plates 352

6.9.2 Couette Flow 3556.9.3 Steady, Laminar Flow in

Circular Tubes 3576.9.4 Steady, Axial, Laminar Flow

in an Annulus 3606.10 Other Aspects of Differential Analysis 362

6.10.1 Numerical Methods 363Key Words and Topics 371References 371Review Problems 371Problems 371

7SIMILITUDE, DIMENSIONAL ANALYSIS, AND MODELING 385

7.1 Dimensional Analysis 3857.2 Buckingham Pi Theorem 3887.3 Determination of Pi Terms 3887.4 Some Additional Comments

About Dimensional Analysis 3957.4.1 Selection of Variables 3957.4.2 Determination of Reference

Dimensions 3977.4.3 Uniqueness of Pi Terms 399

7.5 Determination of Pi Terms by Inspection 400

7.6 Common Dimensionless Groups in Fluid Mechanics 402

7.7 Correlation of Experimental Data 4067.7.1 Problems with One Pi Term 4067.7.2 Problems with Two or More

Pi Terms 4087.8 Modeling and Similitude 411

7.8.1 Theory of Models 4117.8.2 Model Scales 4167.8.3 Practical Aspects of

Using Models 4167.9 Some Typical Model Studies 418

7.9.1 Flow Through Closed Conduits 4187.9.2 Flow Around Immersed Bodies 4217.9.3 Flow with a Free Surface 425

7.10 Similitude Based on Governing Differential Equations 429Key Words and Topics 432References 432Review Problems 432Problems 432

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8VISCOUS FLOW IN PIPES 443

8.1 General Characteristics of Pipe Flow 4448.1.1 Laminar or Turbulent Flow 4458.1.2 Entrance Region and Fully

Developed Flow 4478.1.3 Pressure and Shear Stress 448

8.2 Fully Developed Laminar Flow 4498.2.1 From F ma Applied to a

Fluid Element 4508.2.2 From the NavierStokes

Equations 4558.2.3 From Dimensional Analysis 4568.2.4 Energy Considerations 458

8.3 Fully Developed Turbulent Flow 4608.3.1 Transition from Laminar to

Turbulent Flow 4618.3.2 Turbulent Shear Stress 4638.3.3 Turbulent Velocity Profile 4678.3.4 Turbulence Modeling 4728.3.5 Chaos and Turbulence 472

8.4 Dimensional Analysis of Pipe Flow 4738.4.1 The Moody Chart 4738.4.2 Minor Losses 4808.4.3 Noncircular Conduits 492

8.5 Pipe Flow Examples 4948.5.1 Single Pipes 4958.5.2 Multiple Pipe Systems 507

8.6 Pipe Flowrate Measurement 5138.6.1 Pipe Flowrate Meters 5138.6.2 Volume Flow Meters 518Key Words and Topics 520References 520Review Problems 520Problems 521

9FLOW OVER IMMERSED BODIES 533

9.1 General External Flow Characteristics 5349.1.1 Lift and Drag Concepts 5359.1.2 Characteristics of Flow Past

an Object 5399.2 Boundary Layer Characteristics 544

9.2.1 Boundary Layer Structure and Thickness on a Flat Plate 544

9.2.2 Prandtl/Blasius Boundary Layer Solution 548

9.2.3 Momentum Integral Boundary Layer Equation for a Flat Plate 552

9.2.4 Transition from Laminar to Turbulent Flow 559

9.2.5 Turbulent Boundary Layer Flow 5619.2.6 Effects of Pressure Gradient 5679.2.7 Momentum-Integral Boundary

Layer Equation with Nonzero Pressure Gradient 572

9.3 Drag 5739.3.1 Friction Drag 5739.3.2 Pressure Drag 5759.3.3 Drag Coefficient Data and Examples 578

9.4 Lift 5929.4.1 Surface Pressure Distribution 5929.4.2 Circulation 603Key Words and Topics 607References 607Review Problems 608Problems 608

10OPEN-CHANNEL FLOW 621

10.1 General Characteristics of Open-Channel Flow 622

10.2 Surface Waves 62310.2.1 Wave Speed 62310.2.2 Froude Number Effects 626

10.3 Energy Considerations 62710.3.1 Specific Energy 62810.3.2 Channel Depth Variations 633

10.4 Uniform Depth Channel Flow 63410.4.1 Uniform Flow Approximations 63410.4.2 The Chezy and Manning

Equations 63510.4.3 Uniform Depth Examples 638

10.5 Gradually Varied Flow 64710.5.1 Classification of Surface Shapes 64810.5.2 Examples of Gradually

Varied Flows 64910.6 Rapidly Varied Flow 651

10.6.1 The Hydraulic Jump 65310.6.2 Sharp-Crested Weirs 65910.6.3 Broad-Crested Weirs 66210.6.4 Underflow Gates 665Key Words and Topics 668References 668Review Problems 668Problems 668

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Contents xvii

11COMPRESSIBLE FLOW 679

11.1 Ideal Gas Relationships 68011.2 Mach Number and Speed of Sound 68611.3 Categories of Compressible Flow 68911.4 Isentropic Flow of an Ideal Gas 693

11.4.1 Effect of Variations in Flow Cross-Sectional Area 694

11.4.2 Converging-Diverging Duct Flow 696

11.4.3 Constant-Area Duct Flow 71411.5 Nonisentropic Flow of an Ideal Gas 715

11.5.1 Adiabatic Constant-Area Duct Flow with Friction (Fanno Flow) 715

11.5.2 Frictionless Constant-Area Duct Flow with Heat Transfer (Rayleigh Flow) 730

11.5.3 Normal Shock Waves 73711.6 Analogy Between Compressible

and Open-Channel Flows 74711.7 Two-Dimensional Compressible Flow 748

Key Words and Topics 751References 751Review Problems 752Problems 752

12TURBOMACHINES (E-book only) 759

12.1 Introduction 76012.2 Basic Energy Considerations 76212.3 Basic Angular Momentum

Considerations 76612.4 The Centrifugal Pump 768

12.4.1 Theoretical Considerations 77012.4.2 Pump Performance

Characteristics 77412.4.3 Net Positive Suction

Head (NPSH) 77612.4.4 System Characteristics and

Pump Selection 77812.5 Dimensionless Parameters and

Similarity Laws 78212.5.1 Special Pump Scaling Laws 78512.5.2 Specific Speed 78712.5.3 Suction Specific Speed 787

12.6 Axial-Flow and Mixed-Flow Pumps 788

12.7 Fans 79012.8 Turbines 791

12.8.1 Impulse Turbines 79312.8.2 Reaction Turbines 803

12.9 Compressible Flow Turbomachines 80712.9.1 Compressors 80712.9.2 Compressible Flow Turbines 812Key Words and Topics 814References 815Review Problems 815Problems 815

AUNIT CONVERSION TABLES 824

BPHYSICAL PROPERTIES OF FLUIDS 828

CPROPERTIES OF THE U.S. STANDARD ATMOSPHERE 834

DCOMPRESSIBLE FLOW DATA FOR AN IDEAL GAS 836

EVIDEO LIBRARY (E-book only)

FREVIEW PROBLEMS (E-book only) R-1

GLABORATORY PROBLEMS (E-book only) L-1

ANSWERS ANS-1

INDEX I-1

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TA B L E 1 . 3Conversion Factors from BG and EE Units to SI

To Convert from to Multiply by

AccelerationAreaDensity

Energy Btu JJ 1.356

Force lb N 4.448Length ft m

in. mmile m

Mass lbm kgslug kg

Power W 1.356hp W

Pressure in. Hg

Specific weightTemperature

KVelocity

Viscosity (dynamic)Viscosity (kinematic)Volume flowrate

If more than four-place accuracy is desired, refer to Appendix A.a

6.309 E 5m3sgalmin 1gpm22.832 E 2m3sft3s9.290 E 2m2sft2s4.788 E 1N sm2lb sft24.470 E 1msmihr 1mph23.048 E 1msfts5.556 E 1RTC 1592 1TF 322CF1.571 E 2Nm3lbft36.895 E 3Nm2lbin.2 1psi24.788 E 1Nm2lbft2 1psf23.377 E 3Nm2160 F27.457 E 2

ft lbs1.459 E 14.536 E 11.609 E 32.540 E 23.048 E 1

ft lb1.055 E 35.154 E 2kgm3slugsft31.602 E 1kgm3lbmft39.290 E 2m2ft23.048 E 1ms2fts2

Unitsa

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TA B L E 1 . 4Conversion Factors from SI Units to BG and EE

To Convert from to Multiply by

Acceleration 3.281AreaDensity

Energy J BtuJ

Force N lbLength m ft 3.281

m in.m mile

Mass kg lbm 2.205kg slug

Power WW hp

Pressure in. Hg

Specific weightTemperature

K 1.800Velocity 3.281

2.237Viscosity (dynamic)Viscosity (kinematic)Volume flowrate

If more than four-place accuracy is desired, refer to Appendix A.a

1.585 E 4galmin 1gpm2m3s3.531 E 1ft3sm3s1.076 E 1ft2sm2s2.089 E 2lb sft2N sm2

mihr 1mph2msftsmsR

TF 1.8 TC 32FC6.366 E 3lbft3Nm31.450 E 4lbin.2 1psi2Nm22.089 E 2lbft2 1psf2Nm22.961 E 4160 F2Nm21.341 E 37.376 E 1ft lbs6.852 E 2

6.214 E 43.937 E 1

2.248 E 17.376 E 1ft lb9.478 E 41.940 E 3slugsft3kgm36.243 E 2lbmft3kgm31.076 E 1ft2m2

fts2ms2

Unitsa

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TA B L E 1 . 5Approximate Physical Properties of Some Common Liquids (BG Units)

Specific Dynamic Kinematic Surface Vapor BulkDensity, Weight, Viscosity, Viscosity, Pressure,

Temperature pvv EvvLiquid ( ) ( ) ( ) ( ) ( ) ( ) [ (abs)] ( )

Carbon tetrachloride 68 3.09 99.5Ethyl alcohol 68 1.53 49.3

60 1.32 42.5Glycerin 68 2.44 78.6Mercury 68 26.3 847SAE 30 60 1.77 57.0 Seawater 60 1.99 64.0Water 60 1.94 62.4

aIn contact with air.bIsentropic bulk modulus calculated from speed of sound.cTypical values. Properties of petroleum products vary.

3.12 E 52.26 E 15.03 E 31.21 E 52.34 E 53.39 E 52.26 E 15.03 E 31.26 E 52.51 E 52.2 E 52.5 E 34.5 E 38.0 E 3oilc4.14 E 62.3 E 53.19 E 21.25 E 63.28 E 56.56 E 52.0 E 64.34 E 31.28 E 23.13 E 21.9 E 58.0 E 01.5 E 34.9 E 66.5 E 6Gasolinec1.54 E 58.5 E 11.56 E 31.63 E 52.49 E 51.91 E 51.9 E 01.84 E 36.47 E 62.00 E 5

lbin.2lbin.2lbftft2slb sft2lbft3slugsft3FSNMGR

Modulus,bTension,a

TA B L E 1 . 6Approximate Physical Properties of Some Common Liquids (SI Units)

Specific Dynamic Kinematic Surface Vapor BulkDensity, Weight, Viscosity, Viscosity, Pressure,

Temperature pvv EvvLiquid ( ) ( ) ( ) ( ) ( ) ( ) [ (abs)] ( )

Carbon tetrachloride 20 1,590 15.6Ethyl alcohol 20 789 7.74

15.6 680 6.67Glycerin 20 1,260 12.4Mercury 20 13,600 133SAE 30 15.6 912 8.95 Seawater 15.6 1,030 10.1Water 15.6 999 9.80

aIn contact with air.bIsentropic bulk modulus calculated from speed of sound.cTypical values. Properties of petroleum products vary.

2.15 E 91.77 E 37.34 E 21.12 E 61.12 E 32.34 E 91.77 E 37.34 E 21.17 E 61.20 E 31.5 E 93.6 E 24.2 E 43.8 E 1oilc2.85 E 101.6 E 14.66 E 11.15 E 71.57 E 34.52 E 91.4 E 26.33 E 21.19 E 31.50 E 01.3 E 95.5 E 42.2 E 24.6 E 73.1 E 4Gasolinec1.06 E 95.9 E 32.28 E 21.51 E 61.19 E 31.31 E 91.3 E 42.69 E 26.03 E 79.58 E 4

Nm2Nm2Nmm2sN sm2kNm3kgm3CSNMGR

Modulus,bTension,a

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TA B L E 1 . 7Approximate Physical Properties of Some Common Gases at Standard Atmospheric Pressure (BG Units)

Specific Dynamic Kinematic GasDensity, Weight, Viscosity, Viscosity, Specific

Temperature RGas ( ) ( ) ( ) ( ) ( ) ( ) k

Air (standard) 59 1.40Carbon dioxide 68 1.30Helium 68 1.66Hydrogen 68 1.41Methane (natural gas) 68 1.31Nitrogen 68 1.40Oxygen 68 1.40

aValues of the gas constant are independent of temperature.bValues of the specific heat ratio depend only slightly on temperature.

1.554 E 31.65 E 44.25 E 78.31 E 22.58 E 31.775 E 31.63 E 43.68 E 77.28 E 22.26 E 33.099 E 31.78 E 42.29 E 74.15 E 21.29 E 32.466 E 41.13 E 31.85 E 75.25 E 31.63 E 41.242 E 41.27 E 34.09 E 71.04 E 23.23 E 4 1.130 E 38.65 E 53.07 E 71.14 E 13.55 E 31.716 E 31.57 E 43.74 E 77.65 E 22.38 E 3

ft lbslug Rft2slb sft2lbft3slugsft3FHeat Ratio,bNMGR

Constant,a

TA B L E 1 . 8Approximate Physical Properties of Some Common Gases at Standard Atmospheric Pressure (SI Units)

Specific Dynamic Kinematic GasDensity, Weight, Viscosity, Viscosity, Specific

Temperature RGas ( ) ( ) ( ) ( ) ( ) ( ) k

Air (standard) 15 1.40Carbon dioxide 20 1.30Helium 20 1.66Hydrogen 20 1.41Methane (natural gas) 20 1.31Nitrogen 20 1.40Oxygen 20 1.40

aValues of the gas constant are independent of temperature.bValues of the specific heat ratio depend only slightly on temperature.

2.598 E 21.53 E 52.04 E 51.30 E 11.33 E 02.968 E 21.52 E 51.76 E 51.14 E 11.16 E 05.183 E 21.65 E 51.10 E 56.54 E 06.67 E 14.124 E 31.05 E 48.84 E 68.22 E 18.38 E 22.077 E 31.15 E 41.94 E 51.63 E 01.66 E 11.889 E 28.03 E 61.47 E 51.80 E 11.83 E 02.869 E 21.46 E 51.79 E 51.20 E 11.23 E 0

Jkg Km2sN sm2Nm3kgm3CHeat Ratio,bNMGR

Constant,a

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The break-up of a fluid jet into drops is a function of fluid propertiessuch as density, viscosity, and surface tension. [Reprinted withpermission from American Institute of Physics (Ref. 6) and the AmericanAssociation for the Advancement of Science (Ref. 7).]

7708d_c01_002 8/2/01 11:01 AM Page 2

Fluid mechanics is that discipline within the broad field of applied mechanics concerned withthe behavior of liquids and gases at rest or in motion. This field of mechanics obviously encompasses a vast array of problems that may vary from the study of blood flow in the capillaries 1which are only a few microns in diameter2 to the flow of crude oil across Alaskathrough an 800-mile-long, 4-ft-diameter pipe. Fluid mechanics principles are needed to ex-plain why airplanes are made streamlined with smooth surfaces for the most efficient flight,whereas golf balls are made with rough surfaces 1dimpled2 to increase their efficiency. Nu-merous interesting questions can be answered by using relatively simple fluid mechanicsideas. For example:

How can a rocket generate thrust without having any air to push against in outer space? Why cant you hear a supersonic airplane until it has gone past you? How can a river flow downstream with a significant velocity even though the slope of

the surface is so small that it could not be detected with an ordinary level? How can information obtained from model airplanes be used to design the real thing? Why does a stream of water from a faucet sometimes appear to have a smooth surface,

but sometimes a rough surface? How much greater gas mileage can be obtained by improved aerodynamic design of

cars and trucks?

The list of applications and questions goes on and onbut you get the point; fluid mechanicsis a very important, practical subject. It is very likely that during your career as an engineeryou will be involved in the analysis and design of systems that require a good understandingof fluid mechanics. It is hoped that this introductory text will provide a sound foundation ofthe fundamental aspects of fluid mechanics.

3

1Introduction

Fluid mechanics isconcerned with thebehavior of liquidsand gases at restand in motion.

7708d_c01_02-38 7/5/01 1:37 PM Page 3

One of the first questions we need to explore is, What is a fluid? Or we might ask, What isthe difference between a solid and a fluid? We have a general, vague idea of the difference.A solid is hard and not easily deformed, whereas a fluid is soft and is easily deformed1we can readily move through air2. Although quite descriptive, these casual observations ofthe differences between solids and fluids are not very satisfactory from a scientific orengineering point of view. A closer look at the molecular structure of materials reveals thatmatter that we commonly think of as a solid 1steel, concrete, etc.2 has densely spaced moleculeswith large intermolecular cohesive forces that allow the solid to maintain its shape, and tonot be easily deformed. However, for matter that we normally think of as a liquid 1water, oil,etc.2, the molecules are spaced farther apart, the intermolecular forces are smaller than forsolids, and the molecules have more freedom of movement. Thus, liquids can be easilydeformed 1but not easily compressed2 and can be poured into containers or forced through atube. Gases 1air, oxygen, etc.2 have even greater molecular spacing and freedom of motionwith negligible cohesive intermolecular forces and as a consequence are easily deformed 1andcompressed2 and will completely fill the volume of any container in which they are placed.

Although the differences between solids and fluids can be explained qualitatively onthe basis of molecular structure, a more specific distinction is based on how they deformunder the action of an external load. Specifically, a fluid is defined as a substance that deformscontinuously when acted on by a shearing stress of any magnitude. A shearing stress 1forceper unit area2 is created whenever a tangential force acts on a surface. When common solidssuch as steel or other metals are acted on by a shearing stress, they will initially deform1usually a very small deformation2, but they will not continuously deform 1flow2. However,common fluids such as water, oil, and air satisfy the definition of a fluidthat is, they willflow when acted on by a shearing stress. Some materials, such as slurries, tar, putty, toothpaste,and so on, are not easily classified since they will behave as a solid if the applied shearingstress is small, but if the stress exceeds some critical value, the substance will flow. The studyof such materials is called rheology and does not fall within the province of classical fluidmechanics. Thus, all the fluids we will be concerned with in this text will conform to thedefinition of a fluid given previously.

Although the molecular structure of fluids is important in distinguishing one fluid fromanother, it is not possible to study the behavior of individual molecules when trying to describethe behavior of fluids at rest or in motion. Rather, we characterize the behavior by consideringthe average, or macroscopic, value of the quantity of interest, where the average is evaluatedover a small volume containing a large number of molecules. Thus, when we say that thevelocity at a certain point in a fluid is so much, we are really indicating the average velocityof the molecules in a small volume surrounding the point. The volume is small comparedwith the physical dimensions of the system of interest, but large compared with the averagedistance between molecules. Is this a reasonable way to describe the behavior of a fluid? Theanswer is generally yes, since the spacing between molecules is typically very small. Forgases at normal pressures and temperatures, the spacing is on the order of and forliquids it is on the order of The number of molecules per cubic millimeter is onthe order of for gases and for liquids. It is thus clear that the number of moleculesin a very tiny volume is huge and the idea of using average values taken over this volume iscertainly reasonable. We thus assume that all the fluid characteristics we are interested in1pressure, velocity, etc.2 vary continuously throughout the fluidthat is, we treat the fluid asa continuum. This concept will certainly be valid for all the circumstances considered in thistext. One area of fluid mechanics for which the continuum concept breaks down is in thestudy of rarefied gases such as would be encountered at very high altitudes. In this case thespacing between air molecules can become large and the continuum concept is no longeracceptable.

10211018107 mm.

106 mm,

4 Chapter 1 / Introduction

1.1 Some Characteristics of Fluids

A fluid, such as water or air, de-forms continuouslywhen acted on by shearing stresses ofany magnitude.

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1.2 Dimensions, Dimensional Homogeneity, and Units

1.2 Dimensions, Dimensional Homogeneity, and Units 5

Since in our study of fluid mechanics we will be dealing with a variety of fluid characteristics,it is necessary to develop a system for describing these characteristics both qualitatively andquantitatively. The qualitative aspect serves to identify the nature, or type, of the character-istics 1such as length, time, stress, and velocity2, whereas the quantitative aspect provides anumerical measure of the characteristics. The quantitative description requires both a numberand a standard by which various quantities can be compared. A standard for length might bea meter or foot, for time an hour or second, and for mass a slug or kilogram. Such standardsare called units, and several systems of units are in common use as described in the followingsection. The qualitative description is conveniently given in terms of certain primary quan-tities, such as length, L, time, T, mass, M, and temperature, These primary quantities canthen be used to provide a qualitative description of any other secondary quantity: for example,

and so on, where the symbol is used toindicate the dimensions of the secondary quantity in terms of the primary quantities. Thus,to describe qualitatively a velocity, V, we would write

and say that the dimensions of a velocity equal length divided by time. The primaryquantities are also referred to as basic dimensions.

For a wide variety of problems involving fluid mechanics, only the three basic dimen-sions, L, T, and M are required. Alternatively, L, T, and F could be used, where F is the basicdimensions of force. Since Newtons law states that force is equal to mass times acceleration,it follows that or Thus, secondary quantities expressed in termsof M can be expressed in terms of F through the relationship above. For example, stress,is a force per unit area, so that but an equivalent dimensional equation is

Table 1.1 provides a list of dimensions for a number of common physicalquantities.

All theoretically derived equations are dimensionally homogeneousthat is, the di-mensions of the left side of the equation must be the same as those on the right side, and alladditive separate terms must have the same dimensions. We accept as a fundamental premisethat all equations describing physical phenomena must be dimensionally homogeneous. If this were not true, we would be attempting to equate or add unlike physical quantities,which would not make sense. For example, the equation for the velocity, V, of a uniformlyaccelerated body is

(1.1)

where is the initial velocity, a the acceleration, and t the time interval. In terms ofdimensions the equation is

and thus Eq. 1.1 is dimensionally homogeneous.Some equations that are known to be valid contain constants having dimensions. The

equation for the distance, d, traveled by a freely falling body can be written as

(1.2)

and a check of the dimensions reveals that the constant must have the dimensions of if the equation is to be dimensionally homogeneous. Actually, Eq. 1.2 is a special form ofthe well-known equation from physics for freely falling bodies,

(1.3)d gt 2

2

LT 2

d 16.1t 2

LT 1 LT 1 LT 1

V0

V V0 at

s ML1T 2.s FL2,

s,M FL1 T 2.F MLT 2

V LT 1

density ML3,velocity LT 1,area L2,

.

Fluid characteris-tics can be de-scribed qualitativelyin terms of certain basic quantitiessuch as length,time, and mass.

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in which g is the acceleration of gravity. Equation 1.3 is dimensionally homogeneous andvalid in any system of units. For the equation reduces to Eq. 1.2 and thusEq. 1.2 is valid only for the system of units using feet and seconds. Equations that are restrictedto a particular system of units can be denoted as restricted homogeneous equations, as opposedto equations valid in any system of units, which are general homogeneous equations. Thepreceding discussion indicates one rather elementary, but important, use of the concept ofdimensions: the determination of one aspect of the generality of a given equation simplybased on a consideration of the dimensions of the various terms in the equation. The conceptof dimensions also forms the basis for the powerful tool of dimensional analysis, which isconsidered in detail in Chapter 7.

g 32.2 fts2

6 Chapter 1 / Introduction

TA B L E 1 . 1Dimensions Associated with Common Physical Quantities

FLT MLTSystem System

AccelerationAngleAngular accelerationAngular velocityArea

DensityEnergy FLForce FFrequencyHeat FL

Length L LMass MModulus of elasticityMoment of a force FLMoment of inertia 1area2

Moment of inertia 1mass2Momentum FTPowerPressureSpecific heat

Specific weightStrainStressSurface tensionTemperature

Time T TTorque FLVelocityViscosity 1dynamic2Viscosity 1kinematic2

VolumeWork FL ML2T 2

L3L3L2T 1L2T 1ML1T 1FL2TLT 1LT 1ML2T 2

MT 2FL1ML1T 2FL2M 0L0T 0F 0L0T 0ML2T 2FL3L2T 21L2T 21ML1T 2FL2ML2T 3FLT 1MLT 1ML2FLT 2L4L4ML2T 2ML1T 2FL2

FL1T 2

ML2T 2T 1T 1MLT 2ML2T 2ML3FL4T 2L2L2T 1T 1T 2T 2M 0L0T 0F 0L0T 0LT 2LT 2

General homogen-eous equations arevalid in any systemof units.

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1.2 Dimensions, Dimensional Homogeneity, and Units 7

EXAMPLE1.1

A commonly used equation for determining the volume rate of flow, Q, of a liquid throughan orifice located in the side of a tank is

where A is the area of the orifice, g is the acceleration of gravity, and h is the height of theliquid above the orifice. Investigate the dimensional homogeneity of this formula.

SOLUTIONThe dimensions of the various terms in the equation are ,

, ,

These terms, when substituted into the equation, yield the dimensional form:

or

It is clear from this result that the equation is dimensionally homogeneous 1both sides of theformula have the same dimensions of 2, and the numbers 10.61 and 2 are dimen-sionless.

If we were going to use this relationship repeatedly we might be tempted to simplifyit by replacing g with its standard value of and rewriting the formula as

(1)

A quick check of the dimensions reveals that

and, therefore, the equation expressed as Eq. 1 can only be dimensionally correct if the num-ber 4.90 has the dimensions of Whenever a number appearing in an equation or for-mula has dimensions, it means that the specific value of the number will depend on the sys-tem of units used. Thus, for the case being considered with feet and seconds used as units,the number 4.90 has units of Equation 1 will only give the correct value for when A is expressed in square feet and h in feet. Thus, Eq. 1 is a restricted homogeneousequation, whereas the original equation is a general homogeneous equation that would bevalid for any consistent system of units. A quick check of the dimensions of the various termsin an equation is a useful practice and will often be helpful in eliminating errorsthat is, asnoted previously, all physically meaningful equations must be dimensionally homogeneous.We have briefly alluded to units in this example, and this important topic will be consideredin more detail in the next section.

Q1in ft3s2ft1 2s.

L1 2T 1.

L3T 1 14.902 1L5 22

Q 4.90 A1h32.2 ft s212L3T 11L

3T 12 3 10.61212 4 1L3T 121L3T 12 10.612 1L22 112 2 1LT 221 21L21 2

h height L g acceleration of gravity LT 2 A area L2 Q volumetime L3T 1

Q 0.61 A12gh

1.2.1 Systems of Units

In addition to the qualitative description of the various quantities of interest, it is generallynecessary to have a quantitative measure of any given quantity. For example, if we measurethe width of this page in the book and say that it is 10 units wide, the statement has nomeaning until the unit of length is defined. If we indicate that the unit of length is a meter,and define the meter as some standard length, a unit system for length has been established

7708d_c01_02-38 7/4/01 6:24 AM Page 7

1and a numerical value can be given to the page width2. In addition to length, a unit must beestablished for each of the remaining basic quantities 1force, mass, time, and temperature2.There are several systems of units in use and we shall consider three systems that arecommonly used in engineering.

British Gravitational (BG) System. In the BG system the unit of length is thefoot 1ft2, the time unit is the second 1s2, the force unit is the pound 1lb2, and the temperatureunit is the degree Fahrenheit or the absolute temperature unit is the degree Rankine where

The mass unit, called the slug, is defined from Newtons second law as

This relationship indicates that a 1-lb force acting on a mass of 1 slug will give the mass anacceleration of

The weight, 1which is the force due to gravity, g2 of a mass, m, is given by theequation

and in BG units

Since the earths standard gravity is taken as 1commonly approximated as2, it follows that a mass of 1 slug weighs 32.2 lb under standard gravity.

International System (SI). In 1960 the Eleventh General Conference on Weightsand Measures, the international organization responsible for maintaining precise uniformstandards of measurements, formally adopted the International System of Units as the inter-national standard. This system, commonly termed SI, has been widely adopted worldwideand is widely used 1although certainly not exclusively2 in the United States. It is expectedthat the long-term trend will be for all countries to accept SI as the accepted standard and itis imperative that engineering students become familiar with this system. In SI the unit oflength is the meter 1m2, the time unit is the second 1s2, the mass unit is the kilogram 1kg2, andthe temperature unit is the kelvin 1K2. Note that there is no degree symbol used whenexpressing a temperature in kelvin units. The Kelvin temperature scale is an absolute scaleand is related to the Celsius 1centigrade2 scale through the relationship

Although the Celsius scale is not in itself part of SI, it is common practice to specifytemperatures in degrees Celsius when using SI units.

The force unit, called the newton 1N2, is defined from Newtons second law as

Thus, a 1-N force acting on a 1-kg mass will give the mass an acceleration of 1 Standardgravity in SI is 1commonly approximated as 2 so that a 1-kg mass weighs9.81 N under standard gravity. Note that weight and mass are different, both qualitativelyand quantitatively! The unit of work in SI is the joule 1J2, which is the work done when the

9.81 ms29.807 ms2ms2.

1 N 11 kg2 11 m s22

K C 273.15

1C2

32.2 fts2g 32.174 fts2

w1lb2 m 1slugs2 g 1fts22

w mg

w1 fts2.

1 lb 11 slug2 11 ft s22

acceleration21force mass

R F 459.67

1R2,1F2

8 Chapter 1 / Introduction

Two systems ofunits that arewidely used in engi-neering are theBritish Gravita-tional (BG) Systemand the Interna-tional System (SI).

7708d_c01_008 8/2/01 3:31 PM Page 8

point of application of a 1-N force is displaced through a 1-m distance in the direction of a force. Thus,

The unit of power is the watt 1W2 defined as a joule per second. Thus,

Prefixes for forming multiples and fractions of SI units are given in Table 1.2. Forexample, the notation kN would be read as kilonewtons and stands for Similarly,mm would be read as millimeters and stands for The centimeter is not an acceptedunit of length in the SI system, so for most problems in fluid mechanics in which SI unitsare used, lengths will be expressed in millimeters or meters.

English Engineering (EE) System. In the EE system units for force and massare defined independently; thus special care must be exercised when using this system inconjunction with Newtons second law. The basic unit of mass is the pound mass 1lbm2, theunit of force is the pound 1lb2.1 The unit of length is the foot 1ft2, the unit of time is the second1s2, and the absolute temperature scale is the degree Rankine To make the equationexpressing Newtons second law dimensionally homogeneous we write it as

(1.4)

where is a constant of proportionality which allows us to define units for both force andmass. For the BG system only the force unit was prescribed and the mass unit defined in a

gc

F magc

1R2.

103 m.103 N.

1 W 1 Js 1 N # ms

1 J 1 N # m

1.2 Dimensions, Dimensional Homogeneity, and Units 9

TA B L E 1 . 2Prefixes for SI Units

Factor by Which UnitIs Multiplied Prefix Symbol

tera Tgiga Gmega Mkilo khecto h

10 deka dadeci dcenti cmilli mmicronano npico pfemto fatto a1018

10151012109

m106103102101

1021031061091012

1It is also common practice to use the notation, lbf, to indicate pound force.

In mechanics it isvery important todistinguish betweenweight and mass.

7708d_c01_009 8/2/01 3:31 PM Page 9

consistent manner such that Similarly, for SI the mass unit was prescribed and theforce unit defined in a consistent manner such that For the EE system, a 1-lb forceis defined as that force which gives a 1 lbm a standard acceleration of gravity which is takenas Thus, for Eq. 1.4 to be both numerically and dimensionally correct

so that

With the EE system weight and mass are related through the equation

where g is the local acceleration of gravity. Under conditions of standard gravity the weight in pounds and the mass in pound mass are numerically equal. Also, since a 1-lbforce gives a mass of 1 lbm an acceleration of and a mass of 1 slug an accelerationof it follows that

In this text we will primarily use the BG system and SI for units. The EE system isused very sparingly, and only in those instances where convention dictates its use.Approximately one-half the problems and examples are given in BG units and one-half inSI units. We cannot overemphasize the importance of paying close attention to units whensolving problems. It is very easy to introduce huge errors into problem solutions through theuse of incorrect units. Get in the habit of using a consistent system of units throughout agiven solution. It really makes no difference which system you use as long as you areconsistent; for example, dont mix slugs and newtons. If problem data are specified in SIunits, then use SI units throughout the solution. If the data are specified in BG units, thenuse BG units throughout the solution. Tables 1.3 and 1.4 provide conversion factors for somequantities that are commonly encountered in fluid mechanics. For convenient reference thesetables are also reproduced on the inside of the back cover. Note that in these tables 1andothers2 the numbers are expressed by using computer exponential notation. For example, thenumber is equivalent to in scientific notation, and the number

is equivalent to More extensive tables of conversion factors fora large variety of unit systems can be found in Appendix A.

2.832 102.2.832 E 25.154 1025.154 E 2

1 slug 32.174 lbm

1 ft s2,32.174 ft s2

1g gc2

w mg

gc

gc 11 lbm2 132.174 fts22

11 lb2

1 lb 11 lbm2 132.174 fts22

gc

32.174 ft s2.

gc 1.gc 1.

10 Chapter 1 / Introduction

TA B L E 1 . 3Conversion Factors from BG and EE Units to SI Units

(See inside of back cover.)

TA B L E 1 . 4Conversion Factors from SI Units to BG and EE Units

(See inside of back cover.)

When solving prob-lems it is importantto use a consistentsystem of units,e.g., dont mix BGand SI units.

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1.2 Dimensions, Dimensional Homogeneity, and Units 11

EXAMPLE1.2

Ff

a

F I G U R E E 1 . 2

As you work through a large variety of problems in this text, you will find that unitsplay an essential role in arriving at a numerical answer. Be careful! It is easy to mix unitsand cause large errors. If in the above example the elevator acceleration had been left as

with m and g expressed in SI units, we would have calculated the force as 605 N andthe answer would have been 41% too large!7 fts2

A tank of water having a total mass of 36 kg rests on the floor of an elevator. Determine theforce 1in newtons2 that the tank exerts on the floor when the elevator is accelerating upwardat

SOLUTIONA free-body diagram of the tank is shown in Fig. E1.2 where is the weight of the tankand water, and is the reaction of the floor on the tank. Application of Newtons secondlaw of motion to this body gives

or

(1)

where we have taken upward as the positive direction. Since Eq. 1 can be writtenas

(2)

Before substituting any number into Eq. 2 we must decide on a system of units, and then besure all of the data are expressed in these units. Since we want in newtons we will use SIunits so that

Since it follows that

(Ans)

The direction is downward since the force shown on the free-body diagram is the force ofthe floor on the tank so that the force the tank exerts on the floor is equal in magnitude butopposite in direction.

Ff 430 N 1downward on floor2

1 N 1 kg # m s2Ff 36 kg 39.81 m s2 17 ft s22 10.3048 m ft2 4 430 kg # m s2

Ff

Ff m 1g a2

w mg,

Ff w ma

a F maFf

w

7 ft s2.

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1.3 Analysis of Fluid Behavior

12 Chapter 1 / Introduction

The study of fluid mechanics involves the same fundamental laws you have encountered inphysics and other mechanics courses. These laws include Newtons laws of motion, conser-vation of mass, and the first and second laws of thermodynamics. Thus, there are strongsimilarities between the general approach to fluid mechanics and to rigid-body and deformable-body solid mechanics. This is indeed helpful since many of the concepts and techniques ofanalysis used in fluid mechanics will be ones you have encountered before in other courses.

The broad subject of fluid mechanics can be generally subdivided into fluid statics, inwhich the fluid is at rest, and fluid dynamics, in which the fluid is moving. In the followingchapters we will consider both of these areas in detail. Before we can proceed, however, itwill be necessary to define and discuss certain fluid properties that are intimately related tofluid behavior. It is obvious that different fluids can have grossly different characteristics.For example, gases are light and compressible, whereas liquids are heavy 1by comparison2and relatively incompressible. A syrup flows slowly from a container, but water flows rapidlywhen poured from the same container. To quantify these differences certain fluid propertiesare used. In the following several sections the properties that play an important role in theanalysis of fluid behavior are considered.

1.4 Measures of Fluid Mass and Weight

1.4.1 Density

The density of a fluid, designated by the Greek symbol 1rho2, is defined as its mass perunit volume. Density is typically used to characterize the mass of a fluid system. In the BGsystem has units of and in SI the units are

The value of density can vary widely between different fluids, but for liquids,variations in pressure and temperature generally have only a small effect on the value of

The small change in the density of water with large variations in temperature is illustratedin Fig. 1.1. Tables 1.5 and 1.6 list values of density for several common liquids. The densityof water at is or The large difference between those twovalues illustrates the importance of paying attention to units! Unlike liquids, the densityof a gas is strongly influenced by both pressure and temperature, and this difference willbe discussed in the next section.

999 kgm3.1.94 slugsft360 F

r.

kgm3.slugsft3r

r

@ 4C = 1000 kg/m3

1000

990

980

970

960

9500

Den

sity

,

kg/m

3

20 40 60 80 100Temperature, C

F I G U R E 1 . 1 Density of water as a function of temperature.

The density of afluid is defined asits mass per unitvolume.

7708d_c01_012 8/2/01 3:32 PM Page 12

The specific volume, , is the volume per unit mass and is therefore the reciprocal ofthe densitythat is,

(1.5)

This property is not commonly used in fluid mechanics but is used in thermodynamics.

1.4.2 Specific Weight

The specific weight of a fluid, designated by the Greek symbol 1gamma2, is defined as itsweight per unit volume. Thus, specific weight is related to density through the equation

(1.6)

where g is the local acceleration of gravity. Just as density is used to characterize the massof a fluid system, the specific weight is used to characterize the weight of the system. In theBG system, has units of and in SI the units are Under conditions of standardgravity water at has a specific weight of and Tables 1.5 and 1.6 list values of specific weight for several common liquids1based on standard gravity2. More complete tables for water can be found in Appendix B1Tables B.1 and B.22.

1.4.3 Specific Gravity

The specific gravity of a fluid, designated as SG, is defined as the ratio of the density of thefluid to the density of water at some specified temperature. Usually the specified temperatureis taken as and at this temperature the density of water is or

In equation form, specific gravity is expressed as

(1.7)

and since it is the ratio of densities, the value of SG does not depend on the system of unitsused. For example, the specific gravity of mercury at is 13.55 and the density of mercurycan thus be readily calculated in either BG or SI units through the use of Eq. 1.7 as

or

It is clear that density, specific weight, and specifc gravity are all interrelated, and froma knowledge of any one of the three the others can be calculated.

rHg 113.552 11000 kgm32 13.6 103 kgm3

rHg 113.552 11.94 slugsft32 26.3 slugsft3

20 C

SG r

rH2O@4C

1000 kgm3.1.94 slugsft34 C 139.2 F2,

9.80 kNm3.62.4 lbft360 F1g 32.174 fts2 9.807 ms22,

Nm3.lbft3g

g rg

g

v 1r

v

1.4 Measures of Fluid Mass and Weight 13

TA B L E 1 . 5Approximate Physical Properties of Some Common Liquids (BG Units)

(See inside of front cover.)

TA B L E 1 . 6Approximate Physical Properties of Some Common Liquids (SI Units)

(See inside of front cover.)

Specific weight isweight per unit vol-ume; specific grav-ity is the ratio offluid density to thedensity of water ata certain tempera-ture.

7708d_c01_013 8/2/01 3:32 PM Page 13

1.5 Ideal Gas Law

14 Chapter 1 / Introduction

Gases are highly compressible in comparison to liquids, with changes in gas density directlyrelated to changes in pressure and temperature through the equation

(1.8)

where p is the absolute pressure, the density, T the absolute temperature,2 and R is a gasconstant. Equation 1.8 is commonly termed the ideal or perfect gas law, or the equation ofstate for an ideal gas. It is known to closely approximate the behavior of real gases undernormal conditions when the gases are not approaching liquefaction.

Pressure in a fluid at rest is defined as the normal force per unit area exerted on a planesurface 1real or imaginary2 immersed in a fluid and is created by the bombardment of thesurface with the fluid molecules. From the definition, pressure has the dimension of and in BG units is expressed as 1psf 2 or 1psi2 and in SI units as In SI,

defined as a pascal, abbreviated as Pa, and pressures are commonly specified inpascals. The pressure in the ideal gas law must be expressed as an absolute pressure, whichmeans that it is measured relative to absolute zero pressure 1a pressure that would only occurin a perfect vacuum2. Standard sea-level atmospheric pressure 1by international agreement2is 14.696 psi 1abs2 or 101.33 kPa 1abs2. For most calculations these pressures can be roundedto 14.7 psi and 101 kPa, respectively. In engineering it is common practice to measure pressurerelative to the local atmospheric pressure, and when measured in this fashion it is called gagepressure. Thus, the absolute pressure can be obtained from the gage pressure by adding thevalue of the atmospheric pressure. For example, a pressure of 30 psi 1gage2 in a tire is equalto 44.7 psi 1abs2 at standard atmospheric pressure. Pressure is a particularly important fluidcharacteristic and it will be discussed more fully in the next chapter.

The gas constant, R, which appears in Eq. 1.8, depends on the particular gas and isrelated to the molecular weight of the gas. Values of the gas constant for several commongases are listed in Tables 1.7 and 1.8. Also in these tables the gas density and specific weightare given for standard atmospheric pressure and gravity and for the temperature listed. Morecomplete tables for air at standard atmospheric pressure can be found in Appendix B 1TablesB.3 and B.42.

1 Nm2Nm2.lbin.2lbft2

FL2,

r

p rRT

2We will use to represent temperature in thermodynamic relationships although T is also used to denote the basic dimension oftime.

TA B L E 1 . 7Approximate Physical Properties of Some Common Gases at Standard Atmospheric Pressure(BG Units)

(See inside of front cover.)

TA B L E 1 . 8Approximate Physical Properties of Some Common Gases at Standard Atmospheric Pressure(SI Units)

(See inside of front cover.)

In the ideal gaslaw, absolute pres-sures and tempera-tures must be used.

7708d_c01_014 8/2/01 3:32 PM Page 14

1.6 Viscosity

1.6 Viscosity 15

EXAMPLE1.3

A compressed air tank has a volume of When the tank is filled with air at a gagepressure of 50 psi, determine the density of the air and the weight of air in the tank. Assumethe temperature is and the atmospheric pressure is 14.7 psi 1abs2.

SOLUTIONThe air density can be obtained from the ideal gas law 1Eq. 1.82 expressed as

so that

(Ans)

Note that both the pressure and temperature were changed to absolute values.The weight, of the air is equal to

so that since

(Ans)w 0.276 lb

1 lb 1 slug # ft s2 10.0102 slugsft32 132.2 ft s22 10.84 ft32

w rg 1volume2

w,

r 150 lbin.2 14.7 lbin.22 1144 in.2ft2211716 ft # lbslug # R2 3 170 4602R 4

0.0102 slugsft3

r p

RT

70 F

0.84 ft3.

The properties of density and specific weight are measures of the heaviness of a fluid. Itis clear, however, that these properties are not sufficient to uniquely characterize how fluidsbehave since two fluids 1such as water and oil2 can have approximately the same value ofdensity but behave quite differently when flowing. There is apparently some additional prop-erty that is needed to describe the fluidity of the fluid.

To determine this additional property, consider a hypothetical experiment in which amaterial is placed between two very wide parallel plates as shown in Fig. 1.2a. The bottomplate is rigidly fixed, but the upper plate is free to move. If a solid, such as steel, were placedbetween the two plates and loaded with the force P as shown, the top plate would be displacedthrough some small distance, 1assuming the solid was mechanically attached to the plates2.The vertical line AB would be rotated through the small angle, to the new position We note that to resist the applied force, P, a shearing stress, would be developed at theplate-material interface, and for equilibrium to occur where A is the effective upperP tA

t,AB.db,

da

P P

(a) (b)

Fixed plate

a

b

B

A

B' A

F I G U R E 1 . 2 (a) Defor-mation of material placed between two parallel plates. (b) Forces actingon upper plate.

Fluid motion cancause shearingstresses.

V1.1 Viscous fluids

7708d_c01_015 8/2/01 3:33 PM Page 15

plate area 1Fig. 1.2b2. It is well known that for elastic solids, such as steel, the small angulardisplacement, 1called the shearing strain2, is proportional to the shearing stress, that isdeveloped in the material.

What happens if the solid is replaced with a fluid such as water? We would immediatelynotice a major difference. When the force P is applied to the upper plate, it will movecontinuously with a velocity, U 1after the initial transient motion has died out2 as illustratedin Fig. 1.3. This behavior is consistent with the definition of a fluidthat is, if a shearingstress is applied to a fluid it will deform continuously. A closer inspection of the fluid motionbetween the two plates would reveal that the fluid in contact with the upper plate moves withthe plate velocity, U, and the fluid in contact with the bottom fixed plate has a zero velocity.The fluid between the two plates moves with velocity that would be found to varylinearly, as illustrated in Fig. 1.3. Thus, a velocity gradient, is developedin the fluid between the plates. In this particular case the velocity gradient is a constant since

but in more complex flow situations this would not be true. The experimentalobservation that the fluid sticks to the solid boundaries is a very important one in fluidmechanics and is usually referred to as the no-slip condition. All fluids, both liquids andgases, satisfy this condition.

In a small time increment, an imaginary vertical line AB in the fluid would rotatethrough an angle, so that

Since it follows that

We note that in this case, is a function not only of the force P 1which governs U 2 but alsoof time. Thus, it is not reasonable to attempt to relate the shearing stress, to as is donefor solids. Rather, we consider the rate at which is changing and define the rate of shearingstrain, as

which in this instance is equal to

A continuation of this experiment would reveal that as the shearing stress, is increasedby increasing P 1recall that 2, the rate of shearing strain is increased in directproportionthat is,

t PAt,

g#

U

b

du

dy

g#

limdtS0

db

dt

g#,

db

dbt,db

db U dt

b

da U dt

tan db db da

b

db,dt,

dudy Ub,

dudy,u Uyb,u u 1y2

t,db

16 Chapter 1 / Introduction

b

U

B'B

P

u

Fixed plate

y

A

a

F I G U R E 1 . 3 Behavior of a fluidplaced between two parallel plates.

Real fluids, eventhough they may bemoving, alwaysstick to the solidboundaries thatcontain them.

V1.2 No-slip condition

7708d_c01_02-39 8/30/01 6:47 AM Page 16 mac45 Mac 45:1st Shift:

or

This result indicates that for common fluids such as water, oil, gasoline, and air the shearingstress and rate of shearing strain 1velocity gradient2 can be related with a relationship of theform

(1.9)

where the constant of proportionality is designated by the Greek symbol 1mu2 and is calledthe absolute viscosity, dynamic viscosity, or simply the viscosity of the fluid. In accordancewith Eq. 1.9, plots of versus should be linear with the slope equal to the viscosityas illustrated in Fig. 1.4. The actual value of the viscosity depends on the particular fluid,and for a particular fluid the viscosity is also highly dependent on temperature as illustratedin Fig. 1.4 with the two curves for water. Fluids for which the shearing stress is linearlyrelated to the rate of shearing strain (also referred to as rate of angular deformation2 aredesignated as Newtonian fluids I. Newton (16421727). Fortunately most common fluids,both liquids and gases, are Newtonian. A more general formulation of Eq. 1.9 which appliesto more complex flows of Newtonian fluids is given in Section 6.8.1.

Fluids for which the shearing stress is not linearly related to the rate of shearing strainare designated as non-Newtonian fluids. Although there is a variety of types of non-Newtonianfluids, the simplest and most common are shown in Fig. 1.5. The slope of the shearing stressvs rate of shearing strain graph is denoted as the apparent viscosity, For Newtonian fluidsthe apparent viscosity is the same as the viscosity and is independent of shear rate.

For shear thinning fluids the apparent viscosity decreases with increasing shear ratethe harder the fluid is sheared, the less viscous it becomes. Many colloidal suspensions andpolymer solutions are shear thinning. For example, latex paint does not drip from the brushbecause the shear rate is small and the apparent viscosity is large. However, it flows smoothly

map.

dudyt

m

t m du

dy

t du

dy

t g#

1.6 Viscosity 17

Dynamic viscosityis the fluid propertythat relates shear-ing stress and fluidmotion.

She

arin

g st

ress

,

Crude oil (60 F)

1Water (60 F)

Water (100 F)

Air (60 F)

Rate of shearing strain, du__dy

F I G U R E 1 . 4 Linear varia-tion of shearing stress with rate ofshearing strain for common fluids.

V1.3 Capillary tubeviscometer

7708d_c01_017 8/2/01 3:33 PM Page 17

onto the wall because the thin layer of paint between the wall and the brush causes a largeshear rate and a small apparent viscosity.

For shear thickening fluids the apparent viscosity increases with increasing shear ratethe harder the fluid is sheared, the more viscous it becomes. Common examples of this typeof fluid include water-corn starch mixture and water-sand mixture 1quicksand2. Thus, thedifficulty in removing an object from quicksand increases dramatically as the speed of removalincreases.

The other type of behavior indicated in Fig. 1.5 is that of a Bingham plastic, which isneither a fluid nor a solid. Such material can withstand a finite shear stress without motion1therefore, it is not a fluid2, but once the yield stress is exceeded it flows like a fluid 1hence,it is not a solid2. Toothpaste and mayonnaise are common examples of Bingham plasticmaterials.

From Eq. 1.9 it can be readily deduced that the dimensions of viscosity are Thus, in BG units viscosity is given as and in SI units as Values of viscosityfor several common liquids and gases are listed in Tables 1.5 through 1.8. A quick glance atthese tables reveals the wide variation in viscosity among fluids. Viscosity is only mildlydependent on pressure and the effect of pressure is usually neglected. However, as previouslymentioned, and as illustrated in Fig. 1.6, viscosity is very sensitive to temperature. Forexample, as the temperature of water changes from 60 to the density decreases byless than 1% but the viscosity decreases by about 40%. It is thus clear that particular attentionmust be given to temperature when determining viscosity.

Figure 1.6 shows in more detail how the viscosity varies from fluid to fluid and howfor a given fluid it varies with temperature. It is to be noted from this figure that the viscosityof liquids decreases with an increase in temperature, whereas for gases an increase intemperature causes an increase in viscosity. This difference in the effect of temperature onthe viscosity of liquids and gases can again be traced back to the difference in molecularstructure. The liquid molecules are closely spaced, with strong cohesive forces betweenmolecules, and the resistance to relative motion between adjacent layers of fluid is relatedto these intermolecular forces. As the temperature increases, these cohesive forces are reducedwith a corresponding reduction in resistance to motion. Since viscosity is an index of thisresistance, it follows that the viscosity is reduced by an increase in temperature. In gases,however, the molecules are widely spaced and intermolecular forces negligible. In this caseresistance to relative motion arises due to the exchange of momentum of gas moleculesbetween adjacent layers. As molecules are transported by random motion from a region of

100 F

N # sm2.lb # sft2FTL2.

1large dudy2

18 Chapter 1 / Introduction

Bingham plastic

Rate of shearing strain, dudy

She

arin

g st

ress

,

ap

1

Shear thinning

Newtonian

Shear thickening F I G U R E 1 . 5 Variation of shearing stress with rate of shearing strain for several types of fluids, including common non-Newtonianfluids.

The various typesof non-Newtonianfluids are distin-guished by howtheir apparent vis-cosity changes withshear rate.

V1.4 Non-Newtonian behavior

7708d_c01_018 8/2/01 3:34 PM Page 18

low bulk velocity to mix with molecules in a region of higher bulk velocity 1and vice versa2,there is an effective momentum exchange which resists the relative motion between the layers.As the temperature of the gas increases, the random molecular activity increases with acorresponding increase in viscosity.

The effect of temperature on viscosity can be closely approximated using two empiricalformulas. For gases the Sutherland equation can be expressed as

(1.10)

where C and S are empirical constants, and T is absolute temperature. Thus, if the viscosityis known at two temperatures, C and S can be determined. Or, if more than two viscositiesare known, the data can be correlated with Eq. 1.10 by using some type of curve-fittingscheme.

For liquids an empirical equation that has been used is

(1.11)

where D and B are constants and T is absolute temperature. This equation is often referredto as Andrades equation. As was the case for gases, the viscosity must be known at leastfor two temperatures so the two constants can be determined. A more detailed discussion ofthe effect of temperature on fluids can be found in Ref. 1.

m De BT

m CT 3 2

T S

1.6 Viscosity 19

Viscosity is verysensitive to temper-ature.

4.0

2.0

1.0864

2

1 10-1864

2

1 10-2864

2

1 10-3864

2

1 10-4864

2

1 10-586-20 0 20 40 60 80 100 120

Temperature, C

Dyn

amic

vis

cosi

ty,

N

s

/m2

SAE 10W oil

Glycerin

Water

Air

Hydrogen F I G U R E 1 . 6Dynamic (absolute) viscosityof some common fluids as afunction of temperature.

7708d_c01_019 8/2/01 3:34 PM Page 19

20 Chapter 1 / Introduction

EXAMPLE1.4

A dimensionless combination of variables that is important in the study of viscous flowthrough pipes is called the Reynolds number, Re, defined as where is the fluid den-sity, V the mean fluid velocity, D the pipe diameter, and the fluid viscosity. A Newtonianfluid having a viscosity of and a specific gravity of 0.91 flows through a 25-mm-diameter pipe with a velocity of Determine the value of the Reynolds num-ber using 1a2 SI units, and 1b2 BG units.

SOLUTION(a) The fluid density is calculated from the specific gravity as

and from the definition of the Reynolds number

However, since it follows that the Reynolds number is unitlessthatis,

(Ans)

The value of any dimensionless quantity does not depend on the system of units usedif all variables that make up the quantity are expressed in a consistent set of units. Tocheck this we will calculate the Reynolds number using BG units.

(b) We first convert all the SI values of the variables appearing in the Reynolds number toBG values by using the conversion factors from Table 1.4. Thus,

m 10.38 N # sm22 12.089 1022 7.94 103 lb # sft2 D 10.025 m2 13.2812 8.20 102 ft

V 12.6 ms2 13.2812 8.53 fts

r 1910 kgm32 11.940 1032 1.77 slugsft3

Re 156

1 N 1 kg # ms2 156 1kg # ms22N

Re rVDm

1910 kgm32 12.6 ms2 125 mm2 1103 mmm2

0.38 N # sm2

r SG rH2O@4C 0.91 11000 kgm32 910 kgm3

2.6 ms.0.38 N # sm2

m

rrVDm

Quite often viscosity appears in fluid flow problems combined with the density in theform

This ratio is called the kinematic viscosity and is denoted with the Greek symbol 1nu2. Thedimensions of kinematic viscosity are and the BG units are and SI units are Values of kinematic viscosity for some common liquids and gases are given in Tables 1.5through 1.8. More extensive tables giving both the dynamic and kinematic viscosities forwater and air can be found in Appendix B 1Tables B.1 through B.42, and graphs showingthe variation in both dynamic and kinematic viscosity with temperature for a variety of fluidsare also provided in Appendix B 1Figs. B.1 and B.22.

Although in this text we are primarily using BG and SI units, dynamic viscosity isoften expressed in the metric CGS 1centimeter-gram-second2 system with units of

This combination is called a poise, abbreviated P. In the CGS system, kinematicviscosity has units of and this combination is called a stoke, abbreviated St.cm2s,dyne # scm2.

m2s.ft2sL2T,n

n m

r

Kinematic viscosityis defined as the ratio of the absoluteviscosity to the fluiddensity.

7708d_c01_02-38 7/4/01 6:24 AM Page 20

1.6 Viscosity 21

and the value of the Reynolds number is

(Ans)

since The values from part 1a2 and part 1b2 are the same, as ex-pected. Dimensionless quantities play an important role in fluid mechanics and the sig-nificance of the Reynolds number as well as other important dimensionless combina-tions will be discussed in detail in Chapter 7. It should be noted that in the Reynoldsnumber it is actually the ratio that is important, and this is the property that wehave defined as the kinematic viscosity.

mr

1 lb 1 slug # fts2.

156 1slug # fts22lb 156

Re 11.77 slugsft32 18.53 ft s2 18.20 102 ft2

7.94 103 lb # sft2

EXAMPLE1.5

The velocity distribution for the flow of a Newtonian fluid between two wide, parallel plates(see Fig. E1.5) is given by the equation

where V is the mean velocity. The fluid has a viscosity of When and determine: (a) the shearing stress acting on the bottom wall, and (b) the shearing stress acting on a plane parallel to the walls and passing through the centerline (midplane).

h 0.2 in.V 2 ft s0.04 lb # sft2.

u 3V

2 c1 a

y

hb

2

d

h

h

y u

F I G U R E E 1 . 5

SOLUTIONFor this type of parallel flow the shearing stress is obtained from Eq. 1.9,

(1)

Thus, if the velocity distribution is known, the shearing stress can be determinedat all points by evaluating the velocity gradient, For the distribution given

(2)

(a) Along the bottom wall so that (from Eq. 2)

du

dy

3V

h

y h

du

dy

3Vy

h2

dudy.u u1y2

t m du

dy

7708d_c01_020 8/2/01 3:35 PM Page 21

1.7.1 Bulk Modulus

An important question to answer when considering the behavior of a particular fluid is howeasily can the volume 1and thus the density2 of a given mass of the fluid be changed whenthere is a change in pressure? That is, how compressible is the fluid? A property that iscommonly used to characterize compressibility is the bulk modulus, defined as

(1.12)

where dp is the differential change in pressure needed to create a differential change involume, of a volume The negative sign is included since an increase in pressure willcause a decrease in volume. Since a decrease in volume of a given mass, will resultin an increase in density, Eq. 1.12 can also be expressed as

(1.13)

The bulk modulus 1also referred to as the bulk modulus of elasticity2 has dimensions ofpressure, In BG units values for are usually given as 1psi2 and in SI units as

Large values for the bulk modulus indicate that the fluid is relativelyincompressiblethat is, it takes a large pressure change to create a small change in volume.As expected, values of for common liquids are large 1see Tables 1.5 and 1.62. For example,at atmospheric pressure and a temperature of it would require a pressure of 3120 psito compress a unit volume of water 1%. This result is representative of the compressibilityof liquids. Since such large pressures are required to effect a change in volume, we concludethat liquids can be considered as incompressible for most practical engineering applications.

60 FEv

Nm2 1Pa2.lbin.2EvFL2.

Ev dp

drr

m rV,V.dV,

Ev dp

dVV

Ev,

22 Chapter 1 / Introduction

1.7 Compressibility of Fluids

and therefore the shearing stress is

(Ans)

This stress creates a drag on the wall. Since the velocity distribution is symmetrical,the shearing stress along the upper wall would have the same magnitude and direction.

(b) Along the midplane where it follows from Eq. 2 that

and thus the shearing stress is

(Ans)

From Eq. 2 we see that the velocity gradient (and therefore the shearing stress)varies linearly with y and in this particular example varies from 0 at the center of thechannel to at the walls. For the more general case the actual variation will,of course, depend on the nature of the velocity distribution.

14.4 lbft2

tmidplane 0

du

dy 0

y 0

14.4 lbft2 1in direction of flow2

tbottomwall

m a3V

hb

10.04 lb # sft22 132 12 ft s210.2 in.2 11 ft 12 in.2

Liquids are usuallyconsidered to beimcompressible,whereas gases aregenerally consid-ered compressible.

7708d_c01_02-38 7/4/01 6:24 AM Page 22

As liquids are compressed the bulk modulus increases, but the bulk modulus near atmosphericpressure is usually the one of interest. The use of bulk modulus as a property describingcompressibility is most prevalent when dealing with liquids, although the bulk modulus canalso be determined for gases.

1.7.2 Compression and Expansion of Gases

When gases are compressed 1or expanded2 the relationship between pressure and densitydepends on the nature of the process. If the compression or expansion takes place underconstant temperature conditions 1isothermal process2, then from Eq. 1.8

(1.14)

If the compression or expansion is frictionless and no heat is exchanged with the surroundings1isentropic process2, then

(1.15)