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Library of Congress Cataloging-in-Publication Data
Mott, Robert L.
Applied fluid mechanics/Robert L. Mott, Joseph A. Untener. — Seventh edition. pages cm
Includes bibliographical references and index.
ISBN-13: 978-0-13-255892-1
ISBN-10: 0-13-255892-0
1. Fluid mechanics. I. Untener, Joseph A. II. Title.
TA357.M67 2015
620.1'06—dc23 2013026227
ISBN 10: 0-13-255892-0
ISBN 13: 978-0-13-255892-1
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CONTENTS
Preface xi
Acknowledgments xv
1 The Nature of Fluids and the Study
of Fluid Mechanics 1
The Big Picture 1
1.1 Objectives 3
1.2 Basic Introductory Concepts 3
1.3 The International System of Units (SI) 4
1.4 The U.S. Customary System 4
1.5 Weight and Mass 5
1.6 Temperature 6
1.7 Consistent Units in an Equation 6
1.8 The Definition of Pressure 8
1.9 Compressibility 10
1.10 Density, Specific Weight, and Specific Gravity 11
1.11 Surface Tension 14
References 15
Internet Resources 15
Practice Problems 15
Computer Aided Engineering Assignments 18
2 Viscosity of Fluids 19
The Big Picture 19
2.1 Objectives 20
2.2 Dynamic Viscosity 21
2.3 Kinematic Viscosity 22
2.4 Newtonian Fluids and Non-Newtonian Fluids 23
2.5 Variation of Viscosity with Temperature 25
2.6 Viscosity Measurement 27
2.7 SAE Viscosity Grades 32
2.8 ISO Viscosity Grades 33
2.9 Hydraulic Fluids for Fluid Power Systems 33
References 34
Internet Resources 35
Practice Problems 35
Computer Aided Engineering Assignments 37
3 Pressure Measurement 38
The Big Picture 38
3.1 Objectives 39
3.2 Absolute and Gage Pressure 39
3.3 Relationship between Pressure and Elevation 40
3.4 Development of the Pressure–Elevation Relation 43
3.5 Pascal’s Paradox 45
3.6 Manometers 46
3.7 Barometers 51
3.8 Pressure Expressed as the Height of a Column of Liquid 52
3.9 Pressure Gages and Transducers 53
References 55
Internet Resources 55
Practice Problems 55
4 Forces Due to Static Fluids 63
The Big Picture 63
4.1 Objectives 65
4.2 Gases Under Pressure 65
4.3 Horizontal Flat Surfaces Under Liquids 66
4.4 Rectangular Walls 67
4.5 Submerged Plane Areas— General 69
4.6 Development of the General Procedure for Forces on Submerged Plane Areas 72
4.7 Piezometric Head 73
4.8 Distribution of Force on a Submerged Curved Surface 74
4.9 Effect of a Pressure above the Fluid Surface 78
4.10 Forces on a Curved Surface with Fluid Below It 78
4.11 Forces on Curved Surfaces with Fluid Above and Below 79
Practice Problems 80
Computer Aided Engineering Assignments 92
5 Buoyancy and Stability 93
The Big Picture 93
5.1 Objectives 94
5.2 Buoyancy 94
5.3 Buoyancy Materials 101
5.4 Stability of Completely Submerged Bodies 102
5.5 Stability of Floating Bodies 103
5.6 Degree of Stability 107
Reference 108
Internet Resources 108
Practice Problems 108
Stability Evaluation Projects 116
6 Flow of Fluids and Bernoulli’s Equation 117
The Big Picture 117
6.1 Objectives 118
6.2 Fluid Flow Rate and the Continuity Equation 118
6.3 Commercially Available Pipe and Tubing 122
6.4 Recommended Velocity of Flow in Pipe and Tubing 124
6.5 Conservation of Energy—Bernoulli’s Equation 127
6.6 Interpretation of Bernoulli’s Equation 128
6.7 Restrictions on Bernoulli’s Equation 129
6.8 Applications of Bernoulli’s Equation 129
6.9 Torricelli’s Theorem 137
6.10 Flow Due to a Falling Head 140 References 142
Internet Resources 142 Practice Problems 143
Analysis Projects Using Bernoulli’s Equation and Torricelli’s Theorem 153
7 General Energy Equation 154
The Big Picture 154
7.1 Objectives 155
7.2 Energy Losses and Additions 156
7.3 Nomenclature of Energy Losses and Additions 158
7.4 General Energy Equation 158
7.5 Power Required by Pumps 162
7.6 Power Delivered to Fluid Motors 165 Practice Problems 167
8 Reynolds Number, Laminar Flow, Turbulent Flow, and Energy Losses Due to Friction 178
The Big Picture 178
8.1 Objectives 181
8.2 Reynolds Number 181
8.3 Critical Reynolds Numbers 182
8.4 Darcy’s Equation 183
8.5 Friction Loss in Laminar Flow 183
8.6 Friction Loss in Turbulent Flow 184
8.7 Use of Software for Pipe Flow Problems 190
8.8 Equations for the Friction Factor 194
8.9 Hazen–Williams Formula for Water Flow 195
8.10 Other Forms of the Hazen–Williams Formula 196
8.11 Nomograph for Solving the Hazen–Williams Formula 196
References 198
Internet Resources 198
Practice Problems 198
Computer Aided Engineering Assignments 204
9 Velocity Profiles for Circular Sections and Flow in Noncircular Sections 205
The Big Picture 205
9.1 Objectives 206
9.2 Velocity Profiles 207
9.3 Velocity Profile for Laminar Flow 207
9.4 Velocity Profile for Turbulent Flow 209
9.5 Flow in Noncircular Sections 212
9.6 Computational Fluid Dynamics 216
References 218
Internet Resources 218
Practice Problems 218
Computer Aided Engineering Assignments 224
10 Minor Losses 225
The Big Picture 225
10.1 Objectives 227
10.2 Resistance Coefficient 227
10.3 Sudden Enlargement 228
10.4 Exit Loss 231
Gradual Enlargement 231
Sudden Contraction 233
Gradual Contraction 236
Entrance Loss 237
10.9 Resistance Coefficients for Valves and Fittings 238
10.10 Application of Standard Valves 244
10.11 Pipe Bends 246
10.12 Pressure Drop in Fluid Power Valves 248
10.13 Flow Coefficients for Valves Using CV 251
10.14 Plastic Valves 252
10.15 Using K-Factors in PIPE-FLO® Software 253
References 258
Internet Resources 258 Practice Problems 258
Computer Aided Analysis and Design Assignments 263
11 Series Pipeline Systems 264
The Big Picture 264
11.1 Objectives 265
11.2 Class I Systems 265
11.3 Spreadsheet Aid for Class I Problems 270
11.4 Class II Systems 272
11.5 Class III Systems 278
11.6 PIPE-FLO® Examples for Series Pipeline Systems 281
11.7 Pipeline Design for Structural Integrity 284
References 286
Internet Resources 286
Practice Problems 286
Computer Aided Analysis and Design Assignments 295
12 Parallel and Branching Pipeline Systems 296
The Big Picture 296
12.1 Objectives 298
12.2 Systems with Two Branches 298
12.3 Parallel Pipeline Systems and Pressure Boundaries in PIPE-FLO® 304
12.4 Systems with Three or More Branches— Networks 307
References 314
Internet Resources 314 Practice Problems 314
Computer Aided Engineering Assignments 317
13 Pump Selection and Application 318
The Big Picture 318
13.1 Objectives 319
13.2 Parameters Involved in Pump Selection 320
13.3 Types of Pumps 320
13.4 Positive-Displacement Pumps 320
13.5 Kinetic Pumps 326
13.6 Performance Data for Centrifugal Pumps 330
13.7 Affinity Laws for Centrifugal Pumps 332
13.8 Manufacturers’ Data for Centrifugal Pumps 333
13.9 Net Positive Suction Head 341
13.10 Suction Line Details 346
13.11 Discharge Line Details 346
13.12 The System Resistance Curve 347
13.13 Pump Selection and the Operating Point for the System 350
13.14 Using PIPE-FLO® for Selection of Commercially Available Pumps 352
13.15 Alternate System Operating Modes 356
13.16 Pump Type Selection and Specific Speed 361
13.17 Life Cycle Costs for Pumped Fluid Systems 363 References 364
Internet Resources 365 Practice Problems 366
Supplemental Problem (PIPE-FLO® Only) 367
Design Problems 367
Design Problem Statements 368
Comprehensive Design Problem 370
14 Open-Channel Flow 372
The Big Picture 372
14.1 Objectives 373
14.2 Classification of Open-Channel Flow 374
14.3 Hydraulic Radius and Reynolds Number in Open-Channel Flow 375
14.4 Kinds of Open-Channel Flow 375
14.5 Uniform Steady Flow in Open Channels 376
14.6 The Geometry of Typical Open Channels 380
14.7 The Most Efficient Shapes for Open Channels 382
14.8 Critical Flow and Specific Energy 382
14.9 Hydraulic Jump 384
14.10 Open-Channel Flow Measurement 386
References 390
Digital Publications 390
Internet Resources 390
Practice Problems 391
Computer Aided Engineering Assignments 394
15 Flow Measurement 395
The Big Picture 395
15.1 Objectives 396
15.2 Flowmeter Selection Factors 396
15.3 Variable-Head Meters 397
15.4 Variable-Area Meters 404
15.5 Turbine Flowmeter 404
15.6 Vortex Flowmeter 404
15.7 Magnetic Flowmeter 406
15.8 Ultrasonic Flowmeters 408
15.9 Positive-Displacement Meters 408
15.10 Mass Flow Measurement 408
15.11 Velocity Probes 410
15.12 Level Measurement 414
15.13 Computer-Based Data Acquisition and Processing 414
References 415
Internet Resources 415
Review Questions 416 Practice Problems 416
Computer Aided Engineering Assignments 417
16
Forces Due to Fluids in Motion 418
The Big Picture 418
16.1 Objectives 419
16.2 Force Equation 419
16.3 Impulse–Momentum Equation 420
16.4 Problem-Solving Method Using the Force Equations 420
16.5 Forces on Stationary Objects 421
16.6 Forces on Bends in Pipelines 423
16.7 Forces on Moving Objects 426 Practice Problems 427
17 Drag and Lift 432
The Big Picture 432
17.1 Objectives 434
17.2 Drag Force Equation 434
17.3 Pressure Drag 435
17.4 Drag Coefficient 435
17.5 Friction Drag on Spheres in Laminar Flow 441
17.6 Vehicle Drag 441
17.7 Compressibility Effects and Cavitation 443
17.8 Lift and Drag on Airfoils 443 References 445
Internet Resources 446
Practice Problems 446
18 Fans, Blowers, Compressors, and the Flow of Gases 450
The Big Picture 450
18.1 Objectives 451
18.2 Gas Flow Rates and Pressures 451
18.3 Classification of Fans, Blowers, and Compressors 452
18.4 Flow of Compressed Air and Other Gases in Pipes 456
18.5 Flow of Air and Other Gases Through Nozzles 461
References 467
Internet Resources 467
Practice Problems 468
Computer Aided Engineering Assignments 469
19 Flow of Air in Ducts 470
The Big Picture 470
19.1 Objectives 472
19.2 Energy Losses in Ducts 472
19.3 Duct Design 477
19.4 Energy Efficiency and Practical Considerations in Duct Design 483
References 484
Internet Resources 484
Practice Problems 484
Appendices 488
Appendix A Properties of Water 488
Appendix B Properties of Common Liquids 490
Appendix C Typical Properties of Petroleum Lubricating Oils 492
Appendix D Variation of Viscosity with Temperature 493
Appendix E Properties of Air 496
Appendix F Dimensions of Steel Pipe 500
Appendix G Dimensions of Steel, Copper, and Plastic Tubing 502
Appendix H Dimensions of Type K Copper Tubing 505
Appendix I Dimensions of Ductile Iron Pipe 506
Appendix J Areas of Circles 507
Appendix K Conversion Factors 509
Appendix L Properties of Areas 511
Appendix M Properties of Solids 513
Appendix N Gas Constant, Adiabatic Exponent, and Critical Pressure Ratio for Selected Gases 515
Answers to Selected Problems 516
Index 525
PREFACE
INTRODUCTION
The objective of this book is to present the principles of fluid mechanics and the application of these principles to practical, applied problems. Primary emphasis is on fluid properties; the measurement of pressure, density, viscosity, and flow; fluid statics; flow of fluids in pipes and noncircular conduits; pump selection and application; open-channel flow; forces developed by fluids in motion; the design and analysis of heating, ventilation, and air conditioning (HVAC) ducts; and the flow of air and other gases.
Applications are shown in the mechanical field, including industrial fluid distribution, fluid power, and HVAC; in the chemical field, including flow in materials processing systems; and in the civil and environmental fields as applied to water and wastewater systems, fluid storage and distribution systems, and open-channel flow. This book is directed to anyone in an engineering field where the ability to apply the principles of fluid mechanics is the primary goal. Those using this book are expected to have an understanding of algebra, trigonometry, and mechanics. After completing the book, the student should have the ability to design and analyze practical fluid flow systems and to continue learning in the field. Students could take other applied courses, such as those on fluid power, HVAC, and civil hydraulics, following this course. Alternatively, this book could be used to teach selected fluid mechanics topics within such courses.
APPROACH
The approach used in this book encourages the student to become intimately involved in learning the principles of fluid mechanics at seven levels:
1. Understanding concepts.
2. Recognizing how the principles of fluid mechanics apply to their own experience.
3. Recognizing and implementing logical approaches to problem solutions.
4. Performing the analyses and calculations required in the solutions.
5. Critiquing the design of a given system and recommending improvements.
6. Designing practical, efficient fluid systems.
7. Using computer-assisted approaches, both commercially available and self-developed, for design and analysis of fluid flow systems.
This multilevel approach has proven successful for several decades in building students’ confidence in their ability to analyze and design fluid systems.
Concepts are presented in clear language and illustrated by reference to physical systems with which the reader should be familiar. An intuitive justification as well as a mathematical basis is given for each concept. The methods of solution to many types of complex problems are presented in step-by-step procedures. The importance of recognizing the relationships among what is known, what is to be found, and the choice of a solution procedure is emphasized.
Many practical problems in fluid mechanics require relatively long solution procedures. It has been the authors’ experience that students often have difficulty in carrying out the details of the solution. For this reason, each example problem is worked in complete detail, including the manipulation of units in equations. In the more complex examples, a programmed instruction format is used in which the student is asked to perform a small segment of the solution before being shown the correct result. The programs are of the linear type in which one panel presents a concept and then either poses a question or asks that a certain operation be performed. The following panel gives the correct result and the details of how it was obtained. The program then continues.
The International System of Units (Système International d’Unités, or SI) and the U.S. Customary System of units are used approximately equally. The SI notation in this book follows the guidelines set forth by the National Institute of Standards and Technology (NIST), U.S. Department of Commerce, in its 2008 publication The International System of Units (SI) (NIST Special Publication 330), edited by Barry N. Taylor and Ambler Thompson.
COMPUTER-ASSISTED PROBLEM SOLVING AND DESIGN
Computer-assisted approaches to solving fluid flow problems are recommended only after the student has demonstrated competence in solving problems manually. They allow more comprehensive problems to be analyzed and give students tools for considering multiple design options while removing some of the burden of calculations. Also, many employers expect students to have not only the skill to use software, but the inclination to do so, and using software within the course effectively nurtures
this skill. We recommend the following classroom learning policy.
Users of computer software must have solid understanding of the principles on which the software is based to ensure that analyses and design decisions are fundamentally sound. Software should be used only after mastering relevant analysis methods by careful study and using manual techniques.
Computer-based assignments are included at the end of many chapters. These can be solved by a variety of techniques such as:
■ The use of a spreadsheet such as Microsoft® Excel
■ The use of technical computing software
■ The use of commercially available software for fluid flow analysis
Chapter 11, Series Pipeline Systems, and Chapter 13, Pump Selection and Application, include example Excel spreadsheet aids for solving fairly complex system design and analysis problems.
New, powerful, commercially available software: A new feature of this 7th edition is the integration of the use of a major, internationally renowned software package for piping system analysis and design, called PIPE-FLO®, produced and marketed by Engineered Software, Inc. (often called ESI) in Lacey, Washington. As stated by ESI’s CEO and president, along with several staff members, the methodology used in this textbook for analyzing pumped fluid flow systems is highly compatible with that used in their software. Students who learn well the principles and manual problem solving methods presented in this book will be well-prepared to apply them in industrial settings and they will also have learned the fundamentals of using PIPE-FLO® to perform the analyses of the kinds of fluid flow systems they will encounter in their careers. This skill should be an asset to students’ career development.
Students using this book in classes will be informed about a unique link to the ESI website where a specially adapted version of the industry-scale software can be used. Virtually all of the piping analysis and design problems in this book can be set up and solved using this special version. The tools and techniques for building computer models of fluid flow systems are introduced carefully starting in Chapter 8 on energy losses due to friction in pipes and continuing through Chapter 13, covering minor losses, series pipeline systems, parallel and branching systems, and pump selection and application. As each new concept and problem-solving method is learned from this book, it is then applied to one or more example problems where students can develop their skills in creating and solving real problems. With each chapter, the kinds of systems that students will be able to complete expand in breadth and depth. New supplemental problems using PIPE-FLO® are in the book so students can extend and demonstrate their abilities in assignments, projects, or self-study. The integrated companion software,
PUMP-FLO®, provides access to catalog data for numerous types and sizes of pumps that students can use in assignments and to become more familiar with that method of specifying pumps in their future positions.
Students and instructors can access the special version of PIPE-FLO® at this site: http://www.eng-software.com/appliedflluidmechanics
FEATURES NEW TO THE SEVENTH EDITION
The seventh edition continues the pattern of earlier editions in refining the presentation of several topics, enhancing the visual attractiveness and usability of the book, updating data and analysis techniques, and adding selected new material. The Big Picture begins each chapter as in the preceding two editions, but each has been radically improved with one or more attractive photographs or illustrations, a refined Exploration section that gets students personally involved with the concepts presented in the chapter, and brief Introductory Concepts that preview the chapter discussions. Feedback from instructors and students about this feature has been very positive. The extensive appendixes continue to be useful lea r ning and p roblem-so lv ing to ols and s e ver a l have been updated or expanded.
The following list highlights some of the changes in this edition:
■ A large percentage of the illustrations have been upgraded in terms of realism, consistency, and graphic quality. Full color has been introduced enhancing the appearance and effectiveness of illustrations, graphs, and the general layout of the book.
■ Many photographs of commercially available products have been updated and some new ones have been added.
■ Most chapters include an extensive list of Internet resources that provide useful supplemental information such as commercially available products, additional data for problem solving and design, more in-depth coverage of certain topics, information about fluid mechanics software, and industry standards. The resources have been updated and many have been added to those in previous editions.
■ The end-of-chapter references have been extensively revised, updated, and extended.
■ Use of metric units has been expanded in several parts of the book. Two new Appendix tables have been added that feature purely metric sizes for steel, copper, and plastic tubing. Use of the metric DN-designations for standard Schedules 40 and 80 steel pipes have been more completely integrated into the discussions, example problems, and end-of chapter problems. Almost all metric-based problems use these new tables for pipe or tubing designations, dimensions, and flow areas. This should give students strong foundations on which to build a career in the global industrial scene in which they will pursue their careers.
■ Many new, creative supplemental problems have been added to the end-of-chapter set of problems in several
chapters to enhance student learning and to provide more variety for instructors in planning their courses.
■ Graphical tools for selecting pipe sizes are refined in Chapter 6 and used in later chapters and design projects.
■ The discussion of computational fluid mechanics included in Chapter 9 has been revised with attractive new graphics that are highly relevant to the study of pipe flow.
■ The use of K-factors (resistance coefficients) based on the equivalent-length approach has been updated, expanded, and refined according to the latest version of the Crane Technical Paper 410 (TP 410).
■ Use of the flow coefficient CV for evaluating the relationship between flow rate and pressure drop across valves has been expanded in Chapter 10 with new equations for use with metric units. It is also included in new parts of Chapter 13 that emphasize the use of valves as control elements.
■ The section General Principles of Pipeline System Design has been refined in Chapter 11.
■ Several sections in Chapter 13 on pump selection and application have been updated and revised to provide more depth, greater consistency with TP 410, a smoother development of relevant topics, and use of the PIPE-FLO ® software.
INTRODUCING PROFESSOR JOSEPH A. UNTENER—NEW CO-AUTHOR OF THIS BOOK
We are pleased to announce that the seventh edition of Applied Fluid Mechanics has been co-authored by:
Robert L. Mott and Joseph A. Untener
Professor Untener has been an outstanding member of the faculty in the Department of Engineering Technology at the
University of Dayton since 1987 when he was hired by Professor Mott. Joe’s first course taught at UD was Fluid Mechanics, using the 2nd edition of this book, and he continues to include this course in his schedule. A gifted instructor, a strong leader, a valued colleague, and a wise counselor of students, Joe is a great choice for the task of preparing this book. He brings fresh ideas, a keen sense of style and methodology, and an eye for effective and attractive graphics. He initiated the major move toward integrating the PIPE-FLO® software into the book and managed the process of working with the leadership and staff of Engineered Software, Inc. His contributions should prove to be of great value to users of this book, both students and instructors.
DOWNLOAD INSTRUCTOR RESOURCES FROM THE INSTRUCTOR RESOURCE CENTER
This edition is accompanied by an Instructor’s Solutions Manual and a complete Image Bank of all figures featured in the text. To access supplementary materials online, instructors need to request an instructor access code. Go to www. pearsonhighered.com/irc to register for an instructor access code. Within 48 hours of registering, you will receive a confirming email including an instructor access code. Once you have received your code, locate your text in the online catalog and click on the ‘Instructor Resources’ button on the left side of the catalog product page. Select a supplement, and a login page will appear. Once you have logged in, you can access the instructor material for all Pearson textbooks. If you have any difficulties accessing the site or downloading a supplement, please contact Customer Service at http://247pearsoned.custhelp.com/.
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ACKNOWLEDGMENTS
We would like to thank all who helped and encouraged us in the writing of this book, including users of earlier editions and the several reviewers who provided detailed suggestions: William E. Cole, Northeastern University; Gary Crossman, Old Dominion University; Charles Drake, Ferris State University; Mark S. Frisina, Wentworth Institute of Technology; Dr. Roy A. Hartman, P. E., Texas A & M University; Dr. Greg E. Maksi, State Technical Institute at Memphis; Ali Ogut, Rochester Institute of Technology; Paul Ricketts, New Mexico State University; Mohammad E. Taslim, Northeastern University at Boston; Pao-lien Wang, University of North Carolina at Charlotte; and Steve Wells, Old Dominion University. Special thanks go to our colleagues of the University of Dayton, the late Jesse Wilder, David Myszka, Rebecca Blust, Michael Kozak, and James Penrod, who used earlier editions of this book in class and offered helpful suggestions. Robert Wolff, also of the University of Dayton, has provided much help in the use of the SI system of units, based on his
REVIEWERS
Eric Baldwin
Bluefield State College
Randy Bedington Catawba Valley Community College
Chuck Drake Ferris State
Ann Marie Hardin Blue Mountain Community College
long experience in metrication through the American Society for Engineering Education. Professor Wolff also consulted on fluid power applications. University of Dayton student, Tyler Runyan, provided significant input to this edition by providing student feedback on the text, rendering some illustrations, and generating solutions to problems using PIPE-FLO®. We thank all those from Engineered Software, Inc. (ESI), for their cooperation and assistance in incorporating the PIPE-FLO ® software into this book. Particularly, we are grate ful for the collaboration by Ray Hardee, Christy Bermensolo, and Buck Jones of ESI. We are grateful for the expert professional and personal service provided by the editorial and marketing staff of Pearson Education. Comments from students who used the book are also appreciated because the book was written for them.
Robert L. Mott and Joseph A. Untener
Francis Plunkett Broome Community College
Mir Said Saidpour Farmingdale State College-SUNY
Xiuling Wang Calumet Purdue
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THE NATURE OF FLUIDS AND THE STUDY OF FLUID MECHANICS
THE BIG PICTURE
As you begin the study of fluid mechanics, let’s look at some fundamental concepts and look ahead to the major topics that you will study in this book. Try to identify where you have encountered either stationary or moving, pressurized fluids in your daily life. Consider the water system in your home, hotels, or commercial buildings. Think about how your car’s fuel travels from the tank to the engine or how the cooling water flows through the engine and its cooling system. When enjoying time in an amusement park, consider how fluids are handled in water slides or boat rides. Look carefully at construction equipment to observe how pressurized fluids are used to actuate moving parts and to drive the machines. Visit manufacturing operations where automation equipment, material handling devices, and production machinery utilize pressurized fluids.
On a larger scale, consider the chemical processing plant shown in Fig. 1.1. Complex piping systems use pumps to transfer fluids from tanks and move them through various processing systems. The finished products may be stored in other tanks and then transferred to trucks or railroad cars to be delivered to customers.
Listed here are several of the major concepts you will study in this book:
■ Fluid mechanics is the study of the behavior of fluids, either at rest (fluid statics) or in motion (fluid dynamics).
■ Fluids can be either liquids or gases and they can be characterized by their physical properties such as density, specific weight, specific gravity, surface tension, and viscosity.
■ Quantitatively analyzing fluid systems requires careful use of units for all terms. Both the SI metric system of units and the U.S. gravitational system are used in this book. Careful distinction between weight and mass is also essential.
■ Fluid statics concepts that you will learn include the measurement of pressure, forces exerted on surfaces due to fluid pressure, buoyancy and stability of floating bodies.
■ Learning how to analyze the behavior of fluids as they flow through circular pipes and tubes and through conduits with other shapes is important.
■ We will consider the energy possessed by the fluid because of its velocity, elevation, and pressure.
■ Accounting for energy losses, additions, or purposeful removals that occur as the fluid flows through the
FIGURE 1.1 Industrial and commercial fluid piping systems, like this one used in a chemical processing plant, involve complex arrangements requiring careful design and analysis. (Source: Nikolay Kazachok/Fotolia)
components of a fluid flow system enables you to analyze the performance of the system.
■ A flowing fluid loses energy due to friction as it moves along a conduit and as it encounters obstructions (like in a control valve) or changes its direction (like in a pipe elbow).
■ Energy can be added to a flowing fluid by pumps that create flow and increase the fluid’s pressure.
■ Energy can be purposely removed by using it to drive a fluid motor, a turbine, or a hydraulic actuator.
■ Measurements of fluid pressure, temperature, and the fluid flow rate in a system are critical to understanding its performance.
Exploration
Now let’s consider a variety of systems that use fluids and that illustrate some of the applications of concepts learned from this book. As you read this section, consider such factors as:
■ The basic function or purpose of the system
■ The kind of fluid or fluids that are in the system
■ The kinds of containers for the fluid or the conduits through which it flows
■ If the fluid flows, what causes the flow to occur? Describe the flow path.
■ What components of the system resist the flow of the fluid?
■ What characteristics of the fluid are important to the proper performance of the system?
1. In your home, you use water for many different purposes such as drinking, cooking, bathing, cleaning, and watering lawns and plants. Water also eliminates wastes from the home through sinks, drains, and toilets. Rain water, melting snow, and water in the ground must be managed to conduct it away from the home using gutters, downspouts, ditches, and sump pumps. Consider how the water is delivered to your home. What is the ultimate source of the water—a river, a reservoir, or natural groundwater? Is the water stored in tanks at some points in the process of getting it to your home? Notice that the water system needs to be at a fairly high pressure to be effective for its uses and to flow reliably through the system. How is that pressure created? Are there pumps in the system? Describe their function and how they operate. From where does each pump draw the water? To what places is the water delivered? What quantities of fluid are needed at the delivery points? What pressures are required? How is the flow of water controlled? What materials are used for the pipes, tubes, tanks, and other containers or conduits?
As you study Chapters 6–13, you will learn how to analyze and design systems in which the water flows in a pipe or a tube. Chapter 14 discusses the cases of open-channel flow such as that in the gutters that catch the rain from the roof of your home.
2. In your car, describe the system that stores gasoline and then delivers it to the car’s engine. How is the windshield washer fluid managed? Describe the cooling system and the nature of the coolant. Describe what happens when you apply the brakes, particularly as it relates to the hydraulic fluid in the braking system. The concepts in Chapters 6–13 will help you to describe and analyze these kinds of systems.
3. Consider the performance of an automated manufacturing system that is actuated by fluid power systems such as the one shown in Fig. 1.2. Describe the fluids, pumps, tubes, valves, and other components of the system. What is the function of the system? How does the fluid accomplish that function? How is energy introduced to the system and how is it dissipated away from the system?
4. Consider the kinds of objects that must float in fluids such as boats, jet skis, rafts, barges, and buoys. Why do they float? In what position or orientation do they float? Why do they maintain their orientation? The principles of buoyancy and stability are discussed in Chapter 5.
5. What examples can you think of where fluids at rest or in motion exert forces on an object? Any vessel containing a fluid under pressure should yield examples. Consider a swimming pool, a hydraulic cylinder, a dam or a retaining wall holding a fluid, a high-pressure washer system, a fire hose, wind during a tornado or a hurricane, and water flowing through a turbine to generate power. What other examples can you think of? Chapters 4, 16, and 17 discuss these cases.
FIGURE 1.2 Typical piping system for fluid power.
6. Think of the many situations in which it is important to measure the flow rate of fluid in a system or the total quantity of fluid delivered. Consider measuring the gasoline that goes into your car so you can pay for just what you get. The water company wants to know how
much water you use in a given month. Fluids often must be metered carefully into production processes in a factory. Liquid medicines and oxygen delivered to a patient in a hospital must be measured continuously for patient safety. Chapter 15 covers flow measurement.
There are many ways in which fluids affect your life. Completion of a fluid mechanics course using this book will help you understand how those fluids can be controlled. Studying this book will help you learn how to design and analyze fluid systems to determine the kind of components that should be used and their size.
1.1 OBJECTIVES
After completing this chapter, you should be able to:
1. Differentiate between a gas and a liquid.
2. Define pressure.
3. Identify the units for the basic quantities of time, length, force, mass, and temperature in the SI metric unit system and in the U.S. Customary unit system.
4. Properly set up equations to ensure consistency of units.
5. Define the relationship between force and mass.
6. Define density, specific weight, and specific gravity and the relationships among them.
7. Define surface tension.
1.2 BASIC INTRODUCTORY CONCEPTS
■ Pressure Pressure is defined as the amount of force exerted on a unit area of a substance or on a surface. This can be stated by the equation
When a liquid is held in a container, it tends to take the shape of the container, covering the bottom and the sides. The top surface, in contact with the atmosphere above it, maintains a uniform level. As the container is tipped, the liquid tends to pour out.
When a gas is held under pressure in a closed container, it tends to expand and completely fill the container. If the container is opened, the gas tends to expand more and escape from the container.
In addition to these familiar differences between gases and liquids, another difference is important in the study of fluid mechanics. Consider what happens to a liquid or a gas as the pressure on it is increased. If air (a gas) is trapped in a cylinder with a tight-fitting, movable piston inside it, you can compress the air fairly easily by pushing on the piston. Perhaps you have used a hand-operated pump to inflate a bicycle tire, a beach ball, an air mattress, or a basketball. As you move the piston, the volume of the gas is reduced appreciably as the pressure increases. But what would happen if the cylinder contained water rather than air? You could apply a large force, which would increase the pressure in the water, but the volume of the water would change very little. This observation leads to the following general descriptions of liquids and gases that we will use in this book:
1. Gases are readily compressible.
Fluids are subjected to large variations in pressure depending on the type of system in which they are used. Milk sitting in a glass is at the same pressure as the air above it. Water in the piping system in your home has a pressure somewhat greater than atmospheric pressure so that it will flow rapidly from a faucet. Oil in a fluid power system is typically maintained at high pressure to enable it to exert large forces to actuate construction equipment or automation devices in a factory. Gases such as oxygen, nitrogen, and helium are often stored in strong cylinders or spherical tanks under high pressure to permit rather large amounts to be held in a relatively small volume. Compressed air is often used in service stations and manufacturing facilities to operate tools or to inflate tires. More discussion about pressure is given in Chapter 3.
■ Liquids and Gases Fluids can be either liquids or gases.
2. Liquids are only slightly compressible.
More discussion on compressibility is given later in this chapter. We will deal mostly with liquids in this book.
■ Weight and Mass An understanding of fluid properties requires a careful distinction between mass and weight. The following definitions apply:
Mass is the property of a body of fluid that is a measure of its inertia or resistance to a change in motion. It is also a measure of the quantity of fluid.
We use the symbol m for mass in this book.
Weight is the amount that a body of fluid weighs, that is, the force with which the fluid is attracted toward Earth by gravitation.
We use the symbol w for weight.
The relationship between weight and mass is discussed in Section 1.5 as we review the unit systems used in this book. You must be familiar with both the International System of Units, called SI, and the U.S. Customary System of units.
■ Fluid Properties The latter part of this chapter presents other fluid properties: specific weight , density , specific gravity, and surface tension. Chapter 2 presents an additional property, viscosity, which is a measure of the ease with which a fluid flows. It is also important in determining the character of the flow of fluids and the amount of energy that is lost from a fluid flowing in a system as discussed in Chapters 8–13.
1.3 THE INTERNATIONAL SYSTEM OF UNITS (SI)
In any technical work the units in which physical properties are measured must be stated. A system of units specifies the units of the basic quantities of length, time, force, and mass. The units of other terms are then derived from these.
The ultimate reference for the standard use of metric units throughout the world is the International System of Units (Système International d’Unités), abbreviated as SI. In the United States, the standard is given in the 2008 publication of the National Institute of Standards and Technology (NIST), U.S. Department of Commerce, The International System of Units (SI) (NIST Special Publication 330), edited by Barry N. Taylor and Ambler Thompson (see Reference 1). This is the standard used in this book.
The SI units for the basic quantities are
length = meter (m)
time = second (s)
mass = kilogram (kg) or N # s2/m
force = newton (N) or kg # m/s2
An equivalent unit for force is kg # m/s2, as indicated above. This is derived from the relationship between force and mass,
F = ma
where a is the acceleration expressed in units of m/s2 Therefore, the derived unit for force is
F = ma = kg # m/s2 = N
Thus, a force of 1.0 N would give a mass of 1.0 kg an acceleration of 1.0 m/s2. This means that either N or kg # m/s2 can be used as the unit for force. In fact, some calculations in this book require that you be able to use both or to convert from one to the other.
Similarly, besides using the kg as the standard unit mass, we can use the equivalent unit N # s2/m. This can be derived again from F = ma:
m = F a = N m/s2 = N # s 2 m
Therefore, either kg or N # s2/m can be used for the unit of mass.
TABLE 1.1 SI unit prefixes
Prefix SI symbol Factor
1.3.1 SI Unit Prefixes
Because the actual size of physical quantities in the study of fluid mechanics covers a wide range, prefixes are added to the basic quantities. Table 1.1 shows these prefixes. Standard usage in the SI system calls for only those prefixes varying in steps of 103 as shown. Results of calculations should normally be adjusted so that the number is between 0.1 and 10 000 times some multiple of 103.* Then the proper unit with a prefix can be specified. Note that some technical professionals and companies in Europe often use the prefix centi, as in centimeters, indicating a factor of 10−2. Some examples follow showing how quantities are given in this book.
0.004 23 m 4.23 * 10 - 3 m, or 4.23 mm (millimeters) 15 700 kg 15.7 * 103 kg, or 15.7 Mg (megagrams) 86 330 N 86.33 * 103 N, or 86.33 kN (kilonewtons)
1.4 THE U.S. CUSTOMARY SYSTEM
Sometimes called the English gravitational unit system or the pound-foot-second system, the U.S. Customary System defines the basic quantities as follows:
length = foot (ft)
time = second (s)
force = pound (lb)
mass = slug or lb s2/ft
Probably the most difficult of these units to understand is the slug because we are more familiar with measuring in
*Because commas are used as decimal markers in many countries, we will not use commas to separate groups of digits. We will separate the digits into groups of three, counting both to the left and to the right from the decimal point, and use a space to separate the groups of three digits. We will not use a space if there are only four digits to the left or right of the decimal point unless required in tabular matter.
terms of pounds, seconds, and feet. It may help to note the relationship between force and mass,
F = ma
where a is acceleration expressed in units of ft/s2. Therefore, the derived unit for mass is
m = F a = lb ft/s2 = lb s 2 ft = slug
This means that you may use either slugs or lb s2/ft for the unit of mass. In fact, some calculations in this book require that you be able to use both or to convert from one to the other.
1.5 WEIGHT AND MASS
A rigid distinction is made between weight and mass in this book. Weight is a force and mass is the quantity of a substance. We relate these two terms by applying Newton’s law of gravitation stated as force equals mass times acceleration, or
F = ma
When we speak of weight w, we imply that the acceleration is equal to g, the acceleration due to gravity. Then Newton’s law becomes
➭ Weight–Mass Relationship
w = mg (1–2)
In this book, we will use g = 9.81 m/s2 in the SI system and g = 32.2 ft/s2 in the U.S. Customary System. These are the standard values on Earth for g to three significant digits. To a greater degree of precision, we have the standard values g = 9.806 65 m/s2 and g = 32.1740 ft/s2. For highprecision work and at high elevations (such as aerospace operations) where the actual value of g is different from the standard, the local value should be used.
1.5.1 Weight and Mass in the SI Unit System
For example, consider a rock with a mass of 5.60 kg suspended by a wire. To determine what force is exerted on the wire, we use Newton’s law of gravitation (w = mg):
w = mg = mass * acceleration due to gravity
Under standard conditions, however, g = 9.81 m/s2. Then, we have
w = 5.60 kg * 9.81 m/s2 = 54.9 kg # m/s2 = 54.9 N
Thus, a 5.60 kg rock weighs 54.9 N.
We can also compute the mass of an object if we know its weight. For example, assume that we have measured the weight of a valve to be 8.25 N. What is its mass? We write
w = mg
m = w g = 8.25 N 9.81 m/s2 = 0.841 N # s 2 m = 0.841 kg
1.5.2 Weight and Mass in the U.S. Customary Unit System
For an example of the weight–mass relationship in the U.S. Customary System, assume that we have measured the weight of a container of oil to be 84.6 lb. What is its mass? We write
w = mg
1.5.3 Mass Expressed as lbm (Pounds-Mass)
In the analysis of fluid systems, some professionals use the unit lbm (pounds-mass) for the unit of mass instead of the unit of slugs. In this system, an object or a quantity of fluid having a weight of 1.0 lb has a mass of 1.0 lbm. The pound-force is then sometimes designated lbf. It must be noted that the numerical equivalence of lbf and lbm applies only when the value of g is equal to the standard value.
This system is avoided in this book because it is not a coherent system. When one tries to relate force and mass units using Newton’s law, one obtains
F = ma = lbm(ft/s2) = lbm ft/s2
This is not the same as the lbf.
To overcome this difficulty, a conversion constant, commonly called gc , is defined having both a numerical value and units. That is,
gc = 32.2 lbm lbf/(ft/s2) = 32.2 lbm ft/s2 lbf
Then, to convert from lbm to lbf, we use a modified form of Newton’s law:
F = m(a > gc)
Letting the acceleration a = g, we find
F = m(g > gc)
For example, to determine the weight of material in lbf that has a mass of 100 lbm, and assuming that the local value of g is equal to the standard value of 32.2 ft/s2, we have
w = F = m g gc = 100 lbm
32.2 ft/s2
32.2
This shows that weight in lbf is numerically equal to mass in lbm provided g = 32.2 ft/s2
If the analysis were to be done for an object or fluid on the Moon, however, where g is approximately 1/6 of that on Earth, 5.4 ft/s2, we would find w = F = m g gc = 100 lbm 5.4 ft/s2
32.2 lbm ft/s2 lbf = 16.8 lbf
This is a dramatic difference.
In summary, because of the cumbersome nature of the relationship between lbm and lbf, we avoid the use of lbm in this book. Mass will be expressed in the unit of slugs when problems are in the U.S. Customary System of units.
1.6 TEMPERATURE
Temperature is most often indicated in °C (degrees Celsius) or °F (degrees Fahrenheit). You are probably familiar with the following values at sea level on Earth:
Water freezes at 0 C and boils at 100 C.
Water freezes at 32 F and boils at 212 F.
Thus, there are 100 Celsius degrees and 180 Fahrenheit degrees between the same two physical data points, and 1.0 Celsius degree equals 1.8 Fahrenheit degrees exactly. From these observations we can define the conversion procedures between these two systems as follows:
Given the temperature TF in °F, the temperature TC in °C is
TC = (TF - 32) > 1.8
Given the temperature TC in °C, the temperature TF in °F is
TF = 1.8TC + 32
For example, given TF = 180 F, we have
TC = (TF - 32) > 1.8 = (180 - 32) > 1.8 = 82.2 C
Given TC = 33 C, we have
TF = 1.8TC + 32 = 1.8(33) + 32 = 91.4 F
In this book we will use the Celsius scale when problems are in SI units and the Fahrenheit scale when they are in U.S. Customary units.
1.6.1 Absolute Temperature
The Celsius and Fahrenheit temperature scales were defined according to arbitrary reference points, although the Celsius scale has convenient points of reference to the properties of water. The absolute temperature, on the other hand, is defined so the zero point corresponds to the condition where all molecular motion stops. This is called absolute zero.
In the SI unit system, the standard unit of temperature is the kelvin, for which the standard symbol is K and the reference (zero) point is absolute zero. Note that there is no degree symbol attached to the symbol K. The interval between points on the kelvin scale is the same as the interval used for the Celsius scale. Measurements have shown that the freezing point of water is 273.15 K above absolute zero. We can then make the conversion from the Celsius to the kelvin scale by using
TK = TC + 273.15
For example, given TC = 33 C, we have
TK = TC + 273.15 = 33 + 273.15 = 306.15 K
It has also been shown that absolute zero on the Fahrenheit scale is at - 459.67 F. In some references you will find
another absolute temperature scale called the Rankine scale, where the interval is the same as for the Fahrenheit scale. Absolute zero is 0 R and any Fahrenheit measurement can be converted to °R by using TR = TF + 459.67
Also, given the temperature in °F, we can compute the absolute temperature in K from TK = (TF + 459.67) > 1.8 = TR > 1.8
For example, given TF = 180 F, the absolute temperature in K is
The analyses required in fluid mechanics involve the algebraic manipulation of several terms. The equations are often complex, and it is extremely important that the results be dimensionally correct. That is, they must have their proper units. Indeed, answers will have the wrong numerical value if the units in the equation are not consistent. Table 1.2 summarizes standard and other common units for the quantities used in fluid mechanics.
A simple straightforward procedure called unit cancellation will ensure proper units in any kind of calculation, not only in fluid mechanics, but also in virtually all your technical work. The six steps of the procedure are listed below.
Unit-Cancellation Procedure
1. Solve the equation algebraically for the desired term.
2. Decide on the proper units for the result.
3. Substitute known values, including units.
4. Cancel units that appear in both the numerator and the denominator of any term.
5. Use conversion factors to eliminate unwanted units and obtain the proper units as decided in Step 2.
6. Perform the calculation.
This procedure, properly executed, will work for any equation. It is really very simple, but some practice may be required to use it. We are going to borrow some material from elementary physics, with which you should be familiar, to illustrate the method. However, the best way to learn how to do something is to do it. The following example problems are presented in a form called programmed instruction. You will be guided through the problems in a step-by-step fashion with your participation required at each step.
To proceed with the program you should cover all material under the heading Programmed Example Problem, using an opaque sheet of paper or a card. You should have a blank piece of paper handy on which to perform the requested operations. Then successively uncover one panel at a time down to the heavy line that runs across the page. The first panel presents a problem and asks you to perform
TABLE 1.2 Units for common quantities used in fluid mechanics in SI units and U.S. Customary units
flow rate (W) w/time
Mass flow rate (M) M/time
Specific weight (g) w/V N/m3 or kg/m2·s2 lb/ft3
Density (r) M/V kg/m3 or N·s2/m4 slugs/ft3
some operation or to answer a question. After doing what is asked, uncover the next panel, which will contain information that you can use to check your result. Then continue with the next panel, and so on through the program.
Remember, the purpose of this is to help you learn how to get correct answers using the unit-cancellation method. You may want to refer to the table of conversion factors in Appendix K.
PROGRAMMED EXAMPLE PROBLEM
Example Problem 1.1
Imagine you are traveling in a car at a constant speed of 80 kilometers per hour (km/h). How many seconds (s) would it take to travel 1.5 km?
For the solution, use the equation
s = vt
where s is the distance traveled, y is the speed, and t is the time. Using the unit-cancellation procedure outlined above, what is the first thing to do?
The first step is to solve for the desired term. Because you were asked to find time, you should have written
t = s v
Now perform Step 2 of the procedure described above.
Step 2 is to decide on the proper units for the result, in this case time. From the problem statement the proper unit is seconds. If no specification had been given for units, you could choose any acceptable time unit such as hours.
Proceed to Step 3.
The result should look something like this:
t = s v = 1.5 km 80 km/h
For the purpose of cancellation it is not convenient to have the units in the form of a compound fraction as we have above. To clear this to a simple fraction, write it in the form t = 1.5 km 1 80 km h
This can be reduced to
t = 1.5 km # h 80 km
After some practice, equations may be written in this form directly. Now perform Step 4 of the procedure.
The result should now look like this:
t = 1.5 km # h 80 km
This illustrates that units can be cancelled just as numbers can if they appear in both the numerator and the denominator of a term in an equation.
Now do Step 5.
The answer looks like this:
t = 1.5 km # h 80 km * 3600 s 1 h
The equation in the preceding panel showed the result for time in hours after kilometer units were cancelled. Although hours is an acceptable time unit, our desired unit is seconds as determined in Step 2. Thus, the conversion factor 3600 s/1 h is required.
How did we know we have to multiply by 3600 instead of dividing?
The units determine this. Our objective in using the conversion factor was to eliminate the hour unit and obtain the second unit. Because the unwanted hour unit was in the numerator of the original equation, the hour unit in the conversion factor must be in the denominator in order to cancel.
Now that we have the time unit of seconds we can proceed with Step 6.
The correct answer is t = 67.5 s.
1.8 THE DEFINITION OF PRESSURE
Pressure is defined as the amount of force exerted on a unit area of a substance. This can be stated by the equation
➭ Pressure p = F A (1–3)
Two important principles about pressure were described by Blaise Pascal, a seventeenth-century scientist:
■ Pressure acts uniformly in all directions on a small volume of a fluid.
■ In a fluid confined by solid boundaries, pressure acts perpendicular to the boundary.
These principles, sometimes called Pascal’s laws, are illustrated in Figs. 1.3 and 1.4.
Using Eq. (1–3) and the second of Pascal’s laws, we can compute the magnitude of the pressure in a fluid if we know the amount of force exerted on a given area.
FIGURE 1.4 Direction of fluid pressure on boundaries.
FIGURE 1.3 Pressure acting uniformly in all directions on a small volume of fluid.
Fluid surface
Example Problem 1.2
(a) Furnace duct(b) Pipe or tube(c) Heat exchanger (a pipe inside another pipe) (d) Reservoir
(e) Swimming pool(f) Dam(g) Fluid power cylinder
Figure 1.5 shows a container of liquid with a movable piston supporting a load. Compute the magnitude of the pressure in the liquid under the piston if the total weight of the piston and the load is 500 N and the area of the piston is 2500 mm2
Solution It is reasonable to assume that the entire surface of the fluid under the piston is sharing in the task of supporting the load. The second of Pascal’s laws states that the fluid pressure acts perpendicular to the piston. Then, using Eq. (1–3), we have
p = F A = 500 N 2500 mm2 = 0.20 N/mm2
Fluid
FIGURE 1.5 Illustration of fluid pressure supporting a load.
The standard unit of pressure in the SI system is the N/m2, called the pascal (Pa) in honor of Blaise Pascal. The conversion can be made by using the factor 103 mm = 1 m. We have p = 0.20 N mm2 * (103 mm)2 m2 = 0.20 * 106 N/m2 = 0.20 MPa
Note that the pressure in N/mm 2 is numerically equal to pressure in MPa. It is not unusual to encounter pressure in the range of several megapascals (MPa) or several hundred kilopascals (kPa).
Pressure in the U.S. Customary System is illustrated in the following example problem.
Example Problem 1.3
A load of 200 pounds (lb) is exerted on a piston confining oil in a circular cylinder with an inside diameter of 2.50 inches (in). Compute the pressure in the oil at the piston. See Fig. 1.4.
Solution To use Eq. (1–3), we must compute the area of the piston:
Although the standard unit for pressure in the U.S. Customary System is pounds per square foot (lb/ft2), it is not often used because it is inconvenient. Length measurements are more conveniently made in inches, and pounds per square inch (lb/in2), abbreviated psi, is used most often for pressure in this system. The pressure in the oil is 40.7 psi. This is a fairly low pressure; it is not unusual to encounter pressures of several hundred or several thousand psi.
The bar is another unit used by some people working in fluid mechanics and thermodynamics. The bar is defined as 105 Pa or 105 N/m2. Another way of expressing the bar is 1 bar = 100 * 103 N/m2, which is equivalent to 100 kPa. Because atmospheric pressure near sea level is very nearly this value, the bar has a convenient point of physical reference. This, plus the fact that pressures expressed in bars yield smaller numbers, makes this unit attractive to some practitioners. You must realize, however, that the bar is not a part of the coherent SI system and that you must carefully convert it to N/m2 (pascals) in problem solving.
1.9 COMPRESSIBILITY
Compressibility refers to the change in volume (V) of a substance that is subjected to a change in pressure on it. The usual quantity used to measure this phenomenon is the bulk modulus of elasticity or, simply, bulk modulus, E: ➭ Bulk Modulus
and 68 F (20 C)
Because the quantities V and V have the same units, the denominator of Eq. (1–4) is dimensionless. Therefore, the units for E are the same as those for the pressure.
As stated before, liquids are very slightly compressible, indicating that it would take a very large change in pressure
Example Problem 1.4
to produce a small change in volume. Thus, the magnitudes of E for liquids, as shown in Table 1.3, are very high (see Reference 7). For this reason, liquids will be considered incompressible in this book, unless stated otherwise.
The term bulk modulus is not usually applied to gases, and the principles of thermodynamics must be applied to determine the change in volume of a gas with a change in pressure.
Compute the change in pressure that must be applied to water to change its volume by 1.0 percent.
Solution The 1.0-percent volume change indicates that V > V = - 0.01. Then, the required change in pressure is
TABLE 1.3 Values for bulk modulus for selected liquids at atmospheric pressure
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“They were a bad lot, the pair of them,” cried Jean, who was evidently highly delighted by the disappearance of father and daughter; “only for some reason or other there was no saying a word against them to old M. Bayre. Why, we could all have told him how Nini Portelet came over here one day when Marie had left monsieur’s child alone in the cottage, and how she took it over to St Luke’s with her. It was when you and your friends were over here, monsieur, that it happened. And Marie didn’t put herself out about it, but just borrowed the child of a St Luke’s fisherman, and got her money from old Monsieur Bayre as usual. Ah, they were a pair of beauties! She gave the child back to its mother yesterday, and I guessed somehow that they might be missing to-day. As a matter of fact, I know they crossed over to St Luke’s before it was light.”
Bayre entered the deserted cottage, where the disordered state of the living-room spoke of a sudden departure. Among the displaced articles of furniture, not good enough to be worth any attempt to take away, there were certain signs, cynically left without disguise, of the robbery committed at the château on the previous night.
There was a tablecloth, heavily and handsomely fringed, in which, without doubt, some of the booty had been hastily wrapped up by Marie. There were a few plated articles which had been inadvertently carried away with the silver. And there was the iron box about which so much fuss had been made.
Yes, lying bent and broken among the ashes on the hearth, after having evidently been forced open with a bent poker which lay near, was the very box which Marie had dropped out of the window to her father; and lying on the uneven tiles of the floor, at a little distance from it, was a heap of papers which Bayre at once judged to have been its contents.
He picked these up and began to examine them. To his astonishment and perplexity, the very first of these to attract his attention was one in which the words “my nephew and namesake Bartlett Bayre,” were the first to catch his eye.
Further inspection proved this to be a will made and signed by his uncle only nine months previously, and in it he found that he himself was not only left a legacy of ten thousand pounds, but was appointed guardian of the testator’s infant son and heir.
Bayre started to his feet, so much amazed at what he had read that for the moment he seemed scarcely able to think or even to see.
His uncle, only nine short months ago, had been so kindly disposed towards him that he had made him a handsome legacy! How then had it happened, unless indeed the old man’s mind had become unhinged, that he had shown his nephew, from the first sight of him, nothing but aversion of the strongest kind?
The thing was so strange that Bayre could not trust himself to consider it thoroughly at that time. Hastily gathering up all the rest of the papers which he could find, he decided, after a moment’s hesitation, not to take them back to the château, but to carry them with him to London, and to communicate with Olwen from there, telling her of his find, and asking her advice as to whether he should send them to her or to his uncle’s solicitors.
They would, he thought, be better judges of his uncle’s real state of mind than he could be; and in any case the will could not be of much value, as his uncle had undoubtedly altered his dispositions long since.
So utterly absorbed was he in the strange events which had happened, and in this last, perhaps the strangest discovery of all, that the journey to London seemed only half the length of the journey away from it.
He had sent no word as to the day he was returning, so that when he entered the sitting-room at the Diggings at ten o’clock at night he found Southerley and Repton smoking together by the fire, in a state of gloom and abstraction, and with the supper-table laid for only two.
“Hallo!” said Repton, sulkily. “You, is it?”
But Southerley only scowled and said nothing.
“Yes, it’s me,” replied Bayre, with ungrammatical cheeriness. “How are you, eh? Have you got any bottled stout? And how’s the—”
But Repton sprang up with a yell and a tragic uplifting of the arm.
“Don’t dare to pronounce that evil brat’s name here,” cried he, sepulchrally. “Unless you want to be chucked out of window.”
“But why not?” persisted Bayre, who felt a redoubled interest in the child whose guardian it had certainly once been his uncle’s intention that he should be.
Repton pointed to Southerley with a tragic forefinger.
“Ask him!” said he in a hollow voice.
Southerley growled a little, and then moved sulkily in his chair.
“Oh, the child’s right enough, as children go, I suppose,” said he. “The trouble of it is that Miss Merriman has grown so much attached to the wretched little animal that there’s no talking to her, no getting her attention, no interesting her in anything but its miserable little mewlings and pukings.”
“That’s the worst of the domestic women you’re so fond of, Bayre,” went on Repton; “when there’s a child about they won’t pay the least attention to anything or anybody else. Whenever we go, as, of course, being two out of its three fathers, we’re bound to go, to inspect the child, and see that it’s properly fed and clothed and educated—” Bayre interrupted with a mocking laugh, but Repton went steadily and stodgily on: “Whenever we seek to do our duty, as I say, Miss Merriman makes fun of us, and says, ‘Did its nice ickle papas tum to see if its bockle was too warm-warm?’ And such stuff as that. Now you’re come back I hope you’ll try to bring this young woman to reason, and—”
“I hope you won’t try to do anything of the sort,” growled Southerley in a saturnine manner from his chair. “That would be just the last straw, for you to interfere. For we know you like domestic women, and so no doubt you’d worm yourself into her confidence, and—”
“And we should be nowhere!” added Repton. “That’s true.”
“Certainly we’re nowhere already,” went on Southerley, meditatively. “I’m only hoping you’ll be nowhere too!”
“You needn’t trouble your heads about me,” said Bayre, airily. “I’ve not the least wish to enter the lists, I assure you.”
The words were scarcely out of his mouth when there was a soft knock at the door, and to the rage and consternation of two out of the three young men, beautiful Miss Merriman, who had not been once to the Diggings since Bayre went away, peeped into the room and smiled a gracious “How do you do?” to that fortunate young man.
“Oh, Mr Bayre,” cried she, sweetly, “could you come downstairs a moment? I have something to ask you about the little boy, and whether you’ve heard anything of his parents.”
Bayre having, of course, expressed his ready assent, she retreated with a smile evenly distributed among them all, and left the three young men together. Repton made as if to stab the too lucky Bayre with the bread-knife.
“Villain,” he said, “you deserve to die. But first you shall interview the lady, and we’ll listen outside to see that you don’t take a mean advantage of your visit to the baby’s native haunts.”
Southerley, who was more uneasy than Repton, looked up sullenly.
“Oh, let him go,” said he, in a sort of despair “May as well be put out of one’s misery at once. Go and ask her to marry you, and, for goodness’ sake, get it quickly over.”
From which Bayre, as he went downstairs, with a brand-new suspicion concerning Miss Merriman in his mind, opined that poor Southerley was as false to his ideal of a woman of genius as the only possible lady-love as he, Bayre, was false to his.
CHAPTER XXI.
PARENTS AND GUARDIANS
B found Miss Merriman in the dining-room, which, with a woman’s taste, she had managed to make very pretty. The shabby leather sofa was covered by a piece of handsome tapestry of subdued tints, and the cottage piano stood out from the wall in the modern manner, adorned with a handsome embroidered back, and with a vase of flowers on one corner.
There was a work-basket, too, in the room, and there was a most dainty cot, in which the baby boy lay asleep.
The gas was not alight, but a little table lamp, with a pretty shade, was standing on a small table which had a woman’s work upon it, too far from the cot for the light to fall upon the sleeping infant.
“I thought you’d like to see him,” said Miss Merriman, as she bent over the cot, looking a very Juno in her plain dress of navy serge, cut just low enough at the neck to show the full beauties of a superb white throat.
But it was not so much her physical beauty which attracted Bayre as a certain tender look in her eyes which he thought amounted to self-betrayal.
With a certain air of unaccustomed responsibility the young man said, watching her the while,—
“Yes, indeed. I have very strong reasons for wishing to see the little chap. I’ve found out something about him.”
She looked up quickly, anxiously.
“Ah!”
“I’ve found out, I think beyond a doubt, that he’s not only my first cousin, but that I’m his guardian.”
The answer which this announcement drew from the lady would have been surprising enough but for Bayre’s own suspicions.
“His guardian! He can’t have a guardian till his father’s dead?”
Bayre took her quickly upon her words.
“Who is his father, then?” he asked.
She bit her lip, feeling that she had betrayed herself.
“How should I know?” said she. “I meant only that neither you nor anyone else can be guardian to the child until you can prove that he’s an orphan. Is he an orphan?”
“I think not,” said Bayre, rather drily. And then he added, after a pause: “Would you like me to say what I think?”
A look of fear came into her great ox eyes. She grasped the rail of the cot firmly for a few moments, and then said, in a very dignified and touching manner, “I think, if you want to do your best for the child—and I’m sure that you do—you had better say as little as possible till you know more than you do.”
“Very well,” said he, gently.
There was a pause, and then she said, in a very low voice, “I’m glad you’ve come back. It was getting rather difficult for me. Those two friends of yours, good fellows, dear fellows, but—”
“Well?”
“They don’t know, and they don’t guess, and it makes things difficult.”
“Do you want them to guess?” asked Bayre.
“No, no,” cried Miss Merriman, quickly. “I don’t want anyone to guess anything, not even you. And you must remember that I’ve made no admissions, none whatever. I’ve taken care of this child, who has three fathers and no mother, purely out of good-nature. You understand?”
“I do.”
“But you’ll tell your friends, won’t you?—and especially Mr Southerley, who has been very kind”—and Miss Merriman looked down with a heightened colour—“that while I’m most grateful to them I feel that they are doing more than they ought. I don’t want their flowers; I don’t want their sweets. They’re spending a fortune in things of that sort just because they look upon me as a disinterested philanthropist, which I’m not, who has taken charge of this child from abstract motives of kindness—which I’ve not.”
Bayre looked at the sideboard indicated by the lady, and there he saw such a fine show of flowers, and of bon-bons in elegant wrappers, as would have set up a florist or confectioner in business in a small way.
He looked at her and smiled.
“They’ve been so very lavish,” he said, “that one wonders whether it was all gratitude, or something else, which prompted such profusion.”
Miss Merriman’s beautiful face puckered into lines of distress.
“That’s just what I’m afraid of,” admitted she, sadly. “I don’t mind with Mr Repton; he’s very nice, but he takes things lightly, doesn’t he? But Mr Southerley—”
Her voice faltered, and Bayre began to look rather grave.
“Shall I hint to him that there’s—an obstacle?” asked he, in a low voice.
But she refused emphatically.
“Certainly not. How can you say there’s an obstacle when you know nothing whatever about me except that I’ve good-naturedly relieved you all of a burden?” she said firmly. “No. What I want you to do is to tell them that—that—”
“I’ll tell them that you’re engaged to be married,” said Bayre, with a happy thought. “That will put an end to any aspirations either of them might have without letting them into any secrets.”
“You don’t know any of my secrets,” retorted Miss Merriman, sharply.
Bayre gave her one look and then bowed without speaking. She had to be content with that; for although she began to interrogate him quickly as to what he knew, or guessed, she changed her mind before he could make any reply, and telling him haughtily that he could invent what he pleased about her, she let him go.
Bayre felt himself to be in a difficulty. Certainly he did not know very much of absolute knowledge, but he could guess a good deal; and if his suspicions were correct there was an end to Southerley’s hopes. Between a chivalrous wish to respect the secret of a lady, a secret, too, which he could not be said to have more than guessed at, and his wish to spare his friend the pain of useless longing, Bayre found himself placed in a dilemma.
The consequence was that when he re-entered the common sitting-room there was just enough uneasiness discernible in his look and manner to fill both his friends with anxiety.
Of course this anxiety took an insulting form.
“Well, have you cut us out?” asked Repton, mockingly, looking at him askance from his armchair.
“Not that I know of,” said Bayre, quietly.
“What did she want you for?” growled Southerley in a dictatorial tone.
“Oh, to ask if I had found out anything about the child, of course.”
“And have you?”
“I think so. It’s my uncle’s child, and my first cousin, I have every reason to believe.”
“Then,” cried Repton, springing up in the delight of an interesting discovery, “we’ve only got to wring its neck for you and you’ll be heir to all the old gentleman’s property!”
“I don’t know so much about that,” said Bayre, laughing. “At the same time I’m awfully grateful to you for the suggestion that you’re so ready to oblige me.”
“Oh, well,” said Repton, “it cuts two ways, you know. Of course you’d have to keep Southerley and me out of the proceeds, and handsomely too. I’d let you off with a yacht and a cottage at Deal. But I don’t know what Southerley might want; a house in Park Lane, perhaps, to live in when he’d married Miss Merriman.”
“Hold your tongue, you fool!” said Southerley, in a deep bass voice.
“Well,” said Repton, “I know you won’t be satisfied unless you do marry her. I never saw any fellow so gone on any woman as you are on her. The way the conversation finds its way round to Miss Merriman every ten minutes, even if it starts at the differential calculus—(it never does, by-the-bye, and I haven’t the remotest idea what the differential calculus is)—is perfectly sickening.”
“What rot!” growled Southerley, with a restless turn in his chair.
Bayre looked at him out of the corners of his eyes.
“I hope that’s not true,” said he, “for I happen to know that she’s engaged.”
Southerley started to his feet.
“How do you know?” he asked angrily. “How should you know more than we do about it? unless—”
Repton took up his speech when he dropped it.
“By Jove!” cried he, “unless you’re engaged to her yourself?”
Although Bayre excused himself with vehemence, showed them the absurdity of the suggestion seeing that he had met the lady less often than they had, yet he did not feel sure that he succeeded in convincing them. And there remained a certain shadow over the intercourse of the three during the next few days. One reason for this was his extreme reticence about his visit to the islands. He did not say enough about anything or anybody to satisfy their minds. He was not engaged to Miss Eden, so he said; he was not reconciled to his uncle. On the whole, Repton and Southerley were of opinion that he was either a liar or that he had wasted his time. So that he had more time to himself than usual during the next few days, and he made use of it to devour at his leisure the manuscript novel Olwen had entrusted to his care.
As he read sympathetically, of course, two things became manifest to him. The one was that the olive-skinned hero with the brown eyes and the wavy black hair had been inspired by the girl’s conception of himself; and the other was that, amid all the traces of girlish inexperience and inexpertness with which the tale abounded, there was yet a saving grace, a charm of vivacity and of freshness which, as he was old enough to know, are the commonest marks of real ability in a beginner.
The first discovery touched him the most. But the second had a pathetic interest also; for he recognised the fact that, with all her disadvantages as compared with himself in the way of actual experience of life, there was something in the girl’s manuscript which his own more solid productions lacked, a something which made it not improbable that he would be more successful in disposing of her work than he was in disposing of his own.
Full of his impressions of her tale, he sat down to write to her on the third day after his return to town. He treated the matter of the novel very guardedly indeed; spoke well of it, warned her not to be too hopeful, remarked that her hero, while not unheroic, was very unlike a real man. Thus Bayre thought he would put her off the scent of his own intuition that the hero was meant for his own portrait. He added that he did not despair of selling the work, and that he would
set about it at once. But she must not expect to set up a carriage out of the proceeds.
And then he turned to graver matters. Suspecting her complicity in the abduction of his infant cousin, and resenting her want of confidence in him over the matter, he said nothing about the child and nothing about Miss Merriman. But he told of his discovery of the broken iron box and its contents, and of the will which his uncle had made eight months previously. He asked her advice as to whether he should send these papers to her for his uncle, or to Mr Bayre’s solicitors. Perhaps she, he said, was in a better position than he to decide whether old Mr Bayre was in a fit state to be troubled with matters of business. For he reminded her that the old gentleman was evidently suffering from weakness of memory, as he had professed to have no remembrance whatever of the iron box.
He did not deny that he had read enough of the will to learn, to his surprise, how differently his uncle had thought of him a few months before, but he admitted that the document could have none but a sentimental interest now.
“If only,” went on poor Bayre, “he had continued in the same mind towards me, perhaps some day I might have been able to offer you something better than love in a villa one-brickthick. However, I don’t mean to give up hope. Heaven keep you out of the way of another Monsieur Blaise! Remember, you have promised to write. So keep your promise unless you want me to throw up my berth here and come over again to find out why you don’t.—Yours,
“B B .”
He was finishing this letter in his own room, by the light of a couple of inferior candles, when there came a thump at the door, and without waiting for permission Southerley put his head in.
“Hallo, what’s up?” asked Bayre, perceiving that the usually somewhat phlegmatic red face of the stalwart pressman was the colour of whitey-brown paper, and that his eyes had an unusual look.
“May I come in?” asked Southerley, hoarsely, when he was well inside and had shut the door carefully behind him. “I want to ask you something.” Then his eyes fell on the letter, which Bayre was elaborately trying to hide with a transparent assumption of carelessness. “You’re writing letters, I see?”
Bayre tried to look as if he had forgotten the fact.
“Miss Eden?” went on Southerley in a mysterious voice.
“H’m,” nodded Bayre, shamefacedly.
It is a humiliating thing to have it found out that you are over head and ears in love with a woman! But Southerley took it very nicely
“That’s all right!” he said with a sigh of relief in proportion to his size.
“What do you mean?”
“Why, look here. I haven’t been quite sure that you were not sweet upon the girl downstairs. But you wouldn’t be carrying on with both of them at once, now, would you?”
“Good heavens, no, man! And how do you know that either of them would so much as look at me?”
Southerley sighed again and wiped his face.
“Oh, well, well, women are odd creatures!” he observed frankly. “Anyhow, since you’ve given me your word it’s all right I—I want you to do something for me.”
“Well, what?”
Southerley began to pant heavily as he sat with his hands on his knees on one of Bayre’s boxes.
“I want you to propose for me to Miss Merriman.”
“Good heavens, man, are you mad?”
“Something very like it sometimes since I’ve seen so much of that girl,” said the giant, slowly. “I can’t tell you the effect she has upon me.”
“Effect! Rubbish! Haven’t you often said your ideal of woman is a gen—”
“Oh, woman of genius be blowed!” cried Southerley, impatiently “One says those things before one’s hit, just because one must always be talking of women, even if it’s only talking balderdash. But I tell you it’s serious with me now. I must know how she feels, I must, I must.”
“But haven’t I told you—” began Bayre.
“Told me fiddlesticks! You’ve said she’s engaged. Well, somehow I don’t believe she is. She wears no ring. Besides, how should you know? She didn’t tell you in so many words she was engaged, did she?”
“N-n-no,” admitted his friend.
“Has she ever said she cared about anybody?”
“N-n-no.”
“Then you just go and ask her this minute if she can care for me!”
And Southerley plunged across the room, hauled his friend out of his chair and flung him at the door. There Bayre, however, planted himself, and protested,—
“If you must be such a confounded fool as to want to propose to her after what I’ve told you,” said he, surlily, “why don’t you do it yourself?”
“Because I can’t,” gasped the timid little lad of six feet three in a deep bass voice. “Look here, do you think I haven’t tried? I’ve been down those blessed stairs four times this evening! Four times, mind you, and I’ve got as far as the door, and I’ve heard her singing to that brat. And I tell you the sound of her voice made me feel so queer that I couldn’t go in, because I knew the words would stick in my throat and I should make a fool of myself.”
“You are bad!” remarked Bayre, critically, as he contemplated the giant’s moist face.
“Well, get on, if you don’t want to be kicked downstairs,” retorted Southerley, beginning to get irritated by his friend’s unaccountable perverseness.
Bayre raised his eyebrows and turned slowly.
“It won’t be of any use,” said he, as he opened the door and went downstairs.
CHAPTER XXII.
A RUNAWAY
B felt very nervous over his errand, and when Miss Merriman cried “Come in,” in answer to his knock, he was almost as awkward as Southerley himself would have been, and she gave him a searching look as he crossed the room like a sly schoolboy
She was sitting near the fire, and the baby, in a state of great glee, was turning out the contents of her work-basket on the rug at her feet. Bayre felt that he was called upon to explain his appearance with promptitude.
“I hope I’m not disturbing you,” he said, “but I’ve been sent here by —by somebody else—by Southerley, in fact, with a message which I hardly dare to give.”
Before he was half-way through the speech the lady had looked away; and from the expression of her face he could guess that she had an uneasy suspicion as to the nature of his errand.
“Then why give it,” said she, quickly, in a slightly tremulous voice, “if it’s of no use, and if it’s painful to you?”
“Because I must; because I’ve promised. Forgive me if I’m clumsy over it. The fact is the fellow’s lost his head; I think perhaps he knows there’s not much hope for him; I myself have told him there’s not. But he persists in hoping, hoping, or rather he’s got into such a state that he can’t rest till he’s got a definite answer, even if it’s the wrong one. He’s in love with you, head over ears in love, and he wants to know if you could ever care for him.”
Although he knew that she must have guessed what was coming, Miss Merriman pretended to feel surprised. But it was a poor, worried sort of pretence, without either nature or sincerity.
“Why, it’s absurd,” she said quickly. “What does he know of me? I never heard anything so ridiculous.”
And then there was a short pause, during which she sat very still.
“You’re not offended?” said Bayre, gently.
“Offended!” She just got out the word and then broke down into a flood of tears.
Bayre was appalled. To see a woman cry was a dreadful thing at any time; but to feel that he had opened the floodgates himself, and when he ought to have known better, was a thought of unspeakable horror.
“Forgive me,” he said hoarsely. “And don’t, oh, don’t! You make me feel a brute, and yet I couldn’t help myself. I’ll tell him—I’ll go and tell him—” He was flying to the door, impelled thereto not only by the woman’s tears but by the yells of the small child, who was on his feet by Miss Merriman’s knee, screaming in sympathy after the manner of his kind.
Miss Merriman recovered herself sufficiently to speak.
“No,” she cried imperiously. “Don’t tell him anything. You’re not to tell him anything. Let him think what he likes until—”
“Until what?”
“Never mind.”
She waved her hand in farewell without looking at him, and Bayre made his way reluctantly enough upstairs, where he found Southerley in waiting on the half-landing.
“No good, of course?” said the big man, trembling like a leaf.
Bayre shook his head.
“Any reason?”
“No. Sorry. I did my best.”
Southerley took it very quietly; he just nodded and went upstairs softly whistling, with his hands in his pockets. Then he went out at once, without seeing either of the others again, and he did not come back until long after they were both in bed.
And he alone of the three made no remark whatever when Susan informed them on the following evening that Miss Merriman had gone away and had taken the child with her.
Repton gave a long whistle.
“Well, I’m blest!” he exclaimed tersely Bayre was indignant. Surely he had a right to know where she was taking the child, he who claimed not only to be the infant’s cousin but to have more than a fanciful claim to be its guardian! Miss Merriman was surely carrying a woman’s privileges too far.
“Cousin or no cousin, it’s abominable,” said Repton, indignantly “We’ve had all the trouble of the journey from Guernsey, all the expense of milk and biscuits, sausage rolls and bananas for the brat, and flowers and sweets for her. And now we’re left in the lurch like this! It’s infamous. I’m hurt in my very tenderest feelings. I shall advertise.”
“What! For the price of the flowers and the bon-bons?” laughed Bayre.
“Of course not. But I have a third share in the proprietorship of that infant. And it may be worth money some day Besides, I ought to have been consulted.”
All this time Southerley never moved a muscle. But that he was hard hit it was impossible not to see. His eyes looked glassy and his ruddy skin livid.
“Cheer up, old man!” cried Repton, giving him a ferocious thump on the back. “She wasn’t worth troubling about, a woman who could go without a word after that last box from Fuller’s—the one with the gold ribbon and the picture of the two cupids in a basket. Thank goodness, she’ll never be able to look at those two pink cupids without a self-reproachful thought of you and me!”
But even this thought did not appear to have a consoling effect upon Southerley, who shook him off impatiently and went out again without a word.
“Fool!” cried Repton, contemptuously, “to care so much for a woman who didn’t care two pins about him.”
But Bayre, who remembered Miss Merriman’s tears, was less harsh in his judgment.
“I have an idea,” said he, slowly, “that she didn’t dare to care!”
But he would not proffer any solution of this enigmatic remark.
And before the day was out he had something to divert his attention in the shape of a letter from Miss Eden.
A surprising letter it was, and tantalising, too, for it was evidently written in a sort of breathless way, while the writer was at a white heat of emotion, and it told him just enough to make him want to know more.
It was as follows:—
“D M B ,—I got your letter I have said nothing about it. I think you had better keep the papers yourself for a little while—those, I mean, that you found in the iron box. I will write to you again in a day or two, perhaps. I am afraid this letter is disjointed, but I have had a sort of shock, and I have not got over it yet. Do not be alarmed: we are all well here, or as well as you could expect, remembering the state in which you left us all. The Vazons have not come back and we have heard nothing more of them. We think they must be still in the islands, but they are not at Creux. Nini has come to stay here; she is a trustworthy girl, and I am very glad to have her, for I should not like to be here quite alone.
“Now I am going to tell you something which will surprise you. I have found out who the woman is shut up here. I cannot tell you more now, except this—that she is not here against her will.—Yours sincerely,
“O E .”
Bayre was on thorns to know more, and he could not understand why, having told him so much, she could not have trusted him with the whole of the secret. Was it something she did not like to trust to paper? Was it his young wife whom old Mr Bayre was keeping concealed at the château? And if so, was she in her right mind?
He wrote at once, begging Olwen to let him know more, but yet expressing himself guardedly, for fear the letter should fall into other hands than hers.
He could not rest for thinking about this, wondering whether his uncle knew of the discovery made by Olwen, and whether, in that case, he would make any difference in his treatment of her.
His anxiety grew as day after day passed and no answer came to his second letter. He could not get another holiday or he would have gone back to Creux without delay. In his distress he thought of writing to Madame Nicolas, his landlady at St Luke’s, to ask whether she had heard anything of old Mr Bayre and his household.
The good woman answered almost by return of post, but the information she had to give was exceedingly vague, and was rather
in the nature of gossip than of anything definite.
She had not seen old Monsieur Bayre lately, neither had anyone she knew. But she had heard that he was ill, that there had been changes in his household, and that the young lady had gone away to London and was singing somewhere under the name of Señora Pia, or some such title. As Madame Nicolas did not even mention the Vazons by name, it seemed probable that they had kept quiet and had not made any attempt to turn the tables upon their late master.
This letter, vague as it was, filled Bayre with anxiety and distress. He knew there must be some foundation for this story about Olwen, and it tallied too well with her silence for him to neglect the clue.
“Singing in London under the name of Señora Pia!” This was vague indeed. He seized the newspapers and studied their columns with eager scrutiny. But it was not until the third day after the receipt of this letter that, after having read on the first page of the Daily Telegraph all the names of all the ladies and gentlemen who were advertised in the music publishers’ announcements as singing songs in different parts of England, the name “Signora Beata” attracted his attention and made him decide to set off that very evening on what might be a wild-goose chase after all.
“Signora Beata” was to sing “Those Sweet, Sweet Eyes of Thine” and another ballad with an equally vapid title at the Bromley Institute. And as it was not very far away, Bayre thought it worth while to take the journey on this very slender clue.
The hall was crowded. Bayre got a programme and found that Signora Beata did not appear before the fourth number in the programme. He had to sit through a new loyal song, rendered lustily by the baritone but conspicuous for its loving adhesion to one note. He had to hear a glee, and he had to endure a recitation.
Then came the turn of number four, and it was as much as he could do not to start out of his seat with surprise when Signora Beata appeared and proved to be, not indeed Olwen Eden, but another old friend in the person of Miss Merriman.
She looked magnificent in a dress of cream satin, which showed off her beautiful neck and the exquisite poise of her head to great advantage. She wore no jewels, but half-a-dozen roses of different colours were arranged on the front of her dress, and another was
placed upright on one side of her head and worn as an aigrette. Long white suède gloves completed the costume, and Bayre thought that he had never seen so beautiful a woman, and was glad Southerley was not there to have his chains further riveted.
He became quite anxious to hear her sing, and was not in the least surprised at the burst of applause which greeted her as soon as she came to the front of the platform. It seemed to him that if her voice proved to be as superb as her appearance she was wasting herself at Bromley.
But with the first bars of the song came not exactly disenchantment, but a decidedly modified appreciation of the beauty’s art. She had a good voice, not in the first rank, but pleasant to listen to; the weakness of her performance lay in the fact that her voice had not been sufficiently cultivated, and that she was possessed by an overpowering nervousness which, while it rather added to her charm as a woman, decidedly marred her efforts as a singer.
In brief she had, though singing as a “professional,” scarcely got beyond the stage of “gifted amateur.”
But her beauty, her modesty, her statuesque grace, carried all before them, and the audience applauded her as if she had been Patti herself.
Bayre began now to understand that Madame Nicolas had mixed up in her mind what she had heard about one woman with what she had heard about another, and he resolved, now that he was in for it, to run Signora Beata to earth.
He found the rest of the concert tedious, except when the beauty was on the platform, and as soon as her last appearance on the programme was made he slipped out of his seat and went outside to wait for her.
She fled out of the building so quickly and so quietly, however, that he was not able to speak to her, and he got into the same train, but not into the same carriage, and then when he had seen her enter an omnibus he got on the top of the vehicle, determined to track her down.
She alighted finally at that part of London which used to be known as Brompton, but which has since, by the profuse use of the name
“Egerton” instead of the older and homelier ones, purged away the Brompton taint and become something far higher in the social scale.
Here Bayre followed the lady into a side street where little cards over the door announced “Apartments,” and at one of these she stopped and proceeded to open it with a latch-key.
Bayre stopped too at the foot of the steps and looked up.
She heard the footsteps stop and looked round quickly. An exclamation broke from her lips.
“You’ve followed me!” she cried.
“Yes—Signora.”
She started, hesitated, then shut the door again and came down the steps.
“Why have you done this?” she said passionately.
“Why have you run away, without a word, with my cousin, my ward, Miss Merriman?”
“Your ward!” She laughed derisively. “Don’t talk nonsense. You know that that is a mere farce. You know well enough that I’m his mother.”
“Yes, and you’re something else,” replied Bayre, coolly. “You’re my aunt, Mrs Bartlett Bayre.”
She met his eyes, and then looked down; but she made no attempt to contradict him.
“Come inside,” she said suddenly, “and we can talk better. You must know everything now.”
CHAPTER XXIII.
A PHOTOGRAPH
B followed Miss Merriman into the house, and into the little ground-floor sitting-room, where she turned up the gas and showed the folding doors open into the adjoining room, where a maid sat reading a novelette by the light of a candle beside the baby’s cot.
“Wait here a moment, I always go and kiss my baby the moment I come in; nothing can interfere with that ceremony,” she said, with a pretty defiance which Bayre liked.
And as she disappeared through the folding-doors, which she shut after her, her attitude seemed to say that now she had once owned that that baby was hers she would brandish him in the eyes of the world and snap her fingers at destiny.
Bayre heard the soft whisperings of the two women, the mysterious cooings and cawings they made over the sleeping child. And when Miss Merriman swept majestically back into the room again, dressed in a plain grey tea-gown, with one of her roses pinned in it, he remembered his old ideal of the simple, domesticminded woman, and he sympathised with Southerley’s adoration of this beautiful creature.
“Now,” she said defiantly, “perhaps you’ll explain why you have followed me, why you have come.”
Bayre was rather amused, and rather resentful.
“You must remember,” said he, “that whatever suspicions I may have had concerning your relationship to the child, all that I absolutely knew was that he was my uncle’s son, and that therefore it was a personal duty of mine to know what became of him. My friends too, Repton and Southerley—” She interrupted him with a quick gesture.
“Surely,” she said, panting a little, “you can’t pretend they have a right to know anything whatever about me!”
She was standing on one side of the table and Bayre was on the other He leaned upon it to look earnestly into her face.
