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Texts in Applied Mathematics 56

Mark H. Holmes

Introduction to

the Foundations

of Applied

Mathematics

Second Edition

Texts in Applied Mathematics

Volume 56

Editors-in-chief

A. Bloch, University of Michigan, Ann Arbor, USA

C. L. Epstein, University of Pennsylvania, Philadelphia, USA

A. Goriely, University of Oxford, Oxford, UK

L. Greengard, New York University, New York, USA

Series Editors

J. Bell, Lawrence Berkeley National Lab, Berkeley, USA

R. Kohn, New York University, New York, USA

P. Newton, University of Southern California, Los Angeles, USA

C. Peskin, New York University, New York, USA

R. Pego, Carnegie Mellon University, Pittsburgh, USA

L. Ryzhik, Stanford University, Stanford, USA

A. Singer, Princeton University, Princeton, USA

A. Stevens, Universität Münster, Münster, Germany

A. Stuart, University of Warwick, Coventry, UK

T. Witelski, Duke University, Durham, USA

S. Wright, University of Wisconsin, Madison, USA

The mathematization of all sciences, the fading of traditional scientific boundaries,

the impact of computer technology, the growing importance of computer modeling

and the necessity of scientific planning all create the need both in education and

research for books that are introductory to and abreast of these developments.

The aim of this series is to provide such textbooks in applied mathematics for

the student scientist. Books should be well illustrated and have clear exposition

and sound pedagogy. Large number of examples and exercises at varying levels

are recommended. TAM publishes textbooks suitable for advanced undergraduate

and beginning graduate courses, and complements the Applied Mathematical

Sciences (AMS) series, which focuses on advanced textbooks and research-level

monographs.

More information about this series at http://www.springer.com/series/1214

Mark H. Holmes

Introduction to the

Foundations of Applied

Mathematics

Second Edition

123

Mark H. Holmes

Department of Mathematical Sciences

Rensselaer Polytechnic Institute

Troy, NY, USA

ISSN 0939-2475

ISSN 2196-9949 (electronic)

Texts in Applied Mathematics

ISBN 978-3-030-24260-2

ISBN 978-3-030-24261-9 (eBook)

https://doi.org/10.1007/978-3-030-24261-9

Mathematics Subject Classification (2010): Primary: 76Axx, 76Bxx, 76Dxx 74Bxx, 74Dxx, 74Hxx,

74Jxx 74Axx, 34D05, 34E05, 34E10, 34E13, 35C06, 35C07, 35F50, 60J60, 60J65

1st edition: © Springer Science+Business Media, LLC 2009

© Springer Nature Switzerland AG 2019

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of

the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,

broadcasting, reproduction on microfilms or in any other physical way, and transmission or information

storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology

now known or hereafter developed.

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication

does not imply, even in the absence of a specific statement, that such names are exempt from the relevant

protective laws and regulations and therefore free for general use.

The publisher, the authors, and the editors are safe to assume that the advice and information in this book

are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or

the editors give a warranty, express or implied, with respect to the material contained herein or for any

errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional

claims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Switzerland AG.

The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Colette, Matthew, and Marianna

Preface to the Second Edition

The principal changes are directed to improving the presentation of the material.

This includes rewriting and reorganizing certain sections, adding new examples,

and reorganizing and embellishing the exercises. The added examples range from

the relatively minor to the more extensive, such as the added material for water

waves. This edition also provided an opportunity to update the references.

Another reason for this edition concerns the changes in publishing over the last

decade. The improvements in digital books and the interest in students for having

an ebook version were motivating reasons for working on a new edition.

Finally, the one or two typos in the first edition were also corrected, and thanks

go to Ash Kapila, Emily Fagerstrom, Jan Medlock, and Kevin DelBene for finding

them.

Troy, NY, USA

March 2019

Mark H. Holmes

vii

Preface to the First Edition

FOAM. This acronym has been used for over 50 years at Rensselaer to designate

an upper-division course entitled, Foundations of Applied Mathematics. This

course was started by George Handelman in 1956, when he came to Rensselaer

from the Carnegie Institute of Technology. His objective was to closely integrate

mathematical and physical reasoning, and in the process enable students to obtain

a qualitative understanding of the world we live in. FOAM was soon taken over by

a young faculty member, Lee Segel. About this time a similar course, Introduction

to Applied Mathematics, was introduced by Chia-Ch’iao Lin at the Massachusetts

Institute of Technology. Together Lin and Segel, with help from Handelman,

produced one of the landmark textbooks in applied mathematics, Mathematics

Applied to Deterministic Problems in the Natural Sciences. This was originally

published in 1974, and republished in 1988 by the Society for Industrial and Applied

Mathematics, in their Classics Series.

This textbook comes from the author teaching FOAM over the last few years. In

this sense, it is an updated version of the Lin and Segel textbook. The objective

is definitely the same, which is the construction, analysis, and interpretation of

mathematical models to help us understand the world we live in. However, there

are some significant differences. Lin and Segel, like many recent modeling books, is

based on a case study format. This means that the mathematical ideas are introduced

in the context of a particular application. There are certainly good reasons why

this is done, and one is the immediate relevance of the mathematics. There are

also disadvantages, and one pointed out by Lin and Segel is the fragmentary

nature of the development. However, there is another, more important reason for

not following a case studies approach. Science evolves, and this means that the

problems of current interest continually change. What does not change as quickly

is the approach used to derive the relevant mathematical models, and the methods

used to analyze the models. Consequently, this book is written in such a way as to

establish the mathematical ideas underlying model development independently of a

specific application. This does not mean applications are not considered, they are,

and connections with experiment are a staple of this book.

ix

x

Preface to the First Edition

The first two chapters establish some of the basic mathematical tools that are

needed. The model development starts in Chap. 3, with the study of kinetics. The

goal of this chapter is to understand how to model interacting populations. This

does not account for the spatial motion of the populations, and this is the objective

of Chaps. 4 and 5. What remains is to account for the forces in the system, and this is

done in Chap. 6. The last three chapters concern the application to specific problems

and the generalization of the material to more geometrically realistic systems. The

book, as well as the individual chapters, is written in such a way that the material

becomes more sophisticated as you progress. This provides some flexibility in how

the book is used, allowing consideration for the breadth and depth of the material

covered.

The principal objective of this book is the derivation and analysis of mathematical

models. Consequently, after deriving a model, it is necessary to have a way to

solve the resulting mathematical problem. A few of the methods developed here

are standard topics in upper-division applied math courses, and in this sense there is

some overlap with the material covered in those courses. Examples are the Fourier

and Laplace transforms, and the method of characteristics. On the other hand, other

methods that are used here are not standard, and this includes perturbation approximations and similarity solutions. There are also unique methods, not found in

traditional textbooks, that rely on both the mathematical and physical characteristics

of the problem.

The prerequisite for this text is a lower-division course in differential equations.

The implication is that you have also taken two or three semesters of calculus, which

includes some component of matrix algebra. The one topic from calculus that is

absolutely essential is Taylor’s theorem, and for this reason a short summary is

included in the appendix. Some of the more sophisticated results from calculus,

related to multidimensional integral theorems, are not needed until Chap. 8.

To learn mathematics you must work out problems, and for this reason the

exercises in the text are important. They vary in their difficulty, and cover most

of the topics in the chapter. Some of the answers are available, and can be found at

www.holmes.rpi.edu. This web page also contains a typos list.

I would like to express my gratitude to the many students who have taken my

FOAM course at Rensselaer. They helped me immeasurably in understanding the

subject, and provided much-needed encouragement to write this book. It is also a

pleasure to acknowledge the suggestions of John Ringland, and his students, who

read an early version of the manuscript.

Troy, NY, USA

March 2009

Mark H. Holmes

Contents

1 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

Examples of Dimensional Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.1

Maximum Height of a Projectile . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.2

Drag on a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.3

Toppling Dominoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.4

Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3

Theoretical Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3.1

Pattern Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4

Similarity Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4.1

Dimensional Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4.2

Similarity Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.5

Nondimensionalization and Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.5.1

Projectile Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.5.2

Weakly Nonlinear Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.5.3

Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1

3

5

6

14

16

16

20

22

23

25

27

27

31

34

34

2

49

49

53

53

56

60

64

66

69

76

77

77

79

Perturbation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1

Regular Expansions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

How to Find a Regular Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2.1

Given a Specific Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2.2

Given an Algebraic or Transcendental Equation . . . . . . . . . . .

2.2.3

Given an Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

Scales and Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

Introduction to Singular Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5

Introduction to Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5.1

Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6

Examples Involving Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6.1

Example 1: Layer at Left End . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6.2

Example 2: Layer at Right End. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

xii

Contents

2.6.3

Example 3: Boundary Layer at Both Ends. . . . . . . . . . . . . . . . . .

Multiple Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.7.1

Regular Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.7.2

Multiple Scales Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

84

85

88

92

3

Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.1

Radioactive Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.2

Predator-Prey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.3

Epidemic Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Kinetic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.1

The Law of Mass Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.2

Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.3

Steady States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.4

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.5

End Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

Modeling Using the Law of Mass Action . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3.1

Michaelis-Menten Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3.2

Disease Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3.3

Reverse Mass Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4

General Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

Steady States and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5.1

Reaction Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5.2

Geometric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5.3

Perturbation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6

Solving the Michaelis-Menten Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6.1

Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6.2

Quasi-Steady-State Approximation . . . . . . . . . . . . . . . . . . . . . . . . .

3.6.3

Perturbation Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.7

Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.7.1

Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.8

Modeling with the QSSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.9

Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

103

103

104

104

105

107

109

111

111

113

114

115

116

118

119

123

123

124

126

134

134

135

137

143

145

148

151

151

4

Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

Random Walks and Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.1

Calculating w(m, n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.2

Large n Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3

Continuous Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3.1

What Does D Signify?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4

Solutions of the Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.1

Point Source Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.2

A Step Function Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . .

165

165

167

170

172

174

175

178

178

183

2.7

Contents

xiii

4.5

Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5.1

Transformation of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5.2

Convolution Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5.3

Solving the Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6

Continuum Formulation of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6.1

Balance Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6.2

Fick’s Law of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6.3

Reaction-Diffusion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.7

Random Walks and Diffusion in Higher Dimensions . . . . . . . . . . . . . . . .

4.7.1

Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8

Langevin Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8.1

Properties of the Random Forcing . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8.2

Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

186

187

189

191

194

195

196

203

205

207

211

213

219

220

5

Traffic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2

Continuum Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.1

Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.2

Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3

Balance Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.1

Velocity Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4

Constitutive Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4.1

Constant Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4.2

Linear Velocity: Greenshields Law . . . . . . . . . . . . . . . . . . . . . . . . .

5.4.3

General Velocity Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4.4

Flux and Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4.5

Reality Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.5

Constant Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.5.1

Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6

Density Dependent Velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6.1

Small Disturbance Approximation . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6.2

Method of Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6.3

Rankine-Hugoniot Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6.4

Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6.5

Expansion Fan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6.7

Additional Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.7

Cellular Automata Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

233

233

233

234

236

237

238

239

241

241

242

244

244

245

248

252

253

255

260

262

264

270

271

276

282

6

Continuum Mechanics: One Spatial Dimension . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2

Frame of Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2.1

Material Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2.2

Spatial Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

295

295

295

296

297

xiv

Contents

6.2.3

Material Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2.4

End Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3

Mathematical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.4

Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.4.1

Material Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.5

Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.5.1

Material Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.6

Summary of the Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.7

Steady-State Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.8

Constitutive Law for an Elastic Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.8.1

Derivation of Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.8.2

Material Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.8.3

Material Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.8.4

End Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.9

Morphological Basis for Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.9.1

Metals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.9.2

Elastomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.10 Restrictions on Constitutive Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.10.1 Frame-Indifference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.10.2 Entropy Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.10.3 Hyperelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

300

302

304

305

306

307

309

309

311

312

314

316

319

320

321

321

324

325

326

328

332

335

7

Elastic and Viscoelastic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.1

Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.1.1

Method of Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.1.2

Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.1.3

Geometric Linearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2

Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2.1

Mass, Spring, Dashpot Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2.2

Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2.3

Integral Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2.4

Generalized Relaxation Functions . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2.5

Solving Viscoelastic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

345

345

348

350

362

363

363

366

370

372

373

377

8

Continuum Mechanics: Three Spatial Dimensions . . . . . . . . . . . . . . . . . . . . . .

8.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.2

Material and Spatial Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.2.1

Deformation Gradient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.3

Material Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.4

Mathematical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.4.1

General Balance Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.4.2

Direct Notation and Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.5

Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.5.1

Incompressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

389

389

390

392

395

397

400

401

401

402

Contents

9

xv

8.6

Linear Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.6.1

Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.6.2

Differential Form of Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.7

Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.8

Summary of the Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.8.1

The Assumption of Incompressibility. . . . . . . . . . . . . . . . . . . . . . .

8.9

Constitutive Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.9.1

Representation Theorem and Invariants . . . . . . . . . . . . . . . . . . . .

8.10 Newtonian Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.10.1 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.10.2 Viscous Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.11 Equations of Motion for a Viscous Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.11.1 Incompressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.11.2 Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.12 Material Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.12.1 Frame-Indifference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.12.2 Elastic Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.12.3 Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.13 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.13.1 Incompressible Viscous Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.13.2 Elasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

403

404

406

407

407

408

409

413

414

415

415

418

419

420

423

426

426

429

430

431

432

434

Newtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.1

Steady Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.1.1

Plane Couette Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.1.2

Poiseuille Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.2

Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.2.1

Vortex Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.3

Irrotational Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.3.1

Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.4

Ideal Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.4.1

Circulation and Vorticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.4.2

Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.4.3

End Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.5

Boundary Layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.5.1

Impulsive Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.5.2

Blasius Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.6

Water Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.6.1

Interface Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.6.2

Traveling Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.6.3

Wave Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

445

446

446

450

453

455

456

459

461

462

465

469

469

470

471

477

478

479

481

486

xvi

Contents

A Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A.1 Single Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A.1.1 Simplification via Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A.2 Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A.3 Multivariable Versions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

497

497

498

499

500

B Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

B.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

B.2 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

C Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

D Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

D.1 Trace. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

D.2 Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

D.3 Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

D.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

509

509

509

510

510

E Equations for a Newtonian Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

E.1 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

E.2 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

Chapter 1

Dimensional Analysis

1.1 Introduction

Before beginning the material on dimensional analysis, it is worth considering a

simple example that demonstrates what we are doing. One that qualifies as simple

is the situation of when an object is thrown upwards. The resulting mathematical

model for this is an equation for the height x(t) of the projectile from the surface

of the Earth at time t. This equation is determined using Newton’s second law,

F = ma, and the law of gravitation. The result is

d 2x

gR 2

=

−

, for 0 < t,
dt 2
(R + x)2
(1.1)
where g is the gravitational acceleration constant and R is the radius of the Earth.
Finding the solution x of this equation requires two integrations. Each will produce
an integration constant, and we need more information to find these constants. This
is done by specifying the initial conditions. Assuming the projectile starts at the
surface with velocity v0 , then the initial conditions are as follows:
x(0) = 0,
(1.2)
dx
(0) = v0 .
dt
(1.3)
The resulting initial value problem for x consists in finding the solution of (1.1)
that satisfies (1.2) and (1.3). Mathematically, the problem is challenging because
it involves solving a second-order nonlinear differential equation. One option for
finding the solution is simply to use a computer. However, the limitation with this
is that it does not provide much insight into how the solution depends on the terms
in the equation. One of the primary objectives of this text is to use mathematics to
derive a fundamental understanding of how and why things work the way they do,
© Springer Nature Switzerland AG 2019
M. H. Holmes, Introduction to the Foundations of Applied Mathematics,
Texts in Applied Mathematics 56, https://doi.org/10.1007/978-3-030-24261-9_1
1
2
1 Dimensional Analysis
Fig. 1.1 The solution (1.5)
of the projectile problem in a
uniform gravitational field
and so, we are very interested in obtaining at least an approximate solution of this
problem. This is the same point-of-view taken in most physics books and it is worth
looking at how they might address this issue.
Adopting, for the moment, the typical Physics I approach, in looking at the
equation in (1.1) it is not unreasonable to assume R is significantly larger than even
the largest value of x. If true, then we should be able to replace the x + R term with
just R. In this case, the problem reduces to solving
d 2x
= −g, for 0 < t.
dt 2
(1.4)
Integrating and then using the two initial conditions yields
1
x(t) = − gt 2 + v0 t.
2
(1.5)
This solution is shown schematically in Fig. 1.1. We have what we wanted, a
relatively simple expression that serves as an approximation to the original nonlinear
problem. To complete the derivation we should check that the assumption made in
the derivation is satisfied, namely x is much smaller than R. Now, the maximum
height for (1.5) occurs when
dx
= 0.
dt
(1.6)
Solving this equation yields t = v0 /g and from this it follows that the maximum
height is
xM =
v02
.
2g
(1.7)
Therefore, we must require that v 2 /(2g) is much less than R, which we write as
v02 /(2g) R.
It is now time to critique the above derivation. The first criticism is that the
approach is heuristic. The reason is that even though the argument for replacing
x + R with R seems plausible, we simply ignored a particular term in the equation.
1.2 Examples of Dimensional Reduction
3
The projectile problem is not particularly complicated, so dropping a term as we
did is straightforward. However, in the real world where problems can be quite
complicated, dropping a term in one part of the problem can lead to inconsistencies
in another part. A second criticism can be made by asking a question. Specifically,
what exactly is the effect of the nonlinearity on the projectile? Our reduction
replaced the nonlinear gravitational force, which is the right-hand side of (1.1),
with a uniform gravitational field given by −g. Presumably if gravity decreases
with height, then the projectile will be going higher than we would expect based on
our approximation in (1.5). It is of interest to understand quantitatively what this
nonlinear effect is and whether it might interfere with our reduction.
Based on the comments of the previous paragraph we need to make the reduction
process more systematic. The procedure that is used to simplify the problem should
enable us to know exactly what is large or small in the problem, and it should also
enable us to construct increasingly more accurate approximations to the problem.
Explaining what is involved in a systematic reduction occurs in two steps. The first,
which is the objective of this chapter, involves the study of dimensions and how
these can be used to simplify the mathematical formulation of the problem. After
this, in Chap. 2, we develop techniques to construct accurate approximations of the
resulting equations.
1.2 Examples of Dimensional Reduction
The first idea that we explore will, on the surface, seem to be rather simple, but it is
actually quite profound. It has to do with the dimensions of the physical variables, or
parameters, in a problem. To illustrate, suppose we know that the speed s of a ball is
determined by its radius r and the length of time t it has been moving. Implicit in this
statement is the assumption that the speed does not depend on any other physical
variable. In mathematical terms we have that s = f (r, t). The function f is not
specified and all we know is that there is some expression that connects the speed
with r and t. The only possible way to combine these two quantities to produce the
dimension of speed is through their ratio r/t. For example, it is impossible to have
s = αr + βt without α and β having dimensions. This would mean α and β are
physical parameters, and we have assumed there are no others in the problem. This
observation enables us to conclude that based on the original assumptions that the
only function we can have is s = αr/t, where α is a number.
What we are seeing in this example is that the dimensions of the variables in the
problem end up dictating the form of the function. This is a very useful information
and we will spend some time exploring how to exploit this idea. To set the stage, we
need to introduce the needed terminology and notation.
First, there is the concept of a fundamental dimension. As is well known, physical
variables such as force, density, and velocity can be broken down into length L, time
T , and mass M (see Table 1.1). Moreover, length, time, and mass are independent in
the sense that one of them cannot be written in terms of the other two. For these two
4
1 Dimensional Analysis
Table 1.1 Fundamental dimensions for commonly occurring quantities. A quantity with a one in
the dimensions column is dimensionless
Quantity
Acceleration
Angle
Angular acceleration
Angular momentum
Angular velocity
Area
Energy, work
Force
Frequency
Concentration
Length
Mass
Mass density
Momentum
Power
Pressure, stress, elastic modulus
Surface tension
Time
Torque
Velocity
Viscosity (dynamic)
Viscosity (kinematic)
Volume
Wave length
Strain
Dimensions
LT −2
1
T −2
ML2 T −1
T −1
L2
ML2 T −2
MLT −2
T −1
L−3
L
M
ML−3
MLT −1
ML2 T −3
ML−1 T −2
MT −2
T
ML2 T −2
LT −1
ML−1 T −1
L2 T −1
L3
L
1
Quantity
Enthalpy
Entropy
Gas constant
Internal energy
Specific heat
Temperature
Thermal conductivity
Thermal diffusivity
Heat transfer coefficient
Dimensions
ML2 T −2
ML2 T −2 θ −1
ML2 T −2 θ −1
ML2 T −2
L2 T −2 θ −1
θ
MLT −3 θ −1
L2 T −1
MT −3 θ −1
Capacitance
Charge
Charge density
Electrical conductivity
Admittance
Electric potential, voltage
Current density
Electric current
Electric field intensity
Inductance
Magnetic intensity
Magnetic flux density
Magnetic permeability
Electric permittivity
Electric resistance
M −1 L−2 T 4 I 2
TI
L−3 T I
M −1 L−3 T 3 I 2
L−2 M −1 T 3 I 2
ML2 T −3 I −1
L−2 I
I
MLT −3 I −1
ML2 T −2 I −2
L−1 I
MT −2 I −1
MLT −2 I −2
M −1 L−3 T 4 I 2
ML2 T −3 I −2
reasons we will consider L, T , and M as fundamental dimensions. For problems
involving thermodynamics we will expand this list to include temperature (θ ) and
for electrical problems we add current (I ). This gives rise to the following.
Dimensions Notation. Given a physical quantity q, the fundamental dimensions of
q will be denoted as q. In the case of when q is dimensionless, q = 1.
So, for example, from the projectile problem, v0 = L/T , x = L, g = L/T 2 ,
and xM /R = 1.
It is important to understand that nothing is being assumed about which specific
system of units is used to determine the values of the variables or parameters.
Dimensional analysis requires that the equations be independent of the system of
units. For example, both Newton’s law F = ma and the differential equation (1.1)
do not depend on the specific system one selects. For this reason these equations are
said to be dimensionally homogeneous. If one were to specialize (1.1) to SI units
1.2 Examples of Dimensional Reduction
5
and set R = 6378 km and g = 9.8 m/s2 they would end up with an equation that is
not dimensionally homogeneous.
1.2.1 Maximum Height of a Projectile
The process of dimensional reduction will be explained by applying it to the
projectile problem. To set the stage, suppose we are interested in the maximum
height xM of the projectile as shown in Fig. 1.1. For a uniform gravitational field
the force is F = −mg. With this, and given the initial conditions in (1.2) and
(1.3), it is assumed that the only physical parameters that xM depends on are g,
v0 , and the mass m of the projectile. Mathematically this assumption is written
as xM = f (g, m, v0 ). The function f is unknown but we are going to see if the
dimensions can be used to simplify the expression.
The only way to combine g, m, and v0 to produce the correct dimensions is
through a product or ratio. So, our start-off hypothesis is that there are numbers a,
b, and c, so that
xM = ma v0b g c .
(1.8)
Using the fundamental dimensions for these variables the above equation is
equivalent to
L = M a (L/T )b (L/T 2 )c
= M a Lb+c T −b−2c .
(1.9)
Equating the exponents of the respective terms in this equation we conclude
L:
b + c = 1,
T : −b − 2c = 0,
M:
a = 0.
Solving these equations we obtain a = 0, b = 2, and c = −1. This means the only
way to produce the dimensions of length using m, v0 , and g is through the ratio
v02 /g. Given our start-off assumption (1.8), we conclude that xM is proportional to
v02 /g. In other words, the original assumption that xM = f (g, m, v0 ) dimensionally
reduces to the expression
xM = α
v02
,
g
(1.10)
where α is an arbitrary number. With (1.10) we have come close to obtaining our
earlier result (1.7) and have done so without solving a differential equation or using
6
1 Dimensional Analysis
calculus to find the maximum value. Based on this rather minimal effort we can
make the following observations:
• If the initial velocity is increased by a factor of 2, then the maximum height
will increase by a factor of 4. This observation offers an easy method for
experimentally checking on whether the original modeling assumptions are
correct.
• The constant α can be determined by running one experiment. Namely, for a
given initial velocity v0 = v 0 we measure the maximum height xM = x M . With
these known values, α = g x̄M /v 20 . Once this is done, the formula in (1.10) can
be used to determine xM for any v0 .
• The maximum height does not depend on the mass of the object. This is not a
surprise since the differential equation (1.4) and the initial conditions (1.2) and
(1.3) do not depend on the mass.
The steps we have used are the basis for the method of dimensional reduction,
where an expression is simplified based on the fundamental dimensions of the
quantities involved. Given how easy it was to obtain (1.10) the method is very
attractive as an analysis tool. It does have limitations and one is that we do not
know the value of the number α. It also requires us to be able to identify at the
beginning what parameters are needed. The importance of this and how this relies
on understanding the physical laws underlying the problem will be discussed later.
The purpose of the above example is to introduce the idea of dimensional
reduction. What it does not show is how to handle problems with several parameters
and this is the purpose of the next two examples.
1.2.2 Drag on a Sphere
In the design of automobiles, racing bicycles, and aircraft there is an overall
objective to keep the drag on the object as small as possible. It is interesting to
see what insight dimensional analysis might provide in such a situation, but since
we are beginners it will be assumed the object is very simple and is a sphere (see
Fig. 1.2). The modeling assumption that is made is that the drag force DF on the
sphere depends on the radius R of the sphere, the (positive) velocity v of the sphere,
the density ρ of the air, and the dynamic viscosity μ of the air. The latter is a measure
of the resistance force of the air to motion and we will investigate this in Chap. 8.
For the moment all we need is its fundamental dimensions and these are given in
Table 1.1. In mathematical terms the modeling assumption is
DF = f (R, v, ρ, μ),
(1.11)
and we want to use dimensional reduction to find a simplified version of this
expression. As will become evident in the derivation, this requires two steps.
1.2 Examples of Dimensional Reduction
7
Fig. 1.2 Air flow around an object can be visualized using smoke. The flow around a golf ball is
shown in (a) (Brown (1971)) and around a tennis ball in (b) (Bluck (2000)). In both cases the air
is moving from left to right
Find the General Product Solution
Similar to the last example, the first question is whether we can find numbers a, b,
c, and d, so that
DF = R a v b ρ c μd .
(1.12)
Expressing these using fundamental dimensions yields
MLT −2 = La (L/T )b (M/L3 )c (M/LT )d
= La+b−3c−d T −b−d M c+d .
As before, we equate the respective exponents and conclude
L : a + b − 3c − d = 1,
T :
−b − d = −2,
M:
c + d = 1.
(1.13)
8
1 Dimensional Analysis
We have four unknowns and three equations, so it is anticipated that in solving the
above system of equations one of the constants will be undetermined. From the T
equation we have b = 2 − d, and from the M equation c = 1 − d. The L equation
then gives us a = 2 − d. With these solutions, and based on our assumption in
(1.12), we have that
DF = αR 2−d v 2−d ρ 1−d μd
μ d
2 2
= αρR v
,
Rvρ
where d and α are arbitrary numbers. This can be written as
DF = αρR 2 v 2 Π d ,
(1.14)
where
Π=
μ
.
Rvρ
(1.15)
This is the general product solution for how DF depends on the given variables.
The quantity Π is dimensionless, and it is an example of what is known as a
dimensionless product. Calling it a product is a bit misleading as Π involves both
multiplications and divisions. Some avoid this by calling it a dimensionless group.
We will use both expressions in this book.
Determine the General Solution
The formula for DF in (1.14) is not the final answer. The conclusion that is derived
from (1.14) is that the general solution is not an arbitrary power of Π , but it is an
arbitrary function of Π . Mathematically, the conclusion is that the general solution
can be written as
DF = ρR 2 v 2 F (Π ),
(1.16)
where F is an arbitrary function of the dimensionless product Π . Note that because
F is arbitrary, it is not necessary to include the multiplicative number α that appears
in (1.14).
To explain how (1.16) is derived from the general product solution (1.14),
suppose you are given two sets of values for (α, d), say (α1 , d1 ) and (α2 , d2 ). In
this case, their sum
DF = α1 ρR 2 v 2 Π d1 + α2 ρR 2 v 2 Π d2
= ρR 2 v 2 α1 Π d1 + α2 Π d2
1.2 Examples of Dimensional Reduction
9
is also a solution. This observation is not limited to just two sets of values, and, in
fact, it holds for an arbitrary number. In other words,
DF = ρR 2 v 2 α1 Π d1 + α2 Π d2 + α3 Π d3 + · · ·
(1.17)
is a solution, where d1 , d2 , d3 , . . . are arbitrary numbers as are the coefficients α1 ,
α2 , α3 , . . .. To express this in a more compact form, note that the expression within
the parentheses in (1.17) is simply a function of Π . This is the reason for the F (Π )
that appears in (1.16).
With the general solution in (1.16), we have used dimensional analysis to reduce
the original assumption in (1.11), which involves an unknown of four variables,
down to an unknown function of one variable. Although this is a significant
improvement, the result is perhaps not as satisfying as the one obtained for the
projectile example, given in (1.10), because we have not been able to determine F .
However, there are various ways to address this issue, and some of them will be
considered below.
Representation of Solution
Now that the derivation is complete a few comments are in order. First, it is possible
for two people to go through the above steps and come to what looks to be very
different conclusions. For example, the general solution can also be written as
DF =
μ2
H (Π ),
ρ
(1.18)
where H is an arbitrary function of Π . The proof that this is equivalent to (1.16)
comes from the requirement that the two expressions must produce the same result.
In other words, it is required that
μ2
H (Π ) = ρR 2 v 2 F (Π ).
ρ
Solving this for H yields
H (Π ) =
1
F (Π ).
Π2
The fact that the right-hand side of the above equation only depends on Π shows
that (1.18) is equivalent to (1.16). As an example, if F (Π ) = Π in (1.16), then
H (Π ) = 1/Π in (1.18).
Another representation for the general solution is
DF = ρR 2 v 2 G(Re),
(1.19)
10
1 Dimensional Analysis
103
G(Re)
102
101
100
10–1
10–2
100
102
104
106
108
Re
Fig. 1.3 The measured values of the function G(Re) that arises in the formula for the drag on a
sphere, as given in (1.19)
where
Re =
Rvρ
,
μ
(1.20)
and G is an arbitrary function of Re. In fluid dynamics, the dimensionless product
Re is known as the Reynolds number. To transform between the representation in
(1.19), and the one in (1.16), note Re = 1/Π . From the requirement
ρR 2 v 2 G(Re) = ρR 2 v 2 F (Π ),
we obtain
G(Re) = F (1/Re).
Because of its importance in fluids, G has been measured for a wide range of
Reynolds numbers, producing the curve shown in Fig. 1.3. For those who have taken
a course in fluid dynamics, the data in Fig. 1.3 are usually reported for what is called
the drag coefficient CD of a sphere. The two functions G and CD are related through
the equation G = π2 CD .
The reason for the different representations is that there are four unknowns in
(1.12) yet only three equations. This means one of the unknowns is used in the
general solution and, as expressed in (1.14), we used d. If you were to use one of
the others, then a different looking, but mathematically equivalent, expression would
be obtained. The fact that there are multiple ways to express the solution can be used
to advantage. For example, if one is interested in the value of DF for small values of
the velocity, then (1.19) would be a bit easier to use. The reason is that to investigate
the case of small v it is somewhat easier to determine what happens to G for Re near
zero than to expand F for large values of Π . For the same reason, (1.16) is easier
to work with for studying large velocities. One last comment to make is that even
1.2 Examples of Dimensional Reduction
11
though there are choices on the form of the general solution, they all have exactly
the same number of dimensionless products.
Determining F
A more challenging question concerns how to determine the function F in (1.16).
The mathematical approach would be to solve the equations for fluid flow around
a sphere and from this find F . This is an intriguing idea but a difficult one
since the equations are nonlinear partial differential equations (see Sect. 8.11).
There is, however, another more applied approach that makes direct use of (1.16).
Specifically, a sequence of experiments is run to measure F (r) for 0 < r < ∞.
To do this, a sphere with a given radius R0 , and a fluid with known density ρ0 and
viscosity μ0 , are selected. In this case (1.16) can be written as
F (r) =
γ DF
,
v2
(1.21)
where γ = 1/(ρ0 R02 ) is known and fixed. The experiment consists of taking various
values of v and then measuring the resulting drag force DF on the sphere. To
illustrate, suppose our choice for the sphere and fluid give R = 1, ρ0 = 2, and μ0 =
3. Also, suppose that running the experiment using v = 4 produces a measured
drag of DF = 5. In this case r = μ0 /(R0 vρ0 ) = 3/8 and γ DF /v 2 = 5/32. Our
conclusion is therefore that F (3/8) = 5/32. In this way, picking a wide range of v
values we will be able to determine the values for the function F (r). This approach
is used extensively in the real world and the example we are considering has been a
particular favorite for study. The data determined from such experiments are shown
in Fig. 1.3.
A number of conclusions can be drawn from Fig. 1.3. For example, there is a
range of Re values where G is approximately constant. Specifically, if 103 < Re <
105 , then G ≈ 0.7. This is the reason why in the fluid dynamics literature you
will occasionally see the statement that the drag coefficient CD = π2 G for a sphere
has a constant value of approximately 0.44. For other Re values, however, G is not
constant. Of particular interest is the dependence of G for small values of Re. This
corresponds to velocities v that are very small, what is known as Stokes flow. The
data in Fig. 1.3 show that G decreases linearly with Re in this region. Given that
this is a log-log plot, then this means that log(G) = a − b log(Re), or equivalently,
G = α/Reb where α = 10a . Curve fitting this function to the data in Fig. 1.3 it is
found that α ≈ 17.6 and β ≈ 1.07. These are close to the exact values of α = 6π
and β = 1, which are obtained by solving the equations of motion for Stokes flow.
Inserting these values into (1.19), the conclusion is that the drag on the sphere for
small values of the Reynolds number is
DF ≈ 6π μRv.
(1.22)
12
1 Dimensional Analysis
This is known as Stokes formula for the drag on a sphere, and we will have use for
it in Chap. 4 when studying diffusion.
Scale Models
Why all the work to find F ? Well, knowing this function allows for the use of scale
model testing. To explain, suppose it is required to determine the drag on a sphere
with radius Rf for a given velocity vf when the fluid has density ρf and viscosity
μf . Based on (1.16) we have DF = ρf Rf2 vf2 F (Πf ), where
Πf =
μf
.
Rf vf ρf
(1.23)
Consequently, we can determine DF if we know the value of F at Πf . Also, suppose
that this cannot be measured directly as Rf is large and our experimental equipment
can only handle small spheres. We can still measure F (Πf ) using a small value of
R if we change one or more of the parameters in such a way that the value of Πf
does not change. If Rm , μm , ρm , and vm are the values used in the experiment, then
we want to select them so that
μf
μm
=
,
Rm vm ρm
Rf vf ρf
(1.24)
or equivalently
vm =
μm Rf ρf
vf .
μf Rm ρm
(1.25)
This equation relates the values for the full-scale ball (subscript f ) to those for
the model used in the experiment (subscript m). As an example, suppose we are
interested in the drag on a very large sphere, say Rf = 100 m, but our equipment
can only handle smaller values, say Rm = 2 m. If the fluid for the two cases is
the same, so ρm = ρf and μm = μf , then according to (1.25), in our experiment
we should take vm = 50vf . If the experimental apparatus is unable to generate
velocities 50 times the value of vf , then it would be necessary to use a different
fluid to reduce this multiplicative factor.
The result in the above example is the basis of scale model testing used in wind
tunnels and towing tanks (see Fig. 1.4). Usually these tests involve more than just
keeping one dimensionless product constant as we did in (1.24). Moreover, it is
evident in Fig. 1.4 that the models look like the originals, they are just smaller.
This is the basis of geometric similarity, where the lengths of the model are all a
fraction of the original. Other scalings are sometimes used and the most common
are kinematic similarity, where velocities are scaled, and dynamic similarity, where
forces are scaled.
1.2 Examples of Dimensional Reduction
13
Fig. 1.4 Dimensional analysis is used in the development of scale model testing. On the left is a
test in NASA’s Glenn Research Center supersonic wind tunnel (NASA, 2018), and on the right a
towing tank test of a model for a ship hull (el Moctar, 2018)
Endnotes
One question that has not been considered so far is, how do you know to assume
that the drag force depends on the radius, velocity, density, and dynamic viscosity?
The assumption comes from knowing the laws of fluid dynamics, and identifying
the principal terms that contribute to the drag. For the most part, in this chapter
the assumptions will be stated explicitly, as they were in this example. Later in the
text, after the basic physical laws are developed, it will be possible to construct the
assumptions directly. However, one important observation can be made, and that is
the parameters used in the assumption should be independent. For example, even
though the drag on a sphere likely depends on the surface area and volume of the
sphere it is not necessary to include them in the list. The reason is that it is already
assumed that DF depends on the radius R and both the surface area and volume are
determined using R.
The problem of determining the drag on a sphere is one of the oldest in fluid
dynamics. Given that the subject is well over 150 years old, you would think that
whatever useful information can be derived from this particular problem was figured
out long ago. Well, apparently not, as research papers still appear regularly on this
topic. A number of them come from the sports industry, where there is interest in the
drag on soccer balls (Asai et al. 2007), golf balls (Smits and Ogg 2004), tennis balls
(Goodwill et al. 2004), as well as nonspherical-shaped balls (Mehta 1985). Others
have worked on how to improve the data in Fig. 1.3, and an example is the use of
a magnetic suspension system to hold the sphere (Sawada and Kunimasu 2004).
A more novel idea is to drop different types of spheres down a deep mine shaft,
and then use the splash time as a means to determine the drag coefficient (Maroto
et al. 2005). The point here is that even the most studied problems in science and
engineering still have interesting questions that remain unanswered.
14
1 Dimensional Analysis
Fig. 1.5 Schematic of
toppling dominoes, creating a
wave that propagates with
velocity v
1.2.3 Toppling Dominoes
Domino toppling refers to the art of setting up dominoes, and then knocking them
down. The current world record for this is about 4,500,000 dominoes for a team,
and 320,000 for an individual. One of the more interesting aspects of this activity
is that as the dominoes fall it appears as if a wave is propagating along the line of
dominoes. The objective of this example is to examine what dimensional analysis
might be able to tell us about the velocity of this wave. A schematic of the situation
is shown in Fig. 1.5. The assumption is that the velocity v depends on the spacing
d, height h, thickness w, and the gravitational acceleration constant g. Therefore,
the modeling assumption is v = f (d, h, w, g) and we want to use dimensional
reduction to find a simplified version of this expression. As usual, the first step is to
find numbers a, b, c, and e so that
v = d a hb w c g e .
Expressing these using fundamental dimensions yields
LT −1 = La Lb Lc (L/T 2 )e
= La+b+c+e T −2e .
Equating the respective terms we obtain
L : a + b + c + e = 1,
T :
−2e = −1.
Solving these two equations gives us that e =
have that
1
2
and b =
v = αd a h1/2−a−c w c g 1/2
a
w c
d
= α hg
h
h
a c
= α hgΠ1 Π2 ,
1
2
− a − c. With this we
(1.26)
1.2 Examples of Dimensional Reduction
15
where α is an arbitrary number, and the two dimensionless products are
d
,
h
w
Π2 = .
h
Π1 =
The expression in (1.26) is the general product solution. Therefore, the general
solution for how the velocity depends on the given parameters is
v=
(1.27)
hg F (Π1 , Π2 ),
where F is an arbitrary function of the two dimensionless products. An explanation
of how (1.27) follows from (1.26) is very similar to the method used to derive (1.16)
from (1.15). This is discussed in Exercise 1.26.
Dimensional analysis has been able to reduce the original assumption involving a
function of four-dimensional parameters down to one involving two dimensionless
products. This example is also informative as it demonstrates how to obtain the
general solution when more than one dimensionless product is involved. The
question remains, however, if this really applies to toppling dominoes. It does, but
in using this formula it is usually assumed the dominoes are very thin, or more
specifically that w h. This means that it is possible to assume Π2 = 0, and (1.27)
simplifies to
v=
hg G(Π1 ),
(1.28)
where G is an arbitrary function. Some effort has been made to measure G, and the
measurements for a particular type of domino are given in Fig. 1.6. It is seen that for
smaller values of Π1 , G ≈ 1.5. Therefore, as√an approximation we conclude that the
speed at which dominoes topple is v ≈ 1.5 hg. A typical domino has h = 5 cm,
which results in a velocity of v ≈ 1 m/s. To obtain a more explicit formula for
2
G(II1)
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
II1
0.5
0.6
0.7
0.8
Fig. 1.6 Data for toppling dominoes (Stronge and Shu 1988). In these experiments, w = 0.12h,
so the thin domino approximation is appropriate
16
1 Dimensional Analysis
G, however, requires the solution of a challenging mathematical problem, and an
expanded discussion of this can be found in Efthimiou and Johnson (2007).
1.2.4 Endnotes
Based on the previous examples, the benefits of using dimensional reduction
are apparent. However, a word of caution is needed here as the method gives
the impression that it is possible to derive useful information without getting
involved with the laws of physics or potentially difficult mathematical problems.
One consequence of this is that the method is used to comment on situations and
phenomena that are simply inappropriate (e.g., to study psychoacoustic behavior).
The method relies heavily on knowing the fundamental laws for the problem under
study, and without this whatever conclusions made using dimensional reduction
are limited. For example, we earlier considered the drag on a sphere and in the
formulation of the problem we assumed that the drag depends on the dynamic
viscosity. Without knowing the equations of motion for fluids it would not have
been possible to know that this term needed to be included or what units it might
have. By not including it we would have concluded that d = 0 in (1.14) and instead
of (1.16) we would have DF = αρR 2 v 2 where α is a constant. In Fig. 1.3 it does
appear that DF is approximately independent of Re when 103 < Re < 105 .
However, outside of this interval, DF is strongly dependent on Re, and this means
ignoring the viscosity would be a mistake. Another example illustrating the need
to know the underlying physical laws arises in the projectile problem when we
included the gravitational constant. Again, this term is essential and without some
understanding of Newtonian mechanics it would be missed completely. The point
here is that dimensional reduction can be a very effective method for simplifying
complex relationships, but it is based on knowing what the underlying laws are that
govern the systems being studied.
1.3 Theoretical Foundation
The theoretical foundation for dimensional reduction is contained in the Buckingham Pi Theorem. To derive this result, assume we have a physical quantity q that
depends on physical parameters or variables p1 , p2 , . . . , pn . In this context, the
word physical means that the quantity is measurable. Each can be expressed in
fundamental dimensions and we will assume that the L, T , M system is sufficient
for this task. In this case we can write
q = L 0 T t0 M m0 ,
(1.29)
1.3 Theoretical Foundation
17
and
pi = L i T ti M mi .
(1.30)
Our modeling assumption is that q = f (p1 , p2 , . . . , pn ), and that this is
dimensionally homogeneous. This means, as explained earlier, that this formula is
independent of the system of units used to measure q or the pi ’s.
To dimensionally reduce the equation, we will determine if there are numbers
a1 , a2 , . . . , an so that
q = p1a1 p2a2 · · · pnan .
(1.31)
Introducing (1.29) and (1.30) into the above expression, and then equating exponents, we obtain the equations
L:
1 a1 + 2 a2 + · · · + n an = 0 ,
T :
t1 a1 + t2 a2 + · · · + tn an = t0 ,
M : m1 a1 + m2 a2 + · · · + mn an = m0 .
This can be expressed in matrix form as
Aa = b,
(1.32)
where
⎛
1
⎜
A = ⎝ t1
2
···
⎞
n
⎟
t2 · · · tn ⎠ ,
m1 m2 · · · mn
⎛ ⎞
a1
⎜a2 ⎟
⎜ ⎟
a = ⎜ . ⎟,
⎝ .. ⎠
an
⎛
(1.33)
⎞
0
b = ⎝ t0 ⎠ .
m0
(1.34)
The matrix A is known as the dimension matrix. As expressed in (1.33) it is 3 × n
but if we were to have used L, T , M, θ as the fundamental system, then it would
be 4 × n. In other words, the number of rows in the dimension matrix equals the
number of fundamental units needed, and the number of columns equals the number
of parameters that q is assumed to depend on.
With (1.32) we have transformed the dimensional reduction question into a
linear algebra problem. To determine the consequences of this we first consider the
situation that (1.32) has no solution. In this case the assumption that q depends on
18
1 Dimensional Analysis
p1 , p2 , . . ., pn is incomplete and additional parameters are needed. This situation
motivates the following definition.
Dimensionally Complete. The set p1 , p2 , . . ., pn is dimensionally complete for q
if it is possible to combine the pi ’s to produce a quantity with the same dimension
as q. If it is not possible, the set is dimensionally incomplete for q.
From this point on we will assume the pi ’s are complete so there is at least one
solution of (1.32).
To write down the general solution we consider the associated homogeneous
equation, namely Aa = 0. The set of solutions of this equation form a subspace
N(A), known as the null space of A. Letting k be the dimension of this subspace,
then the general solution of Aa = 0 can be written as a = γ1 a1 + γ2 a2 + · · · +
γk ak , where a1 , a2 , . . ., ak is a basis for N(A) and γ1 , γ2 , . . ., γk are arbitrary. It is
understood here that if k = 0, then a = 0. With this, the general solution of (1.32)
can be written as
a = ap + γ1 a1 + γ2 a2 + · · · + γk ak ,
(1.35)
where ap is any vector that satisfies (1.32) and γ1 , γ2 , . . ., γk are arbitrary numbers.
Example (Drag on a Sphere) To connect the above discussion with what we did
earlier, consider the drag on a sphere example. Writing (1.13) in matrix form we
obtain
⎛ ⎞
⎛ ⎞
⎞ a
1
1 1 −3 −1 ⎜ ⎟
b
⎝ 0 −1 0 −1 ⎠⎜ ⎟ = ⎝−2 ⎠ .
⎝c ⎠
1
0 0 1 1
d
⎛
This is the matrix equation (1.32) for this particular example. Putting this in
augmented form, and row reducing, yields the following
⎛
⎛
⎞
⎞
1 1 −3 −1 1
1 0 0 1 2
⎝ 0 −1 0 −1 −2 ⎠ → ⎝ 0 1 0 1 2 ⎠ .
0 0 1 1 1
0 0 1 1 1
From this we conclude that a = 2 − d, b = 2 − d, and c = 1 − d. To be consistent
with the notation in (1.35), set d = γ , so the solution is
⎛ ⎞ ⎛ ⎞
⎛ ⎞
a
2
−1
⎜b ⎟ ⎜2⎟
⎜−1⎟
⎜ ⎟ = ⎜ ⎟ + γ⎜ ⎟,
⎝ c ⎠ ⎝1⎠
⎝−1⎠
d
0
1
1.3 Theoretical Foundation
19
where γ is arbitrary. Comparing this with (1.35) we have that k = 1,
⎛ ⎞
2
⎜2⎟
⎟
ap = ⎜
⎝1⎠ ,
⎛ ⎞
−1
⎜−1⎟
⎟
a1 = ⎜
⎝−1⎠ .
and
0
1
It is now time to take our linear algebra conclusions and apply them to the
dimensional reduction problem. Just as the appearance of d in (1.14) translated into
the appearance of a dimensionless product in the general solution given in (1.16),
each of the γi ’s in (1.35) gives rise to a dimensionless product in the general solution
for the problem we are currently studying. To be specific, writing the ith basis vector
ai in component form as
⎛ ⎞
α
⎜β ⎟
⎜ ⎟
ai = ⎜ . ⎟ ,
⎝ .. ⎠
(1.36)
γ
then the corresponding dimensionless product is
β
γ
Πi = p1α p2 · · · pn .
(1.37)
Moreover, because the ai ’s are independent vectors, the dimensionless products Π1 ,
Π2 , . . ., Πk are independent.
As for the particular solution ap in (1.35), assuming it has components
⎛ ⎞
a
⎜b ⎟
⎜ ⎟
ap = ⎜ . ⎟ ,
⎝ .. ⎠
c
(1.38)
Q = p1a p2b · · · pnc
(1.39)
then the quantity
has the same dimensions as q.
Based on the conclusions of the previous two paragraphs, the general product
solution is q = αQΠ1κ1 Π2κ2 · · · Πkκk , where α, κ1 , κ2 , . . . , κk are arbitrary constants.
The form of the resulting general solution is given in the following theorem.
20
1 Dimensional Analysis
Buckingham Pi Theorem. Assuming the formula q = f (p1 , p2 , . . . , pn ) is
dimensionally homogeneous and dimensionally complete, then it is possible to
reduce it to one of the form q = QF (Π1 , Π2 , . . . , Πk ), where Π1 , Π2 , . . ., Πk
are independent dimensionless products of p1 , p2 , . . ., pn . The quantity Q is a
dimensional product of p1 , p2 , . . ., pn with the same dimensions as q.
According to this theorem, the original formula for q can be reduced from a function
of n variables down to one with k. The value of k, which equals the nullity of
the dimension matrix, ranges from 0 to n − 1 depending on the given quantities
p1 , p2 , . . . , pn . If it happens that k = 0, then the function F reduces to a constant
and the conclusion is that q = αQ, where α is an arbitrary number.
The importance of this theorem is that it establishes that the process used to
reduce the drag on a sphere and toppling dominoes examples can be applied to more
complex problems. It also provides insight into how the number of dimensionless
products is determined. There are still, however, fundamental questions left
unanswered. For example, those with a more mathematical bent might still be
wondering if this result can really be true no matter how discontinuous the original
function f might be. Others might be wondering if the fundamental units used here,
particularly length and time, are really independent. This depth of inquiry, although
quite interesting, is beyond the scope of this text. Those wishing to pursue further
study of these and related topics should consult Penrose (2007) and Bluman and
Anco (2002).
1.3.1 Pattern Formation
The mechanism responsible for the colorful patterns on seashells, butterfly wings,
zebras, and the like has intrigued scientists for decades. An experiment that has
been developed to study pattern formation involves pouring chemicals into one end
of a long tube, and then watching what happens as they interact while moving
along the tube. This apparatus is called a plug-flow reactor and the outcome of a
mathematical model coming from such an experiment is shown in Fig. 1.7. It was
found in these experiments that patterns appear only for certain pouring velocities v.
According to what is known as the Lengyel-Epstein model, this velocity depends on
the concentration U of the chemical used in the experiment, the rate k2 at which the
chemicals interact, the diffusion coefficient D of the chemicals, and a parameter k3
that has the dimensions of concentration squared. The model is therefore assuming
v = f (U, k2 , D, k3 ).
(1.40)
From Table 1.1 we have that v = L/T , U = 1/L3 , D = L2 /T , and k3 =
1/L6 . Also, from the Lengyel-Epstein model one finds that k2 = L3 /T . Using
dimensional reduction we require
v = U a k2b D c k3d .
(1.41)
1.3 Theoretical Foundation
21
Fig. 1.7 Spatial pattern created from a model for a plug-flow reactor. The tube occupies the
interval 0 ≤ x ≤ 100, and starting at t = 0 the chemicals are poured into the left end. As they flow
along the tube a striped pattern develops
Expressing these using fundamental dimensions yields
LT −1 = (L−3 )a (L3 T −1 )b (L2 T −1 )c (L−6 )d
= L−3a+3b+2c−6d T −b−c .
As before we equate the respective terms and conclude
L : −3a + 3b + 2c − 6d = 1
T :
−b − c = −1.
These equations will enable us to express two of the unknowns in terms of the other
two. There is no unique way to do this, and one choice yields b = −1 + 3a + 6d
and c = 2 − 3a − 6d. From this it follows that the general product solution is
v = αU a k23a+6d−1 D 2−3a−6d k3d
= αk2−1 D 2 (U k23 D −3 )a (k26 D −6 k3 )d .
This can be rewritten as
v = αk2−1 D 2 Π1a Π2d ,
(1.42)
where
Π1 =
U k23
,
D3
(1.43)
Π2 =
k26 k3
.
D6
(1.44)
and
22
1 Dimensional Analysis
The dimensionless products Π1 and Π2 are independent, and this follows from the
method used to derive these expressions. Independence is also evident from the
observation that Π1 and Π2 do not involve exactly the same parameters. From this
result it follows that the general form of the reduced equation is
v = k2−1 D 2 F (Π1 , Π2 ).
(1.45)
It is of interest to compare (1.45) with the exact formula obtained from solving
the differential equations coming from the Lengyel-Epstein model. It is found that
v=
k2 DU G(β),
(1.46)
where β = k3 /U 2 and G is a rather complicated square root function (Bamforth
et al. 2000). This result appears to differ from (1.45). To investigate this, note that
β = Π2 /Π12 . Equating (1.45) and (1.46) it follows that
3/2
k U 1/2
F (Π1 , Π2 ) = 2 3/2 G(β)
D
= Π1 G(Π2 /Π12 ).
Because the right-hand side is a function of only Π1 and Π2 , then (1.45) does indeed
reduce to the exact result (1.46).
Dimensional reduction has therefore successfully reduced the original unknown
function of four variables in (1.40) down to one with only two variables. However,
the procedure is not able to reduce the function down to one dimensionless variable,
as given in (1.46). In this problem that level of reduction requires information only
available from the mathematical problem coming from the Lengyel-Epstein model.
An illustration of how this is done can be found in Exercises 1.29 and 1.30. It is
worth noting that the method for deriving the Lengyel-Epstein model is explained
in Sect. 4.6.3.
1.4 Similarity Variables
Dimensions can be used not only to reduce formulas but also to simplify complex
mathematical problems. The degree of simplification depends on the parameters,
and variables, in the problem. One of the more well-known examples is the problem
of finding the mass density u(x, t). In this case the density satisfies the diffusion
equation
D
∂u
∂ 2u
, for
=
2
∂t
∂x
0 < x < ∞,
0 < t,
(1.47)
1.4 Similarity Variables
23
where the boundary conditions are
ux=0 = u0 ,
ux→∞ = 0,
(1.48)
and the initial condition is
ut=0 = 0.
(1.49)
It is assumed that D is positive and u0 is nonzero.
The constant D is called the diffusion coefficient, and its dimensions can be
determined from the terms in the differential equation. To do this, it is useful to
know the following facts, all of which come directly from the definition of the
derivative.
• Given f (t), then
df
dt
=
f
,
t
and
d 2f
dt 2
=
f
.
t2
(1.50)
• Given u(x, t), then
∂u
∂t
=
u
,
t
∂u
∂x
=
u
x
and
∂ 2u
u
.
=
∂t∂x
tx
(1.51)
Some of the consequences, and extensions, of the above formulas are explored in
Exercise 1.27.
Now, the dimensions of the left and right sides of (1.47) must be the same, and
this means Duxx = ut . Because u = M/L3 , then uxx = u/L2 = M/L5
and ut = u/T = M/(T L3 ). From this we have DM/L5 = M/(T L3 ),
and therefore D = L2 /T . In a similar manner, in boundary condition (1.48),
u0 = u = M/L3 . As a final comment, the physical assumptions underlying the
derivation of (1.47) are the subject of Chap. 4. In fact, the solution we are about to
derive is needed in Sect. 4.6.2 to solve the diffusion equation.
1.4.1 Dimensional Reduction
The conventional method for solving the diffusion equation on a semi-infinite
spatial interval is to use an integral transform, and this will be considered in
Chap. 4. It is also possible to find u using dimensional reduction. The approach
is based on the observation that the only dimensional variables, and parameters,
appearing in the problem are u, u0 , D, x, and t. In other words, it must be true that
u = f (x, t, D, u0 ). With this we have the framework for dimensional reduction,
and the question is whether we can find numbers a, b, c, d so that
24
1 Dimensional Analysis
u = x a t b D c (u0 )d .
(1.52)
Using fundamental dimensions,
ML−3 = La T b (L2 /T )c (M/L3 )d
= La+2c−3d T b−c M d ,
and then equating the respective terms gives us
L : a + 2c − 3d = −3,
T :
b − c = 0,
M:
d = 1.
(1.53)
The solution of the above system can be written as d = 1 and b = c = −a/2. Given
the assumption in (1.52), we conclude that the general product solution is
u = αu0
x
√
Dt
a
.
The general solution therefore has the form
u = u0 F (η),
(1.54)
x
η= √ .
Dt
(1.55)
where
In this case, η is called a similarity variable as it is a dimensionless product that
involves the independent variables in the problem.
When working out the drag on a sphere example, we discussed how it is possible
to derive different representations of the solution. For the current example, when
solving (1.53), instead of writing b = c = −a/2, we could just as well state that
a = −2b and c = b. In this case (1.54) is replaced with u = u0 G(ξ ) where
ξ = Dt/x 2 . Although the two representations are equivalent, in the sense that one
can be transformed into the other, it does make a difference which one is used when
deriving a similarity solution. The reason is that (1.47) requires two derivatives with
respect to x, and the resulting formulas are simpler if the similarity variable is a
linear function of x. If you would like a hands on example of why this is true, try
working out the steps below using the representation u = u0 G(ξ ) instead of (1.54).
1.4 Similarity Variables
25
1.4.2 Similarity Solution
Up to this point we have been using a routine dimensional reduction argument. Our
result, given in (1.54), is interesting as it states that the solution has a very specific
dependence on the independent variables x and t. Namely, u can be written as a
function of a single intermediate variable η. To determine F we substitute (1.54)
back into the problem and find what equation F satisfies. With this in mind note,
using the chain rule,
∂u
∂η
= u0 F (η)
∂t
∂t
= u0 F (η) −
x
2D 1/2 t 3/2
η
= −u0 F (η) .
2t
In a similar manner one finds that
∂ 2u
1
.
= u0 F (η)
2
Dt
∂x
Substituting these into (1.47) yields
1
F = − ηF .
2
Also, since 0 < x < ∞, and t > 0, then 0 < η < ∞.
We must also transform the boundary and initial conditions.
ux=0 = u0 : Letting x = 0 in (1.54) yields u0 F (0) = u0 , and from this we
conclude that F (0) = 1.
ux→∞ = 0: Letting x → ∞ in (1.54) yields u0 F (∞) = 0, and from this we
conclude that F (∞)√= 0.
ut=0 = 0: Given that η = x/ Dt, this condition must be dealt with using a
limit. Specifically, the requirement is that
x
= 0.
lim u0 F √
t→0+
Dt
For 0 < x < ∞, the above limit gives us that F (∞) = 0. This is the
same condition we derived for u(∞, t) = 0.
To summarize the above reduction, we have shown that the original diffusion
problem can be replaced with solving
1
F = − ηF , for 0 < η < ∞,
2
(1.56)
26
1 Dimensional Analysis
where
F (0) = 1,
(1.57)
F (∞) = 0.
(1.58)
and
With this, we have transformed a problem involving a partial differential equation
(PDE) into one with an ordinary differential equation (ODE). As required, the
resulting problem for F is only in terms of η. All of the original dimensional
quantities, including the independent variables x and t, do not appear anywhere
in the problem. This applies not just to the differential equation, but also to the
boundary and initial conditions.
The reduced problem is simple enough that it is possible to solve for F . This
can be done by letting G = F , so (1.56) takes the form G = − 12 ηG. The general
solution of this is G = α exp(−η2 /4). Because F = G, we conclude that the
general solution is
F (η) = β + α
η
e−s
2 /4
ds.
(1.59)
0
From (1.57) we have that β = 1 and from (1.58) we get
∞
1+α
e−s
2 /4
ds = 0.
(1.60)
0
Given that
∞
0
e−s
2 /4
ds =
√
π , then
1
F (η) = 1 − √
π
2
=1− √
π
η
e−s
2 /4
ds
0
η/2
e−r dr.
2
(1.61)
0
Expressions like this arise so often that they have given rise to a special function
known as the complementary error function erfc(z). This is defined as
2
erfc(z) ≡ 1 − √
π
z
e−r dr.
2
(1.62)
0
Therefore, we have found that the solution of the diffusion problem is
x
.
u(x, t) = u0 erfc √
2 Dt
(1.63)
1.5 Nondimensionalization and Scaling
27
As the above example demonstrates, using similarity variables and dimensional
analysis provides a powerful tool for solving PDEs. It is, for example, one of the
very few methods known that can be used to solve nonlinear PDEs. Its limitation
is that the problem must have a specific form to work. To illustrate, if the spatial
interval in the above diffusion problem is changed to one that is finite, so 0 < x <
, then dimensional analysis will show that there are two independent similarity
variables. This represents no improvement as we already know it is a function of
two independent variables, so a reduction is not possible. In some cases it is possible
to take advantage of particular properties of the solution so a similarity reduction
is possible, and this is illustrated in Exercises 1.29 and 1.30. Those interested in
pursuing this a bit more should consult Bluman et al. (2010) and Hydon (2000).
1.5 Nondimensionalization and Scaling
Another use we will have for dimensional analysis is to transform a problem into
dimensionless form. The reason for this is that the approximation methods that are
used to reduce difficult problems are based on comparisons. For example, in the
projectile problem we simplified the differential equation by assuming that x was
small compared to R. In contrast there are problems where the variable of interest
is large, or it is slow or that it is fast compared to some other term in the problem.
Whatever the comparison, it is important to know how all of the terms in the problem
compare and for this we need the concept of scaling.
1.5.1 Projectile Problem
The reduction of the projectile equation (1.1) was based on the assumption that x is
not very large, and so x + R could be replaced with just R. We will routinely use
arguments like this to find an approximate solution and it is therefore essential we
take more care in making such reductions. The way this is done is by first scaling the
variables in the problem using characteristic values. The best way to explain what
this means is to work out an example and the projectile problem is an excellent place
to start.
1.5.1.1
Change Variables
The first step in nondimensionalizing a problem is to introduce a change of
variables, which for the projectile problem will have the form
t = tc τ,
x = xc u.
28
1 Dimensional Analysis
In the above formula, xc is a constant and it is a characteristic value of the variable
x. It is going to be determined using the physical parameters in the problem, which
for the projectile problem are g, R, and v0 . In a similar manner, tc is a constant that
has the dimensions of time and it represents a characteristic value of the variable t.
In some problems it will be clear at the beginning how to select xc and tc . However,
it is assumed here that we have no clue at the start what to choose and will not select
them until the problem is studied a bit more. All we know at the moment is that
whatever the choice, the new variables u, s are dimensionless. To make the change
of variables note that from the chain rule
d
dτ d
=
dt
dt dτ
1 d
,
=
tc dτ
(1.64)
and
d2
d
=
2
dt
dt
d
dt
=
1 d2
.
tc2 dτ 2
(1.65)
With this, the projectile equation (1.1) takes the form
1 d2
gR 2
(x
u)
=
−
.
c
tc2 dτ 2
(R + xc u)2
(1.66)
The method requires us to collect the parameters into dimensionless groups. There
is no unique way to do this, and this can cause confusion when first learning the
procedure. For example, to nondimensionalize the denominator in (1.66) one can
factor it as either R(1+xc u/R) or xc (R/xc +u). The first has the benefit of enabling
us to cancel the R in the numerator. Making this choice yields
Π1
d 2u
1
=−
,
2
dτ
(1 + Π2 u)2
(1.67)
where the initial conditions (1.2) and (1.3) are
u(0) = 0,
(1.68)
du
(0) = Π3 .
dτ
(1.69)
In the above, the dimensionless groups are
Π1 =
xc
,
gtc2
(1.70)
1.5 Nondimensionalization and Scaling
29
xc
,
R
tc v0
Π3 =
.
xc
Π2 =
1.5.1.2
(1.71)
(1.72)
The Dimensionless Groups
Our change of variables has resulted in three dimensionless groups appearing in the
transformed problem. There are a few important points that need to be made here.
First, the Π ’s do not involve the variables u, s and only depend on the parameters in
the problem. Second, they are dimensionless and to accomplish this it was necessary
to manipulate the projectile problem so the parameters end up grouped together
to form dimensionless ratios. The third, and last, point is that the above three
dimensionless groups are independent in the sense that it is not possible to write
any one of them in terms of the other two. For example, Π1 is the only one that
contains the parameter g while Π2 is the only one containing R. It is understood
that in making the statement that the three groups are independent that xc and tc can
be selected, if desired, independently of any of the parameters in the problem.
Before deciding on how to select xc and tc , it is informative to look a little closer
at the above dimensionless groups. We begin with Π2 . In physical terms it is a
measure of a typical, or characteristic, height of the projectile compared to the radius
of the Earth. In comparison, Π3 is a measure of a typical, or characteristic, velocity
xc /tc compared to the velocity the projectile starts with. Finally, the parameter group
Π1 measures a typical, or characteristic, acceleration xc /tc2 in comparison to the
acceleration due to gravity in a uniform field. These observations can be helpful
when deciding on how to nondimensionalize a problem as will be shown next.
1.5.1.3
Use Dimensionless Groups to Determine Scaling
It is now time to actually decide on what to take for xc and tc . There are whole
papers written on what to consider as you select these parameters, but we will take
a somewhat simpler path. For our problem we have two parameters to determine,
and we will do this by setting two of the above dimensionless groups equal to one.
What we need to do is decide on which two to pick, and we will utilize what might
be called rules of thumb.
Rule of Thumb 1 Pick the Π ’s that appear in the initial and/or boundary
conditions.
We only have initial conditions in our problem, and the only dimensionless group
involved with them is Π3 . So we set Π3 = 1 and conclude
xc = v0 tc .
(1.73)
30
1 Dimensional Analysis
Rule of Thumb 2 Pick the Π ’s that appear in the reduced problem.
To use this rule it is first necessary to explain what the reduced problem is.
This comes from the earlier assumption that the object does not get very high in
comparison to the radius of the Earth, in other words, Π2 is small. The reduced
problem is the one obtained in the extreme limit of Π2 → 0. Taking this limit in
(1.67)–(1.69), and using (1.73), the reduced problem is
Π1
d 2u
= −1 ,
dτ 2
where
u(0) = 0,
and
du
(0) = 1.
dτ
According to the stated rule of thumb, we set Π1 = 1, and so
xc = v02 /g.
(1.74)
This choice for xc seems reasonable based on our earlier conclusion that the
maximum height for the uniform field case is v02 /(2g).
Combining (1.73) and (1.74), we have that xc = v02 /g and tc = v0 /g. With this
scaling, then (1.67)–(1.69) take the form
d 2u
1
=−
,
2
dτ
(1 + εu)2
(1.75)
where
u(0) = 0,
(1.76)
du
(0) = 1.
dτ
(1.77)
The dimensionless parameter appearing in the above equation is
ε=
v02
.
gR
(1.78)
This parameter will play a critical role in our constructing an accurate approximation
of the solution of the projectile problem. This will be done in the next chapter but
for the moment recall that since R ≈ 6.4 × 106 m and g ≈ 9.8 m/s2 , then ε ≈ 1.6 ×
10−8 v02 . Consequently for baseball bats, sling shots, BB-guns, and other everyday
projectile-producing situations, where v0 is not particularly large, the parameter ε is
very small. This observation is central to the subject of the next chapter.
1.5 Nondimensionalization and Scaling
1.5.1.4
31
Changing Your Mind
Before leaving this example it is worth commenting on the nondimensionalization
procedure by asking a question. Namely, how bad is it if different choices would
have been made for xc and tc ? For example, suppose for some reason one decides
to take Π2 = 1 and Π3 = 1. The resulting projectile problem is
ε
1
d 2u
=−
,
dτ 2
(1 + u)2
(1.79)
where u(0) = 0, du
dτ (0) = 1, and ε is given in (1.78). No approximation has been
made here and therefore this problem is mathematically equivalent to the one given
in (1.75)–(1.77). Based on this, the answer to the question would be that using this
other scaling is not so bad. However, the issue is amenability and what properties of
the solution one is interested in. To explain, earlier we considered how the solution
behaves if v0 is not very large. With the scaling that produced (1.79), small v0
translates into looking at what happens when ε is near zero. Unfortunately, the limit
of ε → 0 results in the loss of the highest derivative in the differential equation and
(1.79) reduces to 0 = −1. How to handle such singular limits will be addressed
in the next chapter but it requires more work than is necessary for this problem. In
comparison, letting ε approach zero in (1.75) causes no such complications and for
this reason it is more amenable to the study of the small v0 limit. The point here
is that if there are particular limits, or conditions, on the parameters that it is worth
taking them into account when constructing the scaling.
1.5.2 Weakly Nonlinear Diffusion
To explore possible extensions of the nondimensionalization procedure we consider
a well-studied problem involving nonlinear diffusion. The problem consists of
finding the concentration c(x, t) of a chemical for 0 < x < . The concentration
satisfies
D
∂ 2c
∂c
− λ(γ − c)c,
=
∂t
∂x 2
(1.80)
where the boundary conditions are
cx=0 = cx= = 0,
(1.81)
c|t=0 = c0 sin(5π x/ ).
(1.82)
and the initial condition is
32
1 Dimensional Analysis
The nonlinear diffusion equation (1.80) is known as Fisher’s equation, and it arises
in the study of the movement of genetic traits in a population. A common simplifying assumption made when studying this equation is that the nonlinearity is weak,
which means that the term λc2 is small in comparison to the others in the differential
equation. This assumption will be accounted for in the nondimensionalization.
Before starting the nondimensionalization process we should look at the fundamental dimensions of the variables and parameters in the problem. First, c is a
concentration, which corresponds to the number of molecules per unit volume, and
so c = L−3 . The units for the diffusion coefficient D were determined earlier, and
it was found that D = L2 /T . As for γ , the γ − c term in the differential equation
requires these two quantities to have the same dimensions, and so γ = c.
Similarly, from the differential equation we have λ(γ − c)c = ∂c
∂t , and from
this it follows that λ = L3 T −1 . Finally, from the initial condition we have that
c0 = c.
Now, to nondimensionalize the problem we introduce the change of variables
x = xc y,
(1.83)
t = tc τ,
(1.84)
c = cc u.
(1.85)
In this context, xc has the dimensions of length and is a characteristic value of the
variable x. Similar statements apply to tc and cc . Using the chain rule as in (1.64)
the above differential equation takes the form
Dcc ∂ 2 u
cc ∂u
− λcc (γ − cc u)u.
=
2
2
tc ∂τ
xc ∂y
It is necessary to collect the parameters into dimensionless groups, and so in the
above equation we rearrange things a bit to obtain
Dtc ∂ 2 u
∂u
− λtc cc (γ /cc − u)u.
=
2
2
∂τ
xc ∂y
(1.86)
In conjunction with this we have the boundary conditions
uy=0 = uy=
/xc
= 0,
(1.87)
and the initial condition is
uτ =0 = (c0 /cc ) sin(5π xc y/ ).
(1.88)
1.5 Nondimensionalization and Scaling
33
The resulting dimensionless groups are
Π1 =
Dtc
,
xc2
(1.89)
Π2 = λtc cc ,
(1.90)
Π3 = γ /cc ,
(1.91)
Π4 = /xc ,
(1.92)
Π5 = c0 /cc .
(1.93)
It is important to note that the five dimensionless groups given above are independent in the sense that it is not possible to write one of them in terms of the other four.
As before, this statement is based on our ability to select, if desired, the scaling
parameters xc , tc , cc independently of each other and the other parameters in the
problem. Also, in counting the dimensionless groups one might consider adding a
sixth. Namely, in the initial condition (1.88) there is Π6 = xc / . The reason it is not
listed above is that it is not independent of the others because Π6 = 1/Π4 .
We have three scaling parameters to specify, namely xc , tc , and cc . Using Rule of
Thumb 1, the Π ’s that appear in the boundary and initial conditions are set equal to
one. In other words, we set Π4 = 1 and Π5 = 1, fro...
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