MATH 414 GMU Frame Indifferent Functions & Free Energy Function Exercises

Description

Solve 6.19 and explain why 6.71 are frame-indifferent. Please show me the how to solve and don’t use any coding to solve.

3 attachmentsSlide 1 of 3attachment_1attachment_1attachment_2attachment_2attachment_3attachment_3

Unformatted Attachment Preview

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... Purchase answer to see full attachment Tags: graphing Mathematical Exercises Frame Indifferent Functions Free Energy Function velocity gradient User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.

Reviews, comments, and love from our customers and community:

This page is having a slideshow that uses Javascript. Your browser either doesn't support Javascript or you have it turned off. To see this page as it is meant to appear please use a Javascript enabled browser.

Peter M.
Peter M.
So far so good! It's safe and legit. My paper was finished on time...very excited!
Sean O.N.
Sean O.N.
Experience was easy, prompt and timely. Awesome first experience with a site like this. Worked out well.Thank you.
Angela M.J.
Angela M.J.
Good easy. I like the bidding because you can choose the writer and read reviews from other students
Lee Y.
Lee Y.
My writer had to change some ideas that she misunderstood. She was really nice and kind.
Kelvin J.
Kelvin J.
I have used other writing websites and this by far as been way better thus far! =)
Antony B.
Antony B.
I received an, "A". Definitely will reach out to her again and I highly recommend her. Thank you very much.
Khadija P.
Khadija P.
I have been searching for a custom book report help services for a while, and finally, I found the best of the best.
Regina Smith
Regina Smith
So amazed at how quickly they did my work!! very happy♥.