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P395, the question 8.9. Please make sure that show all the steps. Hand writing. The material you may use is chapter 8 in the book.

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Texts in Applied Mathematics

56

Editors

J.E. Marsden

L. Sirovich

S.S. Antman

Advisors

G. Iooss

P. Holmes

D. Barkley

M. Dellnitz

P. Newton

Texts in Applied Mathematics

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7.

8.

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Sirovich: Introduction to Applied Mathematics.

Wiggins: Introduction to Applied Nonlinear Dynamical Systems and Chaos.

Hale/Koçak: Dynamics and Bifurcations.

Chorin/Marsden: A Mathematical Introduction to Fluid Mechanics, 3rd ed.

Hubbard/West: Differential Equations: A Dynamical Systems Approach: Ordinary

Differential Equations.

Sontag: Mathematical Control Theory: Deterministic Finite Dimensional Systems,

2nd ed.

Perko: Differential Equations and Dynamical Systems, 3rd ed.

Seaborn: Hypergeometric Functions and Their Applications.

Pipkin: A Course on Integral Equations.

Hoppensteadt/Peskin: Modeling and Simulation in Medicine and the Life Sciences, 2nd ed.

Braun: Differential Equations and Their Applications, 4th ed.

Stoer/Bulirsch: Introduction to Numerical Analysis, 3rd ed.

Renardy/Rogers: An Introduction to Partial Differential Equations.

Banks: Growth and Diffusion Phenomena: Mathematical Frameworks and Applications.

Brenner/Scott: The Mathematical Theory of Finite Element Methods, 2nd ed.

Van de Velde: Concurrent Scientific Computing.

Marsden/Ratiu: Introduction to Mechanics and Symmetry, 2nd ed.

Hubbard/West: Differential Equations: A Dynamical Systems Approach:

Higher-Dimensional Systems.

Kaplan/Glass: Understanding Nonlinear Dynamics.

Holmes: Introduction to Perturbation Methods.

Curtain/Zwart: An Introduction to Infinite-Dimensional Linear Systems Theory.

Thomas: Numerical Partial Differential Equations: Finitc Difference Methods.

Taylor: Partial Differential Equations: Basic Theory.

Merkin: Introduction to the Theory of Stability of Motion.

Naber: Topology, Geometry, and Gauge Fields: Foundations.

Polderman/Willems: Introduction to Mathematical Systems Theory: A Behavioral Approach.

Reddy: Introductory Functional Analysis with Applications to Boundary-Value

Problems and Finite Elements.

Gustafson/Wilcox: Analytical and Computational Methods of Advanced Engineering

Mathematics.

Tveito/Winther: Introduction to Partial Differential Equations: A Computational Approach.

Gasquet/Witomski: Fourier Analysis and Applications: Filtering, Numerical

Computation, Wavelets.

(continued after index)

Mark H. Holmes

Introduction to the

Foundations of Applied

Mathematics

123

Mark H. Holmes

Department of Mathematical Sciences

Rensselaer Polytechnic Institute

110 8th Street

Troy NY 12180-3590

USA

holmes@rpi.edu

Series Editors

J.E. Marsden

Control and Dynamical Systems, 107–81

California Institute of Technology

Pasadena, CA 91125

USA

marsden@cds.caltech.edu

L. Sirovich

Division of Applied Mathematics

Brown University

Providence, RI 02912

USA

lawrence.sirovich@mssm.edu

S.S. Antman

Department of Mathematics

and

Institute for Physical Science

and Technology

University of Maryland

College Park, MD 20742-4015

USA

ssa@math.umd.edu

ISSN 0939-2475

ISBN 978-0-387-87749-5

e-ISBN 978-0-387-87765-5

DOI 10.1007/978-0-387-87765-5

Springer Dordrecht Heidelberg London New York

Library of Congress Control Number: 2009929235

Mathematics Subject Classification (2000): 74-01; 76R50; 76A02; 76M55; 35Q30; 35Q80; 92C45;

74A05; 74A10

c Springer Science+Business Media, LLC 2009

All rights reserved. This work may not be translated or copied in whole or in part without the written

permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,

NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in

connection with any form of information storage and retrieval, electronic adaptation, computer

software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.

The use in this publication of trade names, trademarks, service marks, and similar terms, even if

they are not identified as such, is not to be taken as an expression of opinion as to whether or not

they are subject to proprietary rights.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

To Colette, Matthew and Marianna

Preface

FOAM. This acronym has been used for over fifty 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

vii

viii

Preface

considered, they are, and connections with experiment are a staple of this

book.

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

are needed. The model development starts in Chapter 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 Chapters 4 and 5. What remains is to account

for the forces in the system, and this is done in Chapter 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 Chapter 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, New York

March, 2009

Mark H. Holmes

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

1.5.1 Projectile Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

1.5.3 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

1

1

3

5

6

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33

2

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

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

2.4.1 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5 Multiple Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6 Multiple Scales and Two-Timing . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

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

3.4.1 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

3.4.3 Perturbation Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5 Assorted Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5.1 Elementary and Nonelementary Reactions . . . . . . . . . . .

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

3.6 Steady-States and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6.1 Reaction Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6.2 Geometric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6.3 Perturbation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.7 Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.7.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

4.4.1 Point Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.2 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

4.5.1 Balance Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

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

4.6.1 Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.7 Langevin Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.7.1 Properties of the Forcing . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.7.2 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 Nonconstant Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

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

5.6.4 Expansion Fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6.5 Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6.6 Return of Phantom Traffic Jams . . . . . . . . . . . . . . . . . . .

5.6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

205

205

206

207

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6

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

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2.1 Material Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2.2 Spatial Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.8.2 Material Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.8.3 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

286

289

290

290

293

294

295

298

302

304

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

311

311

313

316

327

328

329

331

335

337

338

342

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.5 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.5.1 Incompressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 . . . . . . . . . . . . . . . . . . . . .

8.9 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

351

351

352

353

356

358

361

362

362

363

364

367

367

368

368

372

374

374

375

378

379

Contents

xiii

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

380

383

385

387

389

390

391

391

394

9

Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.1 Newtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.2 Steady Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

9.2.2 Poiseuille Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.3 Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.3.1 Vortex Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.4 Irrotational Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.4.1 Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.5 Ideal Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

9.5.2 Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.5.3 End Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.6 Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.6.1 Impulsive Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

403

403

404

405

408

411

412

414

417

419

420

423

426

427

427

429

434

A

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

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

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

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

441

441

441

442

B

Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

B.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

B.2 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

C

Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 449

D

Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

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

451

451

451

452

xiv

E

Contents

Equations for a Newtonian Fluid . . . . . . . . . . . . . . . . . . . . . . . . . 453

E.1 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

E.2 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

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 a 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

gR2

d2 x

=−

,

2

dt

(R + x)2

for 0 < t,
(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,
dx
(0) = v0 .
dt
(1.2)
(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, and so, we are very interested in
M.H. Holmes, Introduction to the Foundations of Applied Mathematics,
Texts in Applied Mathematics 56, DOI 10.1007/978-0-387-87765-5 1,
c Springer Science+Business Media, LLC 2009
1
2
1 Dimensional Analysis
x
xM
tM
2tM
t
Figure 1.1 The solution (1.5) of the projectile problem in a uniform gravitational
field.
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
d2 x
= −g, for 0 < t.
(1.4)
dt2
Integrating and then using the two initial conditions yields
1
x(t) = − gt2 + v0 t.
2
(1.5)
This solution is shown schematically in Figure 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
v2
xM = 0 .
(1.7)
2g
Therefore, we must require that v 2 /(2g) is much less than R, which we write
as v02 /(2g)
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