Applied Mathematics E-Book

J. David Logan

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 925

Veröffentlichungsjahr: 2013

Beschreibung

Praise for the Third Edition

“Future mathematicians, scientists, and engineers should find the book to be an excellent introductory text for coursework or self-study as well as worth its shelf space for reference.”
—MAA Reviews

Applied Mathematics, Fourth Edition is a thoroughly updated and revised edition on the applications of modeling and analyzing natural, social, and technological processes. The book covers a wide range of key topics in mathematical methods and modeling and highlights the connections between mathematics and the applied and natural sciences.

The Fourth Edition covers both standard and modern topics, including scaling and dimensional analysis; regular and singular perturbation; calculus of variations; Green’s functions and integral equations; nonlinear wave propagation; and stability and bifurcation. The book provides extended coverage of mathematical biology, including biochemical kinetics, epidemiology, viral dynamics, and parasitic disease. In addition, the new edition features:

• Expanded coverage on orthogonality, boundary value problems, and distributions, all of which are motivated by solvability and eigenvalue problems in elementary linear algebra
• Additional MATLAB® applications for computer algebra system calculations
• Over 300 exercises and 100 illustrations that demonstrate important concepts
• New examples of dimensional analysis and scaling along with new tables of dimensions and units for easy reference
• Review material, theory, and examples of ordinary differential equations
• New material on applications to quantum mechanics, chemical kinetics, and modeling diseases and viruses

Written at an accessible level for readers in a wide range of scientific fields, Applied Mathematics, Fourth Edition is an ideal text for introducing modern and advanced techniques of applied mathematics to upper-undergraduate and graduate-level students in mathematics, science, and engineering. The book is also a valuable reference for engineers and scientists in government and industry.

Leseprobe

Contents

Cover

Half Title page

Title page

Copyright page

Dedication

Preface

Chapter 1: Dimensional Analysis and One-Dimensional Dynamics

1.1 Dimensional Analysis

1.2 Scaling

1.3 Differential Equations

References and Notes

Chapter 2: Two-Dimensional Dynamical Systems

2.1 Phase Plane Phenomena

2.2 Linear Systems

2.3 Nonlinear Systems

2.4 Bifurcations

2.5 Reaction Kinetics

2.6 Pathogens

References and Notes

Chapter 3: Perturbation Methods and Asymptotic Expansions

3.1 Regular Perturbation

3.2 Singular Perturbation

3.3 Boundary Layer Analysis

3.4 Initial Layers

3.5 The WKB Approximation

3.6 Asymptotic Expansion of Integrals

References and Notes

Chapter 4: Calculus of Variations

4.1 Variational Problems

4.2 Necessary Conditions for Extrema

4.3 The Simplest Problem

4.4 Generalizations

4.5 Hamilton’s Principle

4.6 Isoperimetric Problems

References and Notes

Chapter 5: Boundary Value Problems and Integral Equations

5.1 Boundary-Value Problems

5.2 Sturm–Liouville Problems

5.3 Classical Fourier Series

5.4 Integral Equations

5.5 Green’s Functions

5.6 Distributions

References and Notes

Chapter 6: Partial Differential Equations

6.1 Basic Concepts

6.2 Conservation Laws

6.3 Equilibrium Equations

6.4 Eigenfunction Expansions

6.5 Integral Transforms

6.6 Stability of Solutions

6.7 Distributions

References and Notes

Chapter 7: Wave Phenomena

7.1 Waves

7.2 Nonlinear Waves

7.3 Quasi-linear Equations

7.4 The Wave Equation

References and Notes

Chapter 8: Mathematical Models of Continua

8.1 Kinematics and Mass Conservation

8.2 Momentum and Energy

8.3 Gas Dynamics

8.4 Fluid Motions in R3

References and Notes

Chapter 9: Discrete Models

9.1 One-Dimensional Models

9.2 Systems of Difference Equations

9.3 Stochastic Models

9.4 Probability-Based Models

References and Notes

Index

Applied Mathematics

Copyright © 2013 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Logan, J. David (John David)Applied mathematics / J. David Logan. — 4th ed.     pages cm  Includes bibliographical references and index. ISBN 978-1-118-47580-5 (hardback) — ISBN 978-1-118-51492-4 — ISBN 978-1-118-51493-1 — ISBN 978-1-118-51490-0 1. Mathematics—Textbooks. I. Title.  QA37.3.L64 2013  510—dc23  2013001305

To my parentsGeorge Edd Logan (1919–1996)Dorothy Elizabeth Wyatt Logan (1923–2009)

Preface

The fourth edition of Applied Mathematics shares the same goals, philosophy, and style as its predecessors—to introduce key ideas about mathematical methods and modeling, along with the important tools, to mature seniors and graduate students in mathematics, science, and engineering. The emphasis is on how mathematics interrelates with the applied and natural sciences. Prerequisites include a good command of concepts and techniques of calculus, and sophomore-level courses in differential equations and matrices; a genuine interest in applications in some area of science or engineering is a must.

Readers should understand the limited scope of this text. Being a broad introduction to the methods of applied mathematics, it cannot cover every topic in depth. Indeed, each chapter could be expanded into a one, or even several, full-length books. In fact, readers can find elementary books on all of the topics; some of these are cited in the references at the end of the chapters. Secondly, readers should understand the mathematical level of the text. Some books on applied mathematics take a highly practical approach and ignore technical mathematical issues completely, while others take a purely theoretical approach; both of these approaches are valuable and part of the overall body of applied mathematics. Here, we seek a middle ground by providing the physical basis and motivation for the ideas and methods, and we also give a glimpse of deeper mathematical ideas.

There are major changes in the fourth edition. The material has been rearranged and basically divided into two parts. Chapters 1 through 5 involve models leading to ordinary differential equations and integral equations, while Chapters 6 through 8 focus on partial differential equations and their applications. Motivated by problems in the biological sciences where quantitative methods are becoming central, Chapter 9 deals with discrete-time models, which include some material on random processes. Sections reviewing elementary methods for solving systems of ordinary differential equations have been added in Chapters 1 and 2. Many additional examples and figures are included in this edition, and several new exercises appear throughout. Some exercises from the last edition have been revised for better clarity, and many new exercises are included. The length of the text has expanded over 160 pages. The Table of Contents details the specific topics covered.

Note that equations are numbered within sections. Thus, equation label (3.2) refers to the second numbered equation in Section 3 of the current chapter.

My colleagues in Lincoln, who have often used the text in our core sequence in applied mathematics, deserve special thanks. Glenn Ledder, Richard Rebarber, and Tom Shores have provided me with an extensive errata, and they supplied several exercises from graduate qualifying examinations, homework, and course exams. Former students Bill Wolesensky and Kevin TeBeest read parts of the earlier manuscripts and both were often a sounding board for suggestions. I am extremely humbled and grateful to those who used earlier editions of the book and helped establish it as one of the basic textbooks in the area; many have generously given me corrections and suggestions, and many of the typographical errors from the third edition have been resolved. Because of the extensive revision, some new ones, but hopefully not many, have no doubt appeared. I welcome suggestions, comments, and corrections, and contact information is on the book’s website: http://www.unl.edu/~jlogan1/applied-math.htm. Solutions to some of the exercises and an errata will appear when they become available.

My editor at Wiley, Susanne Steitz-Filler, along with Jackie Palmieri, deserves praise for her continued enthusiasm about this new revision and her skill in making it an efficient, painless process. Finally, my wife, Tess, has been a constant source of support for my research, teaching, and writing, and I again take this opportunity to publicly express my appreciation for her encouragement and affection.

Suggestions for use of the text. The full text cannot be covered in a two-semester, 3-credit course, but there is a lot of flexibility built into the text. There is significant independence among chapters, enabling instructors to design special one- or two-semester courses in applied mathematics that meet their specific needs.

Portions of Chapters 1 through 5 can form the basis of a one-semester course involving differential and integral equations and the basic core of applied mathematics. Chapter 4 on the calculus of variations is essentially independent from the others, so it need not be covered. If students have a strong background in differential equations, then only small portions of Chapters 1 and 2 need to be covered.

A second semester, focused around partial differential equations, could cover Chapters 6,7, and 8. Students have the flexibility to take the second semester, as is often done at the University of Nebraska, without having the first, provided small portions of Chapter 5 on Fourier-type expansions is covered.

Chapter 9, like Chapter 3, is independent from the rest of the book and can be covered at any time.

The text, and its translations, have been used in several types of courses: applied mathematics, mathematical modeling, differential equations, mathematical biology, mathematical physics, and mathematical methods in chemical or mechanical engineering.

J. David Logan, Lincoln, NebraskaApril 2013

Chapter 1

Dimensional Analysis and One-Dimensional Dynamics

The techniques of dimensional analysis and scaling are basic in the theory and practice of mathematical modeling. In every physical setting a good grasp of the possible relationships and comparative magnitudes among the various dimensioned parameters nearly always leads to a better understanding of the problem and sometimes points the way toward approximations and solutions. In this chapter we introduce some of the basic concepts from these two topics. A statement and proof of the fundamental result in dimensional analysis, the Pi theorem, is presented, and scaling is discussed in the context of reducing problems to dimensionless form. The notion of scaling points the way toward a proper treatment of perturbation methods, especially boundary layer phenomena in singular perturbation theory as well as algebraic equations with small parameters.

The first part of Section 1.3 is a review of ordinary differential equations. This material may be perused or used as a reference by readers familiar with the basic concepts and elementary solution methods. The last part includes a discussion of stability and bifurcation; it may be less familiar.

1.1 Dimensional Analysis

1.1.1 The Program of Applied Mathematics

Applied mathematics is a broad subject area in the mathematical sciences dealing with those topics, problems, and techniques that have been useful in analyzing real-world phenomena. In a very limited sense it is a set of methods that are used to solve the equations that come out of science, engineering, and other areas. Traditionally, these methods were techniques used to examine and solve ordinary and partial differential equations, and integral equations. At the other end of the spectrum, applied mathematics is applied analysis, or the theory that underlies the methods. But, in a broader sense, applied mathematics is about mathematical modeling and an entire process that intertwines with the physical reality that underpins its origins.

By a mathematical model we mean an equation, or set of equations, that describes some physical problem or phenomenon having its origin in science, engineering, economics, or some other area. By mathematical modeling we mean the process by which we formulate and analyze the model. This process includes introducing the important and relevant quantities or variables involved in the model, making model-specific assumptions about those quantities, solving the model equations by some analytic or numerical method, and then comparing the solutions to real data and interpreting the results. This latter process, confronting the model with data, is often the most difficult part of the modeling process. It involves determining parameter values from the experimental data. This book does not address these important issues, and we refer to texts on data-fitting techniques. This confrontation may lead to revision and refinement until we are satisfied that the model accurately describes the physical situation and is predictive of other similar observations. This process is depicted schematically in Fig. 1.1. Thus, the subject of mathematical modeling involves physical intuition, formulation of equations, solution methods, analysis, and data fitting. A good mathematical model is simple, applies to many situations, and is predictive.

Figure 1.1 Schematic of the modeling process.

In summary, in mathematical modeling the overarching objective is to make quantitative sense of the world as we observe it, often by inventing caricatures of reality. Scientific exactness is sometimes sacrificed for mathematical tractability. Model predictions depend strongly on the assumptions, and changing the assumptions changes the model. If some assumptions are less critical than others, we say the model is robust to those assumptions. They help us clarify verbal descriptions of nature and the mechanisms that make up natural law, and they help us determine which parameters and processes are important, and which are unimportant.

Another issue is the level of complexity of a model. With modern computer technology it is tempting to build complicated models that include every possible effect we can think of, with large numbers of parameters and variables. Simulation models like these have their place, but computer runs do not always allow us to discern which are the important processes and which are not. Of course, the complexity of the model depends upon the data and the purpose, but it is usually a good idea to err on the side of simplicity and then build in complexity as it is needed or desired.

Finally, authors have tried to classify models in several ways—stochastic vs. deterministic, continuous vs. discrete, static vs. dynamic, quantitative vs. qualitative, descriptive vs. explanatory, and so on. In this book we are interested in modeling the underlying reasons for the phenomena we observe (explanatory) rather than fitting the data with formulas (descriptive) as is often done in statistics. For example, fitting measurements of the size of an animal over its lifetime by a regression curve is descriptive, and it gives some information. But describing the dynamics of growth by a differential equation relating growth rates, food assimilation rates, and energy maintenance requirements tells more about the underlying processes involved.

Models are a blend of physical laws, such as conservation of mass, energy, etc., experimental results that lead to constitutive relations, or equations based on experiment, and even ad hoc assumptions when more specific evidence is lacking. The reader is already familiar with many models. In an elementary science or calculus course we learn that Newton’s second law, force equals mass times acceleration, governs mechanical systems like falling bodies; Newton’s inverse-square law of gravitation describes the motion of the planets; Ohm’s law in circuit theory dictates the voltage drop across a resistor in terms of the current; the law of mass action in chemistry describes how fast chemical reactions occur; or the logistic equation, an intuitive ad hoc model of growth and competition in a population.

The first step in modeling is to select the relevant variables (independent and dependent) and parameters that we need to describe the problem. Usually these are based on available experimental data and natural laws. Physical quantities have dimensions like time, distance, degrees, and so on, or corresponding units like seconds, meters, and degrees Celsius. The equations we write down as models must be dimensionally correct. Apples cannot equal oranges. Verifying that each term in our model has the same dimensions is the first task in obtaining a correct equation. Also, checking dimensions can often give us insight into what a term in the model might be. We always should be aware of the dimensions of the quantities, both variables and parameters, in a model, and we should always try to identify the physical meaning of the terms in the equations we obtain. A general rule is to always let the physical problem and the data available drive the mathematics, and not vice-versa.

It would be a limited view to believe that applied mathematics consists only of developing techniques and algorithms to solve problems that arise in a physical or applied context. Applied mathematics deals with all the stages of the modeling process, not merely the formal solution. It is true that an important aspect of applied mathematics is studying, investigating, and developing procedures that are useful in solving mathematical problems: these include analytic and approximation techniques, numerical analysis, and methods for solving differential and integral equations. It is more the case, however, that applied mathematics deals with all phases of the problem. Formulating the model and understanding its origin in empirics are crucial steps. Because there is a constant interplay between the various stages, the scientist, engineer, or mathematician must understand each phase. For example, the solution stage sometimes involves making approximations that lead to a simplification; the approximations often come from a careful examination of the physical reality, which in turn suggests what terms may be neglected, what quantities (if any) are small, and so on. The origins and analysis are equally important. Indeed, physical insight forces us toward the right questions and at times leads us to the theorems and their proofs. In fact, mathematical modeling has been, and remains, one of the main driving forces for mathematics itself.

In the first part of this chapter our aim is to focus on the first phase of the modeling process. Our strategy is to formulate models for various physical systems while emphasizing the interdependence of mathematics and the physical world. Through study of the modeling process we gain insight into the equations themselves. In addition to presenting some concrete examples of modeling, we also discuss two techniques that are useful in developing and interpreting the model equations. One technique is dimensional analysis, and the other is scaling. The former permits us to understand the dimensional (meaning length, time, mass, etc.) relationships of the quantities in the equations and the resulting implications of dimensional homogeneity. Scaling is a technique that helps us understand the magnitude of the terms that appear in the model equations by comparing the quantities to intrinsic reference quantities that appear naturally in the physical situation. A side benefit in scaling differential equations, for example, is in the great economy it affords; more often than not, the number of independent parameters can be significantly reduced.

1.1.2 Dimensional Methods

One of the basic techniques that is useful in the initial, modeling stage of a problem is the analysis of the relevant quantities and how they must relate to each other in a dimensional way. Simply put, apples cannot equal oranges plus bananas; equations must have a consistency to them that precludes every possible relationship among the variables. Stated differently, equations must be dimensionally homogeneous. These simple observations form the basis of the subject known as dimensional analysis. The methods of dimensional analysis have led to important results in determining the nature of physical phenomena, even when the governing equations were not known. This has been especially true in continuum mechanics, out of which the general methods of dimensional analysis evolved.

The cornerstone result in dimensional analysis is known as the Pi theorem. The Pi theorem states that if there is a physical law that gives a relation among a certain number of dimensioned physical quantities, then there is an equivalent law that can be expressed as a relation among certain dimensionless quantities (often noted by π1, π2,…, and hence the name). In the early 1900s, E. Buckingham formalized the original method used by Lord Rayleigh and gave a proof of the Pi theorem for special cases; now the theorem often carries his name. Birkhoff (1950) can be consulted for a bibliography and history.

Example 1.1

(Atomic explosion) To communicate the flavor and power of this classic result, we consider a calculation made by the British applied mathematician G. I. Taylor in the late 1940s to compute the yield of the first atomic explosion after viewing photographs of the spread of the fireball. In such an explosion a large amount of energy E is released in a short time (essentially instantaneously) in a region small enough to be considered a point. From the center of the explosion a strong shock wave spreads outward; the pressure behind it is on the order of hundreds of thousands of atmospheres, far greater than the ambient air pressure whose magnitude can be accordingly neglected in the early stages of the explosion. It is plausible that there is a relation between the radius of the blast wave front r, time t, the initial air density ρ, and the energy released E. Hence, we assume there is a physical law

(1.1)

which postulates a functional relationship among these four dimensioned quantities. The Pi theorem states that there is an equivalent physical law between the independent dimensionless quantities that can be formed from t, r, E, and ρ. We note that t has dimensions of time, r has dimensions of length, E has dimensions of mass · length2 · time−2, and ρ has dimensions of mass · length−3. Hence, the quantity r5ρ/t2E is dimensionless, because all of the dimensions cancel out of this quantity (this is easy to check). It is not difficult to observe, and we shall show it later, that no other independent dimensionless quantities can be formed from t, r, E, and ρ. The Pi theorem then guarantees that the physical law (1.1) is equivalent to a physical law involving only the dimensionless quantities; in this case

(1.2)

because there is only one such quantity, where f is some function of a single variable. From (1.2) it follows that the physical law must take the form (a root of (1.2))

or

(1.3)

where C is a constant. Therefore, just from dimensional reasoning it has been shown that the radius of the wave front depends on the two-fifths power of time. Experiments and photographs of explosions confirm this dependence. The constant C depends on the dimensionless ratio of the specific heat at constant pressure to the specific heat at constant volume. By fitting the curve (1.3) to experimental data of r versus t, the initial energy yield E can be computed, since C and ρ are known quantities. (See Exercise 3.) Although this calculation is only a simple version of the original argument given by Taylor, we infer that dimensional reasoning can give crucial insights into the nature of a physical process and is an invaluable tool for the applied mathematician, scientist, or engineer.

Example 1.2

(Height of a projectile) Let us imagine going outside and asking how high we can throw a baseball vertically upward. What would the maximum height h depend on? We may conjecture that it depends on the mass m of the ball, the acceleration of gravity g, and the velocity v with which we propel it. For now let’s ignore the air resistance, which could be another issue. We can learn a lot by dimensional methods. Assume a physical law of the form

relating the four quantities we selected. All of them can be written in terms of fundamental dimensions M (mass), L (length), and T (time). If Π is a dimensionless quantity that can be formed from m, g, v, and h, then

for some powers a, b, c, and d. This means

Because Π is dimensionless, the dimensions must cancel and all the exponents must be zero; we obtain the homogeneous system of equations

There is only one such independent dimensionless variable (modulo the exponents being multiplied by some constant). We conclude that the physical law can be written in terms of this single dimensionless variable, or

where C is some constant. This means

so the maximum height depends on the square of the velocity and the inverse of gravity. Based on minimal assumptions and dimensional reasoning we have learned a lot. For example, if we double the velocity, the height is quadrupled. How would we determine the constant C? A single experiment that measures h and v suffices (g is known).

As an aside, there is another aspect of dimensional analysis that is important in engineering, namely the design of small laboratory-scale models (say of an airplane or ship) that behave like their real counterparts. A discussion of this important topic is not treated here, but can be found in many engineering texts, especially books on fluid mechanics.

Example 1.3

(Air resistance) Those who bike have certainly noticed that the force F (mass times length, per time-squared) of air resistance appears to be positively related to their speed v (length per time) and cross-sectional area A (length-squared). But force involves mass, so it cannot depend only upon v and A. Therefore, let us add fluid, or air, density ρ (mass per length-cubed), and assume a relation of the form

for some function f to be determined. What are the possible forms of f? To be dimensionally correct, the right side must be a force. What powers of ρ, A, and v would combine to give a force, that is,

where k is some constant without any units (i.e., dimensionless). If we denote mass by M, length by L, and time by T, then the last equation requires

Equating exponents of M, L, and T gives

Consequently, the force must depend upon the square of the velocity, a fact often used in problems involving fluid resistance, and it must be proportional to density of air and area of the object. Again, substantial information can be obtained from dimensional arguments alone.

EXERCISES

1.1.3 The Pi Theorem

The Pi theorem states that it is generally true that a physical law

(1.4)

relating m dimensional quantities q1, q2,…, qm is equivalent to a physical law

that relates the k dimensionless quantities π1, π2,…, πk that can be formed from q1, q2,…, qm. There are tremendous values in a dimensionless formulation of a problem. One, the formula is independent of the set of units used; two, there are fewer dimensionless quantities than the dimensioned ones, and thus there is economy in the formulation; finally, important relations can be discovered among the quantities. Before making a formal statement, we introduce some basic terminology.

First, the m dimensional quantities q1, q2,…, qm, which are like the quantities t, r, ρ, and e in the blast wave example, Example 1.1, are dimensioned quantities. This means that they can be expressed in a natural way in terms of a minimal set of fundamental dimensions L1, L2,…, Ln (n < m), appropriate to the problem being studied. In the blast wave problem, time T, length L, and mass M can be taken to be the fundamental dimensions, since each quantity t, r, ρ, and e can be expressed in terms of T, L, and M. For example, the dimensions of the energy e are ML2T−2. In general, the dimensions of qi, denoted by the square brackets notation [qi], can be written in terms of the fundamental dimensions as

(1.5)

We proceed as follows. If π is a quantity of the form

The powers of the Li must sum to zero, and thus we obtain a homogeneous system of n equations in m unknowns p1,…, pm, given in matrix form by

called the dimension matrix. The elements in the ith column give the exponents for qi in terms of the powers of L1,…, Ln. It follows from a key result in linear algebra that the number of independent solutions is m − r, where r is the rank of A. We recall that the rank of a matrix is the number of linearly independent rows, which, when the matrix is reduced to row echelon form, is the number of nonzero rows. So, the number of independent dimensionless variables that can be formed from q1,…, qm is m − r.

What assumptions are needed to show equivalence of the dimensionless form of the physical law? The fundamental assumption regarding the physical law (1.4) goes back to the simple statement that apples cannot equal oranges. We assume that (1.4) is unit free in the sense that it is independent of the particular set of units chosen to express the quantities q1, q2,…, qm. We are distinguishing the word unit from the word dimension. By units we mean specific physical units like seconds, hours, days, and years; all of these units have dimensions of time. Similarly, grams, kilograms, slugs, and so on are units of the dimension mass. Any fundamental dimension Li has the property that its units can be changed upon multiplication by the appropriate conversion factor λi > 0 to obtain i in a new system of units. We write

The units of derived quantities q can be changed in a similar fashion. If

then

Definition 1.4

Example 1.5

The physical law

(1.6)

relates the distance x a body falls in a constant gravitational field g to the time t. In the cgs system of units, x is given in centimeters (cm), t in seconds, and g in cm/sec2. If we change units for the fundamental quantities x and t to inches and minutes, then in the new system of units

Then

Therefore (1.6) is unit-free.

We can now give a formal statement of the Pi theorem.

Theorem 1.6

(Pi theorem) Let

(1.7)

be a unit-free physical law that relates the dimensional quantities q1, q2,…, qm. Let L1,…, Ln(n < m) be fundamental dimensions with

(1.8)

expressed only in terms of the dimensionless quantities.

The existence of a physical law (1.7) is an assumption. In practice one must conjecture which are the relevant variables in a problem and then apply the machinery of the theorem. The resulting dimensionless physical law (1.8) must be checked by experiment, or whatever, in an effort to establish the validity of the original assumptions.

We prove the Pi theorem after more examples.

Example 1.7

which relates the six quantities, t, r, u, e, k, c. The next step is to determine a minimal set of fundamental dimensions L1,…, Ln by which the six-dimensional quantities can be expressed. A suitable selection would be the four quantities T (time), L (length), Θ (temperature), and E (energy1). Then

and the dimension matrix A is given by

Therefore, the exponents must vanish and we obtain four homogeneous linear equations for α1,…, α6, namely

The coefficient matrix of this homogeneous linear system is just the dimension matrix A. From elementary matrix theory the number of independent solutions equals the number of unknowns minus the rank of A. Each independent solution will give rise to a dimensionless variable. Now the method unfolds and we can see the origin of the rank condition in the statement of the Pi theorem. By standard methods for solving linear systems we find two linearly independent solutions

and

These give two dimensionless quantities

and

Solving for π2 gives

for some function g, or

(1.9)

Example 1.8

(Blood flow in arteries) Used in conjunction with experiments, dimensional analysis can be a powerful tool in obtaining theoretical results in all areas of science and engineering. In this example we study how blood flows in veins and arteries, depending on the radius of the artery r, the length of the artery l, the density and viscosity of blood, ρ and μ, and the pressure change ΔP over that length segment. Our interest is the volumetric flow rate Q (volume of blood per second) and how those quantities affect it. Of course, what we say applies to any context of viscous flow in a pipe.

There are six quantities. It is fairly clear that the dimensions of five of them are

but the viscosity may be less uncertain for many. Therefore we give a brief, heuristic explanation of viscosity that reveals its dimensions. We all have an intuitive idea about viscosity. Honey is very viscous, oil perhaps a little less, and alcohol less yet. Let us imagine a viscous fluid between two infinite plates made of metal, for example. See Fig. 1.2. The lower plane is fixed, and we pull on the upper plate in a direction tangent to it, giving a shear force per unit area. The basic property of a viscous fluid is its ‘stickiness’ at a boundary; it adheres to a boundary, stationary or moving, with the same velocity as the boundary. So, the movement of the upper plate drags along the fluid at some velocity; at the lower boundary the velocity of the fluid is zero. Therefore, as we drag along the plate, the shear force imparts momentum to the fluid. As shown in the figure, the velocity profile decreases from the top plate to zero at the lower plate. Intuitively, it appears that the shear force should therefore be related to the vertical change of the velocity. As a constitutive assumption, we take a linear relationship with proportionality constant μ. That is,

Figure 1.2 A viscous fluid between a fixed lower plate and an upper plate. A shear force on the upper plate transfers momentum downward into the fluid.

Dimensionally,

or

Now we can perform a dimensional analysis with L, T, and M as fundamental dimensions.

We make the assumption there is a relationship among these six quantities, i.e.,

To find the dimensionless variables we set

Therefore,

and so the exponents for the six quantities must satisfy the homogeneous system

The dimension matrix is

The solution can be written in vector form as

These three linearly independent vectors define the three sets of six exponents of the independent pi variables, and we have

Note that various combinations (products and powers) of pi variables are also dimensionless, and we need to make the appropriate choice to obtain what we want, namely a formula for Q. There are two quantities that depend on Q, and two that depend on ΔP. We can eliminate Q by forming the new variable

Then, the Pi theorem guarantees that the physical law is equivalent to

or

where g is an arbitrary function. We can solve for the first second argument in terms of the other two to get

where G is again arbitrary. This gives

At this point it appears we are at an impasse. However, we can perform experiments to get additional information; this is not uncommon. If we hold all of the quantities fixed except Q and ΔP, we find that the volumetric flow rate Q depends linearly on the pressure gradient ΔP. Then the arbitrary function G cannot depend on its last argument and we obtain

where H is arbitrary. Next, by performing another experiment, we can find how H depends on the ratio r/l. Holding all the quantities fixed but r and l, we can plot Qμ/r3ΔP vs. R/l, and we find that H is linear. Hence,

where C is a constant. In conclusion, we have deduced from dimensional reasoning and experiment that the volumetric flow rate of blood depends linearly on the pressure change and the fourth power of the radius of the artery.

It is interesting to note that the theoretical equations of viscous fluid dynamics, which we examine in the last chapter, imply that the volumetric flow rate is

Of course, we used a lot of foresight in making our dimensional calculations, but it is fairly impressive that we reached almost the same result!

One of the fundamental techniques for finding solutions to differential equations has it basis in dimensional analysis. Among the different variables and parameters in the problem it is sometimes possible to identify new variables that reduce the equation to a much simpler one. For example, it may be possible to reduce a partial differential equation to an ordinary differential equation, or transform a second-order differential equation into a first-order equation. This method works because of natural symmetries in the problem itself, for example, invariance under a stretching transformation. A detailed treatment of symmetries and the discovery of variables that simplify a problem can be found in Logan (2008), as well as in a chapter in the first edition of the present text. Here we give a simple example of how dimensionless variables can be used in this type of reduction.

Example 1.9

and the boundary conditions

along with an initial condition

This partial differential equation is called the diffusion equation. As we observe later, it also appears in the context of heat conduction and random walks. The constant D is called the diffusion constant and has dimensions length-squared per time; it is a measure of how fast the toxicant diffuses through the water medium.

Therefore,

The dimension matrix is

In vector form, the solution to the homogeneous system is

where c and e are arbitrary real numbers. Therefore, two dimensionless quantities are

Of course, dimensionless quantities may be chosen in many ways; we can take powers, products, and quotients and still maintain independent quantities. Here, for convenience in subsequent calculations, we take

where F is an unknown function. This is the dependence given by the Pi theorem.

We are thinking now that the dimensionless quantity s is an independent variable. We can substitute this expression into the diffusion equation as follows. By the chain rule,

and

When these expressions are substituted into the diffusion equation, we obtain

or

Amazingly enough, the diffusion equation reduced to a ordinary differential equation for F(s)! This cancellation of the original variables x and t signals the success of the method. The variable s is called a similarity variable and the solutions we obtain subsequently are called similarity solutions.

Using the well known integral formula

we get

Finally we can write the solution as

In terms of the original variables,

1.1.4 Proof of the Pi Theorem

To prove the Pi theorem we demonstrate two propositions.

The proof of (i) has been outlined earlier—the general argument proceeds exactly like the construction of the dimensionless variables in the last several examples. It makes use of the familiar result in linear algebra that the number of linearly independent solutions of a set of n homogeneous equations in m unknowns is m − r, where r is the rank of the coefficient matrix. For, let π be a dimensionless quantity. Then

(1.10)

for some p1, p2,…,pm. In terms of the fundamental dimensions L1,…, Ln,

(1.11)

By the aforementioned theorem in linear algebra, the homogeneous system (1.11) has exactly m − r independent (vector) solutions [p1,…,pm]. (These solutions form a basis for the nullspace, or kernel, of A.) Each solution gives rise to a dimensionless variable via (1.10), and this completes the proof of (i). The independence of the dimensionless variables is in the sense of linear algebraic independence. Note that we can always, for example, multiply a vector by a constant, say 1/2, and get an equivalent dimensionless variable, the square root of the dimensionless quantity.

The proof of (ii) makes strong use of the hypothesis that the law is unit-free. The argument is subtle, but it can be made almost transparent if we examine a particular example.

Example 1.10

Consider the unit-free law

(1.12)

for the distance a particle falls in a gravitational field. If length and time are chosen as fundamental dimensions, a straightforward calculation shows there is a single dimensionless variable given by

(1.13)

is equivalent to (1.12) and is unit-free because f is. Then F is defined by

which is equivalent to (1.13) and (1.12).

(1.14)

with

where L1 and L2 are fundamental dimensions. The dimension matrix

where the exponents p1, p2, p3, p4 satisfy the homogeneous system (1.11), which in this case is

(1.15)

We wish to determine p1, p2, p3, p4, and the form of the two dimensionless variables. Without loss of any generality, we can assume the first two columns of A are linearly independent. This is because we can rearrange the indices on the qj so that the two independent columns appear as the first two. Then columns three and four can be written as linear combinations of the first two, or

(1.16)

(1.17)

for some constants c31, c32, c41, and c42. Substituting into (1.15) gives

The left side is a combination of linearly independent vectors, and therefore the coefficients must vanish,

Therefore, we can solve for p1 and p2 in terms of p3 and p4 and write

The two vectors on the right represent two linearly independent solutions of (1.15); hence, the two dimensionless quantities are

Next define a function G by

The physical law

(1.18)

(1.19)

(1.20)

where

(1.21)

implies

(1.22)

And, because the coefficient matrix

in (1.22) is nonsingular (recall the assumption that the first two columns of the dimension matrix A are linearly independent), the system (1.22) has a unique solution (ln λ1, ln λ2), from which λ1 and λ2 can be determined to satisfy (1.21). Thus (1.19) is an equivalent physical law and the argument is complete.

The general argument for arbitrary m, n, and r can be found in Birkhoff (1950), from which the preceding proof was adapted. This classic book also provides additional examples and historical comments.2

In performing a dimensional analysis on a problem, two judgments are required at the beginning.

1. The selection of the pertinent variables.

2. The choice of the fundamental dimensions.

The first is a matter of experience and may be based on intuition or experiments. There is, of course, no guarantee that the selection will lead to a useful formula after the procedure is applied. Second, the choice of fundamental dimensions may involve tacit assumptions that may not be valid in a given problem. For example, including mass, length, and time but not force in a given problem assumes there is some relation (Newton’s second law) that plays an important role and causes force not to be an independent dimension. As a specific example, a small sphere falling under gravity in a viscous fluid is observed to fall, after a short time, at constant velocity. Since the motion is un-accelerated we need not make use of the proportionality of force to acceleration, and so force can be treated as a separate, independent fundamental dimension. In summary, intuition and experience are important ingredients in applying the dimensional analysis formalism to a specific physical problem.

Table 1.1 gives dimensions for common quantities, including key relations that remind us of their origins. The reader should review the mks units for each of these quantities (e.g., energy is given in Joules), some of which are listed.

Table 1.1 Dimensions of common quantities in mechanics and thermodynamics. Symbols M, L, T, and Θ (temperature) are fundamental dimensions.

Quantity (symbol)

Dimensions

Relation

velocity (

v

)

LT

−1

length per time (

m/s

)

acceleration (

a

)

LT

−2

velocity per time-squared (

m/s

2

)

momentum (

p

)

MLT

−1

mass · velocity

mass density (ρ)

ML

−3

mass per volume

force (

F

)

MLT

−2

mass · acceleration (

N

)

energy (

E

), work (

W

)

ML

2

T

−2

force · distance (Joules)

power (

P

)

ML

2

T

−3

energy per time (Watts)

pressure (

P

), stress (σ)

ML

−1

T

−2

force per area (Pascals)

frequency (ω)

T

−1

per time (Hertz)

internal energy (

U

)

ML

2

T

−2

energy

specific heat (

c

)

L

2

T

−2

Θ

−1

energy per mass per degree

heat flux (ϕ)

MT

−3

energy per time per area

thermal conductivity (

K

)

MLT

−3

Θ

−1

flux per length per degree

diffusivity (

k

)

L

2

T

−1

K/ρc

viscosity (μ)

ML

−1

T

−1

mass per length per time

heat loss coefficient (

h

)

MT

3

Θ

−1

energy per volume · deg · time

EXERCISES

1.2 Scaling

1.2.1 Characteristic Scales

Another procedure useful in formulating a mathematical model is scaling. Roughly, scaling means selecting new, usually dimensionless variables and reformulating the problem in terms of those variables. Not only is the procedure useful, but it often is a necessity, especially when comparisons of the magnitudes of various terms in an equation must be made in order to neglect small terms. This idea is particularly crucial in the application of perturbation methods to identify small and large parameters. Further, scaling usually reduces the number of parameters in a problem, thereby leading to great simplification, and it identifies what combinations of parameters are important.

For motivation let us suppose that time t is a variable in a given problem, measured in units of seconds. If the problem involved the motion of a glacier, clearly the unit of seconds is too fast because significant changes in the glacier could not be observed on the order of seconds. On the other hand, if the problem involved a nuclear reaction, then the unit of seconds is too slow; all of the important action would be over before the first second ticked. Evidently, every problem has an intrinsic time scale, or characteristic timetc, which is appropriate to the given problem. This is the shortest time for discernible changes to be observed in the physical quantities. For example, the characteristic time for glacier motion would be of the order of years, whereas the characteristic time for a nuclear reaction would be of the order of microseconds. Some problems have multiple time scales. A chemical reaction, for example, may begin slowly and the concentration changes little over a long time; then, the reaction may suddenly go to completion with a large change in concentration over a short time. There are two time scales involved in such a process. Other examples are in the life sciences, where multi-scale processes are the norm. Spatial scales vary over as much as 1015 orders of magnitude as we progress from processes involving genes, proteins, cells, organs, organisms, communities, and ecosystems; time scales vary from times that it takes for protein to fold to times for evolution to occur. Several scales can occur in the same problem. Yet another example occurs in fluid flow, where the processes of heat diffusion, advection, and possible chemical reaction all have different scales.

Once a characteristic time has been identified, at least for a part of a process, then a new dimensionless variable can be defined by

If tc is chosen correctly, then the dimensionless time is neither too large nor too small, but rather of order unity. The question remains to determine the time scale tc for a particular problem, and the same question applies to other variables in the problem (e.g., length, concentration, and so on). The general rule is that the characteristic quantities are formed by taking combinations of the various dimensional constants in the problem and should be roughly the same order of magnitude of the quantity itself.

After characteristic scales, which are built up from the parameters in the model, are chosen for the independent and dependent variables, the model can then be reformulated in terms of the new dimensionless variables. The result will be a model in dimensionless form, where all the variables and parameters in the problem are dimensionless. This process is called non-dimensionalization, or scaling a problem. By the Pi theorem, it is guaranteed that we can always non-dimensionalize a consistent, unit-free problem. The payoff is a simpler model that is independent of units and dimensions and that has fewer parameters, which is often a worthwhile economy of complication.

Example 1.11