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Training Manual on Transport and Fluids

*OHN#.EU

'RADUATE3TUDIES IN-ATHEMATICS 6OLUME

!MERICAN-ATHEMATICAL3OCIETY http://dx.doi.org/10.1090/gsm/109

Training Manual on Transport and Fluids

Training Manual on Transport and Fluids

John C. Neu

Graduate Studies in Mathematics Volume 109

American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Steven G. Krantz Rafe Mazzeo Martin Scharlemann

2000 Mathematics Subject Classification. Primary 35–XX, 44–XX, 76–XX.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-109

Library of Congress Cataloging-in-Publication Data Neu, John C., 1952– Training manual on transport and fluids / John C. Neu. p. cm. — (Graduate studies in mathematics ; v. 109) ISBN 978-0-8218-4083-2 (alk. paper) 1. Transport theory—Mathematical models. 2. Fluid mechanics—Mathematical models. I. Title. QC718.5.T7N48 2998 530.138—dc22 2009028296

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2010 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 151413121110 Contents

Preface xi

Part 1. Transport processes: the basic prototypes

Chapter 1. Convection 3 Material volumes and transport theorem 3 Problem 1.1: Convection along a one-dimensional line 5 Problem 1.2: Jacobian of flow map 6 Problem 1.3: Transport theorem by change of variable 8 Problem 1.4: Time derivative of integral over a changing region 9 The convection PDE 10 Problem 1.5: Density carried by incompressible flow 10 Problem 1.6: Transformation of velocities as a flow map 11 Convective flux and boundary conditions 12 Elastic rebounds from a moving wall 13 Problem 1.7: Adiabatic expansion of particles in a box I 15 Problem 1.8: Adiabatic expansion of particles in a box II 15 Problem 1.9: Adiabatic expansion of particles in a box III 16 Problem 1.10: Adiabatic expansion of particles in a box IV 17 Convective derivative 18 Convected scalars 19 Problem 1.11: A convected scalar 19 Gradient of a convected scalar 20 Problem 1.12: Flows that preserve gradients 22 Problem 1.13: Vector analog of a convected scalar 24 Problem 1.14: Vector analogs of a convected density 25 Geometric postscript 29

v vi Contents

Guide to bibliography 30

Chapter 2. Diffusion 31 Brownian motion 31 The diffusion PDE 33 Problem 2.1: Lattice random walk model of diffusion I 35 Problem 2.2: Lattice random walk model of diffusion II 36 Problem 2.3: Lattice random walk model of diffusion III 37 Problem 2.4: Lattice random walk model of diffusion IV 38 Problem 2.5: Lattice random walk model of diffusion V 38 Diffusion in Rn 39 Fick’s law 39 Problem 2.6: Lyapunov functional for a diffusion process 40 Problem 2.7: The Maxwell flux 41 Problem 2.8: Steady state in two dimensions? 42 Melting interfaces 42 Problem 2.9: Nonlinear free boundary problem I 45 Problem 2.10: Nonlinear free boundary problem II 46 Problem 2.11: Nonlinear free boundary problem III 47 Problem 2.12: Nonlinear free boundary problem IV 48 The diffusion kernel 49 Scale invariance of the diffusion kernel 50 Problem 2.13: R2 diffusion kernel from symmetry and independence 53 Problem 2.14: Similarity for melting interfaces 54 Problem 2.15: Source of Brownian particles in two dimensions 56 Problem 2.16: Homogeneous harmonic function in a wedge 58 Guide to bibliography 59

Chapter 3. Local conservation laws 61 Definition 61 Convection-diffusion 61 Problem 3.1: A “dissolving” interface 62 Smoluchowski boundary value problem 65 Problem 3.2: Time-independent equilibria 67 Problem 3.3: Entropy functional 68 Problem 3.4: Free energy decreases 70 Problem 3.5: Equilibrium or nonequilibrium in Coulomb potential 70 Locality 72 Sources 75 The steady diffusion flame 76 Problem 3.6: Absorbing boundary condition for diffusion 80 Problem 3.7: An “evaporation” interface driven by chemical reaction 82 Contents vii

Guide to bibliography 84

Part 2. Superposition

Chapter 4. Superposition of point source 87 One space dimension 87 Problem 4.1: Soviet rod factory 89 Three space dimensions 91 Problem 4.2: Superposition solution by Green’s identity 93 Problem 4.3: The smoke plume 94 Problem 4.4: Far downstream limit 96 Sources in spacetime 98 Problem 4.5: Moving source 101 The hottest time of the day 102 Problem 4.6: “Diffusive” damping 105 Problem 4.7: Work against “diffusive friction” 107 Problem 4.8: Oscillatory point source solution in R3 and descent to R2 109 Problem 4.9: Direct analysis of the R2 integral representation 111 Guide to bibliography 113

Chapter 5. δ-functions 115 δ-functions as “shorthand” 116 Problem 5.1: Correctly representing a point load 118 Problem 5.2: Correlation function of colored noise 120 Problem 5.3: A primer on polarized media 125 Problem 5.4: Exit times for Brownian motion 128 δ-sequences 130 Problem 5.5: Green’s identity and inhomogeneous diffusion PDEs 132 “Oscillatory” δ-sequences 134 Problem 5.6: Cancellation by oscillation 136 Problem 5.7: An oscillatory δ-sequence 138 Guide to bibliography 140

Part 3. Scaling-based reductions in basic fluid mechanics

Chapter 6. Ideal fluid mechanics 143 Ideal fluid 143 Problem 6.1: Tornado season 147 Problem 6.2: Ideal gas equation of state 148 Problem 6.3: Bernoulli’s theorem and flying over Dagoba 150 Problem 6.4: Circulation and vorticity 152 Problem 6.5: Heating due to shock wave 153 viii Contents

Stable rest state of ideal fluid 157 Finite propagation speed and wavefronts 158 Problem 6.6: Sonic boom 160 Incompressible fluid 165 Problem 6.7: Two-dimensional vortex dynamics 166 Problem 6.8: The “ground effect” 169 Problem 6.9: Elliptical vortices 172 The incompressible limit of compressible flow 176 Problem 6.10: 178 Problem 6.11: Internal waves 181 Guide to bibliography 184

Chapter 7. Free surface waves 187 Basic equations 187 Linearized waves 189 Plane waves and the relation 190 Problem 7.1: Internal waves, again 193 197 Problem 7.2: Variable-depth and “slow currents” 200 Problem 7.3: What bottom topography does to vortices 202 Problem 7.4: A cautionary tale about point vortices 204 Guide to bibliography 207

Chapter 8. Solution of the shallow water equations 209 Simple waves 209 Problem 8.1: Simple waves in ideal fluid 212 Problem 8.2: Longitudinal elastic waves 215 Characteristics and Riemann invariants 219 Problem 8.3: Characteristics and Riemann invariants for elastic waves 222 Problem 8.4: Beginning of a 224 Dam break problem 227 Problem 8.5: Wave generator 228 Bores 231 Problem 8.6: “Potential energy” in shallow water theory 234 Problem 8.7: Elastic shocks 235 Characteristics “eaten” by a bore 238 The afterlife of a simple wave 239 Propagation of a weak bore 240 Problem 8.8: Area rule for Burger’s shocks 242 Problem 8.9: Traveling waves in variable-depth ocean 246 Problem 8.10: Tsunami reveals its true self 251 Guide to bibliography 255 Contents ix

Bibliography 257 Index 259

Preface

This text presents models of transport in continuous media and a corre- sponding body of mathematical techniques. Within this text, I have em- bedded a subtext of problems. Topics and problems are listed together in the table of contents. Each problem is followed by a detailed solution em- phasizing process and craftsmanship. These problems and solutions express the practice of applied mathematics as the examination and re-examination of essential ideas in many interrelated examples. Since the science that falls under the headings “transport” or “fluids” is so broad, this introductory text for a one-term advanced undergraduate or beginning graduate course must select a highly specific path. The main requirement is that topics and exercises be logically interconnected and form a self-contained whole. Briefly, the physical topics are: convection and diffusion as the sim- plest models of transport, local conservation laws with sources as a general “frame” of continuum mechanics, ideal fluid as the simplest example of an actual physical medium with mass, momentum and energy transport, and finally, free surface theory. The idea behind this lineup is the progression from purely geometric and kinematic to genuinely physical. The mathematical prerequisites for engaging the practice of this text are: fluency in advanced calculus and vector analysis, and acquaintance with PDEs from an introductory undergraduate course. The mathematical skills developed in this text have two tracks: First, classical constructions of solutions to linear PDEs and related tools, such as the Dirac δ-function, are presented with a relentless sense of connection to

xi xii Preface the geometric-physical situations they articulate. Second, and more essen- tial, is the emphasis on dimensional analysis and scaling. Some topics, such as physical similarity and similarity solutions, are traditional. In addition, there are asymptotic reductions based on scaling, such as incompressible flow as a limit of compressible flow, shallow water theory from the full equa- tions of free surface waves, and further reduction of shallow water theory to an asymptotic model of a tsunami approaching the shore. The art of scaling reduction is introduced in the very first chapter, and the drum beat of examples and problems persists throughout the body of the text. I am grateful to several people who assisted me with the development of this text. Craig Evans asked me to write this book in the first place, and Sergei Gelfand, the editor, reinforced and upheld this request. I thank Tom Beale for discussions and suggestions regarding the title. Katherine Schwarz, a student in my Fall 2004 graduate course on which this book is based, developed the LaTeX version of my lecture notes and figures, many of which are included here. Yu Tanouchi typed the revised and extended version, bringing the book into its present form. Rahel Wachs and Yossi Farjoun made the computerized versions of the figures. Bianca Cerchiai reviewed the book close to its final form and offered many thoughtful suggestions. Finally, I acknowledge my wife, Wanda Krassowska Neu, for her help and support during this work. The period of writing includes the years from 2005 to 2009, during which I received support from the National Science Foundation grant DMS- 0515616. Bibliography

[1] G.I. Barenblatt (1996) Scaling, Self-Similarity, and Intermediate Asymptotics, Cambridge University Press, Cambridge, UK. [2] G.K. Batchelor (1967) An Introduction to Fluid Mechanics, Cambridge University Press, Cambridge, UK. [3] Y.B. Chernyak, R.M. Rose (1995) The Chicken from Minsk and 99 Other Infuri- atingly Challenging Brainteasers from the Great Russian Tradition of Math and Science, Basic Books, New York, NY. [4] A.J. Chorin, J.E. Mardsen (1988) A Mathematical Introduction to Fluid Mechan- ics, Springer, New York, NY. [5] A.J. Chorin, O.H. Hald (2006) Stochastic Tools in Mathematics and Science, Springer, New York, NY. [6] R.V. Churchill, J.W. Brown (1987) Fourier Series and Boundary Value Problems, McGraw-Hill, New York, NY. [7] R. Courant, K.O. Friedrichs (1999) Supersonic Flow and Shock Waves, Springer, New York, NY. [8] R. Courant, D. Hilbert (1953) Methods of Mathematical Physics, Volume I,John Wiley & Sons, New York, NY. [9] R. Courant, D. Hilbert (1989) Methods of Mathematical Physics, Volume II,John Wiley & Sons, New York, NY. [10] A. Erd´elyi (1962) Operational Calculus and Generalized Functions,Holt,Rhine- hart and Winston, New York, NY. [11] L.C. Evans (1998) Partial Differential Equations, American Mathematical Society, Providence, RI. [12] R.P. Feynman, R.B. Leighton, M. Sands (1963) The Feynman Lectures on Physics, Volume I, Addison-Wesley, Reading, MA. [13] R.P. Feynman, R.B. Leighton, M. Sands (1963) The Feynman Lectures on Physics, Volume II, Addison-Wesley, Reading, MA. [14] I.M. Gel’fand, G.E. Shilov (1964) Generalized Functions, Volume I: Properties and Operations, Academic Press, New York, NY and London, UK.

257 258 Bibliography

[15] F. John (1982) Partial Differential Equations, Springer, New York, NY. [16] L.D. Landau, E.M. Lifshitz (1976) Mechanics. Course of Theoretical Physics, Vol- ume 1, 3rd edition, Butterworth-Heinemann, Oxford, UK. [17] L.D. Landau, E.M. Lifshitz (1959) Fluid Mechanics. Course of Theoretical Physics, Volume 6, Pergamon Press, Oxford, UK. [18] J. Lighthill (1978) Waves in Fluids, Cambridge University Press, Cambridge, UK. [19] M.J. Lighthill (1958) Introduction to Fourier Analysis and Generalised Functions, Cambridge University Press, Cambridge, UK. [20] H. Ockendon, J.R. Ockendon (2004) Waves and Compressible Flow, Springer, New York, NY. [21] P.G. Saffman (1992) Vortex Dynamics, Cambridge University Press, Cambridge, UK. [22] J.J. Stoker (1957) Water Waves, Interscience Publishers, New York, NY. [23] N. Wax, ed. (1954) Selected Papers on Noise and Stochastic Processes, Dover, New York, NY. [24] G.B. Whitham (1974) Linear and Nonlinear Waves, John Wiley & Sons, New York, NY. Index

absorbing surface, 80 asymptotic continuity of Riemann acoustic wave, 214 invariant across weak bore, see adiabatic expansion, 14 Riemann invariant air following precursor trough of tsunami, density, 147 255 oversaturated, 151 perturbation analysis of weak bores, 240 , -level, 148 propagating into undisturbed ocean, 233 Airy function, 138 turbulent, 231 arc length, 119 wake of, 233, 240 asymptotic, symbol for, 55 weak, 240 world line of, 238, 241 bore velocity, 232 balance asymptotic velocity of weak bore, 242 between acceleration and pressure force bottom topography, 181, 200 per unit mass, 198 effect on vortices, 202 between centripetal acceleration and slowly varying, 246 pressure force, 148 boundary condition between diffusive transport and absorbing boundary condition for absorption, 82 diffusion, 80, 128 between external and drag , 65 associated with convection PDE, 12 between kinetic and potential energies, bottom, 187 250 dynamical, 188, 190 between load and tension, 119 for periodic heating and cooling, 103 between nonlinearity and kinematic, 187, 189, 199 variable-bottom effects, 246 linearized, 190 of units, 78, 81 looking like initial condition, 163 Bernoulli’s theorem, 150 no penetration, 63, 65, 182 Bessel functions, integral representation of, boundary value problem, Neumann, for 111 pressure field of incompressible flow, Boltzmann 166 constant, 65, 149 Brownian motion, 31, 87 factor, 65 as integral of white noise, 121 from minimization of free energy, 66 average exit time, 128–130 bore, 231 distribution of exit times, 129 absorbs characteristics, 239 homogeneity and isotropy of, 33 as energy sink, 233 Brownian particles, 39, 49

259 260 Index

source of in two dimensions, 56 convection buoyancy, 178, 180 constitutive relation for, 61 force, 65, 178, 180 PDE, 61 Burger’s equation, 242 convection-diffusion as pedagogical model of shocks, 242 in Smoluchowski PDE, 65 in model of tsunami, 251 PDE, 62, 65 Burger’s model, as asymptotic limit of for smoke plume, 94 small-amplitude shallow water wave, convective 242 derivative, 18, 20, 145 Burger’s shock, 242 flux, 12 convolution, 131 capacitance, 73 correlation function, 120 capacitor, 73 of colored noise, 120, 122 characteristic polynomial, 7 Coulomb interactions, singular limits of, characteristics, 219 207 absorbed by a bore, 238 density, 72, 73 crossing a bore, 238 definition of, 220 dam break problem, 227 extrapolated backward in time, 229 characteristics in, see characteristics for elastic waves, 222 δ-function, 115, 116 for initial value problem of shallow water as shorthand, 115, 116 equations, 220 definition of, 115 in dam break problem, 227 differentiation of point source solution, information carried by, 230 124 charge explanation of superposition, 117 point, see point charge identity, 117 transport, 73 δ-sequence, 115, 123, 130 charge density, 72, 125 derived from scaling symmetry, 137 due to superposition of dipoles, 125 discrete, 135 of point dipole, 125, 127 oscillatory, 134, 137, 138 surface charge density, 73 density, 61 on boundary of polar medium, 126 carried by incompressible flow, 10 circular flow, steady, 147 perturbation of, 157 circulation, 147, 152 descent, from R3 to R2, 109, 204 theorem, 202 diffusion collision rate, 76 absorbing boundary condition, see colored noise, 121 boundary condition asymptotic to white noise, 123 coefficient, 32 compressible flow, incompressible limit of, lattice random walk model for, 38 176 constitutive relation for, 61 compression wave, 224 equation, 49 conductivity, see electrical conductivity from a time-dependent point source, 102 conductor, see electrical conductor in Rn,39 conservation PDE, 33, 34, 40, 61 bulk conservation identity, 4, 9 backwards, 132, 133 local conservation laws, 61 inhomogeneous, 118, 132 of energy, see local conservation spherically symmetric, 109 of mass, see local conservation diffusion flame, 76 of momentum, see local conservation boundary value problem for, 77 constitutive relations, 61 diffusion kernel, 49, 118 continuum mechanics, 61, 75 as a δ-sequence, 130, 131 convected from symmetry and independence, 53 density, 10, 19, 22, 153 in R2,54 scalar, 19, 20 in smoke plume solution, 96, 97 vector analogs of, see vector analogs in solution of initial value problem, 52 vector, 25, 28 initial value problem for, 38 vector PDE, 27 superposition of diffusion kernels, 124 Index 261

diffusive density, in position-velocity phase space, damping, 105 14 flux, 39, 88 entropy flux, continuity of, 46 definition of, 66, 68 diffusivity, thermal, 42 functional, 68 dilute solution, 62, 87 of a composite system, 68 dipole per unit mass, 19 charge density of point dipole, see charge equation of state, 144 density equilibrium or nonequilibrium, in Coulomb definition of, 125 potential, 70 moment, 125, 127 equilibrium thermodynamics, 65 point dipole field, 125 Euler constant, 206 potential, 125, 126 Euler equation Dirac δ-function, see δ-function definition of, 146 discrete initial value problem, for lattice incompressible, 178 random walk, 35–37 linearized, 163, 195, 196 dispersion relation, 190, 191 steady, 147 domain of dependence, 161, 163 Fick’s law, 39, 40 of influence, 158, 221 flood wave, 227 drag flow coefficient, 65 about wing, 161 force, 65 supersonic, 160 drift velocity, 65 flow map, 3, 4, 10, 11, 16 fluid electrically conducting, see electrically eigenfunction, 49 conducting fluid ground state, 49 incompressible, see incompressible fluid eigenvalue, 49 stably stratified, 181, 193 problem, 49, 58, 181 flux elastic condition, due to point source in R2, 111 material diffusive, see diffusive flux hard, 216 for convection-diffusion, 62 soft, 216, 219 for smoke plume, 94 medium, 215 magnetic, 25 shock, 235 Maxwell, 41 wave, 215, 222 of charge, 72 wave, simple, 216 of energy, 233 waves, characteristics for, see of particles, 88, 91 characteristics vector, 61 electric field, integral curves of, 73 fluxes, of area and momentum in shallow electric potential field, 125 water theory, see shallow water theory electrical force conductivity, 73 buoyancy, see buoyancy force conductor, 73 drag, see drag force neutrality, 73 friction, see friction force electrically conducting gravity, see gravity force fluid, 25 per unit mass, arising from pressure medium, 72 gradient, 146 electrophysiology, 72 per unit volume, as source of momentum energy density, 75 absorption, in turbulent bore, 233 pressure, see pressure force density, 233, 235 forms, 29 dissipation, in elastic shock, 236 density as three-form, 29 internal, see internal energy one-form, 29 ensemble three-form, 29 averaging, 120 two-form, 29 262 Index

Fourier integral curves, 25, 27 analysis, 135 of electric field, see electric field series, 108, 134 of velocity field, see velocity field free boundary problem, 43, 46 integral representation nonlinear, 45 of moving source solution, 102 quasistatic limit, 43 of solution to diffusion PDE with source, free energy, 66 99 decrease of, 70 integral, over changing region, 9 free surface, 187 interface continuity of pressure across, 194, 195 between dense and light fluid, 193 waves, 187 evaporation interface, 82 friction internal “diffusive friction”, 106, 107 energy, 144 force, 106 wave, 181, 193 streamlines of, 194 Galilean invariance, 160 isotropy, 32 generalized stream function, 202 gradient, of convected scalar, 20 Jacobian determinant, 4, 6, 7 gravity force, 75, 178, 187 joint distribution, 32, 39 Green’s identity, 92, 93 jump condition, 116, 126 in analysis of diffusion PDE with source, induced by moving point source, 102 99, 132 jump conditions ground effect, 169, 207 across a bore, 239 across shock, 154 harmonic function jump discontinuity, 88, 89, 116 circularly symmetric, 42 in , 46 homogeneous, 57, 59 in normal derivative, 159 spherically symmetric, 41, 44 heat latent heat of fusion, 42 content, 42 lattice random walk, as model of diffusion, source, time-periodic, 102 35 transfer, 42, 144, 146 line source, 124 boundary condition, 43, 44 linearization, 157, 161 homogeneity, of Brownian motion, 32 load, compressive, 236 homogeneous function, definition of, 58 local conservation hottest time of the day, 102, 103 of energy, 235 hydrostatic of energy, in shallow water theory, 233 approximation in shallow water theory, of mass, 143 199 of mass, momentum, and energy in ideal equilibrium, 178 fluid, 145 pressure, 178, 198 of momentum, 236 locality, 72 ideal fluid, 143 Lyapunov functional, 41 mechanics, 143 for diffusion process, 40 steady flow, 160 ideal gas, 19 Mach equation of state, 148 cone, 160, 164 velocity distribution in, 148 number, 162 incompressibility, 167 mass incompressible action, 76 flow, 10, 19, 22 balance, 144 fluid, 63, 165, 207 density, 143 incompressible limit material of compressible flow, see compressible cross-section, 28 flow curve, 24, 25, 28, 152 scaling for, 177 interval, 5 independence, statistical, 120 patch, 4, 27 Index 263

planes, 216 in incompressible two-dimensional point, 17, 19, 21 fluid, 20 on surface, 187 limit, 170 region, 4, 6, 8, 143 Poisson’s equation, 125, 166 shrinking to a material point, 18 in polar medium, 127 surface, 10, 25 polar medium volume,3,4,9 charge density in, see charge density shrinking to a material point, 5 Poisson’s equation for, see Poisson’s Maxwell flux, 41 equation mean polarization displacement, 32 uniform, 126 square displacement, 32, 33, 50 vector field, 125 melting polarized medium, 125 interface, 42, 43, 54 potential energy , 42, 43 in shallow water theory, 234 time, 56 per unit mass in ideal fluid, 144 moments, 34, 54 potential, spherically symmetric, 125 higher, 33 pressure, 143 momentum at , see air pressure balance, 144 change, in vortex, 147 density, 236 force, 143 hydrostatic, see hydrostatic pressure Newtonian mechanics, 143 in incompressible fluid, 165 noise propagation colored, see colored noise instantaneous, 166 white, see white noise of information, finite speed, 158 nondimensionalization, 43, 44, 46, 96, 179 normal relaxation time, 44, 72 derivative, 40 rest frame velocity, 9, 43, 63, 159 of material point, 62 normalization condition, 72, 90 of shock, 154 Riemann oscillatory source solution, of diffusion invariant, 219 PDE in R2, 112 asymptotic continuity across weak bore, 241 particle orbits, in phase plane, 17 invariants penetration depth, 105 for elastic waves, 222 perturbation, of density, see density level curves of, 219 phase rod factory lag, 104, 107, 111 Soviet, 89 due to diffusive friction, 106 shift, 112 scale invariance, 50, 180 speed, 191 scaling table, 43, 79, 198 phase plane, 13 for incompressible limit, see particle orbits in, see particle orbits incompressible limit plane wave, 190 for shallow water theory, see shallow deep water limit, 191 water theory linearized, 191 scaling-based reduction, 143, 178, 187 long wave limit, 197 Schr¨odinger PDE, 138 point similarity solution, see similarity solution charge, 125 separation of variables, 48, 103, 183 load, induces jump in derivative, 119 shallow water source, definition of, 115 analogous to two-dimensional ideal fluid, vortex, 167, 207 199 cautionary tale about point vortices, approximation, 207 204 shallow water equations, 211 flows, superposition of, 169 one-dimensional solution of, 209 264 Index

shallow water theory sources, 75 conservation of energy, see local specific volume, 149 conservation state variables, 61, 143 fluxes of area and momentum, 232 stationary wave, 181, 193, 195 hydrostatic approximation, see strain hydrostatic approximation definition of, 215 in variable-depth ocean, 200 flow, 172 local conservation of area and matrix, 22 momentum, 232 stream function potential energy, see potential energy defined by line integral, 167 scaling for, 198 generalized, 203 slow current limit, 200 streamlines volume flux, see volume flux as integral curves of velocity field, 196 shock, 245 definition of, 150 as traveling sink of energy, 237 for steady flow about a wing, 151 elastic, see elastic shock kinks due to jump in normal velocity, 164 energy dissipation in, see energy thin-wing limit, 161 dissipation stress, 143, 215 fitting, 242 and Newton’s third law, 215 according to area rule, 243 imposedonsurface,222 leading edge, 164 stress-strain relation, 235 pre-shock speed of sound, 157 superposition, 92, 98, 118 strength, 156 of diffusion kernels, see diffusion kernels trailing edge, 165 of point source solutions, 87, 92 wave, heating induced by, 153 in R3, 110 shot problem, 222 in spacetime, 98 similar graphs, 51 superposition solution, analyzed by Green’s similarity solution, 51, 52 identity, 93 of Schr¨odinger PDE, 138 surface related to diffusion kernel, 139 charge density, see charge density of Smoluchowski PDE in Coulomb density, 13 potential, 70 simple wave, 209, 219, 231 afterlife of, 239 temperature definition of, 209 absolute, 149 level curves of, 211 field, 102 simple waves jump, across shock, 154 in ideal fluid, 212 post-shock, 156 slow current limit, in shallow water theory, test function, 130, 132 see shallow water theory thermal slow currents, in variable-depth ocean, 200 energy, conservation of, 42 smoke plume, 94 random motion, 144 convection-diffusion PDE for, see thermodynamic equilibrium convection-diffusion PDE at evaporation interface, 82 diffusion kernel in, see diffusion kernel of particles in solution, 65 far downstream limit, 96 relaxation to, 67 flux for, see flux thermodynamics, 75 Smoluchowski equilibrium, see equilibrium boundary value problem, 65 thermodynamics model of particle transport, 65 first law of, 146 sonic boom, 160 tornado, 147 sound, speed of, 157, 162, 176 transport theorem, 4, 9 source tsunami as sum of point sources, 117 and earthquake, 224 line, see line source approaching the shore, 251, 253 moving, 101 beginning of, 224, 252 point, see point source leading edge of, 251 Index 265

, in precursor to tsunami bore, 252 vector analogs of a convected density, 25, 152, 153 of a convected scalar, 24 velocity field, integral curves of, 150 vertical tangent in graph of gas density, 212 in graph of simple wave, 211 volume flux, in shallow water theory, 201 vortex definition of, 147 dynamics, of two-dimensional incompressible flow, 166 elliptical, 172 image, 171 point, see point vortex traveling, 169 vorticity, 152 center of, 169 evolution equation, 202 PDE, 153 scalar, 167 water vapor, 150 water wave, particle kinematics of, 191, 192 wave acoustic, see acoustic wave elastic, see elastic wave equation, for acoustics, 158 flood, see flood wave free surface, see free surface waves generator, 228 induced by wing in supersonic flow, 161 internal, see plane, see plane wave rising as it nears the shore, 249 shock, see shock wave simple, see simple wave stationary, see stationary wave wavefront definition of, 159 in elastic medium, 215 in ideal compressible fluid, 160 stationary, 160 wavenumber, 181 weak solution, 153, 242 white noise, 120 work against diffusive friction, 107 done by collisions, 17 done by pressure force, 144 done by stress, 144

Titles in This Series

111 Simon Brendle, Ricci flow and the sphere theorem, 2010 110 Matthias Kreck, Differential algebraic topology: From stratifolds to exotic spheres, 2010 109 John C. Neu, Training manual on transport and fluids, 2010 108 Enrique Outerelo and Jes´us M. Ruiz, Mapping degree theory, 2009 107 Jeffrey M. Lee, Manifolds and differential geometry, 2009 106 Robert J. Daverman and Gerard A. Venema, Embeddings in manifolds, 2009 105 Giovanni Leoni, A first course in Sobolev spaces, 2009 104 Paolo Aluffi, Algebra: Chapter 0, 2009 103 Branko Gr¨unbaum, Configurations of points and lines, 2009 102 Mark A. Pinsky, Introduction to Fourier analysis and wavelets, 2009 101 Ward Cheney and Will Light, A course in approximation theory, 2009 100 I. Martin Isaacs, Algebra: A graduate course, 2009 99 Gerald Teschl, Mathematical methods in quantum mechanics: With applications to Schr¨odinger operators, 2009 98 Alexander I. Bobenko and Yuri B. Suris, Discrete differential geometry: Integrable structure, 2008 97 David C. Ullrich, Complex made simple, 2008 96 N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, 2008 95 Leon A. Takhtajan, Quantum mechanics for mathematicians, 2008 94 James E. Humphreys, Representations of semisimple Lie algebras in the BGG category O, 2008 93 Peter W. Michor, Topics in differential geometry, 2008 92 I. Martin Isaacs, Finite group theory, 2008 91 Louis Halle Rowen, Graduate algebra: Noncommutative view, 2008 90 Larry J. Gerstein, Basic quadratic forms, 2008 89 Anthony Bonato, A course on the web graph, 2008 88 Nathanial P. Brown and Narutaka Ozawa, C∗-algebras and finite-dimensional approximations, 2008 87 Srikanth B. Iyengar, Graham J. Leuschke, Anton Leykin, Claudia Miller, Ezra Miller, Anurag K. Singh, and Uli Walther, Twenty-four hours of local cohomology, 2007 86 Yulij Ilyashenko and Sergei Yakovenko, Lectures on analytic differential equations, 2007 85 John M. Alongi and Gail S. Nelson, Recurrence and topology, 2007 84 Charalambos D. Aliprantis and Rabee Tourky, Cones and duality, 2007 83 Wolfgang Ebeling, Functions of several complex variables and their singularities (translated by Philip G. Spain), 2007 82 Serge Alinhac and Patrick G´erard, Pseudo-differential operators and the Nash–Moser theorem (translated by Stephen S. Wilson), 2007 81 V. V. Prasolov, Elements of homology theory, 2007 80 Davar Khoshnevisan, Probability, 2007 79 William Stein, Modular forms, a computational approach (with an appendix by Paul E. Gunnells), 2007 78 Harry Dym, Linear algebra in action, 2007 77 Bennett Chow, Peng Lu, and Lei Ni, Hamilton’s Ricci flow, 2006 76 Michael E. Taylor, Measure theory and integration, 2006 75 Peter D. Miller, Applied asymptotic analysis, 2006 74 V. V. Prasolov, Elements of combinatorial and differential topology, 2006 73 Louis Halle Rowen, Graduate algebra: Commutative view, 2006 72 R. J. Williams, Introduction the the mathematics of finance, 2006 TITLES IN THIS SERIES

71 S. P. Novikov and I. A. Taimanov, Modern geometric structures and fields, 2006 70 Se´an Dineen, Probability theory in finance, 2005 69 Sebasti´an Montiel and Antonio Ros, Curves and surfaces, 2005 68 Luis Caffarelli and Sandro Salsa, A geometric approach to free boundary problems, 2005 67 T.Y. Lam, Introduction to quadratic forms over fields, 2004 66 Yuli Eidelman, Vitali Milman, and Antonis Tsolomitis, Functional analysis, An introduction, 2004 65 S. Ramanan, Global calculus, 2004 64 A. A. Kirillov, Lectures on the orbit method, 2004 63 Steven Dale Cutkosky, Resolution of singularities, 2004 62 T. W. K¨orner, A companion to analysis: A second first and first second course in analysis, 2004 61 Thomas A. Ivey and J. M. Landsberg, Cartan for beginners: Differential geometry via moving frames and exterior differential systems, 2003 60 Alberto Candel and Lawrence Conlon, Foliations II, 2003 59 Steven H. Weintraub, Representation theory of finite groups: algebra and arithmetic, 2003 58 C´edric Villani, Topics in optimal transportation, 2003 57 Robert Plato, Concise numerical mathematics, 2003 56 E. B. Vinberg, A course in algebra, 2003 55 C. Herbert Clemens, A scrapbook of complex curve theory, second edition, 2003 54 Alexander Barvinok, A course in convexity, 2002 53 Henryk Iwaniec, Spectral methods of automorphic forms, 2002 52 Ilka Agricola and Thomas Friedrich, Global analysis: Differential forms in analysis, geometry and physics, 2002 51 Y. A. Abramovich and C. D. Aliprantis, Problems in operator theory, 2002 50 Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory, 2002 49 John R. Harper, Secondary cohomology operations, 2002 48 Y. Eliashberg and N. Mishachev, Introduction to the h-principle, 2002 47 A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi, Classical and quantum computation, 2002 46 Joseph L. Taylor, Several complex variables with connections to algebraic geometry and Lie groups, 2002 45 Inder K. Rana, An introduction to measure and integration, second edition, 2002 44 Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, 2002 43 N. V. Krylov, Introduction to the theory of random processes, 2002 42 Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, 2002 41 Georgi V. Smirnov, Introduction to the theory of differential inclusions, 2002 40 Robert E. Greene and Steven G. Krantz, Function theory of one complex variable, third edition, 2006 39 Larry C. Grove, Classical groups and geometric algebra, 2002 38 Elton P. Hsu, Stochastic analysis on manifolds, 2002 37 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, 2001 36 Martin Schechter, Principles of functional analysis, second edition, 2002 35 James F. Davis and Paul Kirk, Lecture notes in algebraic topology, 2001

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/. I have learned a lot from John Neu over the past years, and his book reflects very well his sense of style and purpose. —Walter Craig, McMaster University, Hamilton, Ontario, Canada and Fields Institute for Research in Mathematical Sciences, Toronto, Ontario, Canada This book presents elementary models of transport in continuous media and a corresponding body of mathematical technique. Physical topics include convec- tion and as the simplest models of transport; local conservation laws with sources as the general framework of continuum mechanics; ideal fluid as the simplest model of a medium with mass; momentum and energy transport; and finally, free surface waves, in particular, shallow water theory. There is a strong emphasis on dimensional analysis and scaling. Some topics, such as physical similarity and similarity solutions, are traditional. In addition, there are reductions based on scaling, such as incompressible flow as a limit of compressible flow, and shallow water theory derived asymptotically from the full equations of free surface waves. More and deeper examples are presented as problems, including a series of problems that model a tsunami approaching the shore. The problems form an embedded subtext to the book. Each problem is followed by a detailed solution emphasizing process and craftsmanship. The problems express the practice of applied mathematics as the examination and re-examination of simple but essential ideas in many interrelated examples.

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