Practical Applied Mathematics Modelling, Analysis, Approximation Sam Howison OCIAM Mathematical Institute Oxford University October 10, 2003 2 Contents 1 Introduction 9 1.1 What is modelling/why model? . . . . . . . . . . . . . . . . . 9 1.2 How to use this book . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . 9 I Modelling techniques 11 2 The basics of modelling 13 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 What do we mean by a model? . . . . . . . . . . . . . . . . . 14 2.3 Principles of modelling . . . . . . . . . . . . . . . . . . . . . . 16 2.3.1 Example: inviscid fluid mechanics . . . . . . . . . . . . 17 2.3.2 Example: viscous fluids . . . . . . . . . . . . . . . . . . 18 2.4 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 Units and dimensions 25 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Units and dimensions . . . . . . . . . . . . . . . . . . . . . . . 25 3.2.1 Example: heat flow . . . . . . . . . . . . . . . . . . . . 27 3.3 Electric fields and electrostatics . . . . . . . . . . . . . . . . . 28 4 Dimensional analysis 39 4.1 Nondimensionalisation . . . . . . . . . . . . . . . . . . . . . . 39 4.1.1 Example: advection-diffusion . . . . . . . . . . . . . . 39 4.1.2 Example: the damped pendulum . . . . . . . . . . . . 43 4.1.3 Example: beams and strings . . . . . . . . . . . . . . . 45 4.2 The Navier–Stokes equations . . . . . . . . . . . . . . . . . . . 47 4.2.1 Water in the bathtub . . . . . . . . . . . . . . . . . . . 50 4.3 Buckingham’s Pi-theorem . . . . . . . . . . . . . . . . . . . . 51 3 4 CONTENTS 4.4 Onwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5 Case study: hair modelling and cable laying 61 5.1 The Euler–Bernoulli model for a beam . . . . . . . . . . . . . 61 5.2 Hair modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.3 Cable-laying . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.4 Modelling and analysis . . . . . . . . . . . . . . . . . . . . . . 65 5.4.1 Boundary conditions . . . . . . . . . . . . . . . . . . . 67 5.4.2 Effective forces and nondimensionalisation . . . . . . . 67 6 Case study: the thermistor 1 73 6.1 Thermistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.1.1 A simple model . . . . . . . . . . . . . . . . . . . . . . 73 6.2 Nondimensionalisation . . . . . . . . . . . . . . . . . . . . . . 75 6.3 A thermistor in a circuit . . . . . . . . . . . . . . . . . . . . . 77 6.3.1 The one-dimensional model . . . . . . . . . . . . . . . 78 7 Case study: electrostatic painting 83 7.1 Electrostatic painting . . . . . . . . . . . . . . . . . . . . . . . 83 7.2 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . 84 7.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 86 7.4 Nondimensionalisation . . . . . . . . . . . . . . . . . . . . . . 87 II Mathematical techniques 91 8 Partial differential equations 93 8.1 First-order equations . . . . . . . . . . . . . . . . . . . . . . . 93 8.2 Example: Poisson processes . . . . . . . . . . . . . . . . . . . 97 8.3 Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 8.3.1 The Rankine–Hugoniot conditions . . . . . . . . . . . . 101 8.4 Nonlinear equations . . . . . . . . . . . . . . . . . . . . . . . . 102 8.4.1 Example: spray forming . . . . . . . . . . . . . . . . . 102 9 Case study: traffic modelling 105 9.1 Case study: traffic modelling . . . . . . . . . . . . . . . . . . . 105 9.1.1 Local speed-density laws . . . . . . . . . . . . . . . . . 107 9.2 Solutionswithdiscontinuities: shocksandtheRankine–Hugoniot relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 9.2.1 Traffic jams . . . . . . . . . . . . . . . . . . . . . . . . 109 9.2.2 Traffic lights . . . . . . . . . . . . . . . . . . . . . . . . 109 CONTENTS 5 10 The delta function and other distributions 111 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 10.2 A point force on a stretched string; impulses . . . . . . . . . . 112 10.3 Informal definition of the delta and Heaviside functions . . . . 114 10.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 10.4.1 A point force on a wire revisited . . . . . . . . . . . . . 117 10.4.2 Continuous and discrete probability. . . . . . . . . . . 117 10.4.3 The fundamental solution of the heat equation . . . . . 119 10.5 Balancing singularities . . . . . . . . . . . . . . . . . . . . . . 120 10.5.1 The Rankine–Hugoniot conditions . . . . . . . . . . . . 120 10.5.2 Case study: cable-laying . . . . . . . . . . . . . . . . . 121 10.6 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . 122 10.6.1 Ordinary differential equations . . . . . . . . . . . . . . 122 10.6.2 Partial differential equations . . . . . . . . . . . . . . . 125 11 Theory of distributions 137 11.1 Test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 11.2 The action of a test function . . . . . . . . . . . . . . . . . . . 138 11.3 Definition of a distribution . . . . . . . . . . . . . . . . . . . . 139 11.4 Further properties of distributions . . . . . . . . . . . . . . . . 140 11.5 The derivative of a distribution . . . . . . . . . . . . . . . . . 141 11.6 Extensions of the theory of distributions . . . . . . . . . . . . 142 11.6.1 More variables . . . . . . . . . . . . . . . . . . . . . . . 142 11.6.2 Fourier transforms . . . . . . . . . . . . . . . . . . . . 142 12 Case study: the pantograph 155 12.1 What is a pantograph? . . . . . . . . . . . . . . . . . . . . . . 155 12.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 12.2.1 What happens at the contact point? . . . . . . . . . . 158 12.3 Impulsive attachment . . . . . . . . . . . . . . . . . . . . . . . 159 12.4 Solution near a support . . . . . . . . . . . . . . . . . . . . . . 160 12.5 Solution for a whole span . . . . . . . . . . . . . . . . . . . . . 162 III Asymptotic techniques 171 13 Asymptotic expansions 173 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 13.2 Order notation . . . . . . . . . . . . . . . . . . . . . . . . . . 175 13.2.1 Asymptotic sequences and expansions . . . . . . . . . . 177 13.3 Convergence and divergence . . . . . . . . . . . . . . . . . . . 178 6 CONTENTS 14 Regular perturbations/expansions 183 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 14.2 Example: stability of a spacecraft in orbit . . . . . . . . . . . 184 14.3 Linear stability . . . . . . . . . . . . . . . . . . . . . . . . . . 185 14.3.1 Stability of critical points in a phase plane . . . . . . . 186 14.3.2 Example (side track): a system which is neutrally sta- ble but nonlinearly stable (or unstable) . . . . . . . . . 187 14.4 Example: the pendulum . . . . . . . . . . . . . . . . . . . . . 188 14.5 Small perturbations of a boundary . . . . . . . . . . . . . . . 189 14.5.1 Example: flow past a nearly circular cylinder . . . . . . 189 14.5.2 Example: water waves . . . . . . . . . . . . . . . . . . 192 14.6 Caveat expandator . . . . . . . . . . . . . . . . . . . . . . . . 193 15 Case study: electrostatic painting 2 201 15.1 Small parameters in the electropaint model . . . . . . . . . . . 201 16 Case study: piano tuning 207 16.1 The notes of a piano . . . . . . . . . . . . . . . . . . . . . . . 207 16.2 Tuning an ideal piano . . . . . . . . . . . . . . . . . . . . . . . 209 16.3 A real piano . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 17 Methods for oscillators 219 17.0.1 Poincar´e–Linstedt for the pendulum . . . . . . . . . . . 219 18 Boundary layers 223 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 18.2 Functions with boundary layers; matching . . . . . . . . . . . 224 18.2.1 Matching . . . . . . . . . . . . . . . . . . . . . . . . . 225 18.3 Cable laying . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 19 ‘Lubrication theory’ analysis: 231 19.1 ‘Lubrication theory’ approximations: slender geometries . . . . 231 19.2 Heat flow in a bar of variable cross-section . . . . . . . . . . . 232 19.3 Heat flow in a long thin domain with cooling . . . . . . . . . . 235 19.4 Advection-diffusion in a long thin domain . . . . . . . . . . . 237 20 Case study: continuous casting of steel 247 20.1 Continuous casting of steel . . . . . . . . . . . . . . . . . . . . 247 21 Lubrication theory for fluids 253 21.1 Thin fluid layers: classical lubrication theory . . . . . . . . . . 253 21.2 Thin viscous fluid sheets on solid substrates . . . . . . . . . . 256 CONTENTS 7 21.2.1 Viscous fluid spreading horizontally under gravity: in- tuitive argument . . . . . . . . . . . . . . . . . . . . . 256 21.2.2 Viscous fluid spreading under gravity: systematic ar- gument . . . . . . . . . . . . . . . . . . . . . . . . . . 258 21.2.3 A viscous fluid layer on a vertical wall . . . . . . . . . 261 21.3 Thin fluid sheets and fibres . . . . . . . . . . . . . . . . . . . . 261 21.3.1 The viscous sheet equations by a systematic argument 263 21.4 The beam equation (?) . . . . . . . . . . . . . . . . . . . . . . 266 22 Ray theory and other ‘exponential’ approaches 277 22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 23 Case study: the thermistor 2 281 8 CONTENTS Chapter 1 Introduction Book born out of fascination with applied math as meeting place of physical world and mathematical structures. have to be generalists, anything and everything potentially interesting to an applied mathematician 1.1 What is modelling/why model? 1.2 How to use this book case studies as strands must do exercises 1.3 acknowledgements Have taken examples from many sources, old examples often the best. If you teach a course using other peoples’ books and then write your own this is inevitable. errors all my own ACF, Fowkes/Mahoney, O2, green book, Hinch, ABT, study groups Conventions. Let me introduce a couple of conventions that I use in this book. I use ‘we’, as in ‘we can solve this by a Laplace transform’, to signal the usualpolite fictionthatyou, thereader, andI,theauthor, areengagedon a joint voyage of discovery. ‘You’ is mostly used to suggest that you should get your pen out and work though some of the ‘we’ stuff, a good idea in view 9 10 CHAPTER 1. INTRODUCTION of my fallible arithmetic. ‘I’ is associated with authorial opinions and can mostly be ignored if you like. I have tried to draw together a lot of threads in this book, and in writing itIhaveconstantlyfelttheneedtosidestepinordertopointoutaconnection with something else. On the other hand, I don’t want you to lose track of Marginal notes are the argument. As a compromise, I have used marginal notes and footnotes1 usually directly rel- with slightly different purposes. evant to the current discussion, often be- ingusedtofillinde- tails or point out a feature of a calcula- tion. 1Footnotes are more digressional and can, in principle, be ignored.