oxford university 2005 practical applied mathematics - modelling, analysis, approximation

GetPedia 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 Howtousethisbook 9 1.3 acknowledgements 9 I Modelling techniques 11 2 The basics of modelling 13 2.1 Introduction 13 2.2 Whatdowemeanbyamodel? 14 2.3 Principles of modelling . . . 16 2.3.1 Example:inviscidfluidmechanics 17 2.3.2 Example:viscousfluids 18 2.4 Conservationlaws 21 2.5 Conclusion 22 3Unitsanddimensions 25 3.1 Introduction 25 3.2 Unitsanddimensions 25 3.2.1 Example:heatflow 27 3.3 Electricfieldsandelectrostatics 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:beamsandstrings 45 4.2 TheNavier–Stokesequations 47 4.2.1 Waterinthebathtub 50 4.3 Buckingham’sPi-theorem 51 3 4 CONTENTS 4.4 Onwards 53 5 Case study: hair modelling and cable laying 61 5.1 TheEuler–Bernoullimodelforabeam 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 Effectiveforcesandnondimensionalisation 67 6 Case study: the thermistor 1 73 6.1 Thermistors 73 6.1.1 Asimplemodel 73 6.2 Nondimensionalisation 75 6.3 Athermistorinacircuit 77 6.3.1 Theone-dimensionalmodel 78 7 Case study: electrostatic painting 83 7.1 Electrostaticpainting 83 7.2 Fieldequations 84 7.3 Boundary conditions . 86 7.4 Nondimensionalisation 87 II Mathematical techniques 91 8 Partial differential equations 93 8.1 First-orderequations 93 8.2 Example:Poissonprocesses 97 8.3 Shocks 99 8.3.1 TheRankine–Hugoniotconditions 101 8.4 Nonlinearequations 102 8.4.1 Example:sprayforming 102 9 Case study: traffic modelling 105 9.1 Case study: traffic modelling . . 105 9.1.1 Localspeed-densitylaws 107 9.2 Solutions with discontinuities: shocks and the Rankine–Hugoniot relations 108 9.2.1 Trafficjams 109 9.2.2 Trafficlights 109 CONTENTS 5 10 The delta function and other distributions 111 10.1 Introduction 111 10.2Apointforceonastretchedstring;impulses 112 10.3 Informal definition of the delta and Heaviside functions . . . . 114 10.4Examples 117 10.4.1Apointforceonawirerevisited 117 10.4.2 Continuous and discrete probability. . . 117 10.4.3 The fundamental solution of the heat equation . . . . . 119 10.5Balancingsingularities 120 10.5.1TheRankine–Hugoniotconditions 120 10.5.2Casestudy:cable-laying 121 10.6Green’sfunctions 122 10.6.1Ordinarydifferentialequations 122 10.6.2Partialdifferentialequations 125 11 Theory of distributions 137 11.1 Test functions 137 11.2Theactionofatestfunction 138 11.3Definitionofadistribution 139 11.4Furtherpropertiesofdistributions 140 11.5Thederivativeofadistribution 141 11.6Extensionsofthetheoryofdistributions 142 11.6.1Morevariables 142 11.6.2Fouriertransforms 142 12 Case study: the pantograph 155 12.1Whatisapantograph? 155 12.2Themodel 156 12.2.1Whathappensatthecontactpoint? 158 12.3Impulsiveattachment 159 12.4Solutionnearasupport 160 12.5Solutionforawholespan 162 III Asymptotic techniques 171 13 Asymptotic expansions 173 13.1 Introduction 173 13.2Ordernotation 175 13.2.1Asymptoticsequencesandexpansions 177 13.3Convergenceanddivergence 178 6 CONTENTS 14 Regular perturbations/expansions 183 14.1Introduction 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- blebutnonlinearlystable(orunstable) 187 14.4 Example: the pendulum 188 14.5 Small perturbations of a boundary 189 14.5.1Example:flowpastanearlycircularcylinder 189 14.5.2Example:waterwaves 192 14.6Caveatexpandator 193 15 Case study: electrostatic painting 2 201 15.1Smallparametersintheelectropaintmodel 201 16 Case study: piano tuning 207 16.1Thenotesofapiano 207 16.2Tuninganidealpiano 209 16.3Arealpiano 210 17 Methods for oscillators 219 17.0.1 Poincar´e–Linstedt for the pendulum 219 18 Boundary layers 223 18.1Introduction 223 18.2 Functions with boundary layers; matching 224 18.2.1Matching 225 18.3Cablelaying 226 19 ‘Lubrication theory’ analysis: 231 19.1‘Lubricationtheory’approximations:slendergeometries 231 19.2Heatflowinabarofvariablecross-section 232 19.3Heatflowinalongthindomainwithcooling 235 19.4Advection-diffusioninalongthindomain 237 20 Case study: continuous casting of steel 247 20.1Continuouscastingofsteel 247 21 Lubrication theory for fluids 253 21.1Thinfluidlayers:classicallubricationtheory 253 21.2Thinviscousfluidsheetsonsolidsubstrates 256 CONTENTS 7 21.2.1 Viscous fluid spreading horizontally under gravity: in- tuitiveargument 256 21.2.2 Viscous fluid spreading under gravity: systematic ar- gument 258 21.2.3Aviscousfluidlayeronaverticalwall 261 21.3Thinfluidsheetsandfibres 261 21.3.1 The viscous sheet equations by a systematic argument 263 21.4Thebeamequation(?) 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 usual polite fiction that you, the reader, and I, the author, are engaged on 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 [...]... all kinds of mathematical ideas to ‘real-world’ problems Some of these arise in attempts to explain natural phenomena, for example models for water waves We will see a number of these models as we go through the book Other applications are found in industry, which is a source of many fascinating and non-standard mathematical problems, and a big ‘end-user’ of mathematics You might be surprised to know... dx + q · n dS = 0, dt V ∂V the first term being the time-rate-of-change of the quantity inside V , and the second the net flux of it into V Using Green’s theorem on this latter integral,4 we have ∂P + ∇ · q dx = 0 V ∂t As V is arbitrary, we conclude that ∂P + ∇ · q = 0, ∂t a statement which is often referred to as a conservation law.5 In the heat-flow example above, P = ρcT is the density of internal... the plates with a material of higher permittivity than ε0 ) Based solely on this dimensional analysis, make an order of magnitude guess at the capacitance of (a) an elephant (assumed conducting); (b) a homemade parallel-plate capacitor made from two ten-metre rolls of kitchen foil 30 cm wide separated by cling-film If you walk across a nylon carpet you may become charged with static electricity, to a... the nature of real-world problems that they are large, messy and often rather vaguely stated It is very rarely worth anybody’s while producing a ‘complete solution’ to a problem which is complicated and whose desired outcome is not necessarily well specified (to a mathematician) Mathematics is usually most effective in analysing a relatively small ‘clean’ subproblem where more broad-brush approaches... Many, many vitally important and useful models are intrinsically discrete: think, for example of the question of optimal scheduling of take-off slots from LHR, CDG or JFK This is a vast area with a huge range of techniques, impinging on practically every other area of mathematics, computer science, economics and so on Space (and my ignorance) simply don’t allow me to say any more • ‘Black box’ models such... where the constant µ is called the dynamic viscosity Such fluids are termed Newtonian Our strategy is again to consider a small element of fluid and on the left-hand side, work out the rate of change of momentum ρ V Du dV, Dt while on the right-hand side we have F dS, ∂V the net force on its boundary Then we use the divergence theorem to turn the surface integral into a volume integral and, as V is arbitrary,... processes work just fine, having been designed by engineers who know their job So where does mathematics come in? Some important uses are in quality control and cost control for existing processes, and simulation and design of new ones We may want to understand why a certain type of defect occurs, or what is the ‘rate-limiting’ part of a process (the slowest ship, to be speeded up), or whether a novel idea... where it convective derivative formula comes from dF ∂F = + u · ∇F, dt ∂t and verify that the left-hand side of the Euler momentum equation ∂u + u · ∇u ∂t ρ = −∇p is the acceleration following a fluid particle 3 Potential flow has slip Suppose that a potential flow of an inviscid irrotational flow satisfies the no-slip condition u = ∇φ = 0 at a fixed boundary Show that the tangential derivatives of φ vanish... we would be saying something obviously ludicrous like apples + lawnmowers = light bulbs + whisky For example, is the This is the most basic of the many consistency (error-correcting) checks which answer you should build into your mathematics real/positive/ when To quantify this idea, we’ll use a fairly standard notation for the dimenit obviously should sions of quantities, denoted by square brackets:... are used Examples such as the imperial/metric cock-up (one team using imperial units, another using metric ones) which led to the failure of the Mars Climate Orbiter mission in 1999 prove this wrong How can any scientist seriously use feet and inches in this day and age? 3.2 UNITS AND DIMENSIONS 3.2.1 27 Example: heat flow We are going to see a lot of heat-flow problems in this book (I assume that you have