Oxford University 2005 Practical Applied Mathematics

Oxford University 2005 Practical Applied Mathematics -...

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Unformatted text preview: 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 7....
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