Industrial mathematics : case studies in the diffusion of heat and matter

The focus in this text is on mathematical modelling stimulated by contemporary industrial problems involving heat conduction and mass diffusion. These include continuous metal casting, laser drilling, spontaneous combustion of industrial waste, water filtration and crop irrigation. The industrial problems prove to be an excellent setting for the introduction and reinforcement of modelling skills, equation solving techniques, qualitative understanding of partial differential equations and their dynamical properties. Mathematical topics include setting up partial differential equations and boundary conditions, dimensional analysis, scaling, perturbation expansions, boundary valuer problems, Fourier series, symmetry reductions, Stefan problems and bifurcations.

Preface	p. ix
1 Preliminaries	p. 1
1.1 Heat and diffusion--A bird's eye view	p. 1
1.2 Mathematics in industry	p. 3
1.3 Overview of the case studies	p. 5
1.4 Units and dimensions	p. 7
1.5 Diffusion equations	p. 10
1.6 Heat conduction equations	p. 16
1.7 Boundary conditions	p. 21
1.8 Solving the heat/diffusion equation	p. 27
1.9 Scaling equations	p. 29
1.10 Dimensional analysis	p. 33
1.11 Problems for Chapter 1	p. 38
2 Case Study: Continuous Casting	p. 49
2.1 Introduction to the case study problem	p. 49
2.2 The Boltzmann similarity solution	p. 56
2.3 A moving boundary problem	p. 64
2.4 The pseudo-steady-state approximate solution	p. 69
2.5 Solving the continuous casting case study	p. 70
2.6 Problems for Chapter 2	p. 76
3 Case Study: Water Filtration	p. 85
3.1 Introduction to the case study problem	p. 85
3.2 Stretching transformations	p. 90
3.3 Diffusion from a point source	p. 96
3.4 Solving the water filtration case study	p. 102
3.5 Problems for Chapter 3	p. 106
4 Case Study: Laser Drilling	p. 112
4.1 Introduction to the case study problem	p. 112
4.2 Method of perturbations	p. 118
4.3 Boundary perturbations	p. 123
4.4 Solving the laser drilling case study	p. 130
4.5 Problems for Chapter 4	p. 136
5 Case Study: Factory Fires	p. 142
5.1 Introduction to the case study problem	p. 142
5.2 Bifurcations and spontaneous ignition	p. 148
5.3 Ignition with conduction	p. 155
5.4 Solving the factory fire case study	p. 162
5.5 Problems for Chapter 5	p. 166
6 Case Study: Irrigation	p. 172
6.1 Introduction to the case study problem	p. 172
6.2 The Kirchhoff transformation	p. 176
6.3 Fourier series solutions	p. 178
6.4 Solving the crop irrigation case study	p. 182
6.5 Problems for Chapter 6	p. 186
7 Conclusions	p. 189
7.1 Introduction	p. 189
7.2 A survey of mathematical techniques	p. 191
7.3 Mathematics in some other industries	p. 193
References	p. 195
Index	p. 199

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