Mathematical Models in the Applied Sciences A.C. FOWLER University of Oxford CAMBRIDGE UNIVERSITY PRESS
1 1.1 1.2 1.3 1.4 Preface Part one: Mathematical modeling What is a model? The procedure of modeling Choosing the model Some examples page xni 1 3 3 4 8 9 13 2 2.1 2.2 2.3 2.4 2.5 Part two: Methods Nondimensionalization 19 19 Damped pendulum 21 Shear flow, heat transport, and convection 23 Using numerical estimates: an example from mathematical biology 25 30 31 17 3 3.1 3.2 3.3 3.4 3.5 3.6 Asymptotics Order notation Asymptotic sequences and expansions Convergence versus divergence An algebraic example Laplace's method 35 35 36 37 38 41 42 43 4 4.1 4.2 4.3 Perturbation methods Elementary boundary layer theory Matched asymptotic expansions Interior 1 avers 45 45 47 48 Vll
viii 4.4 A nonlinear example 50 4.5 Nonlinear oscillations 51 4.6 Partial differential equations 53 4.7 55 55 Part three: Classical models 59 5 Heat transfer 61 5.1 The diffusion equation 61 5.2 69 70 6 6.1 6.2 7 7.1 7.2 7.3 7.4 7.5 Viscous flow The Navier-Stokes equation Solid mechanics Stress and strain Linear elasticity Plasticity Viscoelasticity 76 76 86 87 96 96 97 104 109 113 114 8 Electromagnetism 118 8.1 Fundamentals 118 8.2 Maxwell's equations 120 8.3 127 128 Part four: Continuum models 131 9 Enzyme kinetics 133 9.1 Pseudo-steady state hypothesis 134 9.2 Nondimensionalization 135 9.3 Singular perturbation theory 136 9.4 Enzyme-substrate-inhibitor system 138 9.5 143 143 10 The Belousov-Zhabotinskii reaction 145 10.1 Reaction mechanism 146 10.2 Relaxation oscillation analysis 149 10.3 155 157
11 11.1 11.2 11.3 11.4 11.5 11.6 12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 13 13.1 13.2 13.3 13.4 13.5 13.6 13.7 14 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 Spruce budworm infestations Nondimensionalization and scale analysis Ludwig-Jones Holling (LJH) analysis Summary Finite saturation foliage health, revisited Synopsis Chemical reactors Mathematical modeling Thermal runaway More realistic models: heat and mass transfer The case Le»U«l, and y = 0(1) Nonporous pellet Macroscopic modeling Groundwater flow Basic groundwater flow Dam seepage Dupuit approximation Consolidation Solute dispersivity Heterogeneous porous media Convection in a porous medium Linear stability Nonlinear stability Convection A mathematical model Nondimensionalization Stability analysis Nonlinear stability analysis Boundary layer theory ix 161 164 166 167 168 173 175 176 179 181 185 190 191 192 195 198 199 201 204 205 206 211 214 216 219 222 227 227 227 229 232 233 235 236 239 243 248 250 15 River flow 254 15.1 The role of fluid mechanics 254 15.2 The mechanics of drainage basins 254
15.3 15.4 15.5 15.6 15.7 15.8 15.9 16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10 17 17.1 17.2 17.3 17.4 17.5 17.6 17.7 18 18.1 18.2 18.3 18.4 18.5 19 19.1 19.2 19.3 19.4 19.5 Mathematical model The flood hydrograph Acceleration: stability and waves Nonlinear waves Sediment transport Drainage networks One-dimensional two-phase flow Flow regimes A simple two-fluid model Other models Characteristics More on averaging A simple model for annular flow Mathematical model of a thermosyphon A reduced model Part five: Advanced models Alloy solidification Modeling mushy layers A reduced model No convection, similarity solution Convection Modeling queries Ice sheet dynamics Basic equations and the shallow ice app Isothermal flow Steady, nonisothermal flow Drainage, sliding and ice-till coupling Chemosensory respiratory control Respiratory physiology The Grodins model Reducing the model Oscillations and chaos 255 256 258 262 263 265 269 270 273 273 273 275 275 276 276 280 284 289 292 293 299 301 301 303 309 312 314 324 327 328 329 334 338 340 344 346 346 349 353 356 359
20 20.1 20.2 20.3 20.4 20.5 20.6 20.7 Frost heave in freezing soils Primary frost heave models Secondary frost heave Miller model of secondary frost heave Simplifications A reduced model References Index xi 362 362 366 373 374 378 382 386 387 399