Mathematical Models in the Applied Sciences
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1 Mathematical Models in the Applied Sciences A.C. FOWLER University of Oxford CAMBRIDGE UNIVERSITY PRESS
2 Preface Part one: Mathematical modeling What is a model? The procedure of modeling Choosing the model Some examples page xni Part two: Methods Nondimensionalization Damped pendulum 21 Shear flow, heat transport, and convection 23 Using numerical estimates: an example from mathematical biology Asymptotics Order notation Asymptotic sequences and expansions Convergence versus divergence An algebraic example Laplace's method Perturbation methods Elementary boundary layer theory Matched asymptotic expansions Interior 1 avers Vll
3 viii 4.4 A nonlinear example Nonlinear oscillations Partial differential equations Part three: Classical models 59 5 Heat transfer The diffusion equation Viscous flow The Navier-Stokes equation Solid mechanics Stress and strain Linear elasticity Plasticity Viscoelasticity Electromagnetism Fundamentals Maxwell's equations Part four: Continuum models Enzyme kinetics Pseudo-steady state hypothesis Nondimensionalization Singular perturbation theory Enzyme-substrate-inhibitor system The Belousov-Zhabotinskii reaction Reaction mechanism Relaxation oscillation analysis
4 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 River flow The role of fluid mechanics The mechanics of drainage basins 254
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
6 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
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