Mathematical models in the applied sciences

Contents

Preface

Part one: introduction

1. Mathematical modelling
   • What is a model?
   • The procedure of modelling
   • Choosing the model
   • Some examples
   
   Exercises

   Part two: methods

2. Non-dimensionalisation
   • Introduction
   • Damped pendulum
   • Shear flow, heat transport and convection
   • Using numerical estimates: an example from mathematical biology
   • Notes and references
   
   Exercises

3. Asymptotics
   • Order notation
   • Asymptotic sequences and expansions
   • Convergence versus divergence
   • An algebraic example
   • Laplace's method
   • Notes and references
   
   Exercises

4. Perturbation methods
   • Elementary boundary layer theory
   • Matched asymptotic expansions
   • Interior layers
   • A nonlinear example
   • Nonlinear oscillations
   • Partial differential equations
   • Notes and references
   
   Exercises

   Part three: classical models

5. Heat transfer
   • The diffusion equation
   • Notes and references
   
   Exercises

6. Viscous flow
   • The Navier-Stokes equation
   • Notes and references
   
   Exercises

7. Solid mechanics
   • Stress and strain
   • Linear elasticity
   • Plasticity
   • Viscoelasticity
   • Notes and references
   
   Exercises

8. Electromagnetism
   • Fundamentals
   • Maxwell's equations
   • Notes and references
   
   Exercises

   Part four: continuum models

9. Enzyme kinetics
   • Pseudo-steady state hypothesis
   • Nondimensionalisation
   • Singular perturbation theory
   • Enzyme-substrate-inhibitor system
   • Notes and references
   
   Exercises

10. The Belousov-Zhabotinskii reaction
    • Reaction mechanism
    • Relaxation oscillation analysis
    • Notes and references
    
    Exercises

11. Spruce budworm infestations
    • Nondimensionalisation and scale analysis
    • Ludwig-Jones-Holling analysis
    • Summary
    • Finite saturation foliage health, revisited
    • Synopsis
    • Notes and references
    
    Exercises

12. Chemical reactors
    • Mathematical modelling
    • Thermal runaway
    • More realistic models: heat and mass transfer
    • The case Le >> 1, epsilon << 1, and gamma = O(1)
    • Non-porous pellet
    • Macroscopic modelling
    • Notes and references
    
    Exercises

13. Groundwater flow
    • Basic groundwater flow
    • Dam seepage
    • Dupuit approximation
    • Consolidation
    • Solute dispersivity
    • Heterogeneous porous media
    • Notes and references
    
    Exercises

14. Convection in a porous medium
    • Introduction
    • Linear stability
    • Nonlinear stability
    • Convection
    • A mathematical model
    • Nondimensionalisation
    • Stability analysis
    • Nonlinear stability analysis
    • Boundary layer theory
    • Notes and references
    
    Exercises

15. River flow
    • The role of fluid mechanics
    • The mechanics of drainage basins
    • Mathematical model
    • The flood hydrograph
    • Acceleration: stability and waves
    • Nonlinear waves
    • Sediment transport
    • Drainage networks
    • Notes and references
    
    Exercises

16. One-dimensional two-phase flow
    • Introduction
    • 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
    • Notes and references
    
    Exercises

    Part five: advanced models

17. Alloy solidification
    • Introduction
    • Modelling mushy layers
    • A reduced model
    • No convection, similarity solution
    • Convection
    • Modelling queries
    • Notes and references

18. Ice sheet dynamics
    • Basic equations and the shallow ice approximation
    • Isothermal flow
    • Steady, non-isothermal flow
    • Drainage, sliding and ice-till coupling
    • Notes and references

19. Chemosensory respiratory control
    • Respiratory physiology
    • The Grodins model
    • Reducing the model
    • Oscillations and chaos
    • Notes and references

20. Frost heave in freezing soils
    • Introduction
    • Primary frost heave models
    • Secondary frost heave
    • Miller model of secondary frost heave
    • Simplifications
    • A reduced model
    • Notes and references

    References
    Index