Mathematical models in the applied sciences

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
• 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

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
• 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