Continuum models in the applied sciences


There are several courses in the Oxford syllabus which provide a grounding in the techniques of mathematical modelling, the fundamental methodology which provides the key to scientific understanding in industrial, environmental, biological and financial problems. Mathematical ecology and biology (b10) and Mathematical modelling (o11) are two entry level courses which provide an introduction to the subject. The continuum models course develops the art of modelling at a technically higher level, and in a wider range of applications. The first half of the course introduces applications which are based on `classical' models; the second half then treats four separate problems in industry, biology and geophysics, and in so doing covers a variety of modern applied mathematical techniques.

Reading list

A.B. Tayler 1986 Mathematical models in applied mechanics. O.U.P., Oxford.

A.C. Fowler 1997 Mathematical models in the applied sciences. C.U.P., Cambridge. Chapters 2, 5, 6, 10, 12, 14, 16.

Lecture synopsis

  1. Nondimensionalisation and scaling.
  2. Burgers' equation: shocks, boundary layers and travelling waves.
  3. Heat transfer and diffusion: convection and adiabats.
  4. Nonlinear diffusion: blow up and degeneracy.
  5. Stefan problems.
  6. Surface tension.
  7. Viscous fluids.
  8. Models for droplets and ice sheets.
  9. Belousov-Zhabotinskii reaction; chemical mechanism and mathematical model.
  10. B-Z reaction: relaxation oscillations.
  11. Gas-solid reactors. Exothermic and endothermic reactions.
  12. Thermal runaway. Non-porous pellets.
  13. Convection in a porous medium. Linear stability.
  14. Nonlinear stability; boundary layer theory.
  15. One-dimensional two-phase flow.
  16. Density wave oscillations.

Course materials

These can be downloaded as postscript files; currently available are