## Continuum models in the applied sciences

#### Website: http://www.maths.ox.ac.uk/~fowler/courses/contmods.html

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

- Nondimensionalisation and scaling.

- Burgers' equation: shocks, boundary layers and travelling waves.

- Heat transfer and diffusion: convection and adiabats.

- Nonlinear diffusion: blow up and degeneracy.

- Stefan problems.

- Surface tension.

- Viscous fluids.

- Models for droplets and ice sheets.

- Belousov-Zhabotinskii reaction; chemical mechanism and
mathematical model.

- B-Z reaction: relaxation oscillations.

- Gas-solid reactors. Exothermic and endothermic reactions.

- Thermal runaway. Non-porous pellets.

- Convection in a porous medium. Linear stability.

- Nonlinear stability; boundary layer theory.

- One-dimensional two-phase flow.

- Density wave oscillations.

### Course materials

These can be downloaded as postscript files; currently available are