## Chapter 2

### Nondimensionalisation

The process of nondimensionalisation is illustrated, first on the
damped pendulum equation, then in viscous fluid flow, heat transport
and thermal convection; finally on a phenomenological model of
Meinhardt for the formation of branched network structures such as
leaf veins and blood vessels. This model, involving the nonlinear
interaction of an activator, an inhibitor, a substrate and an
indicator variable which is switched on by the activator, formed the
inspiration for Willgoose *et al*'s 1989 model for the formation
of drainage networks in river basins. Further discussion of this is in
chapter 15, pages 265 and 270.
The exercises discuss scaling of models for earthquakes, respiratory
ventilation, snow melt runoff and oscillatory populations (the Lotka
Volterra model). Exercise 2.3 has some omissions (see the corrections page for these). The
stick-slip rheology used in it is a simplified version of a more
accurate `rate and state dependent' law which is nicely reviewed by
C.H. Scholz (Nature **391**, 37-42 (1998). Actually, this exercise
is a good entry point into the perplexing question, how do you
model/predict self-similar/fractal behaviour of continuous systems
obeying deterministic laws? I've no idea what the answer to this is, yet.

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