TABLE OF CONTENTS The variables and parameters in a mathematical model have physical dimensions in general. Most dimensions are obvious, like time, length, mass, temperature etc., while others can be deduced from the rule that all terms in a particular equation must have the same dimensions. This rule stems from the condition that no equation may depend on the units used. The dimensionalities of constants of proportionality directly follow from these considerations. For instance, if a frictional force is introduced with a strength proportional to the velocity of the object under consideration, then the constant of proportionality will have the dimensions of the quotient of force and velocity.
The technique of non-dimensionalising is an extremely powerful tool in mathematical modelling. See, e.g., [48]. Its importance is only fully appreciated through examples, which account for the largest part of this chapter. We summarise some striking advantages:
The number of variables and parameters decreases.
Dimensional analysis may yield insight into the general scaling properties of the system. In the first example below this point is illustrated by deriving Pythagoras' theorem from dimensional analysis.
Mathematical models for describing seemingly different systems may appear to be identical if they are put into dimensionless form.
The reduction of a model (Step 4 in the modelling scheme of 1) is often accomplished by neglecting those terms in the model equations that are much smaller than the other terms. It should be realised, however, that comparing magnitudes only makes sense if the model...
