Identical results would have been obtained if any other row of the dimensional matrix had been eliminated instead of row 3, since each of the three rows is a linear combination of the other two. This can easily be checked as exercise.
There now remains to discuss how to choose the ordering of variables in a dimensional matrix. This order determines the complete set of dimensionless products obtained from the calculation. The rules are as follows:
(1) The dependent variable is, of necessity, in the first column of the dimensional matrix, since it must be present in only one n (the first dimensionless product is thus called the dependent dimensionless variable). As a consequence, this first variable can be expressed as a function of all the others, which is the goal here. For example, in eq. 3.9, the drag F is in the first column of the dimensional matrix since it is clearly the dependent variable.
(2) The other variables are then arranged in decreasing order, based on their potential for experimental variation. Indeed, a maximum amount of information will result from experimentation if those variables with a wide range of experimental variability occur in a single n.
(3) The initial ordering of variables must obviously be changed when the last r columns of the dimensional matrix have a zero determinant. However, one must then still comply as well as possible with the first two rules.
Two ecological applications, already discussed in Section 3.2, will now be treated using the systematic calculation of complete sets of dimensionless products.
The first example reconsiders Ecological application 3.2d, devoted to the model of Kierstead & Slobodkin (1953). This model provided equations for the critical size of a growing phytoplankton patch and the characteristic time after which this critical size becomes operative.
The dimensional matrix of variables involved in the problem includes: length x, time t, diffusion of cells D, and growth rate k. The dependent variables being x and t, they are in the first two columns of the dimensional matrix:
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