# Consequences of Choosing Incorrect Variables in BuckinghamPi

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1. Sep 25, 2015

### 0pt618

This question arose as I was studying mathematical modeling in fluid mechanics. It was posted to math.stackexchange, but there a was a lack of response, probably due to the applied nature of the problem.

One form of the Buckingham Pi Theorem says that for nn variables with kk dimensions, the number of dimensionless quantities (or pi groups) is n−kn−k. This theorem is often applied when the relationship between the set of variables are unknown, and by using the fewer dimensionless quantities, experiments can be simplified to determine the relationship between the set of variables.

However, the first step is to determine which variables are relevant (and hence the number of variables nn). But what if we make a mistake and identify an extra variable that is not relevant? Or if we mistakenly miss variables that are relevant? Then, n-k will be different and the number of dimensionless numbers will change, which should not be the case in modelling.

For example, if we wish to relate the time traveled by a car (https://en.wikipedia.org/wiki/Buckingham_π_theorem#Speed) to some other variables. If in addition to the velocity of the car and distance traveled by the car, we also identify the distance a squirrel has traveled, we will get:
n=4n=4
k=2k=2
Number of dimensionless variables = 4−2=24−2=2, which is not the case.

Does the Pi Theorem require us to identify the correct variables, or does it indicate somehow we have misidentified the relevant variables?

2. Sep 26, 2015

### Orodruin

Staff Emeritus
I think you are over-interpreting the theorem. It states that any physical relationship between n physical quantities and k physical dimensions may be written as a dimensionless function of n-k dimensionless parameter combinations. This does not imply that the function must depend on all of the dimensionless combinations, a constant function is also a function.