# Buckingham Π-Theorem

I'm about to conduct some experiments concerning a welding process. To prepare for this I wanted to do a dimensional analysis of the process. So I read a lot about the Π-Theorem and I was able to create 5 dimensionless quantities ($\Pi_1, \Pi_2, ..., \Pi_5$) out of 9 variables ($x_1, x_2, ..., x_9$).

e.g.: $$p_1 = \frac{\Pi_1}{\Pi_3 \cdot \Pi_5^2 \cdot \Pi_8}$$

My problem is that I am trying to find a relationship between the following:

$$y = f(\Pi_1, \Pi_2, ..., \Pi_5)$$

                                      where


$$[y] = [f(\Pi_1, \Pi_2, ..., \Pi_5)] = 1$$

Is it possible to use the Π-Theorem if the function value y has no dimension?

Thanks for your help!

• Could you clarify your question? Do the brackets $[y]$ mean dimension? The dimension of the function is irrelevant since you can always cancel out dimensions by multiplying by their inverses on both sides. – Alex R. Mar 26 '13 at 23:14
• Yes, with $[y]$ I mean the dimension of y. Are you sure that the dimensions are irrelevant? Because I was wondering if the statement of the $\Pi$-Theorem is still valid for $[y] = 1$. On the wikipedia I read that it is only valid for functions which have a dimension. However I don't trust the wikipedia completely so I asked here. – Portbane Mar 28 '13 at 10:12