# Dimensional analysis-how one variable scales w/ another

## Homework Statement

In PSet 3, Prob. 1(e), you determined the thickness h of a thin film of fluid (of viscosity µ and density ρ) flowing down an incline of angle θ, driven by gravity (acceleration g), via a supplied upstream flow rate Q per unit width (i.e., [Q] = (length)2/time).

(a) Using only dimensional analysis, obtain a possible dimensionless version of the relationship h = f(Q, µ, ρ, g, θ) for some arbitrary function f.

(b) From your result in (a), can you answer the question “If you double the flow rate Q, what happens to the thickness h?”

## The Attempt at a Solution

Using dimensional analysis on these variables, I obtained the following dimensionless groups:
Π1 = Qρ/μ
Π2 = (h^3)g/(Q^2)
Π3 = θ

Applying Buckingham Π theorem:
Π2 = f(Π1, Π3)
(h^3)g/(Q^2) = f(Qρ/μ, θ)

However, to answer part (b), I believe I would need to manipulate the Π groups that the function f depends on such that none includes either Q or h, while the Π2 outside the function should include both (which it does). I know the answer should be that when Q doubles, h increases by a factor of 2^(1/3), but I cannot figure out how to disentangle the Π groups. Thanks!

If you multiply your equation by ##\Pi 1##, you get $$\frac{\rho gh^3}{\mu Q}=g(Q\rho/\mu, \theta)$$