Dimensional analysis-how one variable scales w/ another

[funandgames97]

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?”

Homework Equations

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!

 

[Chestermiller]

If you multiply your equation by $\Pi 1$, you get $$\frac{\rho gh^3}{\mu Q}=g(Q\rho/\mu, \theta)$$
This is about the best you can do. Part (b) can't be answered without making additional assumptions.