I've been tasked to find a model to represent the flow rate of a pipe. I've done this, and gotten the following dimensionless variables as per the Buckingham $\pi$-theorem:
$$\pi_1 = \frac{r^4\cdot\frac{dz}{dp}}{q\cdot\mu}, \pi_2 = \frac{q\cdot\rho}{r\cdot\mu}$$
$$G(\pi_1,\pi_2) = 0$$
What I am stuck on now is trying to show that you can recover $\pi_1$ from a special case of this G function. So far, I'm thinking of using the Implicit Function Theorem (IFT) to represent the function G as the following:
$$\pi_1 = f(\pi_2)$$
Where f is some function. This is where I'm stuck; I don't know what function I can use to show that it is necessarily true that in one special case you can recover the first dimensionless variable. Through use of the BHT, I did manage to get the first part of this question as the following by assuming one real root, shown below:
$$q = \frac{r^4\cdot\frac{dz}{dp}}{\mu\cdot\kappa}, \kappa \in \mathbb{R}$$
And thus I am wondering what kind of function I should try to show that you can in fact recover the first dimensionless variable as a special case of the overall function of the dimensionless variables.
Any tips would be greatly appreciated, I'm truly stuck here.
I would comment rather than answer this, but I don't have enough reputation.
I assume you obtained $G(\Pi_1,\Pi_2)$ from a dimensionless product of the form $\big(\Pi_1\big)^a\big(\Pi_2\big)^b,$ so that $G(\Pi_1,\Pi_2)$ is a function consisting of the two independent products $\Pi_1$ and $\Pi_2$? Well by considering $(a,b)=(1,0)$ we just get the single independent and dimensionless product $\Pi_1$, so you've recovered your $\Pi_1$, right? Perhaps this is much too simplistic of an outlook on it, but I don't really see how you're going to find $f$, since $f$ is supposed to be experimentally determined (unless you were given other information that you haven't stated).