Deriving the scaling law for the Reynolds number

Question

I'm trying to derive a scaling law for the Reynolds number, to get a better understanding of how it changes for microsystem applications, but I'm getting stuck. From textbook tables I should end up with $l^2$, but can't get there. The goal is to find a simplistic relation of Reynolds number and the system dimension.

This is my reasoning so far:

Considering a tube

$$Re = \frac{\rho v 2R}{\mu}$$

$$v = \frac{Q}{A} = \frac{Q}{\pi R^2}$$

From Hagen-Poiseuille: $$Q = \frac{\delta P \pi R^4}{8 \mu L},$$ therefore:

$$v = \frac{\frac{\delta P \pi R^4}{8 \mu L}}{\pi R^2} = \frac{\delta P \pi R^4}{8 \mu L \pi R^2} = \frac{\delta P R^2}{8 \mu L}$$

Replacing $v$ on $Re$:

$$Re = \frac{\frac{\rho \delta P R^2 2R}{8 \mu L}}{\mu} = \frac{\rho \delta P R^3 2}{8 \mu^2 L}$$

From this, if I consider the $\delta P$ as constant with the size scaling, $\rho$ and $\mu$ are material properties that are constant at different scales, I get: $$Re \sim \frac{R^3}{L}$$

I can only think that I could consider it as $\frac{l^3}{l} = l^2$ but doesn't seem right.

What am I missing here?

Answer 1

The textbook is trying to show you the effect of varying only $Re$ while keeping everything else the same while you scale your system. Therefore you need to keep geometric parameters (among others) of your system constant. Aspect ratio of the system, $L/R$, is a geometric parameter, which must therefore be kept constant. It is in this sense that $Re\sim R^2$. In other words, if aspect ratio of the system is kept constant (assuming that is the only geometric parameter in the problem), then $Re\sim R^2$.