Thinking about the fact that raindrops come with a typical size I was wondering how this can be determined. I am pretty sure that the friction with air and the quantity of water in the clouds are involved with the upper limit of the raindrop size but I cannot figure out the details: maybe the friction is the sole responsible, or maybe is the contrary and the clouds are not thick enough to produce raindrops larger than a bit.
Also, what determines the minimum size? Maybe this side of the problem does not involve meteorology and could be explained by surface tension and other static characteristics. Edit: as noted in a comment, this could also be influenced by the dust particle seeding the formation of the raindrop.
For the maximum size I was thinking of computing a length involving air viscosity $\gamma$, air and water density $\rho_{1/2}$ and gravitational acceleration $g$. By dimensional analysis we have:
$$[\gamma]=\frac{[M]}{[T]}\qquad[\rho]=\frac{[M]}{[L]^3}\qquad[g]=\frac{[L]}{[T]^2}$$ And therefore if we want to build a length we will use a product: $$[\gamma^{\alpha}\rho^{\beta}g^{\lambda}]=[M]^{\alpha+\beta}[L]^{-3\beta+\lambda}[T]^{-\alpha-2\lambda}\equiv[L]$$ Which implies $\alpha=-\beta=-2\lambda$ and $\lambda=1+3\beta\;$, so that $\;\lambda=-\frac{1}{5}\;,\;\beta=-\alpha=-\frac{2}{5}$. So the size should be something like: $$d\sim\left(\frac{\gamma^2}{g\rho_{..}^2}\right)^{1/5}\cdot f\left(\frac{\rho_1}{\rho_2}\right)$$ Where $f$ is an unknown function and I cannot understand which density (or product of densities) will go in the dimensional factor.
Clearly dimensional analysis cannot do all the work here, moreover I would regard the resulting size as a typical size rather than an upper bound.
So the question remains: how do we determine the upper (and the lower) bound on the size of a raindrop?
Thanks for any help!
Edit: in the dimensional computation we should at least consider also the surface tension, $\tau$. This has dimension of a force: $[\tau]=\frac{[M][L]}{[T]^2}$. This will add even more freedom in the guess-result for $d_{max}$, thus making some physical reasoning even more needed.
