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\begin{document}

stokes problem is:

sphere radius r, falling in viscous liquid under gravity.

variables are: v,r,g,\( \rho _{s} \),\( \rho _{l} \),\( u=\rho _{l}v \). dimensions are \( m,l,t \)this means 6-3 = 3 dimensionless
variables. \( \Pi _{1}=\frac{\rho _{s}}{\rho _{l}} \), \( \Pi _{2}=\frac{v^{2}}{gr} \)=Frued's number, \( \Pi _{3}=\frac{\rho _{l}Rv}{\mu } \)=Reynolds number. This means: \( \frac{v^{2}}{rg}=f(\frac{\rho _{l}Rv}{u},\frac{\rho _{s}}{\rho _{l}}) \).
Using the fact that a terminal velocity will appear, assume \( g\propto u\Rightarrow  \)\( gf_{g}(v,r,\rho )=uf(v,r,\rho ) \). This
means \( v=\frac{r^{2}g\rho _{l}}{u}\phi (\frac{\rho _{s}}{\rho _{l}}) \). B.C. suggest \( \phi =\frac{\rho _{s}}{\rho _{l}}-1 \)

In the limit reynolds number \(>\)\(>\)1, you get: \( \frac{v^{2}}{rg}=f(\frac{\rho _{s}}{\rho _{l}})=\Pi \frac{\rho _{s}-\rho _{l}}{\rho _{l}} \),

\( \Pi =\frac{8}{3C_{D}} \)in another convention. \( C_{d}=\frac{24}{Reynolds} \). This is the correct limit for a rain drop
falling in the air.

For a human, \( v=\sqrt{Rg\frac{\rho _{h}}{\rho _{air}}}=\sqrt{40*980*10^{3}}=200km/h \).

For spheres in corn syrup, \( Re<<1 \)\( v_{terminal}=\frac{2}{9}\frac{gr^{2}}{u}(\rho _{s}-\rho _{l})\rightarrow u=70g/cm/s \) from experimental data.

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