Stoke's law and its applications

Stoke performed experiments on the motion of small spherical bodies through different fluids and found that the viscous force F acting on a spherical body of radius r depends directly on (i) the radius r of the sphere, (ii) the velocity v of the sphere, and (iii) the coefficient of viscosity $\eta$ of the liquid. Writing $F\propto\eta^x r^y v^z$ and matching dimensions on both sides (using $[F]=MLT^{-2}$, $[\eta]=ML^{-1}T^{-1}$, $[r]=L$, $[v]=LT^{-1}$) gives three simultaneous equations in the exponents x, y and z, whose solution is $x=1$, $y=1$, $z=1$ -- so $F=k\eta r v$, where k is a dimensionless constant that dimensional analysis alone cannot determine. Stoke found EXPERIMENTALLY that $k=6\pi$, giving STOKE'S LAW: $F=6\pi\eta r v$. PRACTICAL APPLICATIONS OF STOKE'S LAW. Because a small sphere's terminal velocity in a viscous fluid depends on its size (via $v_t\propto r^2$, from the previous section), Stoke's law explains several everyday observations: (a) raindrops that are still small have small terminal velocities, so they stay suspended in the air as clouds rather than falling; as they grow bigger their terminal velocities increase and they begin to fall as rain -- this also explains why larger raindrops hurt more than smaller ones when they strike, since they carry more speed (and momentum) on impact; (b) a parachu …