Dimensional analysis proof of Stokes' Law
Suppose that the sphere has radius r and falls through a fluid of viscosity η. Let the terminal velocity be v (Figure 1).

We can calculate the viscous drag F on the sphere by dimensional analysis.

If

F = kr^xη^yv^z

We have MLT^-2 = L^x(ML^-1T^-1)^y(LT^{- 1})^z

Solving this gives x = 1, y = 1 and z = 1.

By other methods k can be shown to be 6π.

Thus giving: