Rayleigh's method, sphere terminal velocity

Rayleigh's method writes the dependent variable as a product of the independent variables, each raised to an unknown power. The powers are found by equating the dimensions on both sides.

Functional relation

$$V = K\, d^{a}\,\rho_s^{\,f}\,\rho_l^{\,b}\,\mu^{c}\,g^{e}$$

Dimensions

$V = LT^{-1}$, $d = L$, $\rho = ML^{-3}$, $\mu = ML^{-1}T^{-1}$, $g = LT^{-2}$.

There are 5 unknown powers and only 3 equations, so express three of them in terms of $c$ and $f$:

$$e = \frac{1-c}{2},\qquad b = -c - f,\qquad a = \frac12 - \frac{3c}{2}$$

Substitute

$$V = K\, d^{\,\frac12-\frac{3c}{2}}\,\rho_s^{\,f}\,\rho_l^{\,-c-f}\,\mu^{c}\,g^{\,\frac{1-c}{2}}$$

Group terms with the same power:

$$V = K\sqrt{gd}\left[\frac{\mu}{\rho_l\, d^{3/2}g^{1/2}}\right]^{c}\left[\frac{\rho_s}{\rho_l}\right]^{f}$$

Answer:

$$\frac{V}{\sqrt{gd}} = \phi\left[\frac{\mu}{\rho_l\,d^{3/2}g^{1/2}},\ \frac{\rho_s}{\rho_l}\right]$$

This says the dimensionless velocity (Froude number) depends on a viscous term (a form of inverse Reynolds number) and on the density ratio. The functions of these groups are found by experiment.