A small spherical ball of radius r falls from rest in a long column...

The rate of heat production is equal to the mechanical power dissipated by the viscous drag force.

Power $P = F \times v$, where $F$ is the viscous drag force and $v$ is the terminal velocity.

According to Stokes' law, the viscous drag force at terminal velocity is $F = 6\pi\eta r v$.

Therefore, the rate of heat production is:

$P = (6\pi\eta r v) \times v = 6\pi\eta r v^2$

The terminal velocity $v$ of a spherical ball of radius $r$ is given by:

$v = \dfrac{2r^2(\rho - \sigma)g}{9\eta}$

This shows that $v \propto r^2$.

Substituting $v \propto r^2$ into the power equation:

$P \propto r \times (r^2)^2$

$P \propto r \times r^4 = r^5$

Thus, the rate of heat production is proportional to $r^5$, which means $n = 5$.

Answer: $5$