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Viscosity – Stokes' law and terminal velocity
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A small sphere of radius 'r' falls from rest in a viscous liquid. As a result, heat is produced due to viscous force. The rate of production of heat when the sphere attains its terminal velocity, is proportional to
$P = Fv_T \propto r\cdot(r^2)^2 = r^5$.
Power $= 6\pi\eta rv_T\cdot v_T = 6\pi\eta rv_T^2$ $v_T \propto r^2$ $\Rightarrow$ Power $\propto r^5$