A small ball of radius ‘r’ is falling in a viscous liquid under gravity. Find the dependency of rate of heat produced in terms of radius ‘r’ after the drop attains terminal velocity.

The forces acting on the sphere are its weight 43πr3ρg downwards, buoyancy force 43πr3σg upwards, and viscous force 6πηrv upwards. The sphere attains the terminal velocity vt when the resultant force on it is zero i.e.,43πr3ρg=43πr3σg+6πηrvt.Solve above equation to get the terminal velocity,vt=2r2(ρ−σ)g9η.The rate of heat generation is equal to the rate of work done by the viscous force which, in turn, is equal to its power. Thus,dQdt=(6πηrvt)vt=8π(ρ−σ)2g2r527η.