Question #42153

A spherical melting ice particle falls through a warm fluid. Due to uniform heat transfer, its radius $r$ decreases at a constant rate $c$ (i.e., $\frac{dr}{dt} = -c$). The particle always falls at its instantaneous terminal velocity. For large radii, the fluid flow is turbulent and the drag force is $F_D = k_1 r^2 v^2$. For small radii, the flow is laminar and the drag force is $F_D = k_2 r v$. The effective weight of the particle (gravity minus buoyancy) is given by $W_{eff} = k_3 r^3$. Assuming the transition between the drag regimes occurs abruptly when the terminal velocities predicted by both models are equal, what is the total distance $H$ the particle falls from an initial radius $r_0$ (where $r_0$ is in the turbulent regime) until it melts completely?