Heat transfer in solidification of cylindrical billet

No, you don't have to write flow equations for solid part of the billet (sounds like a trivial statement), but you will have to take into account vertical velocity $u_z$ of solid billet while solving energy equation. Under certain conditions you may neglect this as well and treat the problem as one of pure conduction to a good approximation. This happens when heat transfer in vertical direction along solid billet due to motion of billet is negligible in comparison with heat transfer in that direction due to conduction. For this you will need low value of $u_z$ and high value of thermal diffusivity $alpha$; in particular Peclet number, $Peequiv frac{u_zL}{alpha}$ has to be small. $L$ is some length scale relevant to the problem at hand, such as billet diameter or billet length. If length of solid billet is large so that temperature within the billet becomes constant with $z$ well before the billet end, then you may take billet diameter as your length scale. If $Pe$ number is small you wouldn't err much if you obtained your temperature profile by treating it as a conduction problem, with appropriate boundary conditions of course.