Solidification of a sphere with constant heat flux at the boundary (Q1101299)

We consider the solidification of a liquid sphere, initially at its fusion temperature, subject to constant heat flux at the boundary. The problem with constant (sub-fusion) temperature at the boundary has been considered by \textit{R. I. Pedroso} and \textit{G. A. Domoto} for large Stefan number \(\beta\) [Int. J. Heat Mass Transfer 16, 1037-1043 (1973)]. Their perturbation scheme broke down within a time \(O(\beta^{-1/2})\) of total solidification. \textit{D. S. Riley}, \textit{F. T. Smith} and \textit{G. Poots} corrected this in a multi-region structure using matched asymptotic expansions [(*) ibid. 17, 1507-1576 (1974)]. Their solution broke down within a minute (exponentially small) time of total solidification. However, for all practical purposes, the relatively straightforward approach of (*) is perfectly adequate and is followed here. They point out that their method can be applied to other boundary conditions, and indeed, for example, for Newton cooling, it is easy to show that their structure is preserved. However, the case of constant heat flux, considered here, is novel in that the breakdown of the ``outer'' solution occurs at \(O(\beta^{-1/4})\) of total solidification, not \(O(\beta^{- 1/2})\), and the resulting different scalings makes the form of the solution different, and in some respects at least, easier, and is worth reporting.