Solidification of a binary mixture with arbitrary heat flux and initial conditions

The problem of solidification of a binary mixture in a semi-infinite region with arbitrarily prescribed initial temperature and composition subject to an arbitrary heat flux at its surface is studied. The liquidus and solidus lines of the phase diagram are used to relate the freezing temperature and the composition of the mixture. Solidification, depending on the prescribed data, could occur immediately or at a later time. In the latter case, there is a period of presolidification. Thus the initial condition of the subsequent solidification cannot be assigned arbitrarily; in particular, it cannot be taken as a uniform state. The conditions for occurrence of these cases are studied and specified. The exact solutions for each of these are found. Existence, uniqueness and convergence of the series solutions are also considered and proved.