Freezing of a Subcooled Liquid Droplet
Résumé
The freezing process of a subcooled liquid droplet occurs in two steps:
(i) A rapid return to the solid-liquid equilibrium temperature; the drop then freezes partially.
(ii) A slower step directed by the heat balance at the solid-liquid interface and the heat conduction toward this interface; the drop then freezes completely.
Three possibilities are considered for step (i):
(I) the center of the drop freezes first;
(II) solid germs are uniformly distributed at random in the droplet;
(III) the outer shells of the droplet freeze first.
The heat conduction problem in the second step is treated with these three possible initial conditions successively. Two different methods of solution are used, a perturbation method and a numerical method. The perturbation problem becomes singular in cases (II) and (III), when the freezing front comes close to the sphere center. The variation with time of the location of the freezing front is constructed as a uniformly valid expansion, following the matched asymptotic expansions analysis of Stewartson and Waechter (Proc. R. Soc. London Ser. A 348, 415 (1976)) and Soward (Proc. R. Soc. London Ser. A 373, 131 (1980)). Results for the evolution of the freezing front and for the final freezing time from both methods of solution are identical within the approximation used. Model (II) gives probably the best estimate for the freezing time, and models (I) and (III) give lower and upper bounds, respectively. Results from models (II) and (III) are found to be very close to one another.