Recent work [11,12] examining stalactite growth as a free boundary problem established a novel geometrical growth law based on the coupling of thin film fluid dynamics and calcium carbonate chemistry. Numerical studies showed an attractor in the space of shapes whose analytical form was determined and found to compare very favorably with that of natural stalactites. Here we address the question of whether there is an analogous ideal shape for dripping icicles.
The growth of icicles is considered as a free-boundary problem. A synthesis of atmospheric heat transfer, geometrical considerations, and thin-film fluid dynamics leads to a nonlinear ordinary differential equation for the shape of a uniformly advancing icicle, the solution to which defines a parameter-free shape which compares very favorably with that of natural icicles. Away from the tip, the solution has a power-law form identical to that recently found for the growth of stalactites by precipitation of calcium carbonate. This analysis thereby explains why stalactites and icicles are so similar in form despite the vastly different physics and chemistry of their formation. In addition, a curious link between the shape so calculated and that found through consideration of only the thin coating water layer is noted.
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