(95g) A Free-Boundary Theory for the Shape of the Ideal Dripping Icicle

The formation of patterns in snow and ice has been a source of fascination since antiquity. As early as 1611, Johannes Kepler [1] sought a physical explanation for the beautiful forms of snowflakes. While attention has been lavished upon snowflakes ever since [2], their wintry cousins, icicles, have remained largely ignored. The basic mechanisms of icicle growth are well known [3-5], but there are few mathematical analyses describing their forms. For instance, icicle surfaces are typically covered with ripples a few centimeters in wavelength, but only recently [6-8] has theoretical work begun to address the underlying dynamic instability that produces them. On a more basic level, the familiar long, slender form of icicles has not been explained quantitatively from a free-boundary perspective. Icicles and stalactites, the iconic structures found in limestone caves [9], can bear a striking resemblance, particularly insofar as they evince a slightly convex carrot-like form that is distinct from a cone. Of course visual similarity does not imply mechanistic similarity, but there is reason to think that a common mathematical structure might link the two phenomena [10], since each involves an evolving solid structure enveloped by a thin layer of moving fluid--water for stalactites, air for icicles--through which a diffusing field is transported. In the case of stalactites, the diffusing substance is carbon dioxide; for icicles the quantity of interest is latent heat of fusion.

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.

References

[1] J. Kepler, A New Year's Gift, or On the Six-Cornered Snowflake (Clarendon Press, Oxford, 1966).

[2] K.G. Libbrecht, The Snowflake: Winter's Secret Beauty (Voyageur Press, Inc., Stillwater, MN, 2003).

[3] L. Makkonen, A model of icicle growth, J. Glaciology 34, 64 (1988).

[4] K. Szilder and E.P. Lozowski, An analytical model of icicle growth, Ann. Glaciol. 19, 141 (1994).

[5] J. Walker, Icicles ensheathe a number of puzzles: just how does the water freeze?, Sci. Am. 258, 114 (1988).

[6] N. Ogawa, and Y. Furukawa, Surface instability of icicles, Phys. Rev. E 66, 041202 (2002).

[7] K. Ueno, Pattern formation in crystal growth under parabolic shear flow, Phys. Rev. E 68, 021603 (2003).

[8] K. Ueno, Pattern formation in crystal growth under parabolic shear flow. II, Phys. Rev. E 69, 051604 (2004).

[9] C. Hill and P. Forti, Cave Minerals of the World, (National Speleological Society, Inc., Huntsville, AL, 1997).

[10] C.A. Knight, Icicles as crystallization phenomena, J. Crystal Growth 49, 193 (1980).

[11] M.B. Short, J.C. Baygents, J.W. Beck, D.A. Stone, R.S. Toomey, and R.E. Goldstein, Stalactite growth as a free boundary problem: A geometric law and its Platonic ideal, Phys. Rev. Lett. 94, 018501 (2005).

[12] M.B. Short, J.C. Baygents and R.E. Goldstein, Stalactite growth as a free-boundary problem, Phys. Fluids 17, 083101 (2005).