A free-boundary theory for the shape of the ideal dripping icicle

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 is noted between the shape so calculated and that found through consideration of only the thin coating water layer.

[1]  Surface instability of icicles. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Kenneth G. Libbrecht,et al.  The Snowflake : Winter's Secret Beauty , 2003 .

[3]  Herbert Levine,et al.  Pattern selection in fingered growth phenomena , 1988 .

[4]  John S. Rowlinson,et al.  Molecular Theory of Capillarity , 1983 .

[5]  C. Knight.  Icicles as crystallization phenomena , 1980 .

[6]  Wettlaufer,et al.  Dynamics of premelted films: Frost heave in a capillary. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[7]  Carol A. Hill,et al.  Cave Minerals of the World , 1976 .

[8]  Raymond E. Goldstein,et al.  Stalactite growth as a free-boundary problem , 2005 .

[9]  James C Baygents,et al.  Stalactite growth as a free-boundary problem: a geometric law and its platonic ideal. , 2005, Physical review letters.

[10]  L. Makkonen.  A model of icicle growth , 1988 .

[11]  Georg Kaufmann,et al.  Stalagmite growth and palaeo-climate: the numerical perspective , 2003 .

[12]  Wolfgang Dreybrodt,et al.  Chemical kinetics, speleothem growth and climate , 1999 .

[13]  K. Ueno.  Pattern formation in crystal growth under parabolic shear flow. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  H. Schlichting.  Boundary Layer Theory , 1955 .

[15]  Lasse Makkonen,et al.  Growth rates of icicles , 1994 .

[16]  Edward P. Lozowski,et al.  An analytical model of icicle growth , 1994 .

[17]  Tim G. Myers,et al.  The flow and solidification of a thin fluid film on an arbitrary three-dimensional surface , 2002 .

[18]  Ice stalactites: comparison of a laminar flow theory with experiment , 1974 .

[19]  W. White.  CAVE MINERALS OF THE WORLD, 2nd Edition.By Carol Hill and Paolo Forti. National Speleological Society, Huntsville, AL, 463 p. ISBN number: 1-879961-07-5. $70.00. , 2001 .

[20]  W. Dreybrodt,et al.  The kinetics of calcite dissolution and precipitation in geologically relevant situations of karst areas: 2. Closed system☆ , 1985 .