Stalactite growth as a free-boundary problem

Stalactites, the most familiar structures found hanging from the ceilings of limestone caves, grow by the precipitation of calcium carbonate from within a thin film of fluid flowing down their surfaces. We have recently shown [M. B. Short, J. C. Baygents, J. W. Beck, D. A. Stone, R. S. Toomey III, and R. E. Goldstein, "Stalactite growth as a free-boundary problem: A geometric law and its Platonic ideal," Phys. Rev. Lett. 94, 018501 (2005)] that the combination of thin-film fluid dynamics, calcium carbonate chemistry, and carbon dioxide diffusion and outgassing leads to a local geometric growth law for the surface evolution which quantitatively explains the shapes of natural stalactites. Here we provide details of this free-boundary calculation, exploiting a strong separation of time scales among that for diffusion within the layer, contact of a fluid parcel with the growing surface, and growth. When the flow rate, the scale of the stalactite, and the chemistry are in the ranges typically found in nature, the local growth rate is proportional to the local thickness of the fluid layer, itself determined by Stokes flow over the surface. Numerical studies of this law establish that a broad class of initial conditions is attracted to an ideal universal shape, whose mathematical form is found analytically. Statistical analysis of stalactite shapes from Kartchner Caverns (Benson, AZ) shows excellent agreement between the average shape of natural stalactites and the ideal shape. Generalizations of these results to nonaxisymmetric speleothems are discussed.