Stalactite Growth as a Free-Boundary Problem: A Geometric Law and Its Platonic Ideal

Abstract

The chemical mechanisms underlying the growth of cave formations such as stalactites are well known, yet no theory has yet been proposed which successfully accounts for the dynamic evolution of their shapes. Here we consider the interplay of thin-film fluid dynamics, calcium carbonate chemistry, and CO2 transport in the cave to show that stalactites evolve according to a novel local geometric growth law which exhibits extreme amplification at the tip as a consequence of the locally-varying fluid layer thickness. Studies of this model show that a broad class of initial conditions is attracted to an ideal shape which is strikingly close to a statistical average of natural stalactites.

Geological pattern formation by growth and dissolution in aqueous systems

“…Many pattern formation processes that are dominated by mineral precipitation and/or dissolution are paralleled by pattern formation by the freezing and/or melting of ice (e.g. Short et al 2005Short et al , 2006. This provides important model systems that can be conveniently used in laboratory experiments.…”

Tubular precipitation structures: materials synthesis under non-equilibrium conditions

“…Another example is the creation of self-propelled micro-and nanoparticles capable of transporting cargo over macroscopic distances [8][9][10][11][12]. Important ingredients of these efforts include non-equilibrium conditions, compartmentalization and reaction-transport coupling [13][14][15][16][17][18][19]. around 0.15 cm 3 g −1 .…”

Sculpting of Rocks by Reactive Fluids

“…Coupling these relationships to the carbonate precipitation system, it is possible to deduce an analytical expression for the growth velocity normal to the surface. Over time, this leads asymptotically to a stalactite shape described using rescaled values, z' and r', that represent the vertical coordinate, z, and the local radius, r, (see Short et al, 2005, for additional details):…”

“…Using the Stokes flow formulation, and inserting relevant physical constants (viscosity and acceleration of gravity), Short et al (2005) derived the simple equations:…”