An asymptotic approach to the crenulation instability

A novel linear stability analysis for the pattern formation of ripple-like corrugations, over the surface of stalactites due to calcite-laden falling films, is presented by using a modified version of the gradient expansion technique. A fully analytical theory combining thin film hydrodynamics, Karman-Pohlhausen solution of the advection-diffusion equation for calcium concentration and an evolution equation for the shape of the liquid-solid interface is developed. Analytical formulas for the selected wavelength and the corresponding phase velocity are provided, which match previous numerical solutions. The obtained easy-to-use mathematical results have a potential for developing a novel set of paleo-reconstruction proxies based on the link between morphological patterns and paleo-flows.