Gravity-driven drainage of a thin film on a stalagmite

Stalagmites in karstic caves may serve as paleoclimate proxies, especially in regions missing glacial ice cores or other continental proxies. Specifically, the laminae revealed in a stalagmite cross-sectional cut can be related to the past upstream flows and soil coverage above the cave. The shape of these laminae, and therefore of the stalagmite top surface, thus change over time in response to variable environmental conditions. However, the effect of the past flows on such shape variability remains poorly understood. Previous models describing stalagmite growth thus involved strong simplifying assumptions regarding the aerodynamics and hydrodynamics of drops impacting stalagmites in caves. For instance, drops were assumed to always land at the apex of the stalagmite, thereby feeding the thin residual film covering it in one central point, while it was recently shown that the drop impacting position is sometimes scattered over several centimeters, which may have a non-negligible effect on stalagmite width [1]. The concave shape exhibited by some stalagmites was also associated with drops splashing at impact, while most drop impacts in caves lead to splashing and cannot, therefore, be related to a particular stalagmite shape [2]. Another assumption of previous stalagmite growth models is that the thin residual film lying on top of the stalagmite remains uniform in time and space, which may not always be accurate.

We thus propose to study the evolution of this residual film in time and space. Starting from an initially dry stalagmite, the film thickens because of the liquid brought by the successive drops, until it reaches a steady state. The thickness of the film at steady-state results from the balance between the incoming flow of drops falling on the stalagmite, and the film depletion through gravity-driven drainage. If this drop inflow is interrupted, only the drainage remains. We are interested in assessing the effect of the main factors influencing the film thickness evolution during these three phases, namely: (i) the underneath stalagmite shape, and (ii) the drop dripping frequency, i.e., the amount of liquid brought over a certain time. To achieve this, we record film thickness measurements during the filling, stationary and sole drainage phases on actual stalagmites, both in caves and in a lab setting. The caves provide a great diversity of shapes while the lab measurements allow to systematically vary the drop dripping frequency. We complete these measurements by a reduced-order modeling of the film thickness in time and space, using Reynolds lubrication equation expressed in curvilinear coordinates to account for the various existing stalagmite profiles. We obtain a good agreement between the experimental measurements and the results provided by the model with a set of parameters representing adequately the stalagmites of our dataset. We finally show that, depending on the stalagmite shape, considering the film as uniform in time and space may remain a valid assumption, but this is not always the case.

[1] Parmentier J. et al, P. R. Soc. A. (2019), https://doi.org/10.1098/rspa.2019.0556
[2] Parmentier J., Terrapon V. and Gilet T., Phys. Rev. Fluids (2023), https://doi.org/10.1103/PhysRevFluids.8.053603