Discrete inflow and drainage dynamics of a thin film over a stalagmite of variable shape

Stalagmites in karstic caves preserve valuable palaeoclimate records through calcium-rich layered deposits, presenting curvature variations both across and within individual stalagmites. Stalagmites always remain covered by a thin water film fed by a discrete inflow of drops, which bring in new ions in solution for the stalagmites to grow. However, the gravity-induced drainage of this film and its response to the stalagmite underneath shape and the discrete drop inflow remain poorly characterised in existing growth models. To address these limitations, we develop a theoretical framework that captures the combined effects of shape curvature and discrete drop inflow on thin film drainage dynamics, starting from Reynolds lubrication theory expressed in curvilinear coordinates. From there, we show that the limiting cases of thickness-dominated and inclination-dominated drainage translate into distinct scaling laws for both the front propagation position and stationary film thickness. We further validate these results by numerically solving the governing equations. Finally, experimental measurements conducted in both cave and lab settings confirm the predicted stationary film thickness. Our findings provide insights into the influence of substrate shape and inflow dynamics on thin film drainage, with implications for stalagmite growth modelling and other gravity-driven surface flows.

💡 Research Summary

This paper addresses a long‑standing gap in stalagmite growth modelling: the dynamics of the thin water film that coats a stalagmite under the combined influence of the stalagmite’s curvature and the discrete inflow of dripping water drops. Existing models typically assume that drops always land at the stalagmite apex and that the residual film remains spatially uniform, thereby neglecting the interplay between intermittent drop feeding and substrate geometry. To overcome these limitations, the authors develop a comprehensive theoretical framework based on Reynolds lubrication theory expressed in a curvilinear (ξ, ζ) coordinate system that follows the local surface of an axisymmetric stalagmite.

The stalagmite surface is idealised as either perfectly flat (Ψ = 0) or as a downward‑opening parabola η(r)=−Ψr² (Ψ > 0), both of which are common in natural specimens. The film thickness h is assumed much smaller than the minimum radius of curvature of the solid, justifying the thin‑film approximation and allowing the velocity field to be essentially one‑dimensional along the surface. By applying no‑slip at the solid–liquid interface and zero normal stress at the free surface, the authors obtain analytical expressions for the pressure distribution (hydrostatic plus a small inclination term) and for the parabolic velocity profile, which contains two distinct contributions: a term proportional to ∂ₓh cos φ (driven by thickness gradients) and a term proportional to sin φ (driven by surface inclination).

Integrating the velocity profile yields the surface‑parallel flux
q = −g h³/(3ν)