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 underlying stalagmite 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 illustrate these regimes and assess the corresponding scaling laws by numerically solving the governing equations. Finally, experimental measurements conducted in both cave and laboratory 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.