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

TL;DR

The study addresses how a thin water film on a stalagmite drains under gravity when fed by discrete drops, with the substrate curvature modulating the dynamics. It develops a curvilinear Reynolds lubrication framework to derive governing equations, solves them numerically for flat and parabolic shapes, and incorporates discrete inflow via a punctual source term. Key contributions include scaling laws for stationary film thickness, $h_s$, such as $h_s\sim R^{1/4} t_0^{-1/4}$ for flat surfaces and $h_s\sim \Psi^{-1/2} t_0^{-1/2}$ at the centre for parabolic shapes (with $h_s\sim t_0^{-1/3}$ away from the centre), and front-propagation scalings $\ell_f$, e.g., $\ell_f|_{flat}\sim t^{1/8}$ (constant volume) and $\ell_f|_{flat}\sim t^{1/2}$ (continuous inflow). Numerical results reveal a transition length $\ell_{sr}$ separating unsteady, drop-dominated regions from quasi-steady regions, while experiments in caves and labs validate the stationary-thickness scalings and demonstrate geometry-driven drainage effects. The framework provides a foundation for linking thin-film drainage to stalagmite growth and palaeoclimate interpretation, with broader relevance to gravity-driven surface flows.

Abstract

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.

Figures (13)

• Figure 1: (a) General phenomenology: a drop of volume $V_{\rm d}$ is shown just before detaching from a stalactite hanging from the cave ceiling. The previous drop fell $t_{\rm{0}}$ earlier (the dripping period) and contributed to the thin film covering the underlying stalagmite. Successive drops from the same stalactite form a thin film on the stalagmite surface. The dashed red rectangle indicates the region magnified in (b). (b) Cross-sectional sketch at a fixed polar angle of an axisymmetric stalagmite (in beige), covered by a thin film of local thickness $h(\xi)$ (in blue). At position $(r, z)$, the stalagmite elevation is $\eta(r)$, and the local inclination angle with respect to the horizontal is $\varphi(\xi)$ (positive when downward). A local curvilinear coordinate system $(\xi, \zeta)$ is defined accordingly. The pressure field in the film is $p(\xi, \zeta)$, with atmospheric pressure $p_{\infty}$, and the local velocity field is $\mathbf{u}(\xi, \zeta)$, with integrated cross-sectional flux $\mathbf{q}(\xi)$. Gravity acts along $\mathbf{g}$. (c) Example of a flat stalagmite ($\Psi = 0$), modelled as a disk of radius $R$, surrounded by a cone of constant opening angle $\varphi_{\raisebox{-1.65pt}{\scriptsize{$\star$}}}$. (d) Example of a stalagmite with a parabolic profile of revolution ($\Psi > 0$), defined over the domain $r \in [0, R]$ by Eq. \ref{['eq:sm-shape']}.
• Figure 2: Numerical film thickness evolution in space (a, c) and time (b, d), for flat ($\Psi' = 0$) and slightly curved parabolic ($\Psi' = 0.5$) stalagmites, for $t_{\rm{0}}' = 1$. Panels (a) and (c) show the evolution of the film thickness (in blue) over the stalagmite (in beige), after 1, 5 and 25 drops were added and have spread into the film. Panels (b) and (d) show the corresponding film thickness evolution with the successive drop impacts at the centre (in $r' = 0$), and very close to the edge of the stalagmite. These radial positions are indicated by black crosses in (a) and (c). The film reaches a stationary state after about 25 drops in the flat case and 3 drops in the inclined case. Panels (e) and (f) represent the difference between the two envelopes of the oscillating signals, i.e., the difference between the maximum and minimum film thickness values at stationary state shown in (b) and (d) for various radial positions, illustrated in the corresponding inset by the black crosses. The radius/domain size was set to $R' = 3$ for both geometries. The profile $h_{\text{s} }'(r')$ just before an impact $\left( t' = kt_{\rm{0}}'^{-}, k \in \mathbb{N} \right)$ is shown in (a), (c), and insets of (e), (f) in dark blue. The legend in the grey box refers to all panels. Primes denote nondimensional variables where explicit normalisation is absent. See also Supplementary Movies 1 and 2 for similar cases ($t_{\rm{0}} = 1$, $R' = 5$ and $\Psi' = 0$ and 1, resp.).
• Figure 3: Numerical film thickness evolution in time for different inflow frequencies over (a) flat ($\Psi' = 0$) and (b) inclined geometries ($\Psi' = 1$), for a domain size $R' = 5$. The dripping period was varied between high ($t_{\rm{0}}' = 0.1$) and low inflows ($t_{\rm{0}}' = 10$). Both graphs show the first 25 drop additions in the film, then from the 125$^{\text{th}}$ to the 130$^{\text{th}}$ drop addition with a 100-drop long gap indicated by the grey shaded area. The film over the flat stalagmite (a) only reaches a stationary state beyond the first 25 drops shown in the graphs, after about 96 drops ($t_{\rm{0}}' = 0.1$), 54 drops ($t_{\rm{0}}' = 1$) and 31 drops ($t_{\rm{0}}' = 10$). The legend in the grey box refers to all panels. Primes denote nondimensional variables where explicit normalisation is absent.
• Figure 4: Numerical film thickness over time for $t_{\rm{0}}' = 1$, at various radial positions $r' \in \left\{ 0, 1, 2, 3, 4 \right\}$, for $R' = 5$ and (a) $\Psi' = 0$, (b) $\Psi' = 0.1$, (c) $\Psi' = 1$ and (d) $\Psi' = 10$. Each panel presents 5 successive drop impacts at stationary state. The inset of panel (d) shows the maximum film thickness reached at the different positions but corresponding to a single drop impact. It highlights how the successive perturbations of individual drops become progressively shifted along the surface. Primes denote nondimensional variables where explicit normalisation is absent.
• Figure 5: Spatio-temporal evolution of the numerical film thickness at stationary state, $h/h_{\text{s,0} }$, in between two successive impacts, for $t_{\rm{0}}' = 1$, $R' = 5$ and four shape factors: (a) $\Psi' = 0$, (b) $\Psi' = 0.1$, (c) $\Psi' = 1$ and (d) $\Psi' = 10$. The colour bar from (a) corresponds to panel (a). The colour bar from (d) is shared among (b) and (c), but it is accompanied with grey level bars showing the portion of the colour bar actually covered by each heat map. Note that all colour bars start at $\sim 0.6$ (i.e., $h/h_{\text{s,0} } < 1$). The dashed line from (d) shows that the front can be estimated as evolving along $\sim \left(r' \right)^{4/3}$. Primes denote nondimensional variables where explicit normalisation is absent.