Convergent computation, scaling-law validity, and forced stabilization of multi-lobed pendant-drop (rosary) solutions of the Young–Laplace equation obtained by the Thomson–Boys curvature construction

The 2005 note by Gentes, Rousseaux, Coullet and De Gennes modernized the graphical Thomson–Boys curvature construction for the axisymmetric Young–Laplace equation and computed pendant-drop profiles with several bulges — a vertical 'rosary' (chapelet) of drop-like lobes — together with a naive scaling law for the lobe radii. The note itself flagged unresolved issues: the computed lobe count depends spuriously on the construction's spatial step, blocking convergence checks and comparison with the naive model; no stability analysis of the computed shapes was performed; the naive rosary model's domain of validity was unknown; and stabilization of rosaries by parametric forcing of the nozzle was suggested but unexplored. The stability component is resolved by pre-existing rigorous results: Wente (1980) proved that a stable pendent drop cannot have a second bulge, and modern monographs resolve pendent-drop stability via volume–pressure curves. The surviving open core is threefold: (i) a step-size-convergent determination of multi-lobed profiles and a diagnosis of the discretization dependence; (ii) the domain of validity of the naive rosary scaling law relative to exact axisymmetric solutions; (iii) whether parametric forcing of the nozzle can stabilize an observable rosary and whether the Thomson–Boys construction with time-dependent effective gravity predicts the forced shapes.

Background

At equilibrium, the shape of a liquid–air interface is governed by the Young–Laplace equation: the pressure jump $\Delta p$ across the interface equals the surface tension $\gamma$ multiplied by the mean curvature, i.e., the sum $K_1+K_2$ of the two principal curvatures. A pendant drop is a drop hanging from a nozzle; its shape — the basis of the standard pendant-drop method for measuring surface tension — results from the balance of surface tension and gravity. With the capillary length $a=\sqrt{\gamma/(\rho g)}$ (the scale at which capillary and hydrostatic pressures balance for density $\rho$ and gravity $g$), the axisymmetric meridian profile satisfies $z/a^2+\Delta p/\gamma=K_1+K_2$, where $z$ is the height, $K_1$ is the curvature of the meridian curve, and $K_2$ is the azimuthal curvature (fixed by the distance from the rotation axis to the point where the surface normal meets that axis).

In an 1886 lecture, W. Thomson (Kelvin) described a graphical construction for solving this equation: the profile is drawn as a succession of very small circular arcs, and after each step of vertical extent $\delta$ (the 'difference of level'), the meridian curvature is updated from the curvature form of the Young–Laplace equation, $K_1^{(n)}=K_1^{(n-1)}+K_2^{(n-1)}-K_2^{(n)}+\delta_n/a^2$, before the next arc is drawn. Boys (1893) accelerated the construction with a rule graduated in curvatures (marks placed at inverse distances, so that a length read off the rule directly gives a curvature) and a tripod. The construction requires initial data — the nozzle radius $R$, the initial contact angle $\theta$, the initial meridian curvature $K_1^{(0)}$ (equivalently the pressure offset $\Delta p$), and the capillary length $a$ — so computing a profile is an initial-value (shooting) problem for the meridian equation. Gentes, Rousseaux, Coullet and De Gennes (2005) programmed the construction to explore the initial conditions and obtained pendant-drop profiles with several bulges: a 'chapelet' or rosary, a vertical string of drop-like lobes of decreasing size. Multi-bulge pendant-drop equilibria were characterized mathematically by Concus and Finn (1979), who proved that the meridian admits a global analytic continuation and takes, at large heights, the form of a succession of nearly circular bulges joined near the axis; Riera and Risler (2002) reformulated the same equation as a dynamical system. The history of such graphical integration methods is documented by Tournès (2003).

An equilibrium surface is statically stable if it is a local minimum of the energy (surface area plus gravitational potential) under the imposed control (liquid volume, or pressure at the nozzle). Classical analysis by Wente (1980) — for pendant drops pinned to a circular opening under volume control (his Problem C) — established, in particular, that a stable pendent drop cannot have a second bulge (Theorem 3.7(b) there); for the prescribed-contact-angle problem (Problem B) he proved that a drop whose profile contains a second inflection point is unstable (Theorem 3.6 there), and the constant-pressure, fixed-opening problem (Problem A) is treated in the same paper (Theorems 2.6 and 3.1). Multi-lobed equilibria are therefore unstable: they exist as solutions of the Young–Laplace equation but are not observable as static configurations, a fact already recalled in the 2005 note from the literature (Concus and Finn, 1979; Finn, 1986). Modern treatments describe this stability exchange through volume–pressure curves, in which turning points (folds) and branch points mark losses of stability (reviewed by Bostwick and Steen, 2015; developed comprehensively for pendent drops by Johns and Narayanan, 2026).

The 2005 note also proposed a naive model of the rosary: treat it as a continuous stack of spherical drops numbered from the bottom, with radii $R_n$ at heights $z_n$. Balancing the capillary pressure $2\gamma/R_n$ of each drop against the hydrostatic pressure of the liquid column above it, and taking the height increment per drop as $2R_n$, yields the scaling law

$R_n=\frac{R_0}{\sqrt{1-n/n_*}},\qquad n_*=\frac{a^2}{2R_0^2},$

where $R_0$ is the bottom radius. The law diverges at $n=n_*$ and is cut off at the nozzle radius $R_b$, giving a predicted total number of drops $n_f=n_*(1-R_0^2/R_b^2)$. The note itself lists the model's crude assumptions: the continuous limit is valid only for many drops (it fails for two or three lobes); the individual drop height is taken as constant although it is really fixed by volume and geometry; and the nozzle cutoff is ad hoc.

In its conclusions the note reports: (i) the number of lobes computed by its implementation of the Thomson construction depends on the chosen spatial step — the lobe count grows with the step — an effect the authors could not interpret, which blocks any convergence check and any quantitative comparison with the naive model; (ii) they had not performed a stability analysis of their computed shapes (while recalling that chapelets are known from the literature to be unstable); and (iii) they suggested, without exploring it, that a parametric forcing of the nozzle — an effective gravity $g(t)$ periodic in time, in the spirit of the vibrational (Kapitza-type) stabilization of gravitationally unstable interfaces and of liquid bridges, where averaging over a fast oscillation of a control parameter yields effective slow dynamics that can be stabilizing — the Kapitza mechanism, as in the vibrational stabilization of an inverted pendulum (e.g., Piriz et al., 2011; Benilov, 2016) — might stabilize a rosary, in which case the Thomson–Boys construction with $g(t)$ should predict the forced shapes.

Problem Statement

Put the multi-lobed pendant-drop (rosary) equilibria of the axisymmetric Young–Laplace equation, as produced by the modernized Thomson–Boys curvature construction, on a quantitative footing. The static stability of the unforced multi-lobed equilibria is already settled (they are unstable), so the open problem consists of three linked parts, for a pendant drop issuing from a nozzle of radius $R$ with initial contact angle $\theta$, initial meridian curvature $K_1^{(0)}$, and capillary length $a$ (a lobe is one bulge of the meridian profile, that is, a local maximum of the profile's distance from the rotation axis as a function of height, and the radius of a lobe is that maximal distance):

(1) Convergent construction and diagnosis of the step-size dependence. The number of lobes obtained by the Thomson–Boys construction, as implemented in the source, grows with the chosen difference of level $\delta$ (the spatial step), and the origin of this dependence is unexplained. Determine whether the geometric construction, as described, is a consistent discretization of the meridian Young–Laplace equation; diagnose whether the observed dependence stems from an inconsistency of the curvature-update/arc-stepping scheme or from extreme sensitivity (ill-conditioning) of the lobe count with respect to the initial data $(R,\theta,K_1^{(0)},a)$; and produce a computation of the multi-lobed profiles whose lobe count and geometry are independent of the step size, verified against an independent integration of the same initial-value problem.

(2) Validity of the rosary scaling law. Establish the domain of validity, over an explicit parameter domain, of the naive scaling law $R_n=R_0/\sqrt{1-n/n_*}$ with $n_*=a^2/(2R_0^2)$ and its nozzle cutoff, relative to the exact (converged) axisymmetric multi-lobed solutions: how the continuous-limit approximation, the constant-drop-height assumption, and the ad hoc cutoff each break down, and what corrected law, if any, the exact solutions satisfy — including in the small-lobe-number regime (two or three lobes) where the continuous limit is manifestly invalid.

(3) Forced stabilization. Determine whether a time-periodic forcing of the nozzle (an effective gravity $g(t)$) can stabilize a rosary into an observable configuration, including whether the Thomson–Boys construction with effective gravity $g(t)$ predicts the stabilized shapes.

Parts (1) and (2) jointly deliver the quantitative comparison between construction, naive model, and exact solutions that the source could not perform because of the discretization artifact; part (3) addresses the stabilization route the source suggested but did not explore, and is motivated precisely by the established instability of the unforced configurations.

References

1. M. Gentes, G. Rousseaux, P. Coullet, P.-G. De Gennes, 'Résolution de l'équation de Young–Laplace par une méthode géométrique utilisant la courbure', Comptes Rendus. Physique 6 (2005) 1027–1033. DOI: 10.1016/j.crhy.2005.11.009, https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2005.11.009/
2. H. C. Wente, 'The stability of the axially symmetric pendent drop', Pacific Journal of Mathematics 88 (1980) 421–470. DOI: 10.2140/pjm.1980.88.421, https://msp.org/pjm/1980/88-2/pjm-v88-n2-p10-s.pdf
3. P. Concus, R. Finn, 'The shape of a pendant liquid drop', Philosophical Transactions of the Royal Society of London A 292 (1979) 307–340. DOI: 10.1098/rsta.1979.0064, https://doi.org/10.1098/rsta.1979.0064
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