Buckling of Liquid Columns

M. Habibi, Y. Rahmani, Daniel Bonn, and N. M. Ribe

Phys. Rev. Lett. 104, 074301 – Published 17 February 2010

Abstract

Under appropriate conditions, a column of viscous liquid falling onto a rigid surface undergoes a buckling instability. Here we show experimentally and theoretically that liquid buckling exhibits a hitherto unsuspected complexity involving three different modes—viscous, gravitational, and inertial—depending on how the viscous forces that resist bending of the column are balanced. We also find that the nonlinear evolution of the buckling exhibits a surprising multistability with three distinct states: steady stagnation flow, “liquid rope coiling,” and a new state in which the column simultaneously folds periodically and rotates about a vertical axis. The transitions among these states are subcritical, leading to a complex phase diagram in which different combinations of states coexist in different regions of the parameter space.

Received 23 July 2009

DOI:https://doi.org/10.1103/PhysRevLett.104.074301

Authors & Affiliations

M. Habibi¹, Y. Rahmani^1,2, Daniel Bonn^2,3, and N. M. Ribe⁴

¹Institute for Advanced Studies in Basic Sciences, Zanjan 45195-1159, Iran
²Van der Waals-Zeeman Institute, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
³Laboratoire de Physique Statistique, École Normale Supérieure, 24, rue Lhomond, 75231 Paris Cedex 05, France
⁴Laboratoire FAST, Université Pierre et Marie Curie-Paris 6, Université Paris-Sud, CNRS, Bâtiment 502, Campus Universitaire, Orsay 91405, France

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Issue

Vol. 104, Iss. 7 — 19 February 2010

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Images

Figure 1
Time sequence of photographs showing three possible states of a viscous column with $ν = 946 cS$ , $Q = 0.19 ml / s$ , $d = 2.6 mm$ , and $H = 14 cm$ . (a)–(d) Folding with rotation, (f) steady coiling, (h) axisymmetric stagnation flow. Panels (e) and (g) show the finite-amplitude perturbations that trigger the transitions between states. Numbers indicate time in ms.Reuse & Permissions
Figure 2
Experimentally determined phase diagram in the space $(ν, Q)$ . In most cases, the hole diameter was $d = 4 mm$ for $ν \geq 1450 cS$ and 2 mm for $ν \leq 1090 cS$ . Experiments with $ν = 1000 cS$ using both hole sizes confirm that the phase diagram is independent of $d$ . The dashed lines indicate the range of values of $Q$ used in Fig. 3.Reuse & Permissions
Figure 3
Cross section $Q = 0.131 \pm 0.006 ml / s$ of the experimental phase diagram (Fig. 2), expanded in the third ( $H$ ) dimension. Red (upper left, symbol $B$ ) indicates capillary breakup, and the remaining colors are as in Fig. 2. Solid line: Numerically calculated coiling cessation surface separating regions where coiling solutions exist (above or to the right) and do not exist (below or to the left). Dashed line: Coiling cessation surface (uppermost portion only) in the limit of zero surface tension.Reuse & Permissions
Figure 4
Angular frequencies of coiling ( $C$ , solid squares), folding ( $F$ , open circles), and rotation of the folding plane ( $R$ , open triangles) as a function of height for an experiment with $d = 2 mm$ , $ν = 946 cS$ , and $Q = 0.132 ml / s$ . Solid line: Numerical prediction of the coiling frequency using the method of 6. Dashed line: Same, but without surface tension.Reuse & Permissions
Figure 5
(a) Cessation of steady coiling in the inertia-free limit. Solid line: Coiling cessation surface $H / d = F (Π)$ separating regions where steady coiling solutions without surface tension exist (above) and do not exist (below). The portions corresponding to the $V$ and $G$ modes are indicated. Dashed line: Coiling onset surface in the absence of surface tension [2]. Symbols: Experimentally measured critical heights [3] for silicone oil with $γ / ρ g d^{2} = 0.0063$ (circles) and 0.014 (squares). Images: Shapes of the upper part of the column for $H / d = 4$ and the values of $Π$ indicated by the arrows. (b) Numerically determined shape of the lower part of a column coiling in the inertial limit with the critical viscosity $ν = ν_{I}$ , for $Π = 1.0$ and $H = 40 (ν^{2} / g)^{1 / 3}$ . The width of each grid square on the bottom surface is $2 (Q^{2} / g H)^{1 / 4}$ .Reuse & Permissions

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