Buckling instability in a chain of sticky bubbles (2024)

(May 31, 2024)

Slender objects, like strands of hair, rope or blades of grass easily buckle when compressed along the axial direction. Buckling can occur on a multitude of length scales from macroscopic, like a rope coiling when hitting the ground[1, 2] as shown in Fig.1(a) for a falling chain; to microscopic, like the bending of flagella while micro-organisms swim[3, 4, 5, 6]. If the stress on the slender object is normal to the cross-section of the object, the object will undergo a regular deformation, for example a rope fed at a constant speed into a cylindrical tube will bend with a characteristic wavelength determined by the friction and bending stiffness of the rope[7]. Interestingly, this phenomenon is not limited to solid materials and can also be seen with viscous jets. Take for instance the familiar example of a stream of honey which coils when it falls onto toast shown in Fig.1(b)[8, 9, 10, 11, 12, 13, 14, 15]. Here the viscosity of the liquid resists the bending of the thread, and a regular coiling is observed[14]. In addition to buckling and coiling due to the compressive stress induced by a barrier, viscous drag can induce buckling of slender structures driven through a viscous liquid, as shown in the work by Gosselin et al. for solid threads[16], and Chakrabarti et al. for gelling structures[11]. Coiling and buckling arises in diverse areas from orogeny in geosciences, to the coiling of DNA structures, and is of common concern to those building architectural structures[17].Furthermore, coiling and buckling of slender fibers has continued to be explored for applications in 3D printing, preparation of metamaterials, and electrospinning on scales ranging from centimetric to nanometric[18, 19, 20, 21, 22, 23, 24].

In addition to the two examples shown in Fig.1(a) and (b), in panel (c) we see the buckling of a chain of air bubbles that rise due to buoyancy, which is the subject of this study. Previous studies have focussed on the shape[25, 26], and trajectories[27, 28, 29, 30] of rising bubbles and jets of bubbles, as well as the role of droplets and bubbles in the dynamics of multiphase systems [31, 32, 33]. Recent work by Atasi et al. is an example of the complexities encountered with jets of bubbles and the role of surfactants in the liquid phase. Here we experimentally investigate the buckling of a chain of adhesive, uniform air bubbles in an aqueous bath. The sticky bubbles are produced at the bottom of the chamber from a small orifice [see schematic in Fig.2(a) and images (b)-(f)]. The adhesion between the bubbles is crucial: by producing the bubbles quickly such that each subsequent bubble is produced before the previous bubble has risen by a distance of one diameter, the two adhere due to short-range depletion forces. Producing multiple bubbles in a row creates a linear chain [Fig.2(b)]. Upon increasing the bubble production speed further, the hydrodynamic drag force increases; and, at some point exceeds the buoyant force for a given length. At this point the linear chain is no longer stable, and buckling is induced by a compressive force due to viscous drag acting on this granular system of sticky bubbles [Fig.2(c)].

Previous work on the buckling of solid and liquid threads[1, 14, 11], found that a bending resistance is critical to the phenomenon. However, with the bubble chain there is no intrinsic cost to bending: the bubbles have no solid-solid friction, nor do the bubbles have the viscous resistance of a liquid thread to bending; yet, a characteristic buckling length emerges. The physical mechanism for the buckling seen in Fig.1(a) and (b) is fundamentally different to that observed in the bubble chain. We first investigate the balance between the hydrodynamic drag and buoyancy to obtain a criterion for the onset of buckling. We then explore the relationship between bubble size, production speed, and viscosity, and determine the terminal velocity of the rising undulated chain of bubbles. Finally, we use hydrodynamic drag to explain the dependence of the buckling amplitude and wavelength on the rate at which the bubble chain is produced. The model, which relies on simple geometric and hydrodynamic arguments, is sufficient to explain the experimental results.

Typical images of the buckling experiment are shown in Fig.2(b)-(f) (see video in the Supplemental Material[34]), with relevant parameters shown in Fig.2(g).A chain of bubbles with radius $R$ is produced at speed $q$, and rises with terminal speed $v$, and a buckling amplitude $A$. The air bubbles are prepared by pushing air though a small glass micropipette with an opening (diameter $\sim$ 10 $\mu$m) into an aqueous bath with surfactant, sodium dodecyl sulfate (SDS) and salt (NaCl) (see Appendix A). The SDS concentrations used in these experiments range from 0.035 M to 0.28 M, and are well above the critical micelle concentration (CMC)[35]. We emphasize that the surfactant serves two purposes in our experiment. First, the surfactant stabilizes the bubbles against coalescence. Second, excess SDS forms micelles in the solutions which controls the adhesion between the bubbles via the depletion interaction[36]. The adhesion stabilizes the chain against breaking apart due to viscous stresses and buoyancy. The differing SDS concentrations are accompanied by a small change in the viscosity of the aqueous bath[37] due to the presence of micelles[38]. The viscosity of the three solutions was measured independently and are 1.5, 1.6 and 2.0mPa$\cdot$s (see Appendix A). The solution has a density $\rho\approx 1\mathrm{\,kg/m}^{3}$. NaCl is added to the solution to screen electrostatic interactions. The pressure through the micropipette is kept constant for the trials by using a syringe and plunger. For the small amounts of air being expelled through the micropipette relative to the air volume in the syringe, we can treat the amount of air as an infinite reservoir with a constant pressure, which results in a constant bubble size[39, 40]. Changing the size of the micropipette orifice creates bubbles with radii ranging from $16~{}\mu\textrm{m}<R<38~{}\mu$m. At these small length scales, the bubbles have a large enough Laplace pressure to be treated as hard spheres. In these experiments, the Reynolds number is Re = $\rho Rv/\mu<$ 1 for all 105 experiments, and viscous forces dominate over inertial ones.

The aqueous bath is contained in a cell with two mirrors set at 22.5^∘ with respect to the back plane of the reservoir so that one camera can simultaneously image two orthogonal planes of the bubble chain [see Fig.2(a)]. The orthogonal views (Fig.2(b)-(f) are used to reconstruct the three-dimensional shape of the chain (Appendix B). The reconstructions are shown in Fig2 below each corresponding experiment in (b)-(f); however, the reconstructions are rotated so that the maximal amplitude of the chain is shown on the left, and the minimal amplitude on the right. It is clear that the buckling takes place predominantly in a two dimensional plane. We do not observe stable helix formation and suspect that this is due to the small compression accessible in the experiment[41] or hydrodynamic stabilization (entrainment) of the two-dimensional structure. There is a symmetry-breaking which sets the buckling plane, and from our experiments we find that this is determined by the angle at which the bubbles emerge from the orifice, or any small oscillations in the pipette.

Critical production speed for buckling onset – We first focus on the critical speed at which the bubble chain buckles. At low bubble production speeds $q$, the adhesive bubbles naturally align in the vertical direction due to the buoyant force $\vec{F}_{b}$. In fact, if stationary, the chain is under tension due to buoyancy. However, since the bubble chain is created with a speed $q$, there is also a drag force acting downwards $\vec{F}_{d}(q)$ which depends on $q$. The tension in the chain switches to a compressive force at a critical speed $q_{c}$ when the magnitude of the drag force of the chain exceeds the magnitude of the buoyant force. With no bending stiffness to the chain, a compressive force acting on the chain is the minimal requirement for the chain to buckle and therefore the chain will buckle at critical speed $q_{c}$, where the chain transitions from being in tension to being in compression, or when $\vec{F}_{b}+\vec{F}_{d}(q)=0$.

The terminal velocity of the chain – As the bubbles are produced, the effect of the increased production speed is only sustained over the first small number of bubbles due to hydrodynamic drag; this is sensible and consistent with the observation that when one tries to push the end of a rope, the buckling happens near the location of the push. Well above the bubble production orifice, each of the bubbles only moves in the vertical direction and does not translate along the arc length of the bubble chain i.e. each bubble only has a vertical component to its velocity. Since the Reynolds number is small for this system and there will be fluid entrained between the spaces of the bubbles, we approximate the buckled chain as a slender ribbon moving upwards through the liquid. Because the chain is moving with a constant speed, we balance the buoyant force acting on the bubbles with the drag force acting on the ribbon to obtain a terminal velocity $v$.

To test Eq.2, we plot the velocity $v$ as a function of $\left(R^{2}q\Delta\rho g/\mu\right)^{1/2}$, for various values of $q$, $R$ and $\mu$. Although the change in viscosity is relatively small (1.5, 1.6 and 2.0mPa$\cdot$s), the drag depends on viscosity and must be taken into account (see Eq.2). As expected, we see in Figure 4 that the terminal speed increases with increasing chain production speed. The expected relationship defined in Eq.2 and the measured data show excellent agreement, with the only free parameter being the drag coefficient for the ribbon $c_{r}$ = 0.72 $\pm$ 0.09. Although the hydrodynamic drag has been modelled with a single drag coefficient $c_{r}$, clearly the assumption proves to be an adequate approximation for the undulating chain with small amplitudes shown in Fig.2, and we find that $c_{r}\approx c_{c}$.

In Fig.5a) we test Eq.3 and plot $A$ as a function of $(q-q_{c})\mu/R\Delta\rho g$. For a variety of bubble sizes, bubble production speeds, and viscosities, the experimentally obtained amplitude closely agrees with the predicted relationship for all but the largest amplitudes where the simple scaling model fails. At large differences between the production speeds we see a deviation from the prediction, indicating that the simple scaling is insufficient to capture the unaccounted non-linear behaviour at the largest production speeds.

The shape of the buckled chain can equivalently be described by the wavelength using simple geometric relationships between $A$, $\lambda$, $v$, and $q$. In order to obtain an analytic solution, we approximate the undulatory ribbon as a sawtooth profile. The approximation introduces an error that is at most $\sim 3.5$%, in comparison to a more rigorous arc-length of a sinusoid, but yields an analytic solution:We relate the time that the chain rises by a distance of $\lambda$, to the arc-length of chain produced in that time. Then the arc-lenght of chain, per unit wavelength is $l_{\lambda}/\lambda=q/v$. For a sawtooth profile, we obtain the simple expression $16(A/\lambda)^{2}=(q/v)^{2}-1$. We note that the ratio $A/\lambda$ vanishes when $q=v$, this is intuitive since $A\rightarrow 0$ and $\lambda\rightarrow\infty$ as the bubble production speed decreases to the velocity at which the chain rises: the chain becomes straight. In Figure5b) we see that this simple relationship is in good agreement with the data with a best fit slope of $~{}1/27$ which differs from simple geometric model by $\sim 1.6$. We attribute the difference due to the sawtooth approximation, as well as differences in the measurement of $A$ and $\lambda$ (see Appendix B).

Conclusions: We have studied the buckling instability that a slender chain of bubbles undergoes when travelling through a viscous bath. The instability arises when the drag force of the rising chain of bubbles exceeds the buoyant forces. After the buckles are formed, the chain moves as a ribbon through the bath and the terminal velocity can be calculated using a simple balance between buoyancy and drag. We find that the model well describes the experiments for the buckling onset, predicted terminal velocities, the amplitude of the buckling, and the relationship between the amplitude and the wavelength. We have studied a system that shows buckling that is akin to many other familiar systems, like honey coiling on toast, the coiling of a falling rope, or the bending of a fiber pushed into a viscous fluid. However, the fundamental origin of the buckling is fundamentally different. Unlike other systems, in which buckling arises from a cost associated with bending, to our knowledge this is the first study of drag-induced buckling with no intrinsic cost to bending – a buckling instability with a characteristic length-scale emerges as a result of hydrodynamics.