Levitation and dynamics of bodies in supersaturated fluids

A body immersed in a supersaturated fluid like carbonated water can accumulate a dynamic field of bubbles upon its surface. If the body is mobile, the attached bubbles can lift it upward against gravity, but a fluid-air interface can clean the surface of these lifting agents and the body may plummet. The process then begins anew, and continues for as long as the concentration of gas in the fluid supports it. In this work, experiments using fixed and free immersed bodies reveal fundamental features of force development and gas escape. A continuum model which incorporates the dynamics of a surface buoyancy field is used to predict the ranges of body mass and size, and fluid properties, for which the system is most dynamic, and those for which body excursions are suppressed. Simulations are then used to probe systems which are dominated by a small number of large bubbles. Body rotations at the surface are critical for driving periodic vertical motions of large bodies, which in turn can produce body wobbling, rolling, and damped surface ’bouncing’ dynamics.


I. INTRODUCTION
A fluid containing dissolved gas may become supersaturated upon a rapid change in temperature or pressure, leading to bubble formation and eventual gas escape.This phenomenon is most commonly observed when opening a can of sparkling water, or other carbonated beverages.When the fluid is under sufficient pressure, gas accumulation is inhibited, preventing bubble formation [1].Upon a rapid reduction of pressure, bubbles form on or near any containing or immersed surfaces, then detach and depart towards the fluid-air interface, leading to the eventual escape of the gas to the environment [2][3][4].In everyday settings, rather than forming directly upon container walls, bubbles commonly form inside small cellulose fibers left behind during cleaning [5,6].These cavities ('lumen') are remarkable sites for consistent and rapid bubble growth and release, since a pocket of gas remains behind after each pinch-off event to seed the next growth.The coalescence of diffusively growing bubbles itself presents a classical modeling challenge [7,8], as is the process by which coalescing bubbles depart from a wall [9][10][11][12].See in particular the reviews by Liger-Belair [13] and Lohse [14].Supersaturated fluids also appear in geophysical settings.Explosive fragmentation of particulate matter in magma can lead to volcanic eruptions.A Strombolian eruption is caused by bubbles which coalesce into rising 'gas slugs', transporting gas and entraining magma flow towards the surface [15][16][17].Such liquids also appear in industrial processes like deacidification and fractionation of oils [18,19], and in biological settings (e.g.blood and tissues during decompression) [20,21].Large scale flows associated with bubble motion depend on the geometry of the container, and can result in peculiar downward bubble motion and even bubble waves and cascades in fluids from stout beers to magma [22][23][24].
When a free body is introduced into a supersaturated environment it presents new sites at which bubbles can nucleate and grow -a field of such bubbles on a body surface can result in surprising dynamics.Using nothing more than carbonated water and raisins (Fig. 1), periodic vertical body motions can be observed for nearly two hours, though the time between excursions slows considerably after the first 20 minutes (see Movie S1).This phenomenon has earned numerous playful names by a variety of fizziologists [25], from dancing raisins, to divers [26,27] and fizz balls [28].Similar body oscillations can also be generated by reactions in chemical gardens [29,30].Recently, this phenomenon has been investigated using peanuts and beer, based on a common practice in Argentinian drinking establishments [31].Different contact angles of individual attached bubbles show the importance of surface roughness on bubble formation [31].Such effects add to a growing list of fundamental scientific findings which have emerged from the kitchen [32,33].
FIG. 1: Raisins in carbonated water present numerous folds conducive to bubble nucleation and growth; these bubbles may then lift the body upward against gravity, only to release it upon arrival at the free surface (see Movie S1).
In this paper we explore these oscillating dynamics using experiments and simulations, and match numerous features to theoretical predictions.Experiments are used to measure the force development on a spherical body fixed in carbonated water, and its oscillatory dynamics when free.Using a discrete bubble model, and a continuum model which incorporates the dynamics of a surface buoyancy field, ranges of body and fluid properties are provided for which the system is most dynamic, and those for which body excursions are suppressed.Simulations are used to explore the dependence on system parameters in a more controlled setting.Body rotations are found to be critical for the onset of periodic vertical motions of large bodies, which in turn can produce body wobbling, rolling, and multi-period surface return in a damped bouncing dynamics.

II. EXPERIMENTS
Mass loss of a supersaturated fluid.We first measured the change in gas concentration in a supersaturated fluid upon depressurization by examining the fluid's mass loss over time.An empty glass vessel with a square cross-section of side length L = 8.9 cm was filled with a just-opened can of Klarbrunn-brand carbonated water which was stored at room temperature (21.6 • C).The fluid was poured into the vessel and then left alone for two hours at the same room temperature.Time, denoted by t, is measured in minutes, and t = 0 corresponds to the time just after depressurization (the moment when the can was opened).The mean mass loss across five experiments at each time is shown in Fig. 2(a) as a thin red curve.In each experiment the initial volume of fluid, V f , was approximately 355 cm 3 (12 oz); the total mass loss, M e (t), over the course of two hours was roughly 0.3% of the initial mass.
Rapid mass loss over the first 20 minutes was due to the growth and detachment of bubbles on the container surface, and diffusive transport of gas from the free surface, as discussed below.The remaining 100 minutes revealed linear behavior due to water evaporation [34].
At long times we observed that dM e /dt ≈ k e for relatively large t, where k e = 4 • 10 −3 g/min (the same value was found using only tap water).The solid blue curve in Fig. 2(a) shows M e (t) − k e t, expected to be the mass loss of CO 2 , and the standard deviation at each time across experiments is shown using error bars.
Denoting the volume-averaged gas concentration at time t by c(t), with units of g/cm 3 , the mass of CO 2 in the vessel is given by c(t)V f (an adjustment due to evaporative volume change is negligible).The mass loss from the empty vessel is written as where c(0) is the initial concentration of CO 2 .The supersaturation ratio, S(t) = (c(t)/c s − 1), measures the gas concentration level compared to a critical value c s , an equilibrium concentration corresponding to a partial pressure of gaseous CO 2 at 1 atm [13,35].At room temperature in water this value is c s = 1.48 g/L [36,37].The gas concentration decreases until S(t) reaches a value below which bubbles no longer form on or near the container walls; in this case, most likely in the lumen of adhered fibrous matter [5,6].We denote this minimum value associated with the container as S mc , and discuss its genesis more fully in §III.The difference S(t) − S mc may then be inferred from the mass loss data, and is plotted in Fig. 2(b).The dynamics of S are modeled by a Riccati equation, Ṡ = −(S − S mc )/T r − q(S − S mc ) 2 with the dot denoting a time derivative, for reasons to be discussed, which results in a predicted evolution of the form where S 0 = S(0) is the supersaturation ratio at t = 0, and χ = qT r (S 0 − S mc ).This function is plotted atop the experimental measurements in Fig. 2(b) as a dashed curve using fitted parameters S 0 − S mc = 1.66, χ = 2.5, and T r = 36.2min.With S 0 − S mc ≈ S 0 , the observations here are in line with carbonation levels reported in other sparkling beverages [4,38], particularly given the substantial gas loss while pouring [4,39].(b) The CO 2 gas supersaturation ratio, minus a stationary minimum value below which bubbles do not form on the container, S(t) − S mc , inferred from the data in (a).The dashed line is a best fit curve using (1).
Surface buoyancy growth with a fixed body.Next, a body was held fixed in the fluid, and we measured the force development due to bubble growth on its surface.A sphere of radius 1 cm composed of Polylactic acid (PLA) was printed with an Ultimaker 3+ 3D-printer using a 0.15mm nozzle.A schematic of the experimental setup is shown in Fig. 3(a).The fluid was depressurized and poured gently into the container, and placed upon the digital scale.The test sphere was then lowered into the carbonated water at different insertion times, denoted by t 0 , and affixed to the table below.As bubbles grew on the surface (see Fig. 3(e)) they imposed an upward vertical force on the stationary sphere, and the opposing downward force was registered by the scale.The sphere was briefly removed from the fluid every 4 minutes, then reinserted while still wet.Videos of bubble growth, coalescence, and arrival at a fluctuating steady state at insertion times t 0 = 1 min and t 0 = 10 min are included as Movies S2-S3.
The scale-registered weight of the system, denoted by F (t), was decomposed as F (t) = F (0) + M e (t)g + B(t; t 0 ).F (0) includes a constant buoyancy force due to the volume of the displaced fluid; M e (t) is the mass change of a body-free fluid previously described; and B(t; t 0 ) is the surface buoyancy force, the contribution to the buoyancy by the bubbles since the insertion time, t 0 .The weight loss due to gas escape from an empty vessel is small, but it is of comparable magnitude to the force on the body for roughly the first 10 minutes, and must be accounted for here.F (t) − F (0) was measured directly, which then provided an indirect measurement of the added buoyancy force, B(t; t 0 ).
FIG. 3: (a) Experimental setup for measuring the surface buoyancy.A spherical 3D-printed body is held fixed in place in carbonated water for 4 minutes, then removed from and reinserted into the fluid.Bubbles form, grow, merge and detach, and the force is registered by the scale.(b) The surface buoyancy force, B(t; t 0 ) during a single experiment.The force increases at a rate λ(t 0 ) which depends on the insertion time t 0 , tapering off to a value B s (t 0 ) (measured later, at t 0 + 4min).The rate λ(t 0 ) is computed using the lines shown.(c) The growth rate, λ(t 0 ).The mean and standard deviation across five experiments are shown as symbols and error bars.The solid curve is based on a model which assumes λ ∝ S 3/2 and using (1).(d) The stabilized surface buoyancy force, B s , measured at t 0 + 4min, along with a linear fit.(e) Bubble configuration on the spherical surface at three different moments after an early insertion time of t 0 = 1 min.
Figure 3(b) shows the surface buoyancy force as a function of time since insertion, t − t 0 , for insertion times up to 60 minutes.Upon insertion the force increased, at a rate which diminished as the insertion time became larger and the fluid became calmer.As discussed in §III, the radius a(t) of an individual bubble is expected to grow roughly as a(t) ≈ , where D is the diffusion constant for CO 2 at room temperature.The associated volumetric growth suggests a total force on the body proportional to (t − t 0 ) 3/2 ; however, once heterogeneous bubble coalescence begins, for the first 30 minutes the growth appears closer to linear in time (see Fig. 3(b)).This growth rate, denoted by λ(t 0 ), is shown in Fig. 3(c).The solid curve corresponds to a prediction λ(t 0 ) = kS(t 0 ) 3/2 , where k is a proportionality constant, revealing an excellent match to the experimentally measured growth rates.Equivalently, defining λ 0 to be the linear growth rate at t = 0, we may write Using S(t) − S mc from the previous experiment with no immersed body (see Fig. 2(b)), the model best fits the data on a logarithmic scale using λ 0 = 58.4dyn/s and S mc = .020.By comparing with the previous experiment, we find S 0 ≈ 1.68.
The surface buoyancy force eventually stabilized to a value B s (t 0 ) measured four minutes after reinsertion time t 0 .This value is shown in Fig. 3(d) along with a linear fit, B s (t 0 ) ≈ 712 dyn-(6.9dyn/min)t.The saturated value B s (t 0 ) diminished far more slowly than did the growth rate λ(t 0 ) -roughly, at later insertion times, bubbles merely take longer to accumulate and grow until reaching a critical size, at which time they pinch off and depart alone.Whether the slow decay was due to surface wetting or other phenomena remains unclear.A simple method for estimating B s is included as Appendix A.
After the buoyancy force stabilized, large fluctuations were observed.They were most apparent at smaller insertion times due to large bubble surface sliding and detachment events, which carry numerous other bubbles away at the same time.This surface cleaning effect has recently seen more specific attention [12].Detachment is expected beyond a critical 'Fritz radius' where buoyancy overwhelms capillary forces [31,40,41].Bubbles have other opportunities to depart during merging events via self-propelled detachment [9].
The same force development measurement was performed using a skewer of 8 Sunmaid raisins, as reported in the Supplemental Information.The initial maximum force per raisin due to bubbles was B s ≈ 100 dyn, and the initial growth rate per raisin was λ 0 ≈ 20 dyn/s.
Levitation and dynamics of a freely moving sphere.In a final set of experiments, the printed sphere was placed into the fluid, free to move (Fig. 4(a)).The body, with radius A = 1 cm and mass m = 4.25 g, was inserted into the fluid just after depressurization and pouring, and its motion was recorded with a Nikon D7000 DSLR camera with a 24 fps framerate for two hours (see Movies S4-S6).The body's vertical position at each time, denoted by z = A Z(t) (with the fluid surface at z = 0), was recovered using an image tracking code written in Matlab.The experiment was performed five times.
A slow decay of the highly oscillatory dynamics is observed upon locally averaging the vertical position.An excursion is defined as having occurred if the body's center of mass drops below one diameter beneath the free surface before returning upward and crossing the same vertical threshold.The times at which the j th excursion begins and ends are denoted respectively by t j (Z(t j ) = −2 and Z ′ (t j ) < 0), and p j (Z(p j ) = −2 and Z ′ (p j ) > 0).The j th excursion time is given by ∆ j := p j − t j , for j ≥ 1.The body undergoes roughly 300-700 excursions during the two hour experiment.A smoothed excursion time, ∆(t), is defined as another moving average using ±5 minutes on either side of time t.The dancing frequency, f := 1/∆, is shown in Fig. 4(e), and on a log-linear scale in the inset, including error bars representing one For bodies with a dense surface bubble coverage, body rotation at the surface plays a critical role.Early experiments, not shown here, constrained the motion to translation along and rotation about the vertical axis only.This dramatically inhibited surface bubble removal and associated dancing.Only a rare event of a large bubble leaving the surface could result in an excursion.Rather, if the body is free to rotate, then upon reaching the surface and losing a portion of the surface-bound bubbles, the body becomes unstable to rotation due to the remaining bubbles on the underbelly of the sphere.Once the body starts to rotate, the body is cleared of a larger number of buoyancy-conferring bubbles and an excursion becomes far more likely (see Movie S4).The previously noted bouncing at the free surface is an additional consequence.Fluctuations from the active fluid, or other immersed bodies (Movie S5), which nudge the rotational instability to develop appear to be important as well.For bodies like raisins, whose trajectories are influenced more by individual bubble growth, lift, and removal, vertical dancing may be observed without need for such rotations.

III. MATHEMATICAL MODEL
Gas escape.There are a number of mechanisms by which gas may escape from the system: formation of bubbles on the container walls and on the immersed body, which eventually exit into the surrounding environment, and by diffusive transport through the fluid-air interface [13].For the first mechanism, assuming simply that the frequency of surface bubble growth and bubble volume upon exit each to be linear in the supersaturation ratio, the gas loss from this process follows Ṡ = −q(S − S mc ) 2 , where q is a constant which encapsulates the number of bubble growth sites and their geometrical features, and S mc is the minimum value below which bubbles no longer form on or near the container walls (discussed below).This quadratic law is supported by the data in Fig. 2, and it suggests that this mechanism of gas loss is dominant for the first one or two minutes of the experiment.
Gas loss due to growth on the immersed body and direct delivery to the surface may be neglected (see the Supplementary Information).
The steady stream of bubbles rising from (near) the container walls drives a large-scale circulation flow on the scale of the container's lateral measure akin to intrisic convection in particle sedimentation [39,42,43].Such a flow is expected to continually replenish the region near the fluid-air interface with the well-mixed concentration from the bulk fluid.
Balancing advection and diffusion near the free surface, the concentration is predicted to Discrete and continuum buoyancy growth models.We consider two models of bubble/buoyancy growth, a discrete model and a continuum model, as each can be more appropriate depending on the body size and surface properties.In the discrete model, each of N bubbles are assumed to grow independently according to the bubble growth law attributed to Scriven, which builds upon the Rayleigh-Plesset equation: for an isolated bubble of radius where D is a diffusion constant (D = 1.85 • 10 −5 cm 2 /s for CO 2 in water [44]), S(t) is the supersaturation ratio, σ is the surface tension of water (σ ≈ 70mN/m = 70dyn/cm at room temperature), and p is the pressure near the bubble (p ≈ 1 atm = 10 6 dyn/cm 2 ) [1, [45][46][47].
The force conferred to the body is the buoyancy experienced by the bubble.In the discrete model, a bubble is removed from the surface when its position on the body exits the fluid, or when it reaches a critical size for pinch-off, a p , at which time it is immediately rebirthed (once in the fluid) with size a 0 , a characteristic scale of surface roughness.
Of particular importance at longer times, there is a supersaturation ratio below which bubbles are overwhelmed by pressure and surface tension and cease to form on the body, S = 2σ/pa 0 .The initial bubble size, a 0 , depends upon the surface and the nature of bubble nucleation there.The length scale of surface roughness on the body may be associated with such initial bubble sizes; the lumen diameter of the fibers left behind during cleaning on the container walls is another.This value for the bubble formation near the surface was denoted S mc , and inferred from the experiments in §II to be S mc ≈ 0.020.This suggests that the surface roughness (or remnant material) scale on the container wall is 2σ/(pS mc ) ≈ 30 µm, which matches the diameter of cellulose fibers [5].
On the body, meanwhile, using the resolution of the 3D-printer of 0.15 mm to estimate a 0 ≈ 0.015cm, a minimal supersaturation ratio for bubble growth on the body, denoted by S mb , may be roughly 0.010.That S mb < S mc indicates that bubbles should continue to form on the body after they cease to form on/near the container walls.Indeed, after a carbonated fluid was left alone for hours until it appeared 'flat' (bubble-free), inserting at that time a 3D-printed body or a raisin resulted in bubble growth on the body and the onset of body rising events.A detailed determination of S mb generally involves not only the surface roughness but also its chemistry, as the contact angle has been shown to be important for bubble nucleation [31].
The second model considered is a continuum model, more appropriate when the body is covered in a large number of bubbles which are continually growing, merging, and detaching.
With the local maximum of the added traction given by B s /(4πA 2 )ẑ, the instantaneous portion of this local contribution is defined as bB s /(4πA 2 )ẑ, where b(x, t) ∈ [0, 1] and x is a point on the body surface.The evolution of this buoyancy fraction is then modeled as which holds pointwise at every position x on the body surface.Spatial variations in b(x, t) are possible, and generically produce a body torque.Holding the supersaturation ratio fixed and neglecting surface tension and pressure, the bubble growth law suggests a bubble radius growth a(t) ∼ (a 2 0 + 2DS(t 0 )(t − t 0 )) 1/2 .Since the contributed buoyancy is proportional to a(t) 3 , this suggests a growth rate λ(t) which scales as S 3/2 for appreciable times, consistent with the measured data.This leads to the model for the growth rate given in (2).
In both models, the instantaneous lifting force and torque in the lab frame are writ- where we have introduced the following dimensionless numbers, Here, V is the body volume, V S (s) is the submerged volume at time s, α(s) := V S (s)/V ∈ [0, 1], and mA 2 I R is the moment of inertia, where I R = 2/5 for a rigid sphere.Hydrodynamic drag and torque coefficients, respectively, are given by C T = 1 + .0183Re |W |, and C R = 1 + 0.0044 Re |Ω|, with Re = ρA 3/2 g 1/2 /µ the Reynolds number [48].Bubbles generally affect the drag on the body but we neglect this detail here.The surface buoyancy fraction b evolves at each point on the body as where g(s) = (S(T s)/S 0 ) 3/2 , and using Eqn.(1), Any points x on the body surface which are outside of the fluid are given the value b = 0; only once those points reenter the fluid does the growth there begin again according to (8).
The body is positively buoyant and floats without bubbles if M < 1.If M > 1, the body can only be lifted upward against gravity if the fizzy lifting number β is sufficiently large; namely β + M −1 − 1 must be positive.This gives a range of masses for which oscillatory dynamics are expected to reside: The 3D-printed bodies are predicted to dance with density ratios ρ s /ρ ∈ (1, 1.20); raisins are expected to dance in a similar range, ρ s /ρ ∈ (1, 1.29).
Dancing frequency.We approximate the solution to (8) by neglecting the early period of rapid gas escape, and assuming S mc ≪ S 0 .Taking g(s) The dimensionless 'charging time' before the total buoyancy overcomes gravity is given by s charge := s c − s 0 such that β⟨b⟩(s c ) + 1/M − 1 = 0, which from ( 10) is given by where For a sufficiently small container the charging time serves as a proxy for the excursion time, with transit from one surface to another playing only a small role.Two curves corresponding to f = 1/∆ ≈ 1/t charge (with t charge := T s charge ) are included in Fig. 3(e), one with b 0 = 0 and one which best fits the data with b 0 = 0.073.The frequency is sensitive to b 0 , which points yet again to the importance of rotations, and how many bubbles are removed upon each surface visit.For insertion times s 0 > s f un , where s f un = (2τ /3) log(2Λτ /(3(1 + χ)L β )), the charging time in (11) is infinite.Using the values from the experiments the associated dimensionless time is t f un := T s f un = 170 min.A separate approximation starting from ( 8) is more FIG.5: Simulations using the discrete bubble model.A body with (M, β, Λ, Re) = (1.02,0.17, 0.003, 3100), and bubbles at the vertices of an inscribed icosahedron (N = 12), undergoes "bouncing" dynamics near the surface.After the bubbles nearest the surface are removed, the center of surface buoyancy rests below the center of mass, resulting in a torque and eventual body rotation.Only after a few returns to the surface to clear off more bubbles does the body begin a large excursion back towards the container floor.See Movie S7.
appropriate at long times, assuming S mc > S mb (i.e.bubbles continue to form on the body after they cease to form along the container walls).In this case, as s 0 /τ → ∞, we have Using the experimental parameters this gives a final dimensional charging time of between 14 min and 2.5 min, for initial coverages b 0 = 0 and b 0 = 0.073, respectively.Once S ≤ S mc , bubbles no longer form along the container and the primary mechanism driving gas escape is removed.The body can continue to form bubbles and perform its low frequency dance, even in an otherwise quiescent fluid.Since we neglect the days-long timescale of pure diffusive transport, and convective diffusion affected by the body motion, this low frequency dancing is predicted to carry on indefinitely.Dancing with a mean frequency of 1.5 min −1 was indeed observed in one experimental run for the last 4 hours of a 5-hour run.
Simulations.To examine the dynamics in a more controlled environment we solve ( 5)-( 8) numerically.Figure 5 shows the dynamics of a body with bubbles at the vertices of a regular icosahedron (N = 12), using the discrete bubble model, with (M, β, Λ, Re) = (1.02, 0.17, 0.003, 3100).The rebirth and pinch-off radii used are a 0 /A = 0.03 and a p /A = 0.24, respectively.The bubbles first grow to sufficient stature to lift the body to the surface, where the topmost bubbles are released.The body begins to descend slowly before undergoing a rotation, returning soon after to the surface.Each visit to the surface includes bubble removal, a body rotation, additional bubble removal, and then a plummeting to Figure 6(a) shows the dimensionless vertical dancing frequency, T f , using the discrete bubble model as a function of the number of bubbles, all placed at the vertices of a regular polyhedron, all with (M, β, Λ) = (1.015,0.17, 0.0016).To isolate the role of bubble position we adjust the maximum bubble size a p /A = (3B s /(4πρgN )) 1/3 so that each body achieves the same maximal surface buoyancy (β) if all bubbles are at their maximal (pinch-off) size.
The frequency increases with N , as bodies with a greater number of bubbles can begin their descent while still maintaining partial surface coverage, and less bubble growth is needed before the body becomes positively buoyant again.
To further explore body rotations, Fig. 6(b) shows the mean body rotation rate as a function of the growth rate, Λ, using both the discrete bubble model (shown as symbols) and the continuum model (as a solid curve).With increasing Λ the body spends more time at the surface, and experiences opportunities to rotate more frequently, if not more rapidly.
The growth is approximately logarithmic in Λ, and for small Λ there is close agreement between the discrete and continuum models.The discontinuity in the discrete model is due to the onset of premature bubble detachment at large bubble growth rates (see Movie S7).
For sufficiently large growth rates (Λ > 0.08) the bubbles grow from their initial size a 0 to the pinch-off size a p on a shorter timescale than the body's excursion time.Consequently, most, if not all, bubbles are removed near the time that the body reaches the surface, nearly eliminating the torques on the body through pinch-off alone and thus dampening rotations.pinch-off and departure at large bubble growth rates.For Λ = 0.016, if β is small, it is common for most of the bubbles to pinch off before the body traverses the full length of the container.For large β, the body emerges completely out of the fluid in a dramatic jump, and all bubbles are removed leaving none to generate a torque.
Wobbling and rolling.For bodies which are large relative to the maximum bubble size, body rotations are commonly observed, as are a related dynamics: wobbling.As a coarse approximation we consider bubbles to have been removed from one half of the spherical surface, which then regrow with rate Λ.Writing the center of surface buoyancy in the body frame as M = M(0) exp(−Λs), with M(0) = (4π) −1 ⟨bX⟩ = (1/4)ẑ, then with q 1 = cos(θ)x + sin(θ)ẑ and q 2 = ŷ, the (dimensionless) torque in the lab frame is Wobbling diminishes either by viscous damping or by bubble (re)growth.A characteristic initial (s = 0) wobbling frequency from the above is For the 3D-printed body, using β = 0.17, the initial frequency is roughly f wobble /T ≈ 1.6 Hz.
A raisin, meanwhile, due to its larger value of β, oscillates with a higher frequency of just over 2.4 Hz.These values are consistent with the experimental observations (see Movies S1, S4, and S5).This effect is similar to that seen in Quincke rotor dynamics, where surface charging is driven by electrohydrodynamics [50][51][52][53][54][55][56].
A transient rolling mode was also observed.Bubbles on the surface of a rolling body begin to grow upon reentry and are larger just before they exit, producing a sustained rolling torque.For this we consider a cylindrical body and a two-dimensional cross-section.
If the body is fixed at a vertical position Z = − cos(θ * ), with θ * ∈ [−π, π], then the steady state lifting distribution assuming ḃ = Λ (and where where η ≤ 15, since part of the body sits outside of the water.A more detailed study of this rotational drag, like that performed by Hunt et al. [57], is needed.

IV. DISCUSSION
Supersaturated fluids present an accessible playground for exploring the dynamics of bodies and their relationship to a complex fluid environment.In the framework proposed by Spagnolie & Underhill [58] this would appear as either a Type I or Type II systemthe body is much larger than the 'obstacles' (be they bubbles or gas molecules), and the fluid exhibits a natural relaxation time.Among the unexpected findings in this system, we have observed a critical dependence of the dynamics on body rotations for large body to bubble size ratios, and multi-period oscillatory dynamics when a single surface interaction is insufficient to clean the body surface of its lifting agents.Another intriguing feature is that bubbles can continue to form on the body long after they cease to form at the container walls when S mc > S mb (when the surface roughness or fibrous material scale is smaller on the container walls than on the body).Raisins inserted into a fluid which was left out for hours and appeared motionless were indeed observed to dance, albeit at a leisurely pace.Relatedly, Pereira et al. [31] identified that a smaller contact angle between a bubble and a surface decreases the energy needed to form bubbles there.Preliminary results also suggest that the body's presence can affect fluid degassing, even potentially slowing it by disturbing the large-scale convective flow responsible for gas escape, reminiscent of how moving boundaries affect flows which promote heat transport [59][60][61][62].
Additional constraints on the dynamics are expected in general.At greater depths, bubbles are less likely to grow due to the increase in hydrostatic pressure.Should a body plummet sufficiently far below the surface, bubble-assisted levitation may vanish and the plummeting will continue unresisted.Surfactants may also adjust the range of bodies possible to levitate in this manner, since their presence can affect the nature of bubble pinchoff and coalescence, in competition with the pressure and surface roughness scale via S mb [63,64].The shape of the container and temperature can also affect the rate of CO 2 loss [65].Another uncharacterized but potentially important feature is wetting.A return to the surface releases bubbles from the body surface, but interaction with the air above may also help to nucleate other bubbles by drawing additional gas into small cavities.Initially dry bodies danced for far longer than initially wet bodies.The interaction with the free surface appears to damp rotations as well, and as we have seen, any inhibition of the rotation of large bodies tends to inhibit vertical dancing.
A number of directions lie ahead based on additional observations not described above.
Preliminary studies suggest a substantial encouragement of excursions for more elongated and asymmetric body shapes.Also, the behavior of multiple bodies in the system can result in stable rafts of bubble-sharing bodies at the surface.Since rotation is critical for triggering an excursion from the surface, and sharing bubbles inhibits rotation, the system transitions from exhibiting oscillatory to overdamped behavior with increased particle volume fraction.
Additional bodies, however, also increase the fluctuations in the system, which can encourage the rolling and plummeting of others.Exploration of the optimal number of dancing partners is under current investigation.A more detailed study of the fluid flow itself will be highly informative regarding the nature of degassing and the effect of the immersed body.
Theoretical advances are also needed.Models of growing arrays of bubbles date back to classical works by Lifschitz and Slyozov [66] and Wagner [67].Transient coarsening kinetics depend on numerous simultaneous mass transfer mechanisms, resulting in overlapping scaling behaviors in time [68][69][70][71].Collective formation and dissolution of bubbles on a regular patterned grid have recently provided some insight on these coupled effects [72][73][74].The additional presence of a flowing environment presents a substantial new challenge.Future work exploring body shape, multi-body dynamics, and fluid-structure interactions is likely to prove... fruitful.and beyond.The force growth be behavior is roughly (t − t 0 ) c(t 0 ) , with c(t 0 ) a slowly varying exponent which is approximately unity for small t 0 and appears to grow to 2 at larger t 0 .
FIG. 8: Force on a 3D-printed sphere of radius A = 1 cm at late insertion times, on a linear scale (a); and on a log-log scale (b).Power law behavior is observed, (t − t 0 ) c(t 0 ) , with slowly varying exponent, c(t 0 ) ≈ 1 for small t 0 , apparently growing to 2 at larger t 0 .

Force development on a skewer of raisins
We performed the same (fixed body position) experiment as described in the main text using a skewer of 8 Sunmaid raisins, but using two cans (24 oz) of water in the same container.
The results are shown in Fig. 9.The skewered 'cylinder' of raisins had approximate length 7cm and diameter 0.8cm.Each raisin was fresh, weighing close to 0.45g; its surface properties appeared to the eye as unchanged over the course of the experiment (significant changes, however, were noted after the raisins were left in the water for 12 hours).To measure the force on the skewer due to bubbles alone, B(t; t 0 ) shown in Fig. 9(a), the weight loss for two cans of water with no immersed bodies was subtracted off.As in the experiment with the 3D-printed body, insertion times began just after depressurizing the fluid t = t 0 = 0 and then once again every four minutes.With the stabilized surface buoyancy force measured to be roughly 800 dyn, this gives a per-raisin value of B s ≈ 100 dyn.An estimated growth rate curve is shown in Fig. 9(a) which uses the same form as that described in the main text, but with fitted parameter λ 0 = 19.7 dyn/min/raisin.Figure 9(b) shows the steady state mean force, measured 4 minutes after insertion.At later insertion times the force growth is slower and smoother, and the stabilized force is non-monotonic in the insertion time, which was observed for some but not all trials using the 3d-printed body as well.This may be due to the cylindrical shape of the raisin skewer, which suffers more dramatically from bubble departure events which may in turn disrupt a greater portion of the exposed body surfaces.A statistical analysis which includes bubble departure and strong nonlocal interactions may be required to account for this effect.
Separately, the bubbles which formed on the raisins, compared to the 3D-printed bodies, were of a rather different character.Due to their rough surfaces, bubbles which formed were fewer and farther between than those observed on the more regular surface of the 3D-printed body, and the lifting force grew over twice as fast.See Movie S8.Using the dimensions of the cubic vessel, S f = 79 cm 2 , and V f = 355 cm 3 , and an observed free bubble speed of U b ≈ 1 cm/s, the boundary layer is estimated to have thickness δ ≈ 65µm, and the timescale for gas escape is estimated to be T r ≈ 26 min, within range of the relaxation timescale which best fit the data in the main paper.In reality there tend to be multiple points on the surface where bubbles nucleate, and there may be many circulation cells, but accounting for these would appear only to add higher order corrections to the estimate above.
The decay of the supersaturation ratio, S(t), where S(t) = c(t)/c s − 1, is desired.Since the velocity field is driven by bubble growth and detachment from the container walls, it diminishes to zero when S = S mc , a minimum value associated with the container walls (or remnant particulate matter).We are thus motivated to model this period of gas loss instead by Ṡ = −(S − S mc )/T r , so that further decay of the gas concentration is instead by other means.Namely, after bubbles are no longer forming on and detaching from the container surface, the process tends towards a purely diffusive process, assumed to be governed by approximately one-dimensional dynamics, c t = Dc zz .Taking the container depth H as the linear dimension of interest, the natural timescale from this equation is H 2 /D; using experimental values H = 4.5 cm and D = 1.85 • 10 −5 cm 2 /s, the timescale of gas loss is on the order of 304 hrs, over 12 days.
The full picture of gas loss involves a change from the mixing performed by large scale circulation to a diffusively-dominated regime.This very rich problem deserves a more substantial investigation.

Discrete and continuum buoyancy growth models
We consider two models of bubble/buoyancy growth, a discrete model and a continuum model, as each can be more appropriate depending on the body size and surface properties.
In the discrete model, each of N bubbles are assumed to grow independently according to the bubble growth law attributed to Scriven, based on the Rayleigh-Plesset equation: for an isolated bubble of radius a(t), we have where D is a diffusion constant (D = 1.85 • 10 −5 cm 2 /s for CO 2 in water [44]), S(t) is the supersaturation ratio, σ is the surface tension of water (σ ≈ 70mN/m = 70dyn/cm at room temperature), and p is the pressure near the bubble (p ≈ 1 atm = 10 6 dyn/cm 2 ) [1,[45][46][47].Bubbles smaller than a critical size, a * = 2σ/pS ≈ 1.4µm/S, vanish due to surface tension, while larger bubbles grow, which points to the importance of the length-scale of surface roughness.If a(0) := a 0 > a * , fixing S on the timescale of bubble growth, we have ȧ ≈ DS/a, or a(t) 2 = a 2 0 + 2DSt.
We denote by x a point in space on the body surface, and the body center in the lab frame as r(t).It is convenient to work in a reference frame of a fixed unit sphere, where the surface is denoted by S 0 with surface points X.We then write x(X, t) = r(t) + AQ(t)X, where Q is an orthogonal orientation matrix, with Q(0) = I.The force conferred by the i th bubble onto the body, with radius a i (t), is then ρg(2π/3)a 3 i (t)ẑ, with ρ the density of water and g the gravitational constant, where we have assumed a hemispherical bubble shape (generally the volume depends on surface roughness and chemistry [31]).The bubbles are assumed to nucleate with a size a 0 , based on the length scale of surface roughness, and pinch off above

FIG. 2 :
FIG. 2: (a) The mass loss in grams of one can of carbonated water during the first 80 minutes after depressurization (thin red line) at time t averaged over four independent experiments, and the mass loss after subtracting a constant linear loss due to evaporation (dark blue line).Error bars show the standard deviation across experiments at each time.(b)The CO 2 gas supersaturation ratio, minus a stationary minimum value below which bubbles do not form on the container, S(t) − S mc , inferred from the data in (a).The dashed line is a best fit curve using (1).

Figure 4 (
b) shows the mean vertical position at time t over a window [t − 5 min, t + 5 min] during one experimental run.The free surface is located at z = 0, and the body is just touching the free surface when Z = −1.For the first 5 minutes the body spends most of its time near the free surface -bubbles cleaned from the body are rapidly replaced.Eventually, the body begins to spend more time in the bulk fluid, and then on the container floor, where the body resides for longer and longer 'charging' times before rising again.The instantaneous vertical position starting at t = 24.7 min is shown in Fig.4(c), revealing a type of damped 'bouncing' dynamics from the free surface.Multiple visits to the fluid-air interface in succession are commonly observed; each visit clears different portions of the body surface, resulting eventually in a longer excursion into the bulk.Figure4(d)shows common late-stage behavior: long periods on the container floor, punctuated by increasingly rare rising events.Rising often ends prematurely due to the detachment and departure of single large bubbles.The late-stage surface forces tend to be dominated by such individuals, and the body's fate can be dictated by their singular activity.

FIG. 4 :
FIG. 4: (a) A freely moving 3D-printed sphere of radius 1 cm and mass 4.25 g in carbonated water, rotates and returns to the free surface.(b) The locally averaged position over a window of ±5 minutes for a single experiment is shown as a dark curve; the standard deviation is shown with error bars.(c) The instantaneous vertical position from (b).The body tends to visit the surface numerous times in succession, clearing off more and more of the surface, before plummeting.(d) At later times, body rising events are often cut short by premature detachment of large individual bubbles.(e) The dancing frequency, f := 1/∆(t), where ∆(t) measures the averaged excursion time (over a ±5 min window) from surface-departure to surface-return.The solid curve shows the mean over three experiments, error bars show the standard deviation across experiments, and the dashed lines are from (11).Inset: the same, on a log-linear scale.
decay from its volume-averaged value inside the fluid to approximately zero outside the fluid across a small boundary layer of size δ = DH/(2U b ), where H is the fluid depth and U b is the velocity of bubble rise (see Supplementary Information).This motivates a model for the supersaturation ratio, Ṡ = −(S − S mc )/T r , where T r = V f δ/(DS f ) is a relaxation timescale, with S f the free surface area and V f the fluid volume.Using the dimensions of the experiment, L = 8.9 cm and H = 4.5 cm, the diffusion constant for CO 2 in water D = 1.85 • 10 −5 cm 2 /s[44], and an observed bubble rise velocity of U b ≈ 1 cm/s, this gives δ = 65µm.Then with S f = 79 cm 2 , and fluid volume V f = 355 cm 3 , the predicted relaxation time T r is 26 min, within range of the best fit value used in Fig.2.Combining the mechanisms above, Ṡ = −(S − S mc )/T r − q(S − S mc ) 2 , produces the expression in(1).
ten as B s F B [b]ẑ and AB s L B [b], respectively.In the continuum model, F B [b] = ⟨b⟩ := (4π) −1 S 0 b(X, t) dS 0 , where dS 0 is the surface area element in the body frame, and L B [b] = (Q⟨bX⟩) × ẑ.Or, defining the center of surface buoyancy in the body frame as M = ⟨bX⟩ = (4π) −1 S 0 Xb(X, t) dS 0 and m = QM the same in the lab frame, we may write L B [b] = (QM) × ẑ.Additional details are given in the Supplementary Information.Equations of motion and dimensionless groups.Taking T := A/g to be a characteristic time and A/T = √ Ag to be a characteristic speed, we write the position of the body centroid as r(t) = A • Z(t)ẑ, the vertical velocity as (A/T )W , and the body rotation rate as T −1 Ω. Force and torque balance, with the dimensionless time s := t/T , are expressed as

FIG. 6 :
FIG. 6: (a) The (dimensionless) dancing frequency, which increases with the number of bubbles, N , for fixed maximal surface lifting force (β = 0.017).(b) The mean rotation rate increases with the growth rate, Λ, in the continuum model (solid line).For the discrete model with N = 12 bubbles (circles) the mean rotation rate increases with Λ until premature bubble detachment becomes important.(c) The mean rotation rate as a function of β, for a selection of growth rates Λ, using the discrete model with N = 12 bubbles.Rotations are damped at large β and Λ due to the body emerging from the fluid.(d)Same as (c) but using the continuum model.

Figure 6 (
Figure 6(c) shows the mean rotation rate instead as a function of β for a selection of growth rates, Λ, using the discrete bubble model, while Fig. 6(d) shows the same using the continuum model.Generally, larger values of β are associated with larger torques, and thus faster body spinning.The discrete model shows again the importance of premature bubble

FIG. 7 :
FIG. 7: (a) A lighter body is pushed further out of the fluid by surface bubbles.(b) A heavier body presents less of its surface to the air, and bubble-cleaning rotations become more important.(c) B s can be reasonably (under-)estimated during the body's brief visit to the surface by measuring the radial distance A ′ where the body reenters the fluid, as viewed from above.

FIG. 9 :
FIG. 9: Surface buoyancy growth for eight skewered Sunmaid raisins.(a) The growth rate for the entire skewer, λ(t 0 ), as a function of the insertion time, t 0 , and on a log-linear scale (inset).Curves are a best fit proportional to S 3/2 , as in the main text.(b) The saturated surface buoyancy force, measured 4 minutes after insertion.

FIG. 10 :
FIG. 10: Schematic of the large-scale circulation model, a cross-section of a cylindrical container with a single bubble nucleation source at the bottom center.The rate of diffusion out from the free surface (top) is dramatically enhanced by the flow in a boundary layer of thickness δ.
Initial time of the j th (empirical) excursion min p jEnd time of the j th (empirical) excursion min ∆ j j th (empirical) excursion time (= p j − t j ) Dimensionless insertion time at which s charge is infinite dimensionless