Levitation and dynamics of bodies in supersaturated fluids

Supersaturated fluids arise after rapid changes in temperature or pressure and drive bubble nucleation, growth, and gas escape, familiar from carbonated beverages and relevant in geophysical, industrial, and biological contexts. When a free body is present, bubbles nucleate on its surface, altering its buoyancy and inducing vertical oscillations (“dancing raisins”). The central questions are: how surface bubble growth and detachment generate lifting forces on bodies, how gas escape evolves in time in a container, and what body/fluid parameters control the onset, frequency, and persistence of oscillatory motions. This study combines controlled experiments and models (discrete bubbles and a continuum surface-buoyancy field) to quantify force development, identify the role of body rotation and surface cleaning at the interface, and delineate parameter regimes for levitation and dancing dynamics.

Prior work has characterized bubble nucleation and growth in supersaturated liquids, including inhibition under pressure and nucleation at surface cavities such as cellulose fibers (lumen) that seed repeated bubble growth. Classical challenges include coalescence of diffusively growing bubbles and wall detachment mechanics, with comprehensive reviews by Liger-Belair and Lohse. Supersaturated systems also occur in magmatic eruptions (gas slugs in Strombolian activity), industrial CO2 processes (deacidification and fractionation), and decompression physiology. Container geometry can induce large-scale flows, downward bubble motion, waves, and cascades (e.g., stout beers). Playful demonstrations (dancing raisins/peanuts/fizz balls; chemical gardens) highlighted surface roughness and contact angle effects on nucleation. Foundational bubble dynamics (Rayleigh, Plesset, Epstein) and later models (Scriven; Lifshitz–Slyozov–Wagner coarsening) inform growth laws and collective effects. Recent studies emphasize coalescence-induced detachment, contact-angle dependence, and collective bubble interactions. These works set the stage for studying how body-borne bubbles in a supersaturated bath translate to time-varying buoyancy, rotations, and vertical excursions.

Experiments

1. Mass loss of supersaturated fluid: A square glass vessel (side L = 8.9 cm, depth H ≈ 4.5 cm) was filled with a freshly opened 355 cm^3 (12 oz) can of carbonated water at 21.6 °C and left for 2 h. The total mass loss M_e(t) was recorded versus time t (min) on a digital scale; evaporative loss was quantified by a constant rate k ≈ 4 × 10^−3 g/min (validated with tap water) and subtracted to infer CO2 loss. This enabled estimating supersaturation S(t) via M_e(t) = V(c(0) − c(t)) + kt, with c_s = 1.48 g/L at 1 atm.
2. Surface buoyancy on a fixed body: A 3D-printed PLA sphere (radius A = 1 cm, printer nozzle 0.15 mm) was inserted at various times t0 post-depressurization, held fixed, and the added buoyancy B(t; t0) from growing/merging/detaching bubbles was inferred from scale readings: F(t) = F(0) + M_e(t)g + B(t; t0). The sphere was briefly removed every 4 min and reinserted wet. Bubble configurations were visually recorded; growth rates λ(t0) were extracted from early-time linear trends of B versus (t − t0).
3. Free-body dynamics: The same sphere (mass m = 4.25 g) was released into the fluid immediately after pouring. Motion was recorded for 2 h at 24 fps (Nikon D7000). Vertical position z = A Z(t) was tracked via MATLAB image analysis across five runs. Excursions were defined between crossings one diameter below the free surface (Z = −2), enabling statistics of excursion times and dancing frequency f(t).
4. Raisin skewer test: A skewer with 8 raisins was tested similarly; per-raisin initial force and growth rate were measured (reported in SI).

Modeling Gas escape model: The time evolution of S(t) is modeled by combining (i) bubble formation/growth on container walls giving a quadratic loss ∝ q(S − S_mc)^2 (q aggregates nucleation-site features), and (ii) convective–diffusive relaxation at the free surface modeled by (S − S_mc)/T_r, with T_r = V δ/(D S_f) and δ ≈ D H/(2 U_b). This yields a Riccati-type form with solution S(t) = S_mc + [(S_0 − S_mc) e^{−t/T_q}] / [1 + x(1 − e^{−t/T_q})], with fitted parameters. Bubble growth and buoyancy models:

• Discrete model: N bubbles grow independently following a Scriven/Rayleigh–Plesset-type diffusion-limited law (ȧ ∝ (−2σ/(p a))^{1/2} with CO2 diffusivity D = 1.85×10^−5 cm^2/s, surface tension σ = 70 dyn/cm, pressure p ≈ 10^6 dyn/cm^2). Bubbles detach at a pinch-off radius a_p or upon exiting the fluid and are reborn at a_0 (set by surface roughness). Bubble formation ceases below S_min ≈ 2σ/(p a_0).
• Continuum model: A surface coverage field b(x,t) ∈ [0,1] evolves pointwise via ḃ = Λ g(t) (1 − b), where Λ is an initial growth-rate scale and g(t) = (S(t)/S_0)^{3/2}. The mean buoyancy fraction F[b] = ⟨b⟩ determines the vertical lifting force; spatial variations in b generate torques via the center of surface buoyancy M = ⟨b X⟩. Equations of motion: Dimensionless vertical velocity W and angular velocity Ω evolve via force/torque balance including gravity, static buoyancy of the body, bubble-induced buoyancy β F[b], and hydrodynamic drag with Re-dependent coefficients. Surface elements outside the fluid reset b = 0 to model surface cleaning at the interface. Parameters are non-dimensionalized using T = √(A/g), with key groups: mass ratio M = m/(ρ V), fizzy lifting number β (bubble-force scale relative to weight), growth-rate parameter Λ, relaxation time τ = T_r/T, and Re. Simulations: Numerical integration of the coupled ODEs for W, Ω, Z, body orientation Q, and either (i) discrete bubble states (positions at regular polyhedron vertices, with growth, advection on the body, pinch-off/rebirth) or (ii) the continuum surface field b(x,t). Parameter sweeps examine the effects of N, Λ, and β on rotation rate and dancing frequency.

Gas loss and supersaturation:

• Total mass loss over 2 h was ≈ 0.3% of initial mass for a 355 cm^3 can; evaporation dominated after ≈ 20 min at k ≈ 4×10^−3 g/min.
• Fitting the S(t) model (Riccati solution) to data (empty vessel) gave S_0 − S_mc = 1.66, x = 2.5, T_q = 36.2 min; inferred S_0 ≈ 1.68.
• Boundary-layer analysis with L = 8.9 cm, H = 4.5 cm, D = 1.85×10^−5 cm^2/s, U_b ≈ 1 cm/s yields δ ≈ 65 μm and T_r ≈ 26 min, consistent with fits.

Surface buoyancy growth on a fixed sphere:

• Early-time growth of buoyant force is approximately linear in time, with growth rate λ(t_0) ∝ S(t_0)^{3/2}. Best fit yielded λ_0 = 58.4 dyn/s and S_mc = 0.020.
• Stabilized surface buoyancy after 4 min behaves approximately linearly with insertion time: B_s(t_0) ≈ 712 dyn − (6.9 dyn/min) t_0 (slow decay relative to λ(t_0)).
• Large fluctuations arise from sliding/merging/detachment events; detachment occurs beyond a Fritz-radius threshold.
• Raisins: initial per-raisin maximum force B ≈ 100 dyn; initial growth rate λ_0 ≈ 20 dyn/s.

Nucleation thresholds and surface roughness:

• Container-wall minimum supersaturation S_mc ≈ 0.020 implies a cavity/roughness scale 2σ/(p S_mc) ≈ 30 μm (consistent with cellulose fiber lumen diameters).
• Body roughness scale from 3D printer resolution a_0 ≈ 0.015 cm implies S_mb ≈ 2σ/(p a_0) ≈ 0.010; since S_mb < S_mc, bubbles can continue forming on the body after container-wall nucleation ceases.

Free-body dynamics (sphere, A = 1 cm, m = 4.25 g):

• Strong early oscillations with frequent surface interactions; over time, the body spends longer charging at the bottom and exhibits damped bouncing at the interface before plummeting.
• Across 2 h, ≈ 300–700 excursions per run; dancing frequency f(t) decreases monotonically, approximately exponentially, consistent with model predictions.
• Body rotation at the surface is critical: constraining rotation suppresses dancing; free rotation promotes efficient surface cleaning and subsequent excursions.

Parameter regimes and dimensionless analysis:

• For the 3D-printed sphere: (M, β, Λ, τ, Re) = (1.015, 0.17, 2.6×10^−3, 6.8×10^3, 3.1×10^3). For a typical raisin: (M, β, Λ, Re) = (1.12, 0.23, 4.9×10^−3, 1.5×10^3).
• Oscillatory dynamics expected for M ∈ (1, 1/(1 − β)) if β < 1; for the printed spheres, densities ρ_body/ρ_fluid ∈ (1, 1.20) and for raisins ∈ (1, 1.29).

Dancing frequency and charging-time theory:

• An approximate solution for b(x,t) leads to a charging time that predicts the excursion frequency (Eq. 11); fits to data suggest sensitivity to residual coverage b_0 (e.g., b_0 ≈ 0.073 best fit), highlighting the importance of rotations and surface cleaning fraction.
• At long times when S → S_mc and container-wall nucleation ceases, the model predicts a constant, low-frequency dancing sustained by body-surface nucleation (observed mean ≈ 1.5 min^−1 late in a 5 h run).

Simulations (discrete and continuum models):

• Discrete model with N = 12 bubbles (icosahedral positions) reproduces surface bouncing, rotational clearing at the surface, and delayed deep excursions.
• Dancing frequency increases with N when total maximal surface buoyancy is held constant across N (partial coverage during descent reduces charging time).
• Mean rotation rate increases approximately logarithmically with Λ; however, at large Λ premature pinch-off dampens rotations (few bubbles remain to generate torque), and at large β bodies can leap clear of the fluid, fully clearing bubbles and suppressing torque.

Wobbling and rolling:

• A reduced model predicts initial wobbling frequencies f_wobble/T ≈ (5β/8)Λ/(2π): ≈ 1.6 Hz for the printed sphere (β = 0.17) and just over 2.4 Hz for a raisin, matching observations.
• A transient rolling mode arises when bubbles regrow preferentially during rotation, sustaining a rolling torque; a scaling balance yields steady rotation rates increasing with Λ and Re.

The findings demonstrate that time-varying buoyancy from surface-bound bubbles can repeatedly overcome gravity and induce periodic excursions in supersaturated fluids, addressing when and why bodies ‘dance’. Rotational degrees of freedom at the free surface are pivotal: partial cleaning triggers instabilities that roll the remaining bubble-laden underside, enabling efficient bubble removal and subsequent plummeting—establishing a feedback loop underpinning oscillations. The gas-escape model (quadratic wall loss plus convective–diffusive relaxation) captures the early evolution of supersaturation and provides parameters that link fluid degassing to force growth (λ ∝ S^{3/2}). The discrete and continuum bubble models bridge regimes dominated by a few large bubbles versus a dense bubbly coverage, respectively, and, together with the equations of motion, map out mass and force scales (M, β, Λ, τ, Re) that support oscillations. Practical implications include predicting the mass/density ranges for which bodies will levitate and oscillate, the role of surface roughness and contact angle (via S_mb), and the late-time persistence of low-frequency dancing even after the container walls cease nucleating bubbles. The results are relevant to industrial degassing, bubbly flows, and geophysical analogs where buoyancy agents accrue and are intermittently shed.

This work quantifies levitation and oscillatory dynamics of bodies in supersaturated fluids through coordinated experiments, theory, and simulations. Key contributions include: (i) a validated gas-escape model explaining early-time supersaturation decay; (ii) measurements linking buoyant force growth to S^{3/2} and characterizing stabilized buoyancy; (iii) identification of body rotation at the free surface as a critical mechanism enabling periodic excursions in large body-to-bubble size ratios; (iv) discrete and continuum models that reproduce bouncing, wobbling, rolling, and frequency evolution; and (v) parameter maps predicting density ranges and conditions for dancing. Future research directions include detailed hydrodynamic modeling of flow–degassing coupling; effects of surfactants, wetting, and hydrostatic pressure with depth; systematic exploration of body shape and multi-body interactions (rafts and cooperative effects); precise rotational drag characterization for partially immersed bodies; and refined models of nucleation/coarsening under flow to capture collective bubble dynamics on complex surfaces.

The models neglect bubble-induced modifications to hydrodynamic drag, detailed coalescence/pinch-off hydrodynamics, and long-time pure diffusion (days) and body-induced convective diffusion effects. Wetting dynamics and air exposure at the interface (which may influence nucleation) are not quantified. Hydrostatic pressure increases at depth and container-shape/temperature effects are acknowledged but not fully explored. The discrete model’s pinch-off/rebirth rules and the continuum model’s ḃ closure are simplified. Surfactant effects, contact-angle variations, and exact roughness distributions (a_0) are not systematically measured. Rotational drag for partially immersed bodies is only coarsely estimated.