Bubble Competition & Merger Model

• The Bubble Competition and Merger Model defines discrete bubble-like objects with state variables that undergo growth, competition, and merger or breakup events.
• It is applied across multiple domains such as multiphase CFD, wet-foam coarsening, Rayleigh–Taylor instability, electrolysis, and reionization to capture essential physical interactions.
• The framework emphasizes pressure-consistent tracking and topology-aware procedures to accurately model interface coalescence, diffusive transport, and competitive dynamics.

Bubble competition and merger model denotes a family of formalisms in which discrete bubble-like entities evolve under a local growth law, compete for volume, pressure relief, or accessible space, and undergo coalescence, breakup, percolation, or loss of identity under explicitly defined rules. In multiphase CFD, the emphasis is on pressure-consistent tracking of disconnected gas regions; in foam coarsening, on diffusive competition between small and large bubbles; in Rayleigh–Taylor instability, on the distinction between bubble-merger and bubble-competition regimes; in electrolysis, on motion reversal and eventual coalescence of interacting hydrogen bubbles; and in reionization studies, on hierarchical merger trees of ionized regions (Anderl et al., 2016, Morgan et al., 1 Aug 2025, Zhang et al., 2017, Bashkatov et al., 2022, Jamieson et al., 2024).

1. Conceptual scope and recurrent structure

The phrase does not denote a single universally standardized equation set. Taken together, the major uses of the term indicate a recurrent modeling pattern: a “bubble” is treated as an identifiable object, endowed with state variables such as volume, radius, pressure, growth rate, or ancestry; “competition” denotes a rule by which neighboring or overlapping objects inhibit or redirect one another’s growth; and “merger” denotes either literal coalescence of interfaces or a hierarchical absorption event in which a smaller object ceases to exist independently (Anderl et al., 2016, Zhang et al., 2017, Bashkatov et al., 2022, Jamieson et al., 2024).

Domain	Bubble object	Merger meaning
Bubbly-flow CFD	Connected gas region	Coalescence or breakup tracked by IDs
Wet foams	Gas bubble in jammed foam	Bubble loss through diffusive competition
Rayleigh–Taylor mixing	Rising light-fluid bubble	Distinct bubble-merger mechanism
Water electrolysis	Detached H$_2$ bubble	Return and coalescence with second bubble
Reionization	Ionized H II region	Percolation and merger-tree absorption

Two distinctions are especially important. First, some models resolve literal topology change, whereas others treat merger only effectively. The enhanced free-surface lattice Boltzmann formulation detects actual coalescence and breakup of connected gas regions (Anderl et al., 2016), whereas the wet-foam coarsening theory describes a decrease in bubble number caused by diffusive gas transfer rather than interface fusion (Morgan et al., 1 Aug 2025). Second, competition may arise from different physical drivers: compressibility and surface tension in bubbly flows, inter-bubble diffusive transport in foams, mode selection in Rayleigh–Taylor growth, buoyancy–thermocapillary force balance in electrolysis, and percolation of ionized fronts during reionization (Zhang et al., 2017, Bashkatov et al., 2022, Jamieson et al., 2024).

2. Topology-aware fluid formulations

A technically explicit competition-and-merger framework appears in the enhanced bubble model for the free-surface lattice Boltzmann method (FSLBM). Standard FSLBM reduces liquid–gas flow to a single-phase free-surface simulation through a volume-of-fluid-like fill level $\varphi \in [0,1]$, with $\varphi=0$ for gas cells, $\varphi=1$ for liquid cells, and $0<\varphi<1$ for interface cells. In bubbly flows this is insufficient because compressed gas pressure in closed bubbles influences the liquid boundary condition and because disconnected gas regions can merge or split. The enhancement therefore introduces a per-bubble data structure containing a unique bubble ID, an initial volume $V^*$, a current volume $V(t)$, and a pressure (Anderl et al., 2016).

The pressure model separates compression from capillarity. The compression contribution is

$p_V=\frac{V^*}{V(t)}\,p_0,$

with $p_0=\rho_0 c_s^2$ and $\rho_0=1$ in the nondimensional lattice setup, while the surface-tension contribution is

$\varphi \in [0,1]$0

This supplies the missing gas-side information required to reconstruct interface distributions in a solver that does not explicitly evolve gas PDFs. Cell mass is represented as

$\varphi \in [0,1]$1

and local mass exchange updates fill levels, thereby allowing cell-type transitions between gas, interface, and liquid states (Anderl et al., 2016).

The merger rule is local: if an interface cell has a neighboring interface cell carrying a different bubble ID, a coalescence event is triggered. If $\varphi \in [0,1]$2 is the set of merging bubbles, the new bubble is assigned

$\varphi \in [0,1]$3

after which its pressure is recomputed from the ideal-gas relation. Breakup is detected geometrically, through a local configuration near a liquid cell in which two interface sections of the same bubble have anti-parallel surface normals, followed by a seed-flood-fill that colors connected subregions and measures daughter volumes. When a parent bubble $\varphi \in [0,1]$4 splits, each daughter inherits the parent pressure at the breakup instant,

$\varphi \in [0,1]$5

and receives

$\varphi \in [0,1]$6

The design goal is efficiency: flood fill is invoked only when connectivity may have changed, and the parallel implementation resolves color mismatches across process boundaries iteratively through ghost-layer exchanges (Anderl et al., 2016).

A complementary, but not topologically resolving, framework is the unified theory for a central bubble interacting with a surrounding bubble cluster. That model assumes spherical bubbles, weak fluid compressibility, adiabatic gas behavior, and finite-speed pressure-wave coupling. Its central technical element is retarded-time evaluation,

$\varphi \in [0,1]$7

implemented by interpolation on the time axis when discrete time steps do not coincide with the emission time. It therefore captures mutual hindrance, delayed acoustic loading, wave reflection, and superposition, but does not model actual interface coalescence or post-merger topology change (Zhang et al., 2023). This suggests a division between pre-merger interaction physics and topology-aware bookkeeping: the former governs approach and competition, the latter the reassignment of identity once connectedness changes.

3. Diffusive competition in wet-foam coarsening

In wet foams, bubble competition is formulated as diffusive gas transport driven by pressure differences. Small bubbles have higher pressure, with $\varphi \in [0,1]$8 by the Young–Laplace law, and therefore lose gas to larger, lower-pressure neighbors. The relevant regime in the cited mean-field theory is a non-adhesive jammed foam, $\varphi \in [0,1]$9, in which gas transfer occurs mainly through thin contact films rather than through the bulk liquid. The exact border-blocking growth law is

$\varphi=0$0

where $\varphi=0$1 is the area of the $\varphi=0$2-th film, $\varphi=0$3 is the gas diffusion coefficient, $\varphi=0$4 is Henry’s constant, and $\varphi=0$5 is the film thickness (Morgan et al., 1 Aug 2025).

The mean-field closure approximates the bubble pressure as

$\varphi=0$6

defines $\varphi=0$7, and approximates the film-area fraction by

$\varphi=0$8

The resulting three-dimensional mean-field growth law is

$\varphi=0$9

This law is explicitly a competition law: growth depends on whether a bubble lies above or below the critical radius $\varphi=1$0, and on an effective film-coverage factor that modulates access to transport pathways (Morgan et al., 1 Aug 2025).

The same paper derives an analytic scaling-state bubble-size distribution as a function of liquid fraction. In the scaling state, the stable coarsening exponent is

$\varphi=1$1

and the osmotic-pressure parameter is related to liquid fraction by

$\varphi=1$2

An explicit probability density $\varphi=1$3 is obtained, with a cutoff $\varphi=1$4 determined by $\varphi=1$5, and the model predicts $\varphi=1$6 for any $\varphi=1$7 (Morgan et al., 1 Aug 2025).

A central consequence is that the disappearance of small bubbles is not singular: because $\varphi=1$8 as $\varphi=1$9, very small bubbles transfer gas inefficiently and can persist as a substantial population. The model reproduces the general presence of many small bubbles for $0<\varphi<1$0, but it does not reproduce the sharp small-radius peak seen in prior simulations and ISS experiments. The authors attribute this discrepancy to omitted physics, especially rattlers, and possibly adhesion and bulk diffusion (Morgan et al., 1 Aug 2025). In this literature, “merger” is therefore best understood as an effective population-level consequence of competition rather than as direct interface fusion.

4. Bubble-competition and bubble-merger in Rayleigh–Taylor instability

The Rayleigh–Taylor literature distinguishes two self-similar mechanisms of bubble evolution. Bubble-merger dominates when the interface is perturbed mainly by short wavelengths $0<\varphi<1$1; bubbles then grow by merging with neighboring smaller bubbles and obey a universal evolution law associated with a lower-bound growth coefficient around

$0<\varphi<1$2

Bubble-competition dominates when the initial perturbation contains influential long-wave components or a broadband spectrum with long wavelengths. In that case the bubble mixing zone is controlled by the dominant mode or wavepacket selected from the initial spectrum (Zhang et al., 2017).

The theory is formulated in terms of the bubble-zone height $0<\varphi<1$3, the dominant bubble diameter $0<\varphi<1$4, the Atwood number

$0<\varphi<1$5

the aspect ratio

$0<\varphi<1$6

the quadratic growth coefficient

$0<\varphi<1$7

the dimensionless initial perturbation amplitude $0<\varphi<1$8, and the linear-growth reduction factor

$0<\varphi<1$9

The main asymptotic results are

$V^*$0

These relations imply that the dominant bubble expands self-similarly with a steady aspect ratio depending only on density ratio, whereas the growth coefficient depends on initial conditions and material properties rather than being universal (Zhang et al., 2017).

The derivation proceeds by selecting a dominant wavelength $V^*$1 through maximization of the height evolution over wavelengths. That selection principle is the operative definition of competition in this context: different initial wavepackets “compete,” and the one producing the greatest growth determines the macroscopic bubble zone. The theory was proposed to explain the longstanding spread of measured or simulated $V^*$2 values,

$V^*$3

across experiments and computations with $V^*$4 and $V^*$5. Viscosity, miscibility, and surface tension enter only through $V^*$6, as they reduce the actual linear growth rate relative to the ideal value $V^*$7 (Zhang et al., 2017).

5. Motion reversal and coalescence in water electrolysis

The expression “bubble competition and merger” is used most directly in the study of hydrogen bubbles generated at a Pt microelectrode in $V^*$8 M H$V^*$9SO$V(t)$0. The experiments impose a cyclic three-phase potential protocol. Phase 1 applies a short cathodic pulse $V(t)$1 of typically $V(t)$2 to $V(t)$3 V for $V(t)$4 to $V(t)$5 ms, producing the first bubble. The potential is then switched off for $V(t)$6 to $V(t)$7 ms, allowing detachment and a controlled rise distance $V(t)$8. Phase 2 applies $V(t)$9 V for $p_V=\frac{V^*}{V(t)}\,p_0,$0 to $p_V=\frac{V^*}{V(t)}\,p_0,$1 ms, producing a second bubble while the first remains nearby. A final waiting time $p_V=\frac{V^*}{V(t)}\,p_0,$2 ms resets the system. This yields highly reproducible bubble pairs, with first-bubble radii such as $p_V=\frac{V^*}{V(t)}\,p_0,$3, and datasets including 105 and 133 repetitions (Bashkatov et al., 2022).

Three scenarios are observed as a function of the first bubble’s separation $p_V=\frac{V^*}{V(t)}\,p_0,$4 from the electrode when the second bubble begins to grow. In scenario I, the first bubble is close enough that it initially accelerates upward because of fluid displacement by the growing second bubble, then decelerates, reaches a maximum distance $p_V=\frac{V^*}{V(t)}\,p_0,$5, reverses direction, returns toward the electrode against buoyancy, and finally coalesces with the second bubble. In scenario II, the first bubble also reverses, but the thermocapillary attraction is too weak to produce collision. In scenario III, the bubbles are sufficiently far apart that only displaced-flow advection remains and no reversal occurs. The transition is controlled by a critical range around

$p_V=\frac{V^*}{V(t)}\,p_0,$6

with the I–II boundary only about $p_V=\frac{V^*}{V(t)}\,p_0,$7 wide in one comparison, while scenario III appears above about

$p_V=\frac{V^*}{V(t)}\,p_0,$8

For the strongest reversal case, the first bubble exhibits an initial velocity peak $p_V=\frac{V^*}{V(t)}\,p_0,$9 and a negative reversal peak $p_0=\rho_0 c_s^2$0 (Bashkatov et al., 2022).

The proposed mechanism is a competition between buoyancy and thermocapillarity. The free terminal rise speed is estimated as

$p_0=\rho_0 c_s^2$1

for $p_0=\rho_0 c_s^2$2, $p_0=\rho_0 c_s^2$3, and $p_0=\rho_0 c_s^2$4, so the initial positive peak can exceed $p_0=\rho_0 c_s^2$5 by a factor of about $p_0=\rho_0 c_s^2$6 to $p_0=\rho_0 c_s^2$7 because the first bubble is advected by the upward displaced flow from the second. The hydrodynamic interpretation is summarized by the creeping thermocapillary migration relation

$p_0=\rho_0 c_s^2$8

When the thermocapillary contribution exceeds the buoyant term, the motion reverses sign. From the observed reverse-speed scale $p_0=\rho_0 c_s^2$9, the required temperature gradient is estimated to be about $\rho_0=1$0, while temperature rises of order $\rho_0=1$1 K over $\rho_0=1$2 correspond to gradients of order $\rho_0=1$3, making the mechanism plausible (Bashkatov et al., 2022).

The principal evidence is optical. High-speed shadowgraphy resolves the three trajectory classes, particle tracking velocimetry shows the upward bow wave and local Marangoni convection, and Toepler’s schlieren imaging visualizes vertical refractive-index gradients that are interpreted mainly as temperature gradients. The decisive observation is that reversal begins when the thermal boundary layer generated by the second bubble reaches the underside of the first bubble. Electrostatic attraction is disfavored because the electric force was previously shown to be negligible beyond about $\rho_0=1$4, far smaller than the $\rho_0=1$5 interaction distances studied here, and because reversal persists in control experiments with the potential switched off after second-bubble creation (Bashkatov et al., 2022).

6. Hierarchical bubble trees during reionization

A merger-tree version of bubble competition appears in simulations of the Epoch of Reionization. Using the THESAN radiation-hydrodynamics suite in a $\rho_0=1$6 volume, the reconstruction is based on the spatially resolved reionization redshift field $\rho_0=1$7, defined for each grid cell as the last redshift when the local ionized hydrogen fraction $\rho_0=1$8 crosses $\rho_0=1$9 from below. The fiducial run uses 81 snapshots from $\varphi \in [0,1]$00 to $\varphi \in [0,1]$01, typically analyzed on $\varphi \in [0,1]$02 Cartesian grids, with a recommended mild smoothing of 125 ckpc (Jamieson et al., 2024).

The algorithm identifies local maxima in $\varphi \in [0,1]$03 as seeds, gives each bubble a queue of neighboring cells sorted by decreasing $\varphi \in [0,1]$04, and expands each bubble into its highest-value neighbor. Adjacency is usually defined with 26 neighbors in 3D. A merger occurs when two or more groups attempt to occupy the same point; the largest bubble by volume is designated the parent, the smaller groups are merged into it, and they permanently cease independent growth. The output is a genuine merger tree rather than a watershed segmentation, with stored quantities including z_form, z_merge, merged_with, cells_merged, parent_cells_merged, counts, and cell_to_bubble (Jamieson et al., 2024).

Three growth stages emerge. The first is an initial expansion phase at $\varphi \in [0,1]$05, when bubbles are tiny and mergers are rare. The second is a merging or percolation phase at $\varphi \in [0,1]$06, when overlap becomes common and the merger rate spikes. The third is a rapid expansion phase at $\varphi \in [0,1]$07, when one bubble dominates and sweeps through the remaining neutral gas. The dominant bubble appears very early, by $\varphi \in [0,1]$08, and its volume fraction

$\varphi \in [0,1]$09

rises rapidly toward unity after a main merging event around $\varphi \in [0,1]$10. The cumulative and instantaneous expansion fractions,

$\varphi \in [0,1]$11

and the effective number of actively growing bubbles,

$\varphi \in [0,1]$12

all encode the same transition from many competitors to one dominant object (Jamieson et al., 2024).

The size statistics identify a characteristic scale. There is a sharp deficit of bubbles with effective radius near $\varphi \in [0,1]$13 cMpc, and merger events show a cutoff around the same scale. The interpretation given is that once bubbles approach this size, competitive overlap becomes so strong that they are absorbed into the main percolating region before remaining as independent objects. The merger ratio

$\varphi \in [0,1]$14

likewise changes from values around $\varphi \in [0,1]$15 for early, near-equal mergers to much smaller values once the main bubble dominates. In this setting, competition is percolative rather than mechanical: many ionized regions race to claim neighboring cells, but only the first claimant retains identity (Jamieson et al., 2024).

7. Generalizations, limitations, and related abstractions

Across these literatures, the competition-and-merger architecture relies on strong modeling assumptions. In FSLBM, the gas phase is not dynamically solved; internal gas motion is neglected, bubble pressure is modeled rather than evolved from a full compressible two-phase system, and topology change depends on geometric detection plus flood fill (Anderl et al., 2016). In the wet-foam mean-field theory, only border-blocking film transport is included, adhesion is set to zero, and the absence of rattlers is likely responsible for discrepancies in the small-bubble distribution (Morgan et al., 1 Aug 2025). The Rayleigh–Taylor theory assumes dominant-mode selection and packages material effects into the single reduction factor $\varphi \in [0,1]$16 (Zhang et al., 2017). The electrolysis model interprets long-range attraction through thermal boundary layers rather than electrostatics, and the bubble-cluster theory assumes spherical bubbles and therefore cannot describe nonspherical jetting or merger morphology (Bashkatov et al., 2022, Zhang et al., 2023). The reionization merger tree is chronological and object-based, but it is not a watershed segmentation and does not assign post-contact independent growth to merged regions (Jamieson et al., 2024).

The same competition-merger template has also been abstracted beyond literal fluid bubbles. In three-agent games, mergers are allowed when they increase both agents’ probabilities of winning; in the fully competitive limit $\varphi \in [0,1]$17, the score distribution becomes stratified and self-similar, with scaling factor

$\varphi \in [0,1]$18

and shocks governed by the Rankine–Hugoniot condition (Rador et al., 2010). In rank-based market models with a fluctuating number of firms, companies split when a market weight reaches $\varphi \in [0,1]$19 and merge on exponential clocks; under the stated assumptions, the system is nonexplosive, diverse, and admits no relative arbitrage over finite horizons (Karatzas et al., 2014). A different but related use of the terminology appears in frustrated magnets, where a field-induced skyrmion crystal evolves into a bubble crystal as spatial or spin anisotropy increases, and the resulting bubble-crystal regime displays a devil’s-staircase ordering wave-vector (Hayami et al., 2016).

A plausible synthesis is that bubble competition and merger models are most powerful when a continuum field alone is insufficient and one must add object identity, ancestry, or topology to describe evolution. Whether the objects are gas pockets, foam cells, RT bubbles, ionized regions, or abstract competitors, the same formal ingredients recur: local growth dynamics, a competition criterion, an event rule for merger or breakup, and a bookkeeping scheme that preserves the relevant conserved quantities while redefining object identity after interaction.