2.1. Implementation

One of the primary challenges in simulating floating bubbles is the high grid resolution required to resolve the thin liquid film in the spherical-cap region (as shown in figure 2). To alleviate this, the weight of the film can be neglected in determining the bubble shape since its thickness, typically a few microns for a millimetric bubble (Lhuissier & Villermaux Reference Lhuissier and Villermaux2012; Shaw & Deike Reference Shaw and Deike2024), is orders of magnitude smaller than the bubble radius $R_{0}$ . For the film weight to matter, the bubble would need to be much larger – of the order of metres (Cohen et al. Reference Cohen, Texier, Reyssat, Snoeijer, Quéré and Clanet2017) – which exceeds the bubble sizes examined in the present work. Assuming a weightless film (spherical cap) thus simplifies the problem significantly in terms of computational efforts.

The second bottleneck is particular to the use of interface capturing techniques in multiphase flow simulations, specifically the volume-of-fluid (VOF) method employed in this study. While we do not need to resolve the liquid film at the top of the bubble, we must prevent the bubble from coalescing with the free surface. Interface capturing methods such as VOF (Hirt & Nichols Reference Hirt and Nichols1981; Popinet Reference Popinet2009), level set (Osher & Sethian Reference Osher and Sethian1988; Gibou, Fedkiw & Osher Reference Gibou, Fedkiw and Osher2018) and phase field (Patel & Stark Reference Patel and Stark2023; Roccon, Zonta & Soldati Reference Roccon, Zonta and Soldati2023) are designed to allow fluid–fluid interfaces to break and coalesce naturally, which is undesirable in our set-up. To mitigate this, we employ a multi-VOF (MVOF) approach in which each phase boundary in the flow domain is captured by a unique volume-fraction field. For example, the configuration in figure 2 can be realised in simulations by utilising two separate volume-fraction indicators $C_{1}{ (\boldsymbol{x},t )}$ and $C_{2}{ (\boldsymbol{x},t )}$ – one for the free surface and another for the bubble interface. The central idea of our MVOF formulation is similar to recent works on non-coalescing fluid interfaces using VOF-based simulations (Karnakov et al. Reference Karnakov, Litvinov and Koumoutsakos2022; Sanjay et al. Reference Sanjay, Lakshman, Chantelot, Snoeijer and Lohse2023; Ray et al. Reference Ray, Han, Yue, Guo, Chao and Cheng2024; Gao et al. Reference Gao, Liu, Yang and Hu2025).

For a system comprising $\mathcal{N}_{b}$ floating bubbles and a free surface, we initialise $\mathcal{N}_{b}+1$ volume-fraction variables $C{ (\boldsymbol{x},t )}\in [0,1]$ , where $C=0$ for gas/air (inside all bubbles and above the free surface) and $1$ for liquid/water. The movement of individual bubbles and the free surface is captured by solving the VOF-advection equation for the corresponding volume-fraction indicator $C_{i}$ :

(2.1) \begin{eqnarray} \frac {\partial C_{i}}{\partial t}+\boldsymbol{u}{\boldsymbol{\cdot }}\boldsymbol{\nabla }C_{i}=0, \end{eqnarray}

with the subscript $i$ ranging from $1$ to $\mathcal{N}_{b}+1$ . The advection velocity $\boldsymbol{u}$ in (2.1) follows from the flow continuity equation

(2.2) \begin{eqnarray} \boldsymbol{\nabla }{\boldsymbol{\cdot }}\boldsymbol{u}=0 \end{eqnarray}

and the momentum conservation equation

(2.3) \begin{eqnarray} \rho \left (\frac {\partial \boldsymbol{u}}{\partial t}+\boldsymbol{u}{\boldsymbol{\cdot }}\boldsymbol{\nabla \!}\boldsymbol{u}\right )=-\boldsymbol{\nabla \!}p + \boldsymbol{\nabla }{\boldsymbol{\cdot }}\big [\mu \big (\boldsymbol{\nabla \!}\boldsymbol{u}+\big (\boldsymbol{\nabla \!}\boldsymbol{u}\big )^{\mathrm T}\big )\big ] + \rho \boldsymbol{g}+ \sum \limits ^{\mathcal{N}_{b}+1}_{i=1}\boldsymbol{\mathcal{F}}^{s}_{i}, \end{eqnarray}

where $\rho$ , $\mu$ , $t$ , $p$ , $\boldsymbol{g}=\langle 0, g \rangle$ , $\boldsymbol{\mathcal{F}}^{s}_{i}$ denote the density, dynamic viscosity, time, pressure, vector for the gravitational acceleration (see figure 2) and surface tension force per unit volume, respectively. The continuum surface force model (Brackbill, Kothe & Zemach Reference Brackbill, Kothe and Zemach1992; Popinet Reference Popinet2018) yields

(2.4) \begin{eqnarray} \boldsymbol{\mathcal{F}}^{s}_{i}=\sigma \kappa _{i}\delta _{i}\boldsymbol{n}_{i}, \end{eqnarray}

where $\sigma$ , $\kappa$ , $\delta$ , $\boldsymbol{n}$ represent the surface tension, interface curvature, interface delta function and interface normal, respectively. The fluid density $\rho$ in (2.3) is calculated using the weighted average,

(2.5) \begin{eqnarray} \rho =\chi \rho _{l}+{\left (1{-}\chi \right )}\rho _{g}, \end{eqnarray}

whereas the dynamic viscosity $\mu$ follows the harmonic mean (Tryggvason, Scardovelli & Zaleski Reference Tryggvason, Scardovelli and Zaleski2011),

(2.6) \begin{eqnarray} \mu ^{-1}=\chi \mu ^{-1}_{l}+{\left (1{-}\chi \right )}\mu ^{-1}_{g}. \end{eqnarray}

The subscripts $l$ and $g$ correspond to liquid/water and gas/air, respectively. For the problem of floating bubbles, the quantity $\chi$ in (2.5)–(2.6) is defined as

(2.7) \begin{eqnarray} \chi =\prod \limits ^{\mathcal{N}_{b}+1}_{i=1}C_{i}. \end{eqnarray}

Essentially, $\chi$ combines all volume-fraction indicators $C_{i}$ into a global phase indicator to compute the correct fluid density and viscosity for the flow field. This procedure couples all $C_{i}$ and automatically prevents bubble interfaces and the free surface from coalescing when their separation approaches the size of a grid cell.

We use the open-source finite volume flow solver Basilisk (Popinet & Collaborators Reference Popinet2013–Reference Popinet2024) to incorporate governing equations (2.1)–(2.3) into our numerical simulations. The volume-fraction field $C_{i}$ in (2.1) is advected using a piecewise-linear geometrical VOF scheme (Scardovelli & Zaleski Reference Scardovelli and Zaleski1999). The interface curvature $\kappa$ needed for the surface tension body force $\boldsymbol{\mathcal{F}}^{s}$ in (2.4) is determined using the height function method (Popinet Reference Popinet2009). Basilisk’s well-balanced implementation of the surface tension force minimises spurious currents in the simulations (Popinet Reference Popinet2018). For the numerical solution of the flow field, Basilisk relies on the classical time-splitting projection method (Chorin Reference Chorin1969) along with second-order schemes for the spatial gradients. The viscous term in the momentum conservation (2.3) is computed implicitly, while the second-order Bell–Colella–Glaz unsplit upwind scheme (Bell, Colella & Glaz Reference Bell, Colella and Glaz1989) is applied to the nonlinear velocity advection term $\boldsymbol{u}{\boldsymbol{\cdot }}\boldsymbol{\nabla \!}\boldsymbol{u}$ (Popinet Reference Popinet2009). All simulations in this study are conducted on dynamic quadtree grids (Popinet Reference Popinet2003, Reference Popinet2009). This is realised using the wavelet-based built-in adaptive mesh refinement functionality within the Basilisk library (van Hooft et al. Reference van Hooft, Popinet, van Heerwaarden, van der Linden, de Roode and van de Wiel2018). The local mesh resolution at each time step is adjusted based on the flow field $\boldsymbol{u}$ , volume-fraction indicator $C_{i}$ and corresponding interface curvature $\kappa$ . A mesh refinement tolerance of $10^{-4}$ is applied in all cases (see van Hooft et al. (Reference van Hooft, Popinet, van Heerwaarden, van der Linden, de Roode and van de Wiel2018) for more details).

2.2. Computational set-up, code validation and grid convergence study

The effectiveness of the MVOF approach in preventing interface coalescence compared with the traditional VOF implementation is demonstrated qualitatively in Appendix A by simulating the collision of two droplets. To further substantiate our MVOF solver, in this subsection, we simulate an axisymmetric isolated floating bubble using dual volume-fraction fields.

For all subsequent axisymmetric simulations of floating bubbles (including § 3), we set the domain size to $40R_{0}$ in both the radial ( $r$ , horizontal) and axial ( $z$ , vertical) directions. Equivalently, we employ a cubic domain of size $40R_{0}$ for three-dimensional simulations in §§ 4–5. The free surface is initially defined as a horizontal plane positioned at $z=0$ , centred midway through the domain height in the $z$ -direction. One or multiple spherical bubbles are placed beneath this free surface and allowed to float and settle into their equilibrium shapes determined by the Bond number $\textit{Bo}$ . Unless stated otherwise, the density and viscosity ratios in this study are fixed at $\rho _{l}{/}\rho _{g}=\mu _{l}{/}\mu _{g}=100$ , providing large enough property contrasts to mimic a liquid–gas system similar to water and air (Reichl, Hourigan & Thompson Reference Reichl, Hourigan and Thompson2005; Patel et al. Reference Patel, Sun, Yang and Zhu2025). In axisymmetric cases (including § 3), the free-slip boundary condition is imposed at domain edges corresponding to $r=40R_{0}$ and $z={\pm }20R_{0}$ , and at $\{x,y,z\}={\pm }20R_{0}$ for three-dimensional cases in §§ 4–5.

Figure 3 illustrates the equilibrium shapes generated by our MVOF simulations for various Bond ( $\textit{Bo}$ ) numbers. As $\textit{Bo}$ increases, the deformation of the bubble interface and the free surface becomes more noticeable. At a high Bond number of $\textit{Bo}=100$ , the bubble adopts a hemispherical shape, consistent with experimental observations (Lhuissier & Villermaux Reference Lhuissier and Villermaux2012; Poulain et al. Reference Poulain, Villermaux and Bourouiba2018; Bartlett et al. Reference Bartlett, Oratis, Santin and Bird2023; Shaw & Deike Reference Shaw and Deike2024). Note that the thickness of the spherical cap above the bubble is not resolved in MVOF simulations following the assumption of a weightless liquid film § 2.1. Bubble shapes from MVOF simulations in figure 3 are computed with the highest mesh refinement level of $\mathcal{L}_{h}=12$ , equivalent to ${\approx} 102$ uniform grid cells per bubble radius $R_{0}$ , with each cell having a size of $\varDelta =R_{0}{/}102.4$ .

Figure 3.

Comparison of the steady-state axisymmetric floating bubble shapes obtained using the Young–Laplace (YL) model and MVOF simulations for different Bond ( $\textit{Bo}$ ) numbers.

We compare our MVOF simulation results with shapes derived from the YL law. The solution procedure, as discussed in Toba (Reference Toba1959) and Lhuissier & Villermaux (Reference Lhuissier and Villermaux2012), involves iteratively solving the balance equations for pressure and surface tension across distinct interfacial regions: the bubble cavity, spherical cap and meniscus (see Appendix B for further details). The interface profiles from both methods, MVOF and YL, exhibit excellent agreement across different Bond numbers, as demonstrated in figure 3. The shape of the spherical cap in the YL model is captured by imposing the appropriate Laplace pressure jump, as outlined in Appendix B. On the other hand, MVOF simulations allow the bubble to press against the free surface by preventing the coalescence (i.e. keeping the bubble interface and the free surface separated), thereby producing the correct deformation of the spherical cap. While the YL model provides an appealing alternative to the MVOF approach for computing bubble shapes, it becomes impractical for scenarios involving two or more floating bubbles, which is the primary focus of our study.

Finally, figure 4 demonstrates that a lower refinement level of $\mathcal{L}_{h}=11$ also yields accurate results, albeit with minor deviations in the regions highlighted by rectangles. These discrepancies amplify upon further lowering the refinement level (not shown here). The deviation in the interface position at the top of the bubble between MVOF and YL is $2.6\,\%$ at $\mathcal{L}_{h}=11$ and $1.4\,\%$ at $\mathcal{L}_{h}=12$ , while the interface height between $\mathcal{L}_{h}=11$ and $\mathcal{L}_{h}=12$ differs by $1.2\,\%$ relative to $\mathcal{L}_{h}=12$ . Based on these observations, we use an adaptive mesh with $\mathcal{L}_{h}=12(\varDelta =R_{0}{/}102.4)$ in our subsequent two-dimensional axisymmetric simulations of coaxial bubbles in § 3, as the interface deformation in these cases is more pronounced than in the single-bubble scenario. For three-dimensional simulations involving bubbles distributed along the free surface in §§ 4–5, we implement a refinement level of $\mathcal{L}_{h}=11(\varDelta =R_{0}{/}51.2)$ .

Figure 4.

Axisymmetric floating bubble shapes obtained from MVOF simulations using different mesh resolutions and their comparison with the shape derived from the YL equations for a Bond number $\textit{Bo}=100$ .

3.1. Numerical findings

Figure 5 presents a series of bubble shapes obtained through axisymmetric MVOF simulations, spanning a wide range of Bond numbers across two decades. In the surface-tension-dominated regime ( $\textit{Bo}\lt 1$ ), interface deformation remains minimal, with bubbles retaining an almost perfect spherical shape and showing only slight perturbations in the contact region of two bubbles and at the free surface. This behaviour is consistent with the case of an isolated floating bubble shown in figure 3 at $\textit{Bo}=0.5$ . As the Bond number increases beyond $\textit{Bo}=1$ , gravitational forces dominate, leading to pronounced interface deformations. At $\textit{Bo}=5$ and $\textit{Bo}=10$ , the bottom red-coloured bubble starts pressing against the top blue-coloured bubble, adopting an oblate shape. The introduction of an additional bubble in the current case produces significant deformation of the free surface and the top bubble compared with cases involving isolated bubbles, even at relatively low $\textit{Bo}$ values. We can already notice the hemispherical shape of the top bubble at $\textit{Bo}=10$ in figure 5, which was previously observed for an isolated bubble at a 10 times larger Bond number of $\textit{Bo}=100$ in figure 3.

Figure 5.

Equilibrium shapes of axisymmetric floating bubble pairs at various Bond ( $\textit{Bo}$ ) numbers. Interface profiles captured with different volume-fraction indicators are shown in distinct colours.

Previous studies (Georgescu, Achard & Canot Reference Georgescu, Achard and Canot2002; Walls et al. Reference Walls, Henaux and Bird2015) on isolated floating air bubbles in water identified a critical Bond number of $\textit{Bo}^{\star }=\rho _{l}gR^{2}_{c}/\sigma \approx 3$ based on the radius of the spherical cap $R_{c}$ (see figure 6 for the definition of $R_{c}$ ), marking the onset of transition from the production jet to film droplets during bubble bursting. No jet droplets are generated once the critical Bond number $\textit{Bo}^{\star }$ is reached (corresponding to $R_{c}\approx 4.2\, \rm { mm}$ for the air–water combination) due to the dominant effects of gravitational forces (Georgescu et al. Reference Georgescu, Achard and Canot2002; Walls et al. Reference Walls, Henaux and Bird2015). However, the formation of Worthington (or Rayleigh) jets persists (Krishnan et al. Reference Krishnan, Hopfinger and Puthenveettil2017). For bubbles with $\rho _{l}gR^{2}_{c}/\sigma \gt \textit{Bo}^{\star }$ , the size of the spherical cap at the top of the bubble strongly influences the bursting process.

Figure 6.

Comparison of axisymmetric equilibrium shapes: an isolated floating bubble at $\textit{Bo}_{\textit{iso}}=5$ ( $\rho _{l}gR^{2}_{c}/\sigma =3.8$ ) versus a pair of coaxial floating bubbles at $\textit{Bo}_{\textit{coax}}=2$ . Here, $R_{c}$ denotes the radius of the spherical cap.

Figure 6 compares an isolated floating bubble at $\textit{Bo}_{\textit{iso}}=5$ (equivalently $\rho _{l}gR^{2}_{c}/\sigma =3.8$ , slightly larger than $\textit{Bo}^{\star }$ ) with a pair of coaxial floating bubbles at $\textit{Bo}_{\textit{coax}}=2$ . Interestingly, both configurations exhibit spherical caps with nearly identical shapes (see the region above the dashed blue line in figure 6). This similarity suggests the possibility of an early onset of the transition from jet to film droplets at a Bond number as low as $\textit{Bo}_{\textit{coax}}=2$ for a pair of coaxial floating bubbles ( ${\approx} 36\,\%$ smaller bubble sizes), as the size of the spherical cap already exceeds the critical threshold. Prior experimental studies (Lhuissier & Villermaux Reference Lhuissier and Villermaux2012; Poulain et al. Reference Poulain, Villermaux and Bourouiba2018; Shaw & Deike Reference Shaw and Deike2024) have shown that the rupture of large, dome-shaped spherical caps, like that shown in figure 3 at $\textit{Bo}=100$ , generates film droplets. Given this, for a floating bubble pair at $\textit{Bo}=10$ in figure 5 with a similarly shaped spherical cap, the atomisation of the spherical cap can generate spray droplets in quantities comparable to those produced by the burst of a much larger isolated floating bubble with $\textit{Bo}=100$ .

To further our discussion, we analyse various geometrical parameters of both an isolated bubble and a pair of floating bubbles. First, we examine the variations in cap height $h_{c}$ , rim radius $R_{r}$ , bubble position along the axis $Z_{b}$ and the bubble aspect ratio $\Delta r/\Delta z$ , as shown in figures 7 and 8. Two schematics illustrating these geometrical parameters are provided in figure 7(a) for an isolated bubble and figure 8(a) for a bubble pair. Measuring different geometrical features is crucial because they are key to various scaling relations that describe dynamic processes tied to bursting bubbles (Puthenveettil et al. Reference Puthenveettil, Saha, Krishnan and Hopfinger2018). For example, the cap height and rim radius are needed to calculate the surface area of the spherical cap, which helps estimate the drainage time of the liquid within the spherical cap. Conversely, comparing the bubble position $Z_{b}$ and aspect ratio $\Delta r/\Delta z$ in figures 7 and 8 enables a more direct measurement of shape changes caused by the additional buoyancy force from the second bubble in a bubble pair.

Figure 7.

Effect of the Bond number $\textit{Bo}$ on various equilibrium shape parameters of axisymmetric isolated floating bubbles: (a) cap height, $h_{c}/R_{0}$ ; (b) rim radius of the spherical cap, $R_{r}/R_{0}$ ; (c) axial location of the bubble, $Z_{b}/R_{0}$ ; and (d) bubble aspect ratio, $\Delta r/\Delta z$ . All shape parameters are illustrated in the inset of (a). Insets in (d) compare equilibrium bubble shapes obtained using the YL model (left) and MVOF simulations (right). Experimental data in (a) and (b) are from Puthenveettil et al. (Reference Puthenveettil, Saha, Krishnan and Hopfinger2018). Note that the origin $z=0$ for axial measurements is defined at the unperturbed free surface. The legends in (b) apply to all plots.

Figure 8.

Effect of the Bond number $\textit{Bo}$ on various equilibrium shape parameters of a pair of coaxial floating bubbles: (a) cap height, $h_{c}/R_{0}$ ; (b) rim radius of the spherical cap (blue circles) and the film in the contact region of two bubbles (red circles), $R_{r}/R_{0}$ ; (c) axial location of the top (blue circles) and bottom (red circles) bubbles, $Z_{b}/R_{0}$ ; and (d) aspect ratios of the top (blue circles) and bottom (red circles) bubbles, $\Delta r/\Delta z$ . All shape parameters are illustrated in the inset of (a). Note that the origin $z=0$ for axial measurements is defined at the unperturbed free surface.

For an isolated bubble, the cap height $h_{c}$ in figure 7(a) increases with the Bond number $\textit{Bo}$ and reaches a plateau at a value approximately equal to the bubble radius ${\approx} R_{0}$ for $\textit{Bo}\gt 10$ . The present YL and MVOF solutions show good agreement with the experimental results of Puthenveettil et al. (Reference Puthenveettil, Saha, Krishnan and Hopfinger2018). In contrast to isolated bubbles, a pair of bubbles in figure 8(a) exhibits a distinct trend in cap height variation. For $\textit{Bo}\lt 1$ , $h_{c}$ increases more rapidly than in the single-bubble case, after which a sharp transition occurs at $\textit{Bo}=1$ , characterised by a subdued growth of $h_{c}$ for $\textit{Bo}\gt 1$ . The slower rise in cap height for bubble pairs with $\textit{Bo}\gt 1$ , relative to $\textit{Bo}\lt 1$ , is due to part of the work done by buoyancy contributing to the radial expansion of the upper bubble. This effect is evident through the oblate bubble shapes observed in figure 5 at $\textit{Bo}=5$ and $10$ . Notably, at $\textit{Bo}=10$ , the dimensionless cap height $h_{c}/R_{0}$ of a bubble pair already surpasses that of an isolated bubble at $\textit{Bo}=100$ .

Figures 7(b) and 8(b) show the trends in the rim radius $R_{r}$ . For a bubble floating near the free surface, $R_{r}$ denotes the radial distance from the axis to the contact point where the spherical cap, meniscus and bubble cavity meet. In the case of bubble pairs, an additional rim radius is defined to account for the size of the contact area between the top and bottom bubbles. In practice, this contact region contains a thin liquid film that is not resolved in the simulations. For isolated floating bubbles, the qualitative variation of $R_{r}$ in figure 7(b) resembles the cap height $h_{c}$ in figure 7(a). The rim radius $R_{r}$ converges towards the theoretical upper bound of $2^{1/3}R_{0}$ as isolated bubbles approach a hemispherical shape for $\textit{Bo}$ exceeding $10$ (see figure 3 and the bottom inset of figure 7 d). As before, our results in figure 7(b) are in excellent agreement with the experimental observations of Puthenveettil et al. (Reference Puthenveettil, Saha, Krishnan and Hopfinger2018). For bubble pairs in figure 8(b), the rim radius $R_{r}$ of the spherical cap exceeds that of the single-bubble counterpart at a given $\textit{Bo}$ , but in the limit $\textit{Bo}\rightarrow 10$ it asymptotically approaches the same upper bound of $2^{1/3}R_{0}$ as isolated bubbles. With increasing $\textit{Bo}$ , the top and bottom bubbles in a pair spread against each other, forming a contact region with a radius $R_{r}$ nearly as large as $R_{0}$ at $\textit{Bo}=10$ . Our results in figure 8(a,b) demonstrate that the presence of an additional bubble substantially modifies both the cap height $h_{c}$ and the rim radius $R_{r}$ , and thereby the surface area of the spherical cap. The cap area of single and coaxial bubbles will be examined in detail later in this subsection.

We point out that the saturation $R_{r}/R_{0}=2^{1/3}$ for the upper bubble in a floating pair (figure 8 b) may not persist in the limit $\textit{Bo}\rightarrow \infty$ . At sufficiently large Bond numbers, both bubbles can rise above the undisturbed free surface and form a larger spherical cap of volume $8\pi R^{3}_{0}/3$ , representing the energetically favourable configuration. This state would instead yield $R_{r}/R_{0}=4^{1/3}$ . Probing the asymptotic regime to verify this hypothesis would require simulations extending to much higher Bond numbers, together with finer grid resolution to accurately resolve the resulting interface curvature, and is therefore left for future investigation.

Figure 7(c,d) illustrates the bubble’s position $Z_{b}/R_{0}$ along the axis and its aspect ratio $\Delta r/\Delta z$ for isolated bubbles. At low Bond numbers ( $\textit{Bo}\lt 1$ ), the position of an isolated bubble shows a slow rise with $\textit{Bo}$ due to dominant surface tension forces that resist its ascent against the free surface. As the Bond number increases ( $1\leqslant \textit{Bo}\leqslant 10$ ), the deformation of the free surface, caused by the bubble’s buoyancy, allows the bubble to reach a higher elevation. Notably, at $\textit{Bo}=7$ , the bubble centre surpasses the unperturbed free-surface level corresponding to $z=0$ , marking the transition from subsurface floating bubbles immersed mainly within the liquid to those significantly exposed to the outer gas phase. At the largest Bond number of $\textit{Bo}=100$ , the bubble centre is positioned $0.5R_{0}$ above the free surface. The variation in aspect ratio $\Delta r/\Delta z$ shown in figure 7(d) also follows a trend similar to $Z_{b}/R_{0}$ in figure 7(c). Between $\textit{Bo}=1$ and $10$ , the bubble undergoes significant elongation. However, for Bond numbers below unity or beyond $10$ , changes in aspect ratio become relatively marginal, forming two plateau branches that connect the $1\leqslant \textit{Bo}\leqslant 10$ transition range in the middle. The aspect ratio of an isolated bubble peaks at a value of $1.8$ , corresponding to highly stretched, hemisphere-like floating bubbles (see bubble shapes in figure 3 and the bottom inset of figure 7 d).

We now compare position $Z_{b}/R_{0}$ and aspect ratio $\Delta r/\Delta z$ of isolated bubbles with our observations of a pair of floating bubbles in figure 8(c,d). The centre position $Z_{b}/R_{0}$ of the top bubble in figure 8(c) rises significantly faster with increasing $\textit{Bo}$ compared with an isolated bubble. This is attributed to the additional buoyancy force exerted by the bottom bubble in a floating pair. Notably, the centre of the top bubble in a pair reaches the free surface at a relatively low Bond number of $\textit{Bo}=2$ , which is approximately one-third of the Bond number required for an isolated bubble. For $\textit{Bo}\gt 2$ , $Z_{b}/R_{0}$ continues to increase and reaches a value as large as $R_{0}$ at $\textit{Bo}=10$ (see the corresponding bubble shape in figure 5), nearly twice the maximum centre height observed for an isolated bubble at $\textit{Bo}=100$ in figure 7(c). In such cases, where the top bubble penetrates significantly into the gas phase, the stability of the spherical cap may become susceptible to perturbations arising from the external flow within the gas phase (e.g. wind near the free surface in the ocean). This could, in turn, affect the dynamics of the bursting process. The bottom bubble in a floating pair also rises towards the free surface due to buoyancy as $\textit{Bo}$ increases.

The centre position $Z_{b}/R_{0}$ of the bottom bubble rises at a relatively faster rate with $\textit{Bo}$ than that of the top bubble, particularly in the range $1\leqslant \textit{Bo}\leqslant 10$ . This behaviour can be explained by the variation in the bubble aspect ratio $\Delta r/\Delta z$ . Since the bottom bubble resides at greater depth, it can sustain larger meridional curvatures, compensating for reduced azimuthal curvature from radial elongation and thereby maintaining the Laplace pressure balance. On the other hand, at higher Bond numbers, free-surface deformation reduces the radial surface-tension force at the contact line and enhances the vertical component, which counteracts the net buoyancy force acting on the top bubble. Following this, $\Delta r /\Delta z$ of the bottom bubble grows more rapidly than that of the top bubble for $1\leqslant \textit{Bo}\leqslant 10$ in figure 8(d). Consequently, the bottom bubble undergoes rapid axial thinning as $\textit{Bo}$ increases, which contributes to a faster elevation of its centre $Z_{b}/R_{0}$ compared with the top bubble. Lastly, in contrast to isolated bubbles, bubble pairs do not exhibit a clear saturation in $\Delta r/\Delta z$ within the current range of Bond numbers. Instead, the aspect ratio of the top bubble declines for $\textit{Bo}\gt 5$ . Nevertheless, both the top and bottom bubbles achieve aspect ratios as large as isolated bubbles but at considerably lower Bond numbers.

Figure 9 presents a comparison of two additional geometric features – the surface area of the spherical cap and the cavity depth – between isolated bubbles and bubble pairs. The surface area of the spherical cap is defined as $\mathcal{S}=\pi { (R^{2}_{r}+\hbar ^{2}_{c} )}$ , where $\hbar _{c}$ represents the cap height measured from the contact point of the spherical cap, meniscus and bubble cavity (note the distinction between cap heights $h_{c}$ and $\hbar _{c}$ ). Our computations of the dimensionless cap area $\mathcal{S}/R^{2}_{0}$ for isolated bubbles in figure 9(ai) show excellent agreement with the nonlinear fit proposed by Kocárková et al. (Reference Kocárková, Rouyer and Pigeonneau2013). It is evident that $\mathcal{S}/R^{2}_{0}$ in figure 9(ai) increases with the Bond number and gradually approaches the theoretical upper limit of $\mathcal{S}/R^{2}_{0}=9.974$ for bubbles with a perfect hemispherical shape, which can be derived from a simple geometric calculation. For a pair of floating bubbles, the cap area increases more rapidly than in isolated bubbles, as shown in figure 9(aii). Similar to the cap height $h_{c}$ in figure 8(a) for bubble pairs, the cap area in figure 9(aii) also grows at different rates for Bond numbers below and above unity. Specifically, the growth in cap area is relatively slower for $\textit{Bo}$ values exceeding unity. Nevertheless, at $\textit{Bo}=10$ , the cap area becomes significantly large, almost matching that of an isolated floating bubble at $\textit{Bo}=100$ . This is consistent with our earlier observations of spherical-cap shapes in isolated bubbles and bubble pairs in figures 3 and 5, respectively.

Figure 9.

Influence of the Bond number $\textit{Bo}$ on cap area $\mathcal{S}=\pi (R^{2}_{r}+\hbar ^{2}_{c})$ and cavity depth $Z_{c}$ for (ai,bi) isolated bubbles and (aii,bii) bubble pairs. In (ai,aii), blue and orange regions indicate capillary- and gravity-driven drainage regimes of the spherical cap, respectively. The fitted curve in (ai) is a nonlinear function of the Bond number $\textit{Bo}$ , as suggested by Kocárková et al. (Reference Kocárková, Rouyer and Pigeonneau2013). Experimental data in (bi) are from Puthenveettil et al. (Reference Puthenveettil, Saha, Krishnan and Hopfinger2018). The legends in (ai,aii) and (bi) apply to all plots.

While bubbles float at the free surface, the liquid trapped within the spherical region – between the bubble and the free surface – gradually drains into the bulk. This drainage process leads to the thinning of the spherical liquid film at the bubble’s top. Eventually, the film becomes sufficiently thin and ruptures spontaneously, triggering the bursting sequence responsible for spray production. The lifetime of floating bubbles at the free surface can range from $\mathcal{O}({\mathrm {ms}})$ to $\mathcal{O}({\mathrm{s}})$ (Krishnan et al. Reference Krishnan, Hopfinger and Puthenveettil2017; Shaw & Deike Reference Shaw and Deike2024). One key factor influencing the rate of film thinning is the surface area of the spherical cap. When the cap radius $R_{c}$ is smaller than $5l_{\sigma }$ , the drainage mechanism is governed by capillary pressure (Lhuissier & Villermaux Reference Lhuissier and Villermaux2012). The characteristic length scale $l_{\sigma }$ corresponds to the capillary length $\sqrt {\sigma /(\rho _{l}g)}$ . In the capillary-driven drainage regime, the film thickness $\xi$ decreases according to the scaling relation (Lhuissier & Villermaux Reference Lhuissier and Villermaux2012; Poulain et al. Reference Poulain, Villermaux and Bourouiba2018; Shaw & Deike Reference Shaw and Deike2024)

(3.1) \begin{eqnarray} \xi {\sim }R_{c}\left [\left (\frac {\mu _{l}}{\sigma t}\right )\frac {\big(R^{2}_{r}+\hbar ^{2}_{c}\big)}{R_{r}}\right ]^{2/3}\!. \end{eqnarray}

Conversely, when the cap radius exceeds $5l_{\sigma }$ , gravity dominates the drainage process (Lhuissier & Villermaux Reference Lhuissier and Villermaux2012), leading to an exponential decay in initial film thickness  $\xi _{0}$ , described by the expression (Kocárková et al. Reference Kocárková, Rouyer and Pigeonneau2013; Bartlett et al. Reference Bartlett, Oratis, Santin and Bird2023)

(3.2) \begin{eqnarray} \frac {\xi }{\xi _{0}}={\exp}\left ({-}\frac {2R^{2}_{0}}{9\big(R^{2}_{r}+\hbar ^{2}_{c}\big)}\times \frac {2\rho _{l}gR_{0}t}{\mu _{l}}\right )\!. \end{eqnarray}

Both film-thinning mechanisms in (3.1) and (3.2) are directly tied to the surface area of the spherical cap. Thus, when a second bubble is present, modifications in the cap area can impact the residence time of isolated bubbles at the free surface. The Bond number ranges corresponding to the capillary-driven (3.1) and gravity-driven (3.2) drainage regimes are marked in blue and orange areas in figure 9(ai,aii). Notably, the critical Bond number for the transition from capillary- to gravity-driven thinning in a floating bubble pair is approximately five times smaller than that of an isolated bubble. While our simulations allow us to relate changes in geometrical features to the lifetime of floating bubbles, they cannot provide the absolute lifetime in either isolated or coaxial configurations. This limitation stems from the numerical approach, as resolving the thin spherical liquid cap together with the bubble itself is computationally intractable. At present, experiments offer the only practical means to measure the bubble lifetime, while inherently accounting for the influence of surfactant concentration, salinity and temperature (Lhuissier & Villermaux Reference Lhuissier and Villermaux2012; Poulain et al. Reference Poulain, Villermaux and Bourouiba2018; Shaw & Deike Reference Shaw and Deike2024).

Spray generated through bubble bursting can take the form of either jet or film droplets. The size and quantity of jet droplets are determined by the breakup of the unstable Worthington jet that emerges from the bubble cavity upon the rupture of the spherical cap. Previous studies (Ghabache et al. Reference Ghabache, Antkowiak, Josserand and Séon2014; Krishnan et al. Reference Krishnan, Hopfinger and Puthenveettil2017; Deike et al. Reference Deike, Ghabache, Liger-Belair, Das, Zaleski, Popinet and Séon2018) have paid particular attention to the velocity scaling of the Worthington jet, which is crucial for estimating the speed of the first jet droplet upon formation. Krishnan et al. (Reference Krishnan, Hopfinger and Puthenveettil2017) introduced a cavity depth model for jet velocity $U_{j}$ based on their experimental observations: $U^{2}_{j}\rho _{l}R_{0}/\sigma \propto (Z_{c}/R_{0})^{2}$ , where $U^{2}_{j}\rho _{l}R_{0}/\sigma$ represents the dimensionless jet velocity (i.e. the jet Weber number). The depth of the bubble cavity at $r=0$ , measured from the undisturbed free surface, is denoted as $Z_{c}$ .

Figure 9(bi) illustrates the variation of $(Z_{c}/R_{0})^{2}$ for isolated bubbles. It is evident that cavity depth does not follow a power-law trend. For $\textit{Bo}\lt 0.4$ , $(Z_{c}/R_{0})^{2}$ reaches a plateau, indicating that jet velocity remains independent of $\textit{Bo}$ and is thus unaffected by gravitational effects. The current numerical results, along with the experimental findings of Puthenveettil et al. (Reference Puthenveettil, Saha, Krishnan and Hopfinger2018), fit a power-law relationship $2.64/\sqrt {\textit{Bo}}$ for $(Z_{c}/R_{0})^{2}$ at intermediate Bond numbers (see figure 9 bi), suggesting a $\textit{Bo}^{-1/2}$ dependence of the jet Weber number (Krishnan et al. Reference Krishnan, Hopfinger and Puthenveettil2017). This is consistent with the observations of Ghabache et al. (Reference Ghabache, Antkowiak, Josserand and Séon2014). The variation in cavity depth with the Bond number for a floating pair of coaxial bubbles differs significantly from that of isolated bubbles, as shown in figure 9(bii). Moreover, contrary to isolated bubbles, where $(Z_{c}/R_{0})^{2}$ plateaus at lower Bond numbers, this behaviour is absent for coaxial bubble pairs.

Beyond the changes in the cavity depth between single and coaxial bubbles, the thin liquid film separating the top and bottom bubbles introduces additional complexity. Following rupture of the spherical cap and prior to the onset of the Worthington jet, one or more capillary waves may emerge at the top of the bubble cavity and propagate downward (Krishnan et al. Reference Krishnan, Hopfinger and Puthenveettil2017; Brasz et al. Reference Brasz, Bartlett, Walls, Flynn, Yu and Bird2018; Deike et al. Reference Deike, Ghabache, Liger-Belair, Das, Zaleski, Popinet and Séon2018). Understanding the stability of the inter-bubble film against such waves is therefore critical to the ensuing jet dynamics. Although a full analysis of this issue is outside the scope of our article, in Appendix C we provide a brief discussion of an idealised scenario in which the spherical and contact-region films rupture simultaneously. To realise this, a pair of coaxial bubbles is initialised using its equilibrium interface profile, excluding the spherical cap and the film in the contact region. We then allow the interface to evolve and compare the dynamics of the Worthington jet and the formation of jet droplets with the post-bursting outcome of the single-bubble counterpart in Appendix C.

3.2. Analytical shape parameters

Following the observations of geometrical features in the previous subsection, we now formulate unified analytical shape parameters for isolated and coaxial floating bubbles in the limit of small bubble sizes. Earlier work by Puthenveettil et al. (Reference Puthenveettil, Saha, Krishnan and Hopfinger2018) addressed the same problem for isolated bubbles, deriving closed-form expressions for various shape parameters in a hierarchical manner, beginning with the rim radius $R_{r}$ . Since $R_{r}$ frequently appears in expressions of other geometrical parameters, it provides a convenient starting point for the derivation.

The variation of the rim radius $R_{r}$ with the bubble size can be obtained by balancing the net buoyancy with the vertical component of the surface tension at the rim. For relatively small bubbles with a weakly deformed free surface, this balance gives

(3.3) \begin{eqnarray} 2\pi \sigma R_{r}{\sin}\,\alpha \approx \rho _{l}g\mathcal{N}_{b}\frac {4}{3}\pi R^{3}_{0}, \end{eqnarray}

where $\alpha$ is the contact angle, as shown in figure 2. This relation holds for both isolated and coaxial floating bubbles, as the factor $\mathcal{N}_{b}$ already accounts for the number of bubbles in the system. The term involving $\alpha$ in (3.3) can be rewritten using the relation ${\sin}\,\alpha =R_{r}/R_{c}$ , where $R_{c}$ is the cap radius (see figure 2). To evaluate $R_{c}$ , we recall our discussion in Appendix B and derive the following equation by balancing the gas pressure at the top of the spherical cap with that at the bottom of the bubble cavity along the axis of the bubble residing at the free surface:

(3.4) \begin{eqnarray} \frac {R_{0}}{R_{c}}=\frac {\textit{Bo}}{16}\frac {Z_{c}}{R_{0}}+\frac {1}{2}\frac {R_{0}}{R_{b}}, \end{eqnarray}

which corresponds to the dimensionless form of (B6). For small bubble sizes, the first term on the right-hand side of (3.4) vanishes and the radius $R_{b}$ of the bubble cavity can be approximated as $R_{0}$ , yielding $R_{c}\approx 2R_{0}$ . Substituting ${\sin}\,\alpha =R_{r}/R_{c}$ and $R_{c}=2R_{0}$ into (3.3) gives

(3.5) \begin{eqnarray} \frac {R_{r}}{R_{0}}=\sqrt {\frac {\mathcal{N}_{b}{\textit{Bo}}}{3}}. \end{eqnarray}

Thus, irrespective of the isolated or coaxial configuration, (3.5) highlights the universal behaviour of the rim radius $R_{r}$ when expressed as a function of the modified Bond number $\mathcal{N}_{b}\textit{Bo}$ .

The prediction of $R_{r}$ in (3.5) can be refined by correcting the buoyancy in (3.3), which assumes the bubbles are fully immersed within the liquid. The exact submerged volume of a bubble floating near the free surface was derived in Puthenveettil et al. (Reference Puthenveettil, Saha, Krishnan and Hopfinger2018). Their analysis showed that retaining only the leading-order terms of the full submerged-volume expression provides good accuracy for estimating $R_{r}$ . Following this approach, we rewrite the buoyancy force on the right-hand side of (3.3) as

(3.6) \begin{eqnarray} \mathcal{B}=\rho _{l}g\left [\big(\mathcal{N}_{b}-1\big)\frac {4}{3}\pi\! R^{3}_{0}+\frac {2}{3}\pi\! R^{3}_{b}\left (1+\sqrt {1-\frac {R^{2}_{r}}{R^{2}_{b}}}\right )\right ]\!. \end{eqnarray}

Here, the first term in brackets represents the volume of the bottom bubble in a coaxial pair, which vanishes for isolated bubbles with $\mathcal{N}_{b}=1$ . The second term corresponds to the submerged volume of the bubble floating at the free surface (Puthenveettil et al. Reference Puthenveettil, Saha, Krishnan and Hopfinger2018). In the limit of $R^{2}_{r}\ll R^{2}_{b}$ , the buoyancy term in (3.3) is recovered from (3.6).

Next, while balancing $\mathcal{B}$ with the vertical surface tension component, $2\pi \sigma R^{2}_{r}/R_{c}$ , we avoid using the approximate value $R_{c}\approx 2R_{0}$ for the cap radius and instead employ the full expression of $R_{c}$ from (3.4). The cavity depth $Z_{c}$ in (3.4) is unknown, but it can be expressed geometrically as

(3.7) \begin{eqnarray} \frac {Z_{c}}{R_{0}} = \frac {R_{b}}{R_{0}} + \sqrt {\frac {R_{b}^{2}}{R_{0}^{2}} -\frac {R^{2}_{r}}{R^{2}_{0}}} - \frac {h_{r}}{R_{0}}, \end{eqnarray}

where $h_{r}$ is the height of the rim measured from the unperturbed free surface. For relatively small bubbles with $h_{r}\ll R_{b}$ , $Z_{c}$ can be approximated by neglecting $h_{r}$ (Puthenveettil et al. Reference Puthenveettil, Saha, Krishnan and Hopfinger2018). Finally, we apply the same buoyancy–surface-tension balance as in (3.3), but now using the buoyancy force $\mathcal{B}$ from (3.6). Substituting $R_{c}/R_{0}$ from (3.4) into the force balance, together with the approximate $Z_{c}$ from (3.7) (with $h_{r}=0$ ) and assuming $R_{b}\approx R_{0}$ , yields the governing equation for $R_{r}$ :

(3.8) \begin{eqnarray} \frac {R^{2}_{r}}{R^{2}_{0}}\left [\frac {\textit{Bo}}{16}\left (1+\sqrt {1-\frac {R^{2}_{r}}{R^{2}_{0}}}\right )+\frac {1}{2}\right ] - \frac {\textit{Bo}}{6} \left [\mathcal{N}_{b}+\frac {1}{2}\left (1+\sqrt {1-\frac {R^{2}_{r}}{R^{2}_{0}}}\right )-1\right ] = 0. \end{eqnarray}

In the limit $\textit{Bo}\ll 1$ , which implies $R^{2}_{r}\ll R^{2}_{0}$ , (3.8) reduces to the scaling relation in (3.5). By defining $\phi =R_{r}^{2}/R_{0}^{2}$ , (3.8) can be recast as a cubic polynomial in $\phi \in [0,1]$ :

(3.9) \begin{eqnarray} a_{3}\phi ^{3}+a_{2}\phi ^{2}+a_{1}\phi +a_{0}=0, \end{eqnarray}

with

(3.10) \begin{align} a_{3}&=9\textit{Bo}^2, \end{align}

(3.11) \begin{align} a_{2}&=144 \textit{Bo}-24 \textit{Bo}^2+576, \end{align}

(3.12) \begin{align} a_{1}&=192 \textit{Bo}+64 \textit{Bo}^2-48 \textit{Bo}^2 \mathcal{N}_{b}-384 \textit{Bo} \mathcal{N}_{b}, \end{align}

(3.13) \begin{align} a_{0}&=64\textit{Bo}^{2}\mathcal{N}_{b}(\mathcal{N}_{b}-1). \\[12pt] \nonumber \end{align}

For isolated bubbles ( $\mathcal{N}_{b}=1$ ), the constant term $a_{0}$ drops out, and the polynomial in (3.9) reduces to a quadratic equation, consistent with the earlier results of Puthenveettil et al. (Reference Puthenveettil, Saha, Krishnan and Hopfinger2018). We point out that the assumption $R_{b}\approx R_{0}$ has been validated experimentally for isolated bubbles up to $\textit{Bo}=10$ (Puthenveettil et al. Reference Puthenveettil, Saha, Krishnan and Hopfinger2018). However, this approximation becomes susceptible to error for coaxial bubbles at $\textit{Bo}\gt 1$ due to enhanced deformation and flattening at the bottom of the top-bubble cavity.

The variation of $R_{r}$ for isolated and coaxial bubbles in figure 10(a) brings together results from our YL model, MVOF simulations and experimental data reported by Puthenveettil et al. (Reference Puthenveettil, Saha, Krishnan and Hopfinger2018), alongside the scaling relation (3.5) and $R_{r}/R_{0}=\sqrt {\phi _{R}^{\textit{max}}}$ , where $\phi _{R}^{\textit{max}}$ denotes the largest real solution of (3.9). We indeed obtain a master curve by plotting $R_{r}/R_{0}$ against the modified Bond number $\mathcal{N}_{b}\textit{Bo}$ emerging from (3.5). For both isolated and coaxial bubbles, numerical and experimental results collapse well onto (3.5) provided that $\mathcal{N}_{b}\textit{Bo}\lt 1$ . This limitation is rooted into the formulation of the buoyancy force used while deriving (3.5). The values of $R_{r}/R_{0}=\sqrt {\phi _{R}^{max }}$ obtained from (3.9) for $\mathcal{N}_{b}=1$ and $2$ are in agreement with (3.5) for small values of $\mathcal{N}_{b}\textit{Bo}$ . Both solutions of (3.9) remain accurate to a good extent for $\mathcal{N}_{b}\textit{Bo}\gt 1$ . At the same time, they diverge from each other for different values of $\mathcal{N}_{b}$ , indicating the breakdown of $\mathcal{N}_{b}\textit{Bo}$ scaling for large bubbles. This breakdown originates in the structure of the coefficients of (3.9), which contain cross-terms involving non-uniform powers of $\mathcal{N}_{b}$ and $\textit{Bo}$ . Even so, numerical results for both isolated and coaxial bubbles still show a reasonably good collapse for $\mathcal{N}_{b}\textit{Bo}\gt 1$ .

Figure 10.

Comparison of unified analytical shape parameters (§ 3.2) for axisymmetric isolated and coaxial floating bubbles with numerical results (§ 3.1) from the YL model and MVOF simulations, together with experimental observations of Puthenveettil et al. (Reference Puthenveettil, Saha, Krishnan and Hopfinger2018). The plots show variations in (a) rim radius, $R_{r}/R_{0}$ ; (b) cap height, $h_{c}/R_{0}$ ; (c) cap area, $\pi (R^{2}_{r}+\hbar ^{2}_{c})/R^{2}_{0}$ ; and (d) cavity depth, $Z_{c}/R_{0}$ , as functions of $\mathcal{N}_{b}\textit{Bo}$ . Here, $\mathcal{N}_{b}$ denotes the number of bubbles, with $\mathcal{N}_{b}=1$ for isolated and $\mathcal{N}_{b}=2$ for coaxial configurations. The experimental data in (c) correspond to the approximate cap area, evaluated as $\pi R^{2}_{r}/R^{2}_{0}$ .

We now derive the expression for the cap height $h_{c}$ , expressed as the sum of the cap height measured from the rim, $\hbar _{c}$ , and the rim height measured from the unperturbed free surface, $h_{r}$ . We first evaluate $\hbar _{c}$ by considering weakly deformed bubbles and a free surface with a small contact angle $\alpha$ :

(3.14) \begin{eqnarray} \frac {\hbar _{c}}{R_{0}}&=&\frac {R_{c} - \sqrt {R^{2}_{c}-R^{2}_{r}}}{R_{0}}\quad ({\rm from\, geometry})\nonumber \\ &=&\frac {R_{r}}{R_{0}}\frac {(1-{\cos}\,\alpha )}{{\sin}\,\alpha } \quad (\because R_{c}=R_{r}/{\sin}\,\alpha )\nonumber \\ &\approx &\frac {R_{r}}{R_{0}}\frac {\alpha }{2} \quad (\because {\cos}\,\alpha \approx 1{-}\frac {\alpha ^2}{2} { \rm and } \,{\sin}\,\alpha \approx \alpha )\nonumber \\ &\approx &\frac {\mathcal{N}_{b}\textit{Bo}}{12} \quad (\because (3.3)\, { \rm and }\, {\sin}\,\alpha \approx \alpha ). \end{eqnarray}

Note that substituting $\alpha$ from the buoyancy–surface-tension balance in the final step eliminates the explicit dependence on $R_r$ . To determine the rim height $h_{r}$ , we require the meniscus profile $z(r)$ . We proceed by invoking the Laplace pressure jump condition at the meniscus (Lo Reference Lo1983; Puthenveettil et al. Reference Puthenveettil, Saha, Krishnan and Hopfinger2018). Assuming $z^{\prime 2}\ll 1$ (low Bond numbers) yields a linear governing equation of the form

(3.15) \begin{eqnarray} r^{2}z^{\prime \prime } + rz^{\prime } = \frac {\rho _{l}g}{\sigma }r^{2}z, \end{eqnarray}

which is the modified Bessel equation of order zero, subjected to the boundary conditions $z=0$ as $r\rightarrow \infty$ and $|z^{\prime }|=R_{r}/\sqrt {R^{2}_{c}-R^{2}_{r}}$ at $r=R_{r}$ . The prime in $z^{\prime }(r)$ denotes differentiation with respect to the radial coordinate $r$ . Finally, by solving (3.15), we arrive at the meniscus profile

(3.16) \begin{eqnarray} z(r)=\frac {l_{\sigma \!}R_{r}}{\sqrt {R^{2}_{c}-R^{2}_{r}}}\times \frac {K_{0}\!\left (r/l_{\sigma }\right )}{K_{1}\!\left (R_{r}/l_{\sigma }\right )}. \end{eqnarray}

Here, $K_{0}$ and $K_{1}$ denote the the modified Bessel functions of the second kind of orders $0$ and $1$ , respectively. Recall that the capillary length $l_{\sigma }$ is given by $l_{\sigma }=\sqrt {\sigma /(\rho _{l}g)}=2R_{0}/\sqrt {\textit{Bo}}$ . The rim height $h_{r}$ follows from evaluating (3.16) at $r=R_{r}$ . For small bubble with $R_{r}\ll l_{\sigma }$ , the modified Bessel functions admit the limiting forms $K_{0}(R_{r}/l_{\sigma })\approx {-}\ln (R_{r}/l_{\sigma })$ and $K_{1}(R_{r}/l_{\sigma })\approx l_{\sigma }/R_{r}$ (Olver et al. Reference Olver, Lozier, Boisvert and Clark2010). Following this, we obtain

(3.17)

Using (3.14), (3.17) and the definition of $R_{r}$ in (3.5), we arrive at the cap height:

(3.18) \begin{eqnarray} \frac {h_{c}}{R_{0}} &=& \frac {\hbar _{c}}{R_{0}} + \frac {h_{r}}{R_{0}} = \frac {\mathcal{N}_{b}\textit{Bo}}{12}\left [1-\ln \left (\frac {\mathcal{N}_{b}\textit{Bo}^{2}}{12}\right )\right ]\!. \end{eqnarray}

Unlike the rim radius in (3.5), the cap height does not follow a simple $\mathcal{N}_{b}\textit{Bo}$ scaling because of the logarithmic term, which introduces a dependence on $\mathcal{N}_{b}\textit{Bo}^2$ . According to (3.18), the cap height scales as $h_{c}/R_{0} \sim \mathcal{N}_{b}\textit{Bo}|\ln (\textit{Bo}^2)|$ for isolated and coaxial bubbles with $\mathcal{N}_{b}\textit{Bo}\ll 1$ , where the logarithmic factor dominates. On the other hand, the difference in cap height between coaxial and isolated bubbles is $\Delta h_{c}/R_{0}=\mathcal{N}_{b}\textit{Bo}\ln 2/12$ , which becomes vanishingly small as $\mathcal{N}_{b}\textit{Bo}$ decreases. Thus, the curves for $\mathcal{N}_{b}=1$ and $2$ should nearly collapse for small bubbles ( $\mathcal{N}_{b}\textit{Bo}\ll 1$ ) while plotting $h_{c}/R_{0}$ versus $\mathcal{N}_{b}\textit{Bo}$ . This is corroborated by analytical, numerical and experimental data of isolated and coaxial bubbles in figure 10(b), which together follow a master-curve-like trend in the small-bubble limit. Nevertheless, the divergence of $h_{c}/R_{0}$ between larger isolated and coaxial bubbles highlights a clear deviation from $\mathcal{N}_{b}\textit{Bo}$ scaling, as anticipated for strongly deformed bubbles.

Having $R_{r}$ and $\hbar _{c}$ at hand, we can write two different expressions for the cap area, $\pi (R^{2}_{r}+\hbar ^{2}_{c})$ , using (3.8), (3.9) and (3.14), given as

(3.19)

and

(3.20)

By discarding the quadratic term $\mathcal{N}^{2}_{b}\textit{Bo}^{2}$ in (3.19) for $\mathcal{N}_{b}\textit{Bo}\ll 1$ , we obtain $\mathcal{S}/R^{2}_{0}\approx \pi \mathcal{N}_{b}\textit{Bo}/3$ , which is consistent with the known estimate $\pi \textit{Bo}/3$ for the dimensionless cap area of small isolated bubbles (Howell Reference Howell1999; Kocárková et al. Reference Kocárková, Rouyer and Pigeonneau2013). Figure 10(c) shows the variation of $\mathcal{S}/R^{2}_{0}$ with $\mathcal{N}_{b}\textit{Bo}$ . A good agreement is observed between analytical, numerical and experimental results for both isolated and coaxial bubbles. The numerical results also confirm that the master-curve scaling with $\mathcal{N}_{b}\textit{Bo}$ remains valid even for relatively large bubbles up to $\mathcal{N}_{b}\textit{Bo}\approx 4$ . Between the two formulations, the cap area expression in (3.20), which incorporates the improved estimate of $R_{r}/R_{0}$ , provides a more accurate prediction than (3.19) when $\mathcal{N}_{b}\textit{Bo}\gt 1$ . Note that the derivations of $R_{r}/R_{0}$ and  $\hbar _{c}/R_{0}$ in (3.20) rely on different formulations of the buoyancy force: $R_{r}/R_{0}$ is obtained using a more accurate treatment of the buoyancy, whereas $\hbar _{c}$ is derived under a simplified force balance. Despite this inconsistency, (3.20) performs better since the dominant contribution to the cap area comes from the term $R^{2}_{r}/R^{2}_{0}$ . The divergence between the cap areas of isolated and coaxial bubbles predicted by (3.20) in figure 10(c) follows the same reasoning already discussed for $R_{r}$ in figure 10(a).

Finally, the last remaining shape parameter is the cavity depth, $Z_{c}/R_{0}$ . We can readily utilise the geometric relation in (3.7) by substituting $R_{r}/R_{0}$ and $h_{r}/R_{0}$ from (3.5) and (3.17), respectively, together with the approximation $R_{b}\approx R_{0}$ . The resulting expression of cavity depth is

(3.21) \begin{eqnarray} \frac {Z_{c}}{R_{0}}=1+\sqrt {1-\frac {\mathcal{N}_{b}\textit{Bo}}{3}}+\frac {\mathcal{N}_{b}\textit{Bo}}{12}\ln\! \left (\frac {\mathcal{N}_{b}\textit{Bo}^{2}}{12}\right )\!. \end{eqnarray}

To maintain continuity with the earlier discussion, we plot the variation of $Z_{c}/R_{0}$ as a function of $\mathcal{N}_{b}\textit{Bo}$ in figure 10(d). Numerical and experimental results for isolated and coaxial bubbles show that, unlike the other shape parameters, $Z_{c}/R_{0}$ does not collapse when plotted against $\mathcal{N}_{b}\textit{Bo}$ . For isolated bubbles, the analytical prediction from (3.21) agrees well with both numerical and experimental measurements. By contrast, (3.21) fails to capture the enhanced deformation of the cavity in coaxial bubbles and produces nearly identical results for $\mathcal{N}_{b}=1$ and $2$ .

To address this, we incorporate $\mathcal{O}(\textit{Bo})$ correction terms into (3.21) from the perturbation expansion derived by Kralchevsky, Ivanov & Nikolov (Reference Kralchevsky, Ivanov and Nikolov1986), yielding

(3.22) \begin{eqnarray} \frac {Z_{c}}{R_{0}}=1+\sqrt {1-\frac {\mathcal{N}_{b}\textit{Bo}}{3}}+\frac {\mathcal{N}_{b}\textit{Bo}}{12}\ln\! \left (\frac {\mathcal{N}_{b}\textit{Bo}^{2}}{12}\right ) + \frac {\textit{Bo}}{4}\mathcal{Z}(\alpha ), \end{eqnarray}

where

(3.23) \begin{eqnarray} \mathcal{Z}(\alpha )=\frac {1}{3}{\sin}^{2}\alpha +\frac {2}{3}\ln\! \left ({\sin}\frac {\alpha }{2}\right )-\frac {1}{2}(1+{\cos}\,\alpha ). \end{eqnarray}

Following (3.5), we use $\alpha ={\sin}^{-1}\sqrt {\mathcal{N}_{b}\textit{Bo}/12}$ in (3.23). The first two terms in (3.22) give the axial distance between the rim and the bottom of the cavity. The fourth term in (3.22) provides a correction to this measurement, while the remaining logarithmic term removes the contribution of the rim height $h_{r}/R_{0}$ from the axial distance to obtain $Z_{c}/R_{0}$ . The corrected cavity depth (3.22) provides a significantly improved estimate for coaxial bubbles, as shown in figure 10(d). However, (3.22) overestimates the cavity deformation for isolated bubbles (not shown here), which is consistent with the findings of Puthenveettil et al. (Reference Puthenveettil, Saha, Krishnan and Hopfinger2018).

4.1. Numerical findings

The coaxial configuration discussed in the previous section is most likely to occur in confined systems, where the lateral dispersion of bubbles floating at the free surface is hindered, leaving insufficient room for newly arriving bubbles to float and causing them to stack vertically (see Liger-Belair et al. Reference Liger-Belair, Polidori and Jeandet2008). However, dilute suspensions of sparsely distributed floating bubbles in open systems tend to self-organise into raft-like configurations (Néel & Deike Reference Néel and Deike2021). Understanding the formation of such structures and the geometry of bubbles within them is equally essential. To this end, in this section, we consider a minimal configuration involving two side-by-side floating bubbles. Unlike coaxial arrangements, this side-by-side configuration requires fully three-dimensional MVOF simulations.

Our previous discussion on coaxial bubbles focused solely on their final equilibrium shapes at the free surface. Here, we place particular emphasis on examining the dynamic process that leads to the clustering of side-by-side bubbles via capillary attraction – commonly known as the Cheerios effect (Vella & Mahadevan Reference Vella and Mahadevan2005; Ho et al. Reference Ho, Pucci and Harris2019). To capture this, we begin with two side-by-side bubbles, separated by an initial centre-to-centre distance $d_{i}$ . These bubbles are initially placed beneath the free surface at a depth of $1.25R_{0}$ , which is measured from their centres. We then allow bubbles to rise to the free surface due to buoyancy, followed by the capillary migration along the free surface. The deformation of the bubble interface and the transient Reynolds number during the initial acceleration are governed by the Morton number $\textit{Mo}=g\mu ^{4}_{l}/\rho _{l}\sigma ^{3}$ . For a given Bond number $\textit{Bo}$ , we adjust the Morton number $\textit{Mo}$ based on the phase diagram provided in Clift, Grace & Weber (Reference Clift, Grace and Weber1978), Park et al. (Reference Park, Park, Lee and Lee2017) and Jia, Xiao & Kang (Reference Jia, Xiao and Kang2019) (see figure 25), ensuring that the Reynolds number remains below $10$ during the bubble rise. A set of various $(\textit{Bo},\textit{Mo})$ combinations used in our work is $\{(1,10^{-4}),(5,9.2\times 10^{-3}),{}(10,1),(25,1),(50,50),(75,100),(100,100)\}$ . A detailed discussion about the choice of $\textit{Mo}$ for each $\textit{Bo}$ , along with the corresponding liquid–gas combination, is provided in Appendix D.

We begin with a qualitative analysis of our findings related to the Cheerios effect. Figure 11 showcases the evolution of bubble interfaces and the free surface in the mid-plane during various stages of capillary migration. Upon reaching the free surface, the bubbles display a noticeable horizontal misalignment of contact points on either side. For instance, see figures 11(aiii) and 11(biv) corresponding to Bond numbers $\textit{Bo}=10$ and $\textit{Bo}=100$ , respectively. This misalignment generates a net horizontal surface tension force. Additionally, buoyancy also contributes by driving the bubbles upward along the asymmetrically deformed free surface. As a result, both bubbles propel towards one another under the influence of the net capillary forces acting in the horizontal direction, as visible in figures 11(aiv–avii) and 11(biv–bvii). The full three-dimensional visualisation in figure 12 shows that the deformation pattern on the outer side of both bubbles still resembles that of an isolated bubble in figure 3. However, the axisymmetric nature of the deformation is disrupted by the formation of a contact region between the two bubbles. The size of this contact area increases with the Bond number. The asymmetric nature of the spherical cap in figure 12 calls for the experimental investigation to understand the drainage pattern of the liquid within the film into the bulk and the subsequent nucleation of the first hole at the onset of the bursting event, which typically occurs at the foot of the cap during the formation of film droplets from an isolated bubble (Lhuissier & Villermaux Reference Lhuissier and Villermaux2012; Poulain et al. Reference Poulain, Villermaux and Bourouiba2018; Shaw & Deike Reference Shaw and Deike2024). While the present study focuses on numerical analysis, such experimental insights remain essential for fully characterising the role of multiple floating bubbles in the bursting process.

Figure 11.

Capillary attraction between two side-by-side floating bubbles at (a) $\textit{Bo}=10$ (top row) and (b) $\textit{Bo}=100$ (bottom row). The instantaneous snapshots show bubble interfaces and the free surface in the midplane of a three-dimensional domain. Interface profiles captured with different volume-fraction indicators are shown in distinct colours. The full three-dimensional view of the bubbles and the free surface in the equilibrium state is shown in figure 12 for both Bond numbers. The Morton number is set to $\textit{Mo}=1$ for $\textit{Bo}=10$ and $\textit{Mo}=100$ for $\textit{Bo}=100$ . Note that $\textit{Mo}$ affects bubble dynamics only during the initial acceleration phase, when bubbles rise towards the free surface.

Figure 12.

Equilibrium shapes of two side-by-side bubbles floating at the free surface in a three-dimensional setting: (a) $\textit{Bo}=10$ (top row) and (b) $\textit{Bo}=100$ (bottom row).

We now turn our attention to quantifying the capillary migration of side-by-side bubbles. Figure 13(a) presents the temporal evolution of the bubble spacing $\mathcal{G}$ for a range of Bond ( $\textit{Bo}$ ) numbers spanning two decades. The initial gap between bubbles is set to $\mathcal{G}=R_{0}$ in all cases. At early times, the bubble pair exhibits a brief repulsion due to the flow field generated by the vertical acceleration of bubbles, causing a slight increase in $\mathcal{G}$ depending on the value of $\textit{Bo}$ . At the lowest Bond number, $\textit{Bo}=1$ , capillary attraction is minimal, and the bubble spacing shows little to no reduction over time. However, starting from $\textit{Bo}=5$ , a clear decay in $\mathcal{G}$ is observed, eventually reaching $\mathcal{G}\approx 0$ as the bubbles come into contact. The bubble gap $\mathcal{G}$ shrinks at an even faster rate as the Bond number is further increased, until $\textit{Bo}=50$ . Interestingly, this trend reverses for larger Bond numbers. Bubbles with $\textit{Bo}=75$ and $100$ exhibit slower capillary migration than the $\textit{Bo}=50$ case. This behaviour can be qualitatively explained by examining the shape of the meniscus. At a low Bond number of $\textit{Bo}=1$ , the free-surface deformation around each bubble is mild. Thus, the presence of the neighbouring bubble is not strongly felt in a side-by-side configuration. As $\textit{Bo}$ increases, meniscus deformation becomes more pronounced, enhancing horizontal capillary forces and accelerating migration. However, floating bubbles with $\textit{Bo}\geqslant 50$ assume a hemispherical shape and localise the free-surface deformation in the region close to the bubble, as shown in figure 3 for $\textit{Bo}=100$ . Thereby slowing capillary migration as the interaction strength between the two nearby bubbles diminishes.

Figure 13.

Time variation of the dimensionless bubble spacing, $\mathcal{G}=(d{-}2R_{0})/R_{0}$ , for different (a) Bond ( $\textit{Bo}$ ) numbers and (b) starting positions. Here, $d$ denotes the centre-to-centre distance between two bubbles, and $\tau _{c}$ is the dimensionless capillary time. The dash-dotted lines in (a) represent extrapolated trends. The dashed brown line in ( $b$ ) corresponds to the dimensionless capillary length $l_{\sigma }/R_{0}=2/\sqrt {\textit{Bo}}$ . The Bond number in (b) is fixed at $\textit{Bo}=10$ . The negative equilibrium values of $\mathcal{G}$ in (a,b) indicate a deviation from spherical shape upon contact between two bubbles. (c) Time variation of the dimensionless volume-averaged bubble velocity, $|\boldsymbol{u}^{\parallel }|\mu _{l}/\sigma$ , during capillary migration for three different combinations of $\textit{Bo}$ and initial centre-to-centre distances $d_{i}$ . The velocity magnitude $|\boldsymbol{u}^{||}|$ refers to the velocity component in the $xy$ -plane parallel to the unperturbed free surface. Insets: (b) effect of $\textit{Bo}$ on the total dimensionless migration time, $\Delta t_{\diamondsuit \bigtriangledown }\sigma /\mu _{l}R_{0}$ , required to form a two-bubble raft; and (c) effect of $\textit{Bo}$ on the maximum dimensionless velocity, $|\boldsymbol{u}^{\parallel }|_{{max}}\mu _{l}/R_{0}$ , observed during capillary migration. The symbols $\Delta t_{\diamondsuit \bigtriangledown }$ and $d_{\diamondsuit }$ in the inset of (b) indicate the time difference between instances marked by $\diamondsuit$ and $\bigtriangledown$ , and the distance at the time instance marked by $\diamondsuit$ in the main plot, respectively.

The trends shown in figure 13(a) can also be analysed alongside the bubble-size density variation observed in the ocean in figure 1. It should be noted, however, that the distribution in figure 1 corresponds to subsurface bubbles. In contrast, the actual surface distribution of floating bubbles in the ocean typically lacks larger bubbles with Bond numbers exceeding $\textit{Bo}=10$ (Néel & Deike Reference Néel and Deike2021). Within this context, bubbles with intermediate Bond numbers, $5\leqslant \textit{Bo}\leqslant 10$ , although occurring at a lower density than smaller bubbles ( $\textit{Bo}\lt 5$ ) in the super-Hinze regime, are more favourable for forming rafts because of their considerably faster capillary migration, as observed in figure 13(a).

We extend our investigation of capillary migration by analysing the influence of initial bubble spacing. Figure 13(b) presents the temporal evolution of the gap $\mathcal{G}$ for various initial spacings at a constant Bond number $\textit{Bo}=10$ . Remarkably, even when $\mathcal{G}$ significantly exceeds the capillary length $l_{\sigma }/R_{0}=2/\sqrt {\textit{Bo}}$ , the bubbles can still attract each other and form a two-bubble raft. Regardless of the starting distance, $\mathcal{G}$ decays at the same rate once it falls below $l_{\sigma }/R_{0}$ , ultimately reaching $\mathcal{G}\approx 0$ within a short duration of approximately $\Delta \tau _{c}\approx 11$ , as shown in figure 13(b). The total time required to form a two-bubble assembly increases exponentially with the initial spacing, as illustrated in the inset of figure 13(b). Larger initial separations result in a prolonged period before the bubbles approach to a gap size of $\mathcal{G}=l_{\sigma }/R_{0}$ , contributing significantly to the overall migration time. For instance, the cases with initial spacings of $\mathcal{G}=1.75$ (grey curve) and $\mathcal{G}=2$ (olive curve) demonstrate noticeably longer delays. Once $\mathcal{G}$ reaches $l_{\sigma }/R_{o}$ , the bubbles rapidly accelerate toward each other under the influence of enhanced capillary forces.

In addition to inter-bubble spacing, we discovered an interesting trend upon analysing the dimensionless volume-averaged migration velocity, $|\boldsymbol{u}^{||}|\mu _{l}/\sigma$ , of bubbles. The main plot in figure 13(c) illustrates the velocity time series for three different cases. A distinct spike in migration velocity is observed as the bubbles approach contact. For a fixed Bond number, the magnitude $|\boldsymbol{u}^{||}|_{{max}}\mu _{l}/\sigma$ of this spike remains nearly the same irrespective of the initial separation between the bubbles, as evident from the green and blue curves in figure 13(c). However, increasing the Bond number to $\textit{Bo}=25$ (red curve in figure 13 c) amplifies the magnitude of the velocity spike. Across multiple simulations with varying Bond numbers, the peak migration velocity was found to follow an approximate linear relationship: $|\boldsymbol{u}^{||}|_{{max}}\mu _{l}/\sigma \propto \textit{Bo}$ , as illustrated in the inset of figure 13(c). The spike in the velocity time series arises from a sharp imbalance of horizontal surface tension forces just before both bubbles assemble, caused by asymmetric free-surface contact angles on opposite sides of each bubble. In the limit of $\mathcal{G}\rightarrow 0$ , higher Bond numbers lead to greater deformation and thus larger contact angle mismatches. This mechanism qualitatively accounts for trends observed in figure 13(c).

4.2. Modelling capillary migration

To support our observations from MVOF simulations, we develop a simplified model to capture the temporal evolution of the centre-to-centre distance, $d$ , during capillary migration. To achieve this, we rely on the linear/superposition approximation devised by Nicolson (Reference Nicolson1949), which has been widely implemented for both light and heavy rigid particles floating at the free surface (Chan et al. Reference Chan, Henry and White1981; Vella & Mahadevan Reference Vella and Mahadevan2005; Dalbe et al. Reference Dalbe, Cosic, Berhanu and Kudrolli2011; Dani et al. Reference Dani, Keiser, Yeganeh and Maldarelli2015). Assuming that the steady-state free-surface deformation from an isolated bubble satisfies a linear governing equation, the combined deformation induced by a pair of bubbles in a side-by-side configuration can be approximated by the superposition of individual deformations. Subsequently, to derive the magnitude of the capillary force acting on a given bubble within the pair, we treat that bubble as a buoyant point particle, placed within the deformation field created by the other bubble in the pair. Modelling a bubble as a point particle implies that it has no free-surface perturbation field of its own. As this buoyant point-like bubble climbs the meniscus formed by the neighbouring bubble, it experiences a force corresponding to the gradient of its potential energy, which serves as the capillary attraction force. Although this framework is applicable to a pair of bubbles with different sizes (as illustrated in figure 14), we focus here on the case of two identical side-by-side floating bubbles. Following this, the capillary force on the $B_{2}$ bubble in figure 14 is given by

(4.1) \begin{eqnarray} \big |\boldsymbol{F}^{\textit{cap}}_{B_{2}}(r)\big |&=&\underbrace {(2 \pi \sigma\! R_{r} {\sin}\,\alpha )_{B_{2},\textit{iso}}}_{{\rm net\,weight}} \times \underbrace {\big |z^{\prime }(r)\big |_{B_{1}, \textit{iso}}}_{{\rm slope\,of\,meniscus}}\nonumber \\ &=&\left (2 \pi \sigma \frac {R^{2}_{r}}{R_{c}} \right )_{B_{2},\textit{iso}} \times \big |z^{\prime }(r)\big |_{B_{1}, \textit{iso}}, \end{eqnarray}

where the prime in $z^{\prime }(r)$ denotes differentiation with respect to the radial coordinate $r$ , representing the slope of the meniscus. The origin, $r=0$ , is defined at the centre of the corresponding bubble (see figure 14).

Figure 14.

Schematic of two side-by-side floating bubbles of unequal size, labelled $B_{1}$ and $B_{2}$ , with a centre-to-centre distance $d$ . Each bubble has a rim radius and a spherical-cap radius, represented by $R_{r}$ and $R_{c}$ , as previously described in figures 6 and 7. The deformation of the free surface surrounding each bubble generates surface tension forces at the contact point between the meniscus and the spherical cap. A surface tension force of magnitude $F^{\sigma }$ acting at an angle $\alpha$ is shown on the left side of the $B_{1}$ bubble as an example. All vertical measurements are referenced from the origin $z=0$ situated at the undisturbed free surface far from the bubbles.

Based on earlier assumptions involving superposition and a point-like bubble, the force $|\boldsymbol{F}^{{cap}}_{B_{2}}(r)|$ depends on the net weight of the $B_{2}$ bubble and the slope of the meniscus generated by the adjacent $B_{1}$ bubble. Both quantities are evaluated in the isolated floating state of each bubble, indicated by the subscript ‘iso’ in (4.1). The net weight of $B_{2}$ , defined as the difference between its weight and buoyancy, is expressed through the vertical component of the surface tension force. Assuming the contact angle $\alpha$ remains constant during capillary migration, the variation in $|\boldsymbol{F}^{{cap}}_{B_{2}}(r)|$ with respect to $r$ is governed solely by the change in the meniscus slope $z^{\prime }(r)$ . A similar expression for the capillary force acting on the $B_{1}$ bubble can be obtained by interchanging the $B_{1}$ and $B_{2}$ subscripts in (4.1). The expression for $z^{\prime }(r)$ required in (4.1) can be readily obtained from the meniscus height profile (3.16), which follows from the linearised Laplace pressure balance (3.15):

(4.2) \begin{eqnarray} z^{\prime }(r)=\frac {R_{r}}{\sqrt {R^{2}_{c}-R^{2}_{r}}}\times \frac {-K_{1}\!\left (r/l_{\sigma }\right )}{K_{1}\!\left (R_{r}/l_{\sigma }\right )}. \end{eqnarray}

Here, $K_{1}$ denotes the modified Bessel function of the second kind of order $1$ , and $l_{\sigma }=\sqrt {\sigma /(\rho _{l}g)}=2R_{0}/\sqrt {\textit{Bo}}$ is the capillary length.

To verify the accuracy of (3.16) and (4.2), we compare the meniscus shape and the corresponding slope obtained from these expressions with our numerical results from the YL model (see § 2.2 and Appendix B). To proceed, we first compute the prefactors involving radii $R_{r}$ and $R_{c}$ in (3.16) and (4.2) using geometrical information available through the equilibrium shape derived from the YL equations. As shown in figure 15(a), the numerical and semi-analytical results are in good agreement. At $\textit{Bo}=5$ , a slight deviation in the semi-analytical meniscus profile appears near the contact point. However, this does not impact the evaluation of the meniscus slope, as evident from the inset of figure 15(b). The semi-analytical solution accurately captures the variation in $z^{\prime }(r)$ up to $\textit{Bo}=10$ (not shown).

Figure 15.

(a) Dependence of the meniscus height $z(r)$ and slope $|z^{\prime }(r)|$ (inset) on the Bond ( $\textit{Bo}$ ) number for an axisymmetric floating bubble. The solid curves represent the YL solution. The semi-analytical solution of $z(r)$ and $|z^{\prime }(r)|$ , indicated by the black dashed line, is obtained from (3.16) and (4.2), respectively. The geometry-dependent prefactors in these expressions are calculated using the YL solution. (b) Time evolution of the bubble spacing $\mathcal{G}$ during capillary migration, comparing MVOF simulation results (solid lines) with predictions from the linear model given in (4.4) (dashed lines), for various Bond numbers and initial spacings. The dash-dotted line in (b) shows the extrapolated trend.

Having established the formulation of the capillary force using (4.1) and (4.2), the final step involves applying a force balance to determine the time rate of change of the bubble separation, $\dot {d}$ . Based on velocity statistics from the MVOF simulations (figure 13 c), we conclude that the Reynolds number associated with capillary migration remains below unity. Consequently, the force balance for the $B_{2}$ bubble in the Stokes regime simplifies to

(4.3) \begin{eqnarray} \big |\boldsymbol{F}^{{cap}}_{B_{2}}(r)\big |=\underbrace {4\pi\! \mu _{l}C_{d}\!R_{0}\left (\frac {\mu _{l}/\mu _{g}+1.5}{\mu _{l}/\mu _{g}+1}\right )\big |\boldsymbol{u}^{\parallel }\big |_{B_{2}}}_{\displaystyle {\big |\boldsymbol{F}^{\textit{drag}}_{B_{2}}\big |}}. \end{eqnarray}

In the force balance (4.3), we have implemented the drag law $|\boldsymbol{F}^{\textit{drag}}_{B_{2}}|$ derived in Batchelor (Reference Batchelor2000) for bubbles and droplets, where $C_{d}$ represents the drag coefficient. For the present liquid–gas system with $\mu _{l}/\mu _{g}=100$ , the bracketed term on the right-hand side of (4.3) evaluates to ${\approx} 1$ . Considering two identical floating bubbles approaching each other with equal instantaneous velocity during capillary migration, we have $|\boldsymbol{u}^{\parallel }|_{B_{1}}=|\boldsymbol{u}^{\parallel }|_{B_{2}}=\dot {d}/2$ . Substituting this into (4.3) and incorporating the capillary force expression from (4.1), we arrive at

(4.4) \begin{eqnarray} \dot{d}= -\frac {\sigma\! R^{2}_{r}}{\mu _{l}C_{d}\!R_{0}\!R_{c}}\left |z^{\prime }(r=d)\right |_{\textit{iso}}\!. \end{eqnarray}

The negative sign signifies the decay in centre-to-centre distance $d$ over time due to capillary attraction.

The time evolution of $d$ from our linear model in (4.4) is obtained through numerical time marching. By appropriately tuning the free parameter $C_{d}$ , which varies with $\textit{Bo}$ , we achieve good agreement with the results from our MVOF simulations. The drag law $|\boldsymbol{F}^{\textit{drag}}|$ in (4.3) was originally derived for an isolated spherical bubble or droplet moving in an unbounded medium. Thus, it is necessary to adjust the value of $C_{d}$ in our model to account for bubble deformation in the floating state and the influence of the free surface. Figure 15(b) compares the predictions from our linear model with MVOF simulations for various initial bubble spacings and Bond numbers. The linear model captures the correct trends up to $\textit{Bo}=10$ . We observe that the model begins to deviate from the MVOF results for $\textit{Bo}=5$ and $10$ as $\mathcal{G}\rightarrow 0$ . This discrepancy likely arises because the linear model does not include the surface tension forces generated by asymmetries in the free-surface contact angles on either side of the bubbles, which can significantly impact short-range interactions between floating bubbles. The value of $C_{d}$ in the model decreases with increasing Bond number, as illustrated in figure 15(b). This trend coincides with the observation that, at higher Bond numbers, increased deformation of the free surface results in a smaller fraction of each bubble being submerged within the liquid during capillary migration. A comparison of the hydrodynamic drag on pairs of floating bubbles and pairs of spheres straddling a liquid–gas interface is provided in Appendix E.

The slope of the meniscus, $|z^{\prime }(r)|$ , can be approximated using an exponential fit of the form $|z^{\prime }(r)|=\lambda \exp (-\gamma r)$ , as illustrated in the inset of figure 15(a). Here, $\lambda$ and  $\gamma$ are constant positive fitting parameters. Notably, applying the limiting form of $K_{1}(r/l_{\sigma })$ in (4.2) for relatively large values of the argument $r/l_{\sigma }$ (Olver et al. Reference Olver, Lozier, Boisvert and Clark2010) also produces an exponential decay of the form ${\exp}(-r/l_{\sigma })$ , but with a variable prefactor. Substituting $|z^{\prime }(r)|=\lambda \exp (-\gamma r)$ into (4.4) and integrating both sides after separating variables yields $\Delta t_{m}=[\exp (\gamma d_{i})-\exp (\gamma d_{\!f})]/(\gamma \varGamma )$ , with $\lambda$ and all constants from (4.4) grouped into $\varGamma$ . The total migration time required for the centre-to-centre bubble distance to decay from an initial value $d_{i}$ to a specified distance $d_{\!f}$ is given by $\Delta t_{m}$ . In the limiting case with a fixed $d_{\!f}=2R_{0}$ ( $\mathcal{G}=0$ ), the total migration time required to form a two-bubble raft exhibits exponential scaling with the initial distance $d_{i}$ , which is consistent with the simulation results presented in the inset of figure 13(b).

5.1. Monolayer arrangement

In this section, we scale up the side-by-side floating bubbles configuration to explore how a moderately large collection of monodisperse floating bubbles self-organises through capillary attraction. Following the observations from § 4.1, we first select the parameter set $\{\textit{Bo}, \textit{Mo}\}=\{10, 1\}$ . At time $t=0$ , we initialise a monolayer of $\mathcal{N}_{b}=15$ randomly distributed spherical bubbles beneath the free surface, ensuring no initial contact with either the free surface or between bubbles. This set-up is computationally demanding for the current MVOF implementation, as it requires solving $16$ distinct volume-fraction indicators alongside the Navier–Stokes equations. For reference, in recent controlled experiments, Néel & Deike (Reference Néel and Deike2021) studied floating-bubble rafts with bubble counts $\mathcal{N}_{b}$ ranging from $\mathcal{O}(10)$ to $\mathcal{O}(100)$ .

After being released, bubbles rise to the free surface and adopt their equilibrium shape, similar to the axisymmetric case shown in the inset of figure 7(a). Once at the free surface, the initially dispersed floating bubbles begin to accumulate by migrating towards nearby bubbles along the free surface. Figure 16 shows instantaneous snapshots of free-surface deformation from our MVOF simulation, illustrating various stages of capillary migration and the emerging patterns. Initially, two interior bubbles at the centre pair up (figure 16 a), resembling the side-by-side configuration observed in § 4.1. Subsequently, bubbles along the periphery assemble into distinct structures: an island on the left and a long chain on the right, as shown in figure 16(b,c). The formation of a floating chain of bubbles is reminiscent of the linear chain assembly of rigid floating particles observed in experiments by Dalbe et al. (Reference Dalbe, Cosic, Berhanu and Kudrolli2011). Over time, the chain of floating bubbles loses its structure and merges with adjacent bubbles to form a raft of $12$ floating bubbles (figure 16 d,e, f). Meanwhile, the remaining three bubbles stay separate from the raft, persisting as an island until time $t\sigma /\mu _{l}R_{0}=86.4$ , as visible in figure 16(f).

Figure 16.

Emergent dynamics of a monodisperse suspension comprising $\mathcal{N}_{b}=15$ floating bubbles, driven by capillary migration along the free surface. The Bond and Morton numbers are fixed at $\textit{Bo}=10$ and $\textit{Mo}=1$ .

Figure 17 tracks the centre-of-mass trajectories of all $15$ bubbles in the $xy$ -plane (parallel to the free surface) over the entire duration of the simulation (until time $t\sigma /\mu _{l}R_{0}=86.4$ ). During the initial rise, hydrodynamic interactions cause the bubbles to move radially outward from the origin. This outward motion is evident in the trajectories of peripheral bubbles, particularly those located in the upper half and lower right quadrant in figure 17. As time elapses, most bubbles eventually undergo capillary migration towards the origin, with those in the outer regions travelling distances comparable to their diameter $2R_{0}$ (see figure 17).

Figure 17.

Trajectories of individual bubble centres in the $xy$ -plane (top view), illustrating their movement as bubbles self-organise into floating clusters at the free surface due to capillary attraction. The simulation parameters are indicated in the plot.

To further quantify the phenomenon of self-organisation, we plot the temporal variation of the Reynolds number ${Re}=2\rho _{l}u_{z}R_{0}/\mu _{l}$ and the capillary number $\textit{Ca}=|\boldsymbol{u}^{\parallel }|\mu _{l}/\sigma$ in figure 18(a) for all $15$ bubbles. Here $u_{z}$ and $|\boldsymbol{u}^{\parallel }|$ denote the volume-averaged vertical velocity and velocity magnitude in the $xy$ -plane parallel to the free surface for a given bubble in the suspension, respectively. During the initial acceleration phase, ${Re}$ rises sharply, followed by a quick drop once bubbles hit the free surface. At later stages, we observe only weak oscillations in ${Re}$ . The temporal variation of the capillary number $\textit{Ca}$ also exhibits a spike at early times for most bubbles as they settle at the free surface, visible though dark red spots in the contour plot shown in the inset of figure 18(a). Later, when bubbles undergo capillary migration, we notice white patches in the contour plot of $\textit{Ca}$ , corresponding to the occurrence of the elevated in-plane velocity magnitude $|\boldsymbol{u}^{\parallel }|$ for short times. This is observed at multiple time instances and for different bubbles in figure 18(a). This short-lived enhancement in $\textit{Ca}$ takes place when two or more bubbles are about to make contact through capillary migration, as described previously in § 4.1.

Figure 18.

(a) Temporal evolution of the Reynolds number ${Re}$ , calculated using the vertical volume-averaged velocity $u_{z}$ of individual bubbles. Inset: colour-coded variation of the capillary number $|\boldsymbol{u}^{\parallel }|\mu _{l}/\sigma$ with dimensionless time $t\sigma /\mu _{l}R_{0}$ for each bubble. Bubbles are indexed by $\mathcal{N}^{k}_{b}$ , consistent with the labels in figure 17. The velocity magnitude $|\boldsymbol{u}^{\parallel }|$ refers to the volume-averaged velocity in the $xy$ -plane parallel to the unperturbed free surface. (b) Time evolution of the polar order parameter $\mathcal{Q}$ and the global bond orientational order parameter $q_{6}$ . Insets: instantaneous snapshots showcasing bubble positions from the top and their orientation vectors $\boldsymbol{e}$ . Bubbles are illustrated as perfect circles of radius $R_{0}$ . The time instances of these snapshots are indicated by black dashed vertical lines in the main plot. The values of the local bond order parameter $|q^{(2)}_{6}|$ at the top of each inset correspond to the bubble with index $\mathcal{N}^{k}_{b}=2$ , which is highlighted in olive colour. The Bond and Morton numbers are fixed at $\textit{Bo}=10$ and $\textit{Mo}=1$ .

Next, figure 18(b) illustrates the evolution of the polar order parameter

(5.1) \begin{eqnarray} \mathcal{Q}(t)=\frac {1}{\mathcal{N}_{b}}\left |\sum \limits _{k=1}^{\mathcal{N}_{b}}\boldsymbol{e}_{k}(t)\right | \end{eqnarray}

and the global sixfold bond orientational parameter

(5.2) \begin{eqnarray} q_{6}(t){:}=\left |\frac {1}{\mathcal{N}_{b}}\sum \limits _{k=1}^{\mathcal{N}_{b}}q^{(k)}_{6}(t)\right | {\rm with }\, q^{(k)}_{6}(t){:}=\frac {1}{6}\!\sum \limits _{j\in \mathcal{N}^{(k)}_{6}} {\rm e}^{i6\theta _{kj}} \!, \end{eqnarray}

both of which are commonly utilised to characterise the collective dynamics of active particles (Evans et al. Reference Evans, Ishikawa, Yamaguchi and Lauga2011; Stark Reference Stark2018). The orientation vector $\boldsymbol{e} _{k}$ of the $k{{\rm th}}$ bubble in (5.1) is defined based on the bubble’s instantaneous volume-averaged velocity vector in the $xy$ -plane. While computing $q^{(k)}_{6}$ for the $k{{\rm th}}$ bubble, the summation in (5.2) is performed over its six nearest neighbouring bubbles $\mathcal{N}^{(k)}_{6}$ . Furthermore, the angle $\theta _{kj}$ in (5.2) is measured between the line connecting the centre of the $k{{\rm th}}$ bubble to one of its six neighbouring bubbles ( $j\in \mathcal{N}^{(k)}_{6}$ ) and a predefined axis.

The physical interpretation of parameters $\mathcal{Q}$ and $q_{6}$ is as follows. The polar order parameter $\mathcal{Q}$ approaches one when all bubbles migrate in the same direction. The local bond order parameter reflects the presence of hexagonal clustering, reaching its maximum value of $|q^{(k)}_{6}|=1$ when a bubble is surrounded by six bubbles in a perfect hexagonal arrangement. Similarly, the global bond parameter $q_{6}$ attains a value of one when a hexagonal lattice forms. However, in bubble rafts, $q_{6}$ can never reach unity because bubbles along the outermost edge of the raft do not fulfil the $|q^{(k)}_{6}|=1$ condition.

In the present case, the polar order parameter $\mathcal{Q}$ fluctuates over time, as seen in figure 18(b). On the other hand, the global bond parameter $q_{6}$ shows a slower overall decay with time, with a sharp increase near the end. Figure 18(b) also includes three snapshots showcasing instantaneous bubble positions in the $xy$ -plane alongside their orientation vectors. Earlier in figure 17, we noted that bubbles initially migrate radially outward. This behaviour is also evident from the orientation vectors of individual bubbles shown in the leftmost inset of figure 18(b). In this state, the bubble suspension has the lowest polar order $\mathcal{Q}$ , as marked by the first vertical dashed line in figure 18(b). The maximum value of $\mathcal{Q}$ occurs when bubbles form chain-like structures, as demonstrated in the central inset corresponding to the second vertical dashed line in figure 18(b). Here, most bubbles within each local cluster move in nearly the same direction. However, the chain-like geometry of these clusters leads to a reduction in the bond parameter $q_{6}$ relative to its initial value. Eventually, bubbles transition into a more compact, raft-like configuration, as shown in the rightmost inset. This configuration yields a higher value of the global bond parameter  $q_{6}$ , as indicated by the third vertical black line. Of all the bubbles, the olive-coloured one with index $\mathcal{N}^{k}_{b}=2$ in figure 18(b) forms an almost perfect hexagonal cluster around itself (see the last inset), achieving $|q^{(2)}_{6}|=0.94$ .

Néel & Deike (Reference Néel and Deike2021) reported that dense suspensions of nearly monodisperse floating bubbles exhibit cyclic transitions between ordered and disordered velocity states. In their experiments, time intervals of elevated polar order were observed, during which bubbles exhibited strongly coordinated motion with nearly identical in-plane velocities. These ordered phases were followed by less ordered states, characterised by a broader distribution of velocities across the suspension. Overall, their measurements showed two successive cycles of ordered and disordered phases in the velocity time series. To facilitate a qualitative comparison with these experimental findings, we examine the temporal evolution of the in-plane velocity components $u_{x}\mu _{l}/\sigma$ and $u_{y}\mu _{l}/\sigma$ , as shown in figure 19. Our simulations reveal that there are indeed intervals in which certain bubbles attain nearly identical values of either $u_{x}$ or $u_{y}$ . For instance, around $\tau _{c}=20$ and $55$ bubbles exhibit clear alignment in one velocity component, as highlighted in figure 19. However, in contrast to the observations of Néel & Deike (Reference Néel and Deike2021), our results do not display simultaneous overlap in both velocity components across the suspension. Instead, the coordination we observe is restricted to a single direction at a time, rather than fully bidirectional alignment. To establish more robust statistics and approach experimental conditions, simulations of denser suspensions would be required. However, the computational cost of resolving such systems within the present MVOF formulation remains prohibitive, limiting the extent of direct comparison with experiments.

Figure 19.

Temporal evolution of the dimensionless in-plane velocity components: (a) $u_{x}\mu _{l}/\sigma$ and (b) $u_{y}\mu _{l}/\sigma$ , during the capillary migration of floating bubbles shown in figure 16. The velocity of each bubble in the swarm is distinguished by a unique colour corresponding to its index $\mathcal{N}^{k}_{b}$ in the colourbar. The bubble index $\mathcal{N}^{k}_{b}$ is consistent with the labels in figure 17. The simulation parameters are $\textit{Bo}=1$ , $\textit{Mo}=10$ and $\mathcal{N}_{b}=15$ . The magenta boxes highlight bubbles with identical velocities.

Earlier in § 4.1, we demonstrated that side-by-side bubbles exhibit enhanced capillary migration as the Bond number increases within the range $10\leqslant \textit{Bo}\leqslant 50$ . Building upon this observation, we now consider a monodisperse suspension of $\mathcal{N}_{b}=15$ floating bubbles of relatively large size at $\{\textit{Bo},\textit{Mo}\}=\{25,1\}$ . The bubbles are initialised beneath the free surface using the same random coordinates employed for the previous suspension at $\textit{Bo}=10$ , thereby enabling a direct comparison between the two cases. Figure 20 displays instantaneous snapshots of free-surface deformation at $\textit{Bo}=25$ up to $t\sigma /\mu _{l}R_{0}=50$ . By $t\sigma /\mu _{l}R_{0}=19$ (figure 20 c), distinct localised clusters consisting of two or more floating bubbles already form. A comparison between figures 16(b) and 20(c) highlights that the swarm of bubbles at $\textit{Bo}=25$ undergoes rapid self-organisation at early times. However, this process subsequently slows, as indicated by the more gradual structural evolution observed in the later frames of figure 20. In the final two snapshots, the bubble chain along the periphery extends itself by merging with the neighbouring cluster, ultimately containing $60\,\%$ of the bubbles in the swarm. Unlike the $\textit{Bo}=10$ case shown in figure 16, where the chain of floating bubbles is relatively short-lived, the chain at $\textit{Bo}=25$ proves to be far more persistent, maintaining its structure over a significantly longer duration of $\Delta t\sigma /\mu _{l}\!R_{0}=25$ , as shown in figure 20(d,e).

Figure 20.

Emergent dynamics of a monodisperse suspension comprising $\mathcal{N}_{b}=15$ floating bubbles, driven by capillary migration along the free surface. The Bond and Morton numbers are fixed at $\textit{Bo}=25$ and $\textit{Mo}=1$ .

5.2. Bilayer arrangement

Motivated by the coaxial and side-by-side bubble pairs investigated in §§ 3–4, we now examine the dynamics of a bilayer bubble swarm, i.e. stacks of bubbles. Specifically, we initialise two vertical layers of perfectly aligned bubbles beneath the free surface, with each layer containing seven bubbles. This gives a total of $\mathcal{N}_{b}=14$ bubbles distributed among seven coaxial pairs. During initialisation, one coaxial pair is placed at the centre, while the remaining six pairs are randomly distributed around it, forming a distorted hexagonal arrangement. This configuration allows us to probe the mechanical stability of the central coaxial pair at early times, under the confinement imposed by the surrounding coaxial pairs. The Bond and Morton numbers are fixed at $\textit{Bo}=10$ and $\textit{Mo}=1$ , respectively.

Snapshots of free-surface deformation and the corresponding subsurface bubble dynamics are shown in figure 21 at different times, where bubbles in the top and bottom layers are coloured orange and olive, respectively. At early times, peripheral pairs tilt outward, as seen in figure 21(aii), closely resembling the initial radial migration observed in a monolayer suspension at $\textit{Bo}=10$ in figure 17. This tilt eventually destabilises the orientation of the peripheral pairs, causing them to topple. In contrast, the central pair retains stability for a longer duration, although with slight distortion (see figure 21 bii). The shape of the central pair in figure 21(bii) closely matches the equilibrium axisymmetric configuration of a coaxial pair at $\textit{Bo}=10$ reported in figure 5. However, the central pair also loses its coaxial arrangement at a later stage, as shown in figure 21(cii). Notably, all coaxial pairs ultimately transpose into side-by-side pairs as evident from figure 21(aii,cii). As time progresses, the swarm reorganises into a compact monolayer structure under the influence of capillary attraction. Interestingly, all bubbles from the bottom layer are located at the core of the raft, while those from the top layer are arranged along the periphery of the raft in figure 21(eii).

Figure 21.

Emergent dynamics of a monodisperse suspension comprising $\mathcal{N}_{b}=14$ floating bubbles, driven by capillary migration along the free surface. The top and bottom rows of snapshots display the free surface and the subsurface bubbles, respectively. The bubbles are initialised in a bilayer configuration of seven bubbles each in the upper and lower layers, represented in orange and olive, respectively. The Bond and Morton numbers are fixed at $\textit{Bo}=10$ and $\textit{Mo}=1$ .

Lastly, we note that although the coaxial state of the central pair in figure 21 is not indefinitely sustained, its transient stability provides further support to our argument for studying coaxial pairs in § 3. Under strong lateral confinement, coaxial states can be realised and maintained for significantly longer durations.