3.1.1. Qualitative observations on viscous effect

Figure 3 shows the breakup sequence for DI and G80 droplets in the bag breakup regime. Before the droplet breaks up into small fragments, it undergoes deformation that can be divided into two distinct characteristic phases. During the first phase, a nearly spherical droplet deforms in the streamwise direction and expands radially. Jackiw & Ashgriz (Reference Jackiw and Ashgriz2021) found that during this phase, there exists a universal time scale, which was referred to as the flow balancing time ( $T_{\textit{bal}}$ ), after which the streamwise deformation of the droplet leads to the constant radial expansion. This phase of deformation continues until the time, defined as initiation time ( $T_i$ ), when the droplet attains the disc type structure (minimum streamwise thickness) (Flock et al. Reference Flock, Guildenbecher, Chen, Sojka and Bauer2012). The second phase governs the bag film development in which the central portion of the disc inflates like a balloon due to the dynamic pressure of the air flow and continues to expand in size. During this phase, the peripheral region of the disc emerges as a toroidal rim. As the bag is attached to the rim, its growth is also associated with the bag development. Jackiw & Ashgriz (Reference Jackiw and Ashgriz2021) provided detailed insights for a droplet breaking up in the bag breakup regime. Once the bag has stretched enough, its rupture starts at the thinnest point on the film, which recedes with a constant retraction speed, and collides with the main toroidal rim to trigger the rim breakup. There have been various mechanisms by which these structures break and produce discrete droplet sizes. The detailed analysis on the sizes produced from these mechanisms have been proposed recently by Jackiw & Ashgriz (Reference Jackiw and Ashgriz2022), and utilized by others (Ade et al. Reference Ade, Chandrala and Sahu2023, Reference Ade, Kirar, Chandrala and Sahu2024; Xu et al. Reference Xu, Wang and Che2023; Qian et al. Reference Qian, Cheng, Sun and Lv2025). For Newtonian liquids, G80 and DI droplets exhibit similar morphological features as described above and shown in figure 3. However, the viscosity brings three noticeable changes in the breakup dynamics: (i) the temporal events of deformation and breakup get delayed. For instance, the initiation time at state-iii for DI is 1.13 as seen in figure 3(a), whereas figure 3(b) shows that viscous G80 drop is taking a longer time (1.31) to reach the same state. (ii) Another noticeable effect of viscosity reflects in terms of the higher stretching extent of the thin film membrane in a bag inflation stage along the streamwise direction. In figure 3 at state-iv, the bag size for G80 is larger as compared with DI. (iii) Finally, when the bag membrane ruptures completely, the toroidal rim is left over in the flow field. This rim experiences higher radial expansion for a liquid with higher viscosity causing it to become thinner before breaking up into the child droplets. It may be clearly observed in figure 3(a) at state-vii that the corrugated rim produces only two characteristic size classes of child droplets for DI. In contrast, as shown in figure 3(b), the rim of a G80 droplet becomes smoother due to stretching and produces droplets by viscous pinching and satellite droplets via neck formation. Niranjan et al. (Reference Niranjan, Bodi and Gadgil2024) have studied the droplet breakup phenomenon in the bag breakup regime for liquids of varying viscosity to show the effect on time scales quantitatively. They also showed that the fragments from the main rim consist of two distinctive bands for inviscid liquids, while three different characteristic sizes result from the rim breakup for viscous droplets. Keshavarz et al. (Reference Keshavarz, Houze, Moore, Koerner and McKinley2020) provided the size estimation of such satellite droplets produced from the stretching of the viscous rim.

Figure 4.

Montage of breakup events for gel samples in the typical bag breakup regime in side view (near view at camera position $P_2$ ) arrangement for (a) S1 at $\textit{We}\approx 13.21$ ; ( $\textit{Oh} = 0.014$ ), (b) S2 at $\textit{We}\approx 15.77$ ; ( $\textit{Oh} = 0.11$ ) and (c) S3 at $\textit{We}\approx 41.08$ ; ( $\textit{Oh} = 0.31$ ).

3.1.2. Deformation and breakup characteristics of gel droplets

Figures 4(a), 4(b) and 4(c) provide a comprehensive morphological picture of the breakup events in the non-Newtonian, shear-thinning gel droplets with varying gellant concentration in the bag breakup regime. Deformation and bag inflation phases in the gel droplets also show qualitatively similar features until breakup as observed for a Newtonian liquid. Since higher gellant content increases the viscosity (from S1 to S3 in the range of $0.05{-}0.15$ wt. $\%$ of Carbopol), the resistance to the breakup of the droplet increases. This results in the delayed onset of the bag breakup regime in terms of Weber number. It is evident from figure 4 that the Weber number corresponding to the bag breakup regime for S3 ( $\approx$ 41) is significantly larger than that for S1 and S2 ( $\approx$ 13–15). Further, the initiation time to achieve a disc structure also gets delayed as we go from S1 to S3 due to the higher gel viscosity, which is similar to the viscous Newtonian liquid G80.

Although certain features observed in the breakup of gel droplets are due to the increased viscosity from the base liquid, the distinctive nature of the gel droplets can be noted during the bag inflation stage. To highlight this distinction, we compare the phenomenon for liquids G80 and S2, since they have approximately the same viscosity values and breakup approximately at the same $\textit{We}$ . Observation of state-iv in figures 3(b) and 4(b) reveals that the streamwise expansion of the bag is significantly lower for gel S2 as compared with G80. Further, comparison of state-iv in gel samples in figures 4(a), 4(b) and 4(c) shows that the streamwise length of the bag decreases with increase in gel viscosity from S1 to S3. This is counterintuitive to the natural effect of viscosity highlighted in § 3.1.1. The enhanced viscosity must have stretched the bag size along the flow direction. However, it is observed that gel samples tend to rupture with reduced extent of bag inflation with increased gellant concentration, even though the viscosity of the samples has increased. In fact, the bag size is even less compared with the relatively low viscosity solvent (DI) as shown in figure 3(a) at state-iv. Another interesting qualitative observation is that the expansion of the toroidal rim is also affected by the reduced inflation of the bag as the bag is anchored at the toroidal rim. Consequently, the rim size at a time of bag bursting gets significantly reduced as the gellant concentration is increased. These parameters will be quantified later in this section.

Figure 5.

Ensemble averaged deformation characteristics (lies within 95 % confidence interval (CI) and details are provided in Appendix A, figure 20) at the onset of deformation just before the bag ruptures in (a) the streamwise direction, (b) the cross-streamwise direction and (c) bag stretching in the streamwise direction from initiation time. Markers larger in sizes with error bar typically show the averaged values at initiation and rupture times for each fluids. Measurement directions for deformation are shown inside a square box while the variation of $T_{in}$ with gellant concentration ( $c$ wt. %) is also shown as an inset in (a).

Figure 5. Long description

The image contains three line graphs labeled (a), (b), and (c), each depicting different aspects of the deformation characteristics of shear-thinning materials. Panel A: The line graph shows the deformation in the streamwise direction. The x-axis is labeled T and the y-axis is labeled L/d0. The graph includes various markers representing different fluids, with larger markers and error bars indicating averaged values at initiation and rupture times. An inset within this panel shows the variation with gellant concentration in weight percent. Panel B: The line graph illustrates the deformation in the cross-streamwise direction. The x-axis is labeled T and the y-axis is labeled 2R/d0. Similar to Panel A, it includes different markers for various fluids, with larger markers and error bars showing averaged values at initiation and rupture times. Panel C: The line graph depicts bag stretching in the streamwise direction from the initiation time. The x-axis is labeled T-Ti and the y-axis is labeled Bag stretching: L/d0. This graph also includes different markers for various fluids, with larger markers and error bars indicating averaged values at initiation and rupture times. The graphs collectively show the deformation characteristics of shear-thinning materials at the onset of deformation just before the bag ruptures.

The initiation phase gets delayed with an increase in viscosity irrespective of the nature of the liquid. Although the streamwise dimension after the initiation time (representing the length of inflated bag in streamwise direction) increases monotonically, two distinct effects of pure viscosity and non-Newtonian rheology are clearly seen in figure 5(a). If we compare DI and G80 liquids, we notice that the bag elongates more and the first rupture event is delayed in the case of G80 as compared with DI without significantly altering the growth rate. This is the characteristic effect of pure viscosity which is also qualitatively shown in figure 3 and explained in § 3.1.1. The gel samples, however, show interesting features depending upon the gellant concentration. Sample S1 follows a similar growth pattern as that of Newtonian liquids possibly due to the relatively lower degree of the shear-thinning nature (higher value of $n$ ). However, its bag gets ruptured early and with lower expansion as compared with the base liquid DI, though S1 has a higher viscosity than DI. If we look at gels of higher viscosity than S1, i.e. S2 and S3, the elongation of the bag in the streamwise direction gets reduced significantly.

Moreover, samples S2 and S3 show a much lower bag growth rate compared with other liquids. Similar observations can be noted for the variation of cross-stream dimension in figure 5(b). Until the initiation time, all the liquids exhibit a nearly linear growth of the cross-stream dimension. Post initiation, although all the liquid show monotonic increase in the cross-stream dimension, viscosity brings a subtle modification. Increase in viscosity brings a plateau region ( $T_{in}\le T\le 2$ for G80) in the growth of the cross-stream length before shooing up (after $T\ge 2$ ), which can be noticed in the comparison of DI and G80 and reported by Niranjan et al. (Reference Niranjan, Bodi and Gadgil2024). This is visible in the case of all gel samples also. Sample S1 follows the trend of viscous, Newtonian liquids. However, the samples S2 and S3 depart from initiation time, follow a plateau region and saturate without showing any increase in the cross-stream length. It is likely that the lower expansion of the bag stops the growth of the rim in the cross-stream direction. To provide a better idea of the bag stretching during the inflation phase (from $T_{in}\le T \le T_{br}$ ), figure 5(c) is plotted from the initiation time which clearly shows the differences described here.

The present study highlights that gels are not able to sustain the liquid stretching to smaller length scales. This might seem to be the result of the non-Newtonian nature of the gels at first. If that is so, samples S2 and S3 should behave like G80 as their viscosity is similar to G80 at high shear rates. However, the present observations indicate that the effective viscosity of the gels does not enhance the axial stretching of the bag. This behaviour contrasts with the established theory for Newtonian viscous liquids, where increasing viscosity generally promotes greater axial elongation. Hence, the inhomogeneity in the gel liquid, which also induces a non-Newtonian character to it, may be the reason that makes the liquid interface too weak to sustain further thinning. Some evidence of this inhomogeneity may be observed from the surface of the stretched bags in figures 3 and 4. We will come back to this aspect with more evidence in a later section.

Figure 6.

(a) Drop radial extent at initiation time, (b) length of the stretched bag just before the bag punctures in the bag breakup regime. The colourbar shows viscosity scale in terms of $\textit{Oh}$ and each liquid droplet is colour coded as per the $\textit{Oh}$ value. Horizontal dashed–dotted line in (a) shows the averaged initiation diameter over all the $\textit{We}$ for sample S3 while it represents the average bag film extension in (b). The dashed line in (b) denotes the average bag size in sample S2. The insets in (a) and (b) show the schematic geometry at the initiation and breakup stations. The vertical bars represent the variability across realizations, quantified using the sample standard deviation. Details are provided in Appendix A, figure 21.

In figure 6(a), cross-stream diameter ( $2R_i$ ) at the initiation time is plotted against the Weber number. It shows that the diameter of the disc at the initiation time increases with $\textit{We}$ in the bag breakup regime for all the samples except S3. For sample S3, this value as insensitive to the Weber number. Although the cross-stream diameter increases for other samples, the increment is typically of the order of 10 % which lies within the limits of an error bar. So, for all practical purposes, the averaged value of $2R_i/d_0\approx 1.8$ is also found to be suitable for all liquids. This constant value of initiation diameter indicates that the evolution of droplet morphology is indeed independent of viscosity and the nature of the liquid during the deformation phase.

The variation of non-dimensional streamwise bag stretching (the length of the bag at the time of the first rupture instant) with Weber number is shown in figure 6(b). In the bag breakup regime, the bag size increases for DI and S1 indicating the similarity in their growth characteristics. Jackiw & Ashgriz (Reference Jackiw and Ashgriz2021) have also reported a similar trend for the bag dimension until the rupture instant in their experimental findings with DI in the bag breakup regime. But as the gellant concentration increases from S1 to S3, the streamwise length of the bag becomes shorter as well as insensitive to Weber number. The length of the bag decreases very sharply with the increase in the gellant concentration. As seen in figure 6(b), a blue dashed and red dash–dot line show the average value of bag size ( $L_{br}$ ) at the breakup for S2 and S3, respectively. This implies that beyond a threshold gellant concentration, approximately $0.05\,\mathrm{wt.}\%$ in the present case, saturation in bag stretching begins to emerge irrespective of the aerodynamic forcing. In other words, the current study indicates that with increasing gellant concentration, the gel progressively loses its ability to sustain bag stretching to smaller film thicknesses, despite an increase in viscosity. If the volume within the thin inflating membrane remains constant, in that case the bag film thickness at the time of rupture depends upon the extended bag size. Since the bag film cannot be thinned below a certain threshold critical thickness for a given gellant concentration, the bag thickness at the rupture time is also governed by the viscosity and the nature of liquids. As this bag bursts to produce a set of child droplets, it is critical to estimate its thickness. In the next subsection, we propose a methodology to estimate average bag thickness at the time of rupture in the bag breakup regime.

Figure 7.

Temporal image sequence of breakup events in Newtonian and non-Newtonian liquids in bag breakup regime from $45^{\circ }$ view arrangement for (a) G80 at $\textit{We}\approx 15$ ; and (b) S2 at $\textit{We}\approx 15.45$ . In each panel, the droplet deforms at the initial spherical shape (state-i) into the disc shape (state-iii) during the deformation period. At the initiation time (state-iii), the shape is equivalent to the disc and a region highlighted between the solid and the dotted curve evolves as a rim, which is modelled as a torus. A central thin region of the disc within the dotted region accounts for the bag film development later as the bag inflates, and approximated as thin hollow semispherical cap.

Figure 7. Long description

The image contains two sets of temporal image sequences labeled as (a) and (b), each showing the breakup events of liquids in the bag breakup regime. Panel A: This sequence consists of seven images labeled (i) to (vii). The images capture the deformation and breakup of a droplet over time. Initially, the droplet is spherical (state-i) and deforms into a disc shape (state-iii). At the initiation time (state-iii), the shape is equivalent to a disc, and a region highlighted between the solid and the dotted curve evolves as a rim, modeled as a torus. A central thin region of the disc within the dotted region accounts for the bag film development later as the bag inflates and is approximated as a thin hollow semispherical cap. The time intervals are marked as 0 ms, 1.50 ms, 5.50 ms, 11.25 ms, 12.50 ms, 14.75 ms, and 18.25 ms. Panel B: This sequence also consists of seven images labeled (i) to (vii). Similar to Panel A, the images capture the deformation and breakup of a droplet over time. The droplet starts as a sphere (state-i) and deforms into a disc shape (state-iii). At the initiation time (state-iii), the shape is equivalent to a disc, and a region highlighted between the solid and the dotted curve evolves as a rim, modeled as a torus. A central thin region of the disc within the dotted region accounts for the bag film development later as the bag inflates and is approximated as a thin hollow semispherical cap. The time intervals are marked as 0 ms, 2.75 ms, 7.75 ms, 12.25 ms, 13.75 ms, 16.75 ms, and 19.50 ms.

3.4.1. Bag breakup mechanism and droplets formation

A montage of bag bursting events observed in the current experiments is shown in figure 13 for sample S2 at a $\textit{We}\approx 15.45$ . A thin film undergoes breakup due to the piercing of the bag membrane generating fine droplets at the bag tip, which corresponds to the thinnest region of the film. The punctured hole is bordered with a rim that recedes accumulating the bag liquid and breaks via ligaments as shown in figure 13(c). During this process, the child droplets form from different parts of the film interface. Figure 14 depicts typical transient variation of fragments generated from various parts of the receding bag, and droplet sizes increase over time as the bag recedes along the curved interface. This pattern is consistent with the fact that the film thickness varies along the bag interface, being thinnest at the tip and thickest at the root. All the gel samples show similar transient variation in child droplet sizes formed from the bag. At the moment of rupture ( $T=0$ ), the droplets produced from the thinnest region near the bag tip are minimal in size. During initial bag recession stage between $0\le T \le 0.2$ , bag recedes very rapidly due to the lowest film thickness. As shown in figure 13(a–e), this rapid retraction of the thin bag film transports liquid into the receding bag rim, causing the rim to thicken sufficiently and subsequently forms thick ligaments which eventually sheds bag child droplets. As a result, the characteristic droplet size in figure 14 increases most rapidly during this short period. As the receding bag film approaches its root near the main rim, the local film thickness progressively increases. According to the Taylor–Culick mechanism, this results in a reduction in the film retraction velocity. Consequently, liquid accumulation within the rim slows, reducing the growth rate of the ligament diameter and, hence, the rate of increase of the characteristic droplet size, as shown in the figure 14 during $0.2\le T \le 0.5$ .

Figure 14.

Temporal evolution of normalized bag child droplets as the bag film continues to recede along the interface for gel samples S1 at $\textit{We} = 13.21$ , S2 at $\textit{We}=15.86$ and S3 at $\textit{We} = 41.08$ . Here $T = 0$ represents the first instance of bag film rupture. Lines are the averaged fragment sizes over the entire duration of bag rupturing.

All the samples exhibit monotonic increase in the bag droplet size with time. Moreover, it is worth noting from figure 14 that a increase in gellant concentration leads to larger child droplet sizes. Sample S3, which has the highest film thickness, results in the largest fragment sizes, followed by S2, while S1 produces the smallest fragments due to its lower film thickness. The lines (solid, dash–dot and dotted) in figure 14 represent the time-averaged bag droplets sizes over the entire duration of bag fragmentation. This average is indicative of the characteristic droplet size formed from the bag during an individual droplet breakup event. It is also worth noting in figure 14 that the average is skewed towards the earlier time instance and this non-dimensional time instant is same (nearly 0.2) for all the gel samples.

Figure 15 illustrates the ruptured gel film during the bag breakup event with figures 15(a), 15(b) and 15(c) corresponding to samples S1, S2 and S3, respectively. A thick rim is formed around the punctured hole having instantaneous thickness larger than the local bag film thickness. The receding bag rim motion follows the curvature of the bag film which imposes a centripetal acceleration, $a_c$ , modelled as (Lhuissier & Villermaux Reference Lhuissier and Villermaux2012; Wang et al. Reference Wang, Dandekar, Bustos, Poulain and Bourouiba2018; Jackiw & Ashgriz Reference Jackiw and Ashgriz2022; Chandra et al. Reference Chandra, Sharma, Basu and Kumar2024)

(3.8) \begin{equation} a_c = \frac {U^2_{\textit{TC}}}{L_b}. \end{equation}

In (3.8), $U_{\textit{TC}}$ is a receding film velocity that is assumed to follow the Taylor–Culick ( $U_{\textit{TC}} =\sqrt {2\sigma /\rho _l \bar {h}_{\!f}}$ ) retraction phenomenon. Here $L_b$ is the radius of the bag (defined in figure 8) at the time of the bag burst. The accelerating rim develops evenly spaced ligaments, which are susceptible to the RP-type instability enabling the corrugated ligaments to stretch and eventually shed as child droplets. The wavelength associated with disturbances on the receding rim ( $\lambda _{rr}$ ) is given as (Lhuissier & Villermaux Reference Lhuissier and Villermaux2012; Wang et al. Reference Wang, Dandekar, Bustos, Poulain and Bourouiba2018; Chandra et al. Reference Chandra, Sharma, Basu and Kumar2024)

(3.9) \begin{equation} \frac {\lambda _{rr}}{d_0}= 4.5 \frac {b}{d_0} .\end{equation}

Wang et al. (Reference Wang, Dandekar, Bustos, Poulain and Bourouiba2018) have described that rim destabilization during the bursting of a surface bubble leads to corrugation of ligaments and follows the universal criterion of instantaneous Bond number, $Bo (= \rho _l b^2 a_c/ \sigma ) = 1$ . Therefore, the thickness of receding rim ( $b$ ) is estimated as follows:

(3.10) \begin{equation} b = \sqrt {\frac {\sigma }{\rho _l a_c}}. \end{equation}

Putting (3.8) into (3.10), the receding rim thickness becomes

(3.11) \begin{equation} \frac {b}{d_0} = 0.71 \sqrt {\left (\frac {\bar {h}_{\!f}}{d_0}\right )\left (\frac {L_b}{d_0}\right )}. \end{equation}

Figure 15.

Recessed bag structure in gel samples in (a) S1, (b) S2 and S3 at $\textit{We} = 12.71$ , $17.46$ and $43.15$ , respectively. Blue and red arrows show bag receded bag rim and corrugated ligament, respectively, while the wavelength of the receding rim, $\lambda _{rr}$ , is shown as a double headed black arrow. The bag film droplets generated via ligament mediated breakup are shown inside the magenta dotted square box. The child droplet’s number shedding frequency from corrugated ligaments diminishes with the increased bag thickness from sample S1 to S3. This shows as the darker and wider bag rim thickness, $b$ (inside the yellow box) identified as blue arrow in the schematics. All images are rescaled to the common scale of S3, and bar in (c) shows 5 mm of length.

Figure 16.

(a) Variation of average bag fragment sizes with $\textit{We}$ in bag breakup regime for gel droplets. In (a), markers denote experimental measurements, while the lines (S1, -; S2, –; S3, .) show the average bag child droplet size and coloured band shows standard deviation combining all average sizes along the $\textit{We}$ . Predictions of the non-dimensional fragment size based on the average bag length at rupture are shown as dashed–cross lines (– $\times$ –), obtained from (3.12). Panel (b) presents the non-dimensional wavelength of the disturbance on the receding bag rim, ( $\lambda _{rr}/d_0$ ). Markers correspond to direct experimental measurements (see figure 15), while the dashed–cross (– $\times$ –) lines are the prediction from (3.9) denote predictions from (3.9). In both panels, marker colour indicates the relative magnitude of $\textit{Oh}$ , as shown by the colour bar in (b).

Figure 16. Long description

The image contains two line graphs labeled as Panel A and Panel B. Panel A shows the variation of average bag fragment sizes with We in the bag breakup regime for gel droplets. The y-axis represents the average bag fragments in units of d_b/d_0, and the x-axis represents We. Markers denote experimental measurements for three different fluids: S1, S2, and S3. The lines show the average bag child droplet size for each fluid, with S1 represented by a red dashed line, S2 by a blue dashed line, and S3 by a green dashed line. Colored bands around the lines indicate the standard deviation combining all average sizes along We. Predictions of the non-dimensional fragment size based on the average bag length at rupture are shown as dashed cross lines. Panel B presents the non-dimensional wavelength of the disturbance on the receding bag rim. The y-axis represents the receding wavelength in units of λ_rr/d_0, and the x-axis represents We. Markers correspond to direct experimental measurements, while the dashed cross lines are predictions from a specific equation. In both panels, marker color indicates the relative magnitude of Oh, as shown by the color bar in Panel B.

It should be noted that the receding rim experiences an inward inertial acceleration due to its curved motion (Lhuissier & Villermaux Reference Lhuissier and Villermaux2012). This acceleration destabilizes it through a Rayleigh–Taylor (RT)-type instability, causing evenly spaced ligaments to form around the rim. The instability wavelength is a geometrical mean of bag thickness and radius. A similar observation can be inferred from (3.11) which suggests that the rim thickness of the receding bag film depends upon the two most important length scales, i.e. average bag film thickness and bag size at the time of the bag burst. We have seen that the film thickness and the bag size are coupled, and for this coupling the relation is given by (3.5) which in turn reduces (3.11) into a single length scale as

(3.12) \begin{equation} \frac {b}{d_0} \approx \frac {1}{10}\left (\frac {1}{L_{b}/d_0}\right )^{1/2}. \end{equation}

Equation (3.12) reveals that the receding rim thickness has an inverse dependency on the bag size at the time of rupture. Chandra et al. (Reference Chandra, Sharma, Basu and Kumar2024) have assumed that at the time of rim destabilization, the diameter of the ligaments formed have an approximately similar magnitude as that of the rim. Hence, the child droplet mean diameter is the same as the ligament diameter when the droplet pinches-off. Since there is a continuous liquid flow from the thin bag film to the receding bag rim and eventually to the ligament, stretching does not thin down the ligament (Lhuissier & Villermaux Reference Lhuissier and Villermaux2012). Thus, the mean child droplet size has the same order of magnitude as that of the thickness of receding bag rim. The variation of average film droplet sizes for all gel samples with $\textit{We}$ in the bag breakup regime is shown as markers in figure 16(a). The mean droplet size from the bag is insensitive to the Weber number just like the behaviour of the bag film thickness. It is evident that S3, which exhibits the largest average film thickness among the gel samples, produces higher drop sizes compared with S2, and S1. The average normalized film droplet sizes are 0.054, 0.073 and 0.093 for samples S1, S2 and S3, respectively. On the other hand, the analytical predictions from (3.12) (based on the average bag size extracted from figure 6 b) for all gel samples show very good agreement with experimental measurements as shown in figure 16(a). Lhuissier & Villermaux (Reference Lhuissier and Villermaux2012) have also provided a relation to estimate mean bag droplet size for bursting surface bubble as

(3.13) \begin{equation} \frac {\bar {d}_b}{d_0} \sim {\left (\frac {\bar {L}_b}{d_0}\right )}^{3/8}{\left (\frac {\bar {h}_{\!f}}{d_0}\right )}^{5/8}. \end{equation}

The predictions from (3.13), obtained using the average bag radius and film thickness extracted from figures 6(b) and 10, yield non-dimensional fragment sizes $\bar {d}_b/d_0$ $\approx$ 0.039, 0.077 and 0.089 for the gel samples S1, S2 and S3, respectively, when a proportionality prefactor of 0.75 is used. These predictions are also in good agreement with the estimates provided by (3.12). It is worth noting, however, that (3.13) requires knowledge of both bag radius and film thickness, and obtaining reliable film-thickness estimates is considerably challenging than determining the bag radius. This restricts the applicability equation (3.13) for predicting average fragment sizes. In contrast, the expression proposed in this study (3.12) requires only the bag size at rupture to estimate the mean bag fragment size, thereby offering a clear practical advantage over the formulation of Lhuissier & Villermaux (Reference Lhuissier and Villermaux2012). Nevertheless, both expressions provide reasonable and consistent estimates of the fragment sizes with the present experiments. Hence, present findings confirm and conclude that the mean child droplet size follows the receding bag rim thickness. Moreover, in figure 16(b), we presented the variation of the experimentally measured receding rim wavelength ( $\lambda _{rr}$ ), as defined in figure 15 with $\textit{We}$ . In the present experiments, the average wavelength remains insensitive to the Weber number, but increases with gellant concentration due to the increase in the value of $b$ . Thus, the magnitude of the average disturbance wavelength also grows from S1 to S3. These findings also show excellent agreement between present experimental data and the predictions from (3.9). Although S1, S2 and S3 possess different absolute bag film thicknesses, they rupture through the same underlying mechanism: the formation and breakup of corrugated ligaments generated during bag receding process. While the bag film thickness primarily controls the droplet size scale and determines the mean diameter, the fluctuations around this mean arise from a universal stochastic fragmentation cascade. In fact, the breakup of child droplets mediated by ligament fragmentation consistently produces a final droplet size distribution that can be described by a single-parameter gamma distribution (Villermaux et al. Reference Villermaux, Marmottant and Duplat2004; Keshavarz et al. Reference Keshavarz, Houze, Moore, Koerner and McKinley2016; Kulkarni et al. Reference Kulkarni, Shirdade, Rodrigues, Radhakrishna and Sojka2023; Varkevisser et al. Reference Varkevisser, Kooij, Villermaux and Bonn2024),

(3.14) \begin{equation} P \left ( x = \frac {d}{\langle d \rangle } \right ) = \frac {n^n}{\varGamma (n)} x^{n-1} e^{-nx}, \end{equation}

Figure 17.

Single parameter gamma distribution of bag fragments in (a) S1 at $\textit{We} = 13.21$ , (b) S2 at $\textit{We} = 13.65$ and (c) S3 at $\textit{We} = 36.87$ . The fragments are captured from side-view images at near view location, and only fragments from the bags are considered at the end of bag rupture. The solid line denotes that fragments are single parameter gamma distributed with the best fitted parameter ( $n$ ), and rescaled by the average droplet size for each liquid. The dashed line shows corresponding fitted gamma distribution with $n=4$ . The number mentioned in the box above and below the markers represents the bag droplet counts within each bin width. The red dash–dot, and dash line show the mean and Sauter mean diameter (SMD) of the distribution. The bands in light red and blue show the standard deviation of the mean and SMD.

where the shape parameter $n$ quantifies the degree of corrugation and governs the breadth of the distribution. Here $\langle d \rangle$ is the mean droplet diameter of bag fragments. Normalizing each droplet dataset by its respective mean removes the influence of absolute ligament thickness, renders droplet sizes in non-dimensional form, and isolates these intrinsic universal breakup statistics. This leads to the self-similarity or universality in the bag droplet distribution and droplet sizes distribution collapse onto the gamma form with $n \approx 4$ . This can be seen in figure17(a–c), which show that bag child droplet size distributions for all three gel samples are well described by a gamma distribution. The solid line represents the gamma prediction with best-fit value of parameter $n$ , while the dashed line in each panel corresponds to a distribution with fixed $n=4$ . The best-fit value ( $n$ ) across S1, S2 and S3 are 3.3, 4.2 and 3.2, respectively. The physical interpretation of $n$ is commonly associated with the corrugation level of the stretched ligament prior to its fragmentation. Whether this corrugation level is identical for Newtonian, viscoelastic and yield-stress fluids has not been established, and no universal value of $n$ has been reported in the literature. Instead, previous studies have consistently shown that gamma-distributed droplet sizes arise from ligament-mediated fragmentation over a range of $n$ values. Among these studies, Keshavarz et al. (Reference Keshavarz, Houze, Moore, Koerner and McKinley2016) reported $n\approx 4$ for viscoelastic polymer solutions, whereas Varkevisser & Bonn (Reference Varkevisser and Bonn2026) obtained values closer to $n\approx 5$ for air-assisted breakup of dilute polymer solutions. Similarly, Rao & Basu (Reference Rao and Basu2025) observed a range $3\lesssim n\lesssim 8$ , with $n=4$ being the most frequently occurring value for DI droplets undergoing fragmentation at extremely high Weber numbers ( $\textit{We} = 900 -4000$ ). Varkevisser et al. (Reference Varkevisser, Kooij, Villermaux and Bonn2024) also reported an optimal fit with $n\approx 4$ for bag breakup of DI, while Kulkarni et al. (Reference Kulkarni, Shirdade, Rodrigues, Radhakrishna and Sojka2023) obtained $n\approx 3.7$ for aqueous glycerol droplets (70 wt. % glycerol) breaking in the bag breakup regime. Collectively, these studies indicate that child droplet sizes is gamma-distributed as a robust feature of ligament-mediated fragmentation, with the corresponding shape parameter typically lying within the range ( $3\lesssim n\lesssim 6$ ), rather than possessing a single universal value.

Table 3.

Typical bag and rim fragment sizes found in the previous work for Newtonian liquids (DI, ethanol and aqueous glycerol mixtures). Dimensions marked as (*) are in microns.

Table 3. Long description

The table presents a comparison of typical bag and rim fragment sizes for Newtonian liquids, specifically DI water, ethanol, and aqueous glycerol mixtures, across various research works. The table includes six columns: Research work, d0, db/d0, dR/d0, We, Liquid, and Experimental set-up. The rows list different research works along with their respective values for the parameters d0, db/d0, dR/d0, We, the type of liquid used, and the experimental set-up. Row 1: Zhao et al. (2011), 4100, 0.028, 0.19-0.59, 9-23, Water, aqueous glycerol, Cross-flow. Row 2: Gao et al. (2013), 2580, 0.029, n/a, 11, Ethanol, Cross-flow. Row 3: Guildenbecher et al. (2017a), 2300, 0.028, 0.24-0.34, 13, Ethanol, Cross-flow. Row 4: Radhakrishna et al. (2021), 3000, 0.025, 0.1-0.2, 14, Aqueous glycerol, Cross-flow. Row 5: Jackiw & Ashgriz (2022), 1900, n.a., 0.16-0.26, 7-15, Water, Suspended drop. Row 6: Kulkarni et al. (2023), 2100, 0.031, 0.17-0.29, 12-20, Water, aqueous glycerol, Cross-flow. Row 7: Varkevisser et al. (2024), n.a., 90-100*, 200-800*, n/a, Water, Flat fan nozzle. Dimensions marked with an asterisk (*) are in microns.

Although the gels investigated in the present work exhibit different rheological properties, the fitted values of $n$ remain within the range reported in the literature. This indicates that the statistical characteristics of ligament fragmentation during the bag recession are relatively insensitive to the rheological differences considered here, despite their influence on ligament dimensions and breakup time scales. Consequently, the ligaments formed during bag recession appear to possess similar levels of corrugation. Nevertheless, whether this behaviour persists over a broader range of rheological conditions remains an open question and requires further investigation that is beyond the scope of the present study.

A comprehensive summary of bag fragment size statistics from previous studies on droplet breakup during secondary atomization of Newtonian liquids is provided in table 3. The normalized bag fragment sizes reported in those works typically range from $0.025{-}0.031$ . These values indicate that the droplets formed from the rupture of thin films in Newtonian liquids are generally smaller than those produced from the rupture of non-Newtonian gel films in the present study. This is consistent with the variation of bag film thickness across different liquids. The present observations conclusively show for non-Newtonian liquids that a characteristic or unique average droplet size emerges from bag rupture, irrespective of flow conditions. In the case of non-Newtonian gels, the gellant concentration primarily governs the resulting average size by influencing the bag stretching dynamics and leading to premature film rupture which, consequently, increases the mean film thickness.

3.4.2. Node and rim breakup

As the bag elongates in both the streamwise and cross-stream directions, the rim grows until it reaches its maximum radial extent, after which instability governs its breakup and discrete nodes begin to form (Zhao et al. Reference Zhao, Liu, Xu and Li2011; Jackiw & Ashgriz Reference Jackiw and Ashgriz2022). These nodes are commonly attributed to RT or RP instabilities acting on the rim following disc initiation. Jackiw & Ashgriz (Reference Jackiw and Ashgriz2022) demonstrated that both mechanisms are plausible and yield reasonable agreement with experimentally measured node sizes. The RP theory better predicts the number of nodes, whereas RT theory estimates their characteristic size more accurately. However, classical instability theory does not clearly specify the instant at which the instability takes control of the rim dynamics.

The node diameter may be estimated from volume conservation over an RT wavelength segment of the rim leading to (Jackiw & Ashgriz Reference Jackiw and Ashgriz2022)

(3.15) \begin{equation} \frac {d_n}{d_0} = \left [ \frac {3}{2} \underbrace {\left (\frac {h_i}{d_0}\right )^2}_{\text{(I)}} \underbrace {\left (\frac {\lambda _{\textit{RT}}}{d_0}\right )}_{\text{(II)}}n \right ]^{1/3} \!. \end{equation}

The node size, in (3.15), depends on two parameters: term-(I), $h_i/d_0$ is the normalized rim thickness at initiation time; term-(II), $\lambda _{\textit{RT}}/d_0$ is the non-dimensional most unstable wavelength for RT instability. Here, factor $n$ represents the fraction of wavelength segment that ultimately form a node ( $n = 0.4$ for DI; Jackiw & Ashgriz Reference Jackiw and Ashgriz2022). For the present experiments, the rim geometry at the initiation times remains nearly invariant across $\textit{We}$ and gel samples, with $2R_i/d_0 \approx 1.8$ and $h_i/d_0 \approx 0.32$ , indicating that node size primarily depends on $\lambda _{\textit{RT}}$ , which is given as (Aliseda et al. Reference Aliseda, Hopfinger, Lasheras, Kremer, Berchielli and Connolly2008)

(3.16) \begin{equation} \frac {\lambda _{\textit{RT}}}{d_0} = 4\pi \left [ \underbrace { \frac {1}{\sqrt {C_D \textit{We}}} \left (\frac {d_0}{2R_i}\right ) }_{\text{Inviscid term}} + \underbrace { \frac {{\textit{Oh}}^{2/3}}{(6 C_D \textit{We})^{1/3}} \left (\frac {h_i}{d_0}\right )^{1/3} \left (\frac {d_0}{2R_i}\right )^{2/3} }_{\text{Viscous term}} \right ]\!. \end{equation}

Here, $C_D$ is drag coefficient on the deforming droplet, $2R_i/d_0$ is cross-stream dimension of the rim at initiation time RT instability hold the control over rim and $\textit{Oh}$ is Ohnesorge number based on the rim minor diameter, $h_i$ . Unlike earlier studies on low-viscosity Newtonian liquids (Zhao et al. Reference Zhao, Liu, Li and Xu2010; Zhang & Liu Reference Zhang and Liu2011; Jackiw & Ashgriz Reference Jackiw and Ashgriz2022), contribution of viscous effects (term-(II)) on $\lambda _{\textit{RT}}$ cannot be neglected here because gels S2 and S3 possess significantly higher viscosity. Moreover, Newtonian rims undergo appreciable radial growth prior to RT dominance, whereas shear-thinning gels exhibit suppressed radial expansion. Consequently, gels retain thicker rims and yield larger RT wavelengths. The mean node sizes for the gel samples are shown as markers in figure 19(a), and they exhibit a decreasing trend with increasing $\textit{We}$ . The theoretical predictions from (3.15), incorporating both inviscid and viscous contributions to $\lambda _{\textit{RT}}$ , are plotted as solid lines and generally underpredict the measure values. Nonetheless, a reasonable agreement is obtained for samples S1 and S2, whereas the prediction deviates significantly for S3. This discrepancy is attributed to the strong $\textit{We}$ -dependency of (3.15). As the viscosity increases, the critical Weber number for bag breakup ( $\textit{We}_{cr}$ ) also increases, which causes S3 gel droplets to break at comparatively higher $\textit{We}$ , where the model performance deteriorates. Samples S1 and S2 exhibit similar critical Weber number values and rupture takes place within a comparable $\textit{We}$ range leading a better agreement with model.

Predictions based solely on the inviscid contribution (dashed lines) underpredict node sizes for all the gel samples, demonstrating that viscous effects are non-negligible and play a dominant role in determining the node droplet scale. Overall, although the framework of Jackiw & Ashgriz (Reference Jackiw and Ashgriz2022) was originally formulated for inviscid Newtonian liquids, it can be effectively extended to viscous, shear-thinning gels when viscous contributions are incorporated – up to a finite concentration range, which in the present study holds satisfactorily up to S2 (0.1 wt. %). However, the present work does not attempt to modify the existing formulation to improve prediction capability for highly viscous gel S3 and further detailed investigations are required to overcome this limitation.

A majority of the secondary fragments originates from the breakup of the residual rim after the complete recession of the bag film. Traditionally, rim breakup has been attributed to RP instability acting on a liquid torus with a final rim thickness $h_{r,f}$ . However, this mechanism overpredicts fragment sizes (Jackiw & Ashgriz Reference Jackiw and Ashgriz2021; Ade et al. Reference Ade, Chandrala and Sahu2023; Niranjan et al. Reference Niranjan, Bodi and Gadgil2024). Jackiw & Ashgriz (Reference Jackiw and Ashgriz2022) demonstrated that collision of the receding bag rim, characterized by a disturbance wavelength $\lambda _{rr}$ , induces premature breakup of the rim as compared with the classical RP instability.

The volume conservation of the wavelength segment of the main rim that ultimately forms rim droplets of diameter $d_r$ , yields (Jackiw & Ashgriz Reference Jackiw and Ashgriz2022)

(3.17) \begin{equation} \frac {d_r}{d_0} = \left [\frac {3}{2}\underbrace {\left (\frac {h_{r,f}}{d_0}\right )^2}_{\text{(I)}} \underbrace {\left (\frac {\lambda _{rr}}{d_0}\right )}_{\text{(II)}}\right ]^{1/3}. \end{equation}

Equation (3.17) shows that average rim size depends on two terms whose combined magnitude decide final size. Term (I) represents the final rim thickness and can be estimated through volume conservation between rim initiation and breakup as $h_{r,f}/d_0 = (h_i/d_0)\sqrt {(2R_i/d_0)/(L_r/d_0)}$ . Since, in the present experiments, the geometric scales at initiation time have found $2R_i/d_0\approx 1.8$ and $h_i/d_0\approx 0.32$ , the final rim thickness primarily depends on its terminal radial extent $L_r$ (refer figure 18) before it disintegrates. Figure 18 shows that increase in gellant concentration decreases the length of the rim thereby producing a thicker final rim than in Newtonian inviscid or viscous counterparts. Here $L_r$ is largest for S1, followed by S2 and S3 among the gels. Interestingly, despite higher viscosity and prolonged rim lifetime $T_{rim}$ (shown in the inset of figure 19 a), the rims of the gels do not elongate as much as the viscous Newtonian liquid expands. This situation is very similar to the deformation stage behaviour, where higher viscosity delays the process but does not enhance the bag expansion. The variation of transient rim expansion is presented as an inset in the figure 19(a), the approximate values of $L_r/d_0$ are 4.3, 3.3 and 3.0 for S1, S2 and S3, respectively. Consequently, term (I) increases with gellant concentration.

Figure 18.

The rim structure after its partial breakup for gel droplets in (a) S1 at $\textit{We} =15.31$ and $\textit{Oh} = 0.013$ , (b) S2 at $\textit{We} =15.31$ and $\textit{Oh} = 0.1$ and (c) S3 at $\textit{We} =36.87$ and $\textit{Oh} = 0.32$ . The dotted circles marked in yellow and blue denote droplets from the nodes ( $d_n$ ) and remaining rim ( $d_r$ ), respectively. The rim length ( $L_r$ ) is highlighted as double headed arrow for each liquids. The scalebar in (c) denotes the 5 mm length.

Figure 19.

Variation of average fragment sizes with $\textit{We}$ in bag breakup regime (a) node, (b) rim. Each marker is coloured with a relative magnitude of $\textit{Oh}$ shown in (b) as colour bar. The inset in (a) shows the temporal radial elongation of the rim from the time bag completely receded to the onset of rim disintegration. The solid lines are the theoretical prediction of the node sizes from (3.15) with the viscous contribution in $\lambda _{\textit{RT}}$ , while the inviscid predictions are shown as dashed lines. The dotted lines in (b) with open markers are rim size predictions from (3.17) for gel samples at $\textit{We}$ highlighted in inset of (a). On the other hand, solid lines show the average of the rim fragments across the $\textit{We}$ for present experiments and are in close agreement with the predicted mean of the rim child droplets using (3.17). The uncertainty shown by the error bars corresponds to the standard deviation of the eventwise means, calculated as described in Appendix C.

Figure 19. Long description

The image contains two panels, each showing a line graph with data points. Panel A shows the average node fragment size as a function of the Weber number (We). The y-axis is labeled ‘Avg. node fragment: <d_n/d_0>’ and ranges from 0 to 0.8. The x-axis is labeled ‘We’ and ranges from 10 to 50. The graph includes three data series labeled S1, S2, and S3, represented by different colored lines and markers. The inset in Panel A shows the temporal radial elongation of the rim from the time the bag completely receded to the onset of rim disintegration. The y-axis is labeled ‘L_rim/d_0’ and ranges from 0 to 6, while the x-axis is labeled ‘t_rim’ and ranges from 0 to 0.6. Panel B shows the average rim fragment size as a function of the Weber number (We). The y-axis is labeled ‘Avg. rim fragment: <d_r/d_0>’ and ranges from 0.10 to 0.50. The x-axis is labeled ‘We’ and ranges from 10 to 50. The graph includes three data series labeled S1, S2, and S3, represented by different colored lines and markers. The color bar on the right side of Panel B indicates the relative magnitude of Oh times 10^3, ranging from 0 to 300. The solid lines in both panels represent theoretical predictions, while the dashed lines represent inviscid predictions. The dotted lines in Panel B with open markers are rim size predictions for gel samples. The error bars correspond to the standard deviation of the eventwise means.

Term-(II), which is similar to (3.9), indicates that the dominant disturbance wavelength ( $\lambda _{rr}$ ) imposed by the receding bag film increases with gellant concentration. Thus, the wavelength ( $\lambda _{rr}$ ) increases from sample S1 to S3. Together, increased rim thickness and enhanced imposed wavelength yield larger rim fragments, as captured by (3.17). Figure 19(b) shows the variation of mean droplet size from the rim with $\textit{We}$ . The markers are the experimental results and highlight that the present observations are inline with the arguments we made above. Analytical predictions from (3.17) (shown as dotted line with open marker at the representative $\textit{We}$ values indicated in inset of figure 19 a) agree well with the measurements. Although predictions are shown at discrete $\textit{We}$ , figure 16(b) demonstrates that $\lambda _{rr}$ is nearly insensitive to $\textit{We}$ for gel samples but increases from sample S1 to S3. However, as $\textit{We}$ increases, rim stretching is expected to increase producing a thinner rim at breakup and correspondingly smaller rim fragments. Hence, the predictions from (3.17) should also follow the trend as expected with $\textit{We}$ as shown in figure 19(b).

It is worth mentioning that the range of $\textit{We}$ covered in the bag breakup regime always has a very narrow span, and the present experiments show negligible practical dependency on $\textit{We}$ and the observed variations lie within experimental uncertainty. This is to say that results are best interpreted as regime-characteristic averages rather than strongly $\textit{We}$ -dependent values. Although the rim-breakup framework proposed by Jackiw & Ashgriz (Reference Jackiw and Ashgriz2022) was developed for Newtonian DI, the present findings demonstrate that the underlying physical arguments remain applicable to shear-thinning hydrogels and provides a good agreement with measurements. In summary, it may be concluded that the heterogeneity associated with the gels alters the time and length scales (particularly the smallest length scales that decide the child droplets) without changing the underlying mechanism of producing variety in the child droplet sizes.

While the present study extends the understanding of secondary droplet breakup of Carbopol-based shear-thinning hydrogels, several important questions remain open and require further investigation. Previous studies have related such heterogeneity to a characteristic correlation length scale for suspensions of known particle sizes (Thiévenaz & Sauret Reference Thiévenaz and Sauret2022; Xu et al. Reference Xu, Wang and Che2023). This length scale acts as a limiting thickness for the bag film and consequently determines the minimum attainable droplet size during fragmentation. However, as (3.7) suggests that a direct application of these approaches and estimation of this length scale in gels remain challenging for two primary reasons. First, the characteristic microgel particle size ( $d_p$ ) and its dependency on gellant concentration ( $c\, \%$ ) and volumetric swelling are not yet sufficiently characterized. Second, hydrous Carbopol microgel particles are inherently soft and deformable, whereas existing geometric models for estimating the critical packing fraction ( $\phi _c$ ) are largely developed for rigid or weakly deformable particles and therefore may not accurately represent the current system. Nevertheless, the heterogeneity-based framework proposed by Xu et al. (Reference Xu, Wang and Che2023) provides a useful basis for interpreting the present observations. Due to current experimental limitations, the characteristic microgel particle size and local film thickness could not be measured directly. Therefore, the proposed interpretation remains phenomenological and requires further validation through direct characterization of the evolving gel microstructure, its breakdown dynamics and the resulting microgel particle distribution on the bag film interface. Such measurements would enable more accurate estimation of the correlation length scale and provide a stronger physical basis for predictive modelling of fragmentation in heterogeneous complex fluids.

Despite these limitations, the present analysis suggests that the fundamental mechanisms governing child-droplet formation remain largely invariant across different liquids, even in the presence of rheological complexity. This observation motivates future investigations on a broader class of complex fluids, including viscoelastic liquids, nanofluids and metalize gels, to examine the universality of the underlying breakup mechanisms. The present work therefore establishes a framework for understanding the secondary breakup dynamics of heterogeneous and structurally evolving complex liquids.