Droplet breakup and size distribution in an airstream - effect of inertia

Figure 3 depicts the droplet dynamics for different values of the normalised released height $(h/D)$ and subjected to an airstream in a cross-flow configuration. Based on the stereo-particle image velocimetry (PIV) measurements of the airstream, as detailed Kirar et al. [27], in figure 4, we illustrate a schematic representation of the relative positions of droplets in various flow zones during the fragmentation process for different dispensing heights, $h/D$. The airstream can be characterised by the potential core, shear and outer regions as depicted in figure 4. Figure 3(a) shows the temporal evolution of drop when it is released into the aerodynamic field from a height of $h/D=0.6$ (slightly above the air nozzle). Due to its minimal potential energy and reduced inertia, the droplet cannot penetrate the airstream and remains positioned at the outer periphery of the airflow. In this outer region of the airstream, the aerodynamic force lacks the necessary intensity to trigger fragmentation. Consequently, the droplet exhibits a vibrational mode. Figure 3(b) depicts the droplet dynamics for $h/D=0.8$. In this scenario, the droplet possesses sufficient inertia to traverse the outer region of the aerodynamic flow field, and it enters the shear zone. In the shear zone, the strong aerodynamic forces lead to the deformation of the droplet from a sphere to a tilted disk, resulting from the pronounced velocity gradients present within the shear zone. Subsequently, the aerodynamic influence induces the formation of a liquid sheet from the tilted disk. This sheet retracts in a direction opposing the prevailing airflow, driven by a negative pressure gradient, and the droplet undergoes fragmentation due to capillary instability [19]. A similar breakup pattern was observed by Kirar et al. [27] in a swirling airflow, which they referred to as “retracting bag breakup”. Increasing the droplet released height further ($h/D=1.9$), it can be seen in figure 3(c) that the droplet’s inertia becomes sufficient to overcome the resistance posed by both the outer region and the shear zone of the aerodynamic field. As a result, approximately half of the droplet takes position within the shear zone, while the remaining half occupies the potential core region of the aerodynamic field. In both these regions, the aerodynamic force is strong enough to counterbalance the surface tension force of the droplet. This interaction causes the droplet to change its shape from a sphere to a disk. Due to the Rayleigh-Taylor instability, air infiltrates the centre part of the formed disk, creating a central bag-like structure and a toroidal rim encircling its periphery. Subsequently, the tip of the bag ruptures due to the exerted aerodynamic forces, while the rim disintegrates due to capillary instability.

It can be seen in figure 3(d) and (e) that the droplet dispensed from $h/D=5.6$ and 8.4 gets sufficient inertia and downward velocity to traverse through the potential core region of the airstream while it undergoes a rapid change in topology from a spherical shape to a disk shape. For $h/D=5.6$ (figure 3d), the outer edge of the disk-shaped drop is elongated into a thin sheet due to drag forces. The central region of the disk is subject to Rayleigh-Taylor instability, which occurs because the lighter air is accelerated against the heavier liquid, leading to the formation of a bag-like structure. The Rayleigh-Taylor instability is predominant at the interface where the liquid disk meets the air, causing the central region to expand outward and form a bag. The stamen that appears at the center of the bag is due to the localized thinning and elongation of the liquid as it is pulled by both aerodynamic forces and surface tension effects, accompanied by a toroidal rim surrounding its periphery $(\tau=2.01)$. The bag eventually ruptures as the dominant aerodynamic forces exceed the surface tension. In contrast, the toroidal rim and stamen fragment due to capillary instability $(\tau=3.02)$ at the lower part of the air nozzle. This entire process is termed the bag-stamen breakup phenomenon. For $h/D=8.4$ (figure 3e), the higher downward velocity enables the deformed disk to cross the potential core region, with part of the disk entering the lower shear zone of the airstream. Here, the outer edge of the disk experiences elongation into a thin sheet due to drag forces, while the central portion undergoes Rayleigh-Taylor instability, extending outward to form a bag. This results in a distinct bag-stamen pattern within the droplet. The bag-stamen structure inclines due to the large velocity gradient within the lower shear layer. During the subsequent period, the bag retracts, due to a negative pressure gradient, and the droplet fragments due to capillary instability. This sequence of events is termed the retracting bag-stamen breakup phenomenon. For the highest value of $h/D$ value considered in this study ($h/D=11.2$), the droplet possesses significant potential energy and greater inertia to migrate into the potential core with a higher downward velocity. The droplet undergoes morphological change from sphere to disk quickly while the inertia effect is sufficiently strong to cross the potential core and lower the shear layer. As a result, the droplet enters the outer region of the lower portion of the flow field and undergoes vibrational breakup. The characteristic time scales for the Rayleigh-Taylor and capillary instabilities align with the observed time scales of the breakup process. The Rayleigh-Taylor instability develops rapidly as the droplet enters the high-speed airstream, forming the bag within milliseconds. The subsequent breakup of the bag, toroidal rim, and stamen due to capillary instability also occurs on a similarly short-time scale, driven by the competing forces of surface tension and aerodynamic drag.

A falling drop exhibits natural oscillations in its shape, transitioning between oblate, spherical and prolate shapes as it descends from a height due to the competition between inertia and surface tension forces. It detects the shape of the droplet as it enters the potential core region, which also influences the drop breakup phenomenon. Thus, in order to gain better insights, we investigate the shape oscillation of the drop during its free fall after release from the needle. The frequency $(f)$ of the natural oscillations of a freely suspended water drop derived using a linear theory in the inviscid limit is given by [37, 38]:

where $n$ denotes the oscillation mode, with the fundamental mode corresponding to $n=2$. The time period $(T_{p})$ of the natural oscillations of a drop is given by $1/f$. In figure 5, we investigate the temporal variation of the drop shape and the aspect ratio $(a/b)$ during its free fall for $h/D=11.2$. Here, $a$ and $b$ are the droplet diameters in $x$ and $y$ directions, respectively, such that $a/b>1$, $a/b=1$ and $a/b<1$ represent oblate, spherical and prolate shapes, respectively. It can be seen that the falling drop exhibits periodic shape oscillations about the spherical shape. These symmetric and periodic oscillations have also been reported in earlier investigations in the context of a falling nonspherical droplet in the air [39, 40, 41]. Inspection of figure 5 reveals that the dimensionless time period of oscillations is $\approx 2.97$, which closely matches with the theoretical prediction of the dimensionless time period obtained using Eq. (1) for the fundamental mode ($(1/f)U\sqrt{\rho_{a}/\rho_{w}}/d_{0}=2.73$). Note that the time period of the shape oscillations of the primary droplet remains unchanged for different release heights, as it solely depends on the liquid properties and the initial size of the droplet (Eq. 1). Our experiments also observe the same time period regardless of the height from which the droplet is released (not shown). However, as expected, the number of shape oscillation cycles decreases with the decrease in the release height. Figure 6 shows the shape of the droplet at the instant when it enters the aerodynamic field (represented by $\tau=0$) for different values of $h/D$. As expected, it can be observed that there is a substantial influence of the descent height, and consequently its inertia of the drop, on both its shape and aspect ratio before encountering the airflow. This, in turn, influences the ensuing droplet breakup phenomenon.

Figure 7 illustrates the morphology of the droplet for various breakup modes corresponding to different values of $h/D$. The droplet exhibits a vibrational mode for $h/D=0.6$ to $0.7$. As discussed above, this phenomenon can be attributed to the position of the drop within the outer region of the aerodynamic field. In the range of $h/D=0.7$ to $1.1$, the drop experiences a retracting bag breakup due to its presence within the shear region of the aerodynamic field. For $h/D=1.1$ to $4.1$, the drop undergoes a bag breakup mode as a result of the combined influences of the shear region and the potential core within the aerodynamic field. As we increase the value of $h/D$ further, between $h/D=4.1$ to $7.8$, the droplet enters a distinct bag-stamen breakup phenomenon driven by its position within the potential core of the aerodynamic field. Subsequently, for $h/D=7.8$ to $10.6$, the droplet transitions to a retracting bag-stamen breakup due to its presence within the potential core and the lower shear layer of the aerodynamic field. For $h/D=10.6$ to $22.1$, the droplet again displays vibrational breakup mode attributed to its exposure to the lower outer region of the aerodynamic field. A close inspection of figure 7 also reveals that the normalised bag inflation in the streamwise direction, defined as $\beta=l/d_{0}$, increases and then decreases as we increase the value of $h/D$.

Figure 8 depicts the droplet trajectories in the aerodynamic field before the onset of fragmentation for different values of $h/D$. For $h/D=0.6$, the droplet possesses minimal inertia, restricting its vertical descent to the outermost region of the aerodynamic field. In this outer region, the droplet travels horizontally along the predominant airstream direction and undergoes a vibrational mode. For $h/D=0.8$, the droplet has sufficient inertia to cross the outer region and migrates vertically downward up to the shear zone of the aerodynamic field. In this case, the droplet travels in both horizontal and vertical upward directions in the shear region due to large velocity gradients. As a result, the droplet exhibits the retracting bag breakup mode. For $h/D=1.9$, the droplet has high inertia to migrate vertically downward and get exposed to both the shear and potential core zone of the aerodynamic field. In this case, the droplet travels horizontally along the airstream and exhibits bag breakup mode. For $h/D=5.6$, the droplet has higher inertia to migrate vertically downward till the potential core region of the aerodynamic field. In this region, the droplet travels horizontally due to the drag of the airstream and vertically downward due to the inertia effect while undergoing a bag-stamen fragmentation. For $h/D=8.4$, the droplet possesses more substantial inertia to travel vertically downward and get exposed to the potential core and lower shear region of the aerodynamic field. In this case, the droplet exhibits retracting bag-stamen breakup and generates a trajectory similar to the bag-stamen mode. For $h/D=11.2$, the droplet travels vertically downward with a high velocity and first enters the potential core. Later, due to the high inertia effect, the droplet further crosses the potential and lower shear zone and enters the lower outer region of the flow field. In this case, the droplet trajectory follows less migration in the predominant airstream direction and more travel vertically downward due to a more substantial inertia effect. It is important to note that for the vibrational, retracting bag, and bag breakup modes, the trajectory of the droplet ends above the central axis of the nozzle within the aerodynamic field. In contrast, for the bag stamen and retracting bag-stamen modes, the trajectory of the droplet concludes below the central axis of the nozzle.

As discussed in §III.1, the deformation of the droplet undergoes significant variations during its interaction with the airstream when released from different heights, primarily due to changes in inertia, even though the Weber number for the airstream is maintained fixed (${\it We}=12.1$). Due to the intricate interplay between the inertia of the droplet and the aerodynamic flow field, the Weber number $({\it We})$ calculated based on the air velocity at the nozzle exit fails to adequately describe the various droplet breakup modes associated with different release heights. Our experimental results demonstrate that the release height $(h/D)$ has a profound impact on the radial deformation of the droplet, which, in turn, influences both the breakup morphology and the resulting sizes of the child droplets. Consequently, it becomes imperative to employ an effective Weber number $({\it We}_{{\it eff}})$ that considers the effect of the radial deformation of the droplet in the analytical model (discussed in the following section) to estimate the size distribution of the resulting child droplets [22, 27]. Figure 9 depicts the temporal evolution of the radial deformation of the droplet, denoted as $\alpha=d_{c}/d_{0}$, for different values of $h/D$. Here, $d_{c}$ represents the diameter of the disk-shaped droplet, which is illustrated in the inset of figure 9. For $h/D=0.8$, the droplet experiences minimal radial deformation, primarily due to its residence in the shear zone of the airstream, where it encounters weak aerodynamic forces. For $h/D=1.9$, the droplet displays more substantial radial deformation as it interacts with both the shear and potential core regions of the airstream, subjecting it to stronger aerodynamic forces. For $h/D=5.6$, the droplet undergoes the most extensive radial deformation as it migrates into the potential core of the airflow, exposing it to the strongest aerodynamic forces.

Considering the radial deformation of the drop for various $h/D$ values, we derive an expression for the effective Weber number $({\it We}_{{\it eff}})$ based on the analyses presented in Refs. [11, 22, 27]. We start our analysis by examining the aerodynamic field around the drops exposed to the airstream [11]. Let $U=(u_{r},u_{y})$ represent the axisymmetric velocity field near the drop. The continuity equation for the airflow around the drop, assuming steady, inviscid, and incompressible flow, is given by:

Due to the aerodynamic pressure imbalance between the aft and rear sides of the drop, the drop deforms into a flat disk aligned with the airstream direction. This flow can be modeled as a stagnation point flow, where $u_{y}=-\gamma y$, with $\gamma$ being the stretching rate, $y$ the distance from the drop surface on the front side, and $r$ a direction pointing radially outward. Thus, $u_{r}=r\gamma/2$. The pressure field $P_{a}(r,y)$ around the droplet can be obtained from the steady, inviscid, and incompressible momentum equations in the $r$ and $y$ directions, which are given by

By substituting the expressions of $u_{y}$ and $u_{r}$ into the above equations and integrating the resultant equations, we obtain

Therefore, the pressure of air at the drop surface is

Now, the pressure inside the disk-shaped droplet ($P_{l}$) can be evaluated as:

Here, $R$ represents the radius of the disk-shaped droplet, $\sigma$ denotes for the surface tension at the liquid-gas interface, and $h$ denotes the thickness of the disk-shaped droplet. We employ the continuity equation for inviscid and incompressible fluid in order to evaluate the velocity field $(u_{l})$ inside the droplet, which is given by

Now, substituting $h(t)={d_{o}^{3}/6R^{2}}$ (obtained using the global mass conservation) into the above equation, and integrating the resultant equation yields

The momentum equation for the droplet is given by

Non-dimensionalizing and integrating Eq. (9) from $r=0$ to $r=R$ and using Eq. (6), we obtain

where $\alpha(t)=2R/d_{0}=d_{c}/d_{0}$, and $f=\gamma d_{0}/U$ denotes the stretching factor. Now, solving Eq. (10) using the initial conditions, given by $\alpha(0)=1$ and $\mathrm{d\alpha}/\mathrm{d\tau}=0$, we get [27]

In Eq. (11), the radial expansion of a droplet $\alpha=d_{c}/d_{0}$ can be evaluated for a given value of ${\it We}$ at different dimensionless times $\tau$. However, this equation does not consider the falling inertia of the droplet, which can significantly impact the breakup dynamics and the resultant size distribution. Therefore, it is essential to incorporate the effective Weber number (${\it We}_{{\it eff}}$) in Eq. (11). Thus, the modified equation considering the falling inertia of the droplet becomes

Here, $f$ represents the stretching factor, which is approximately $2\sqrt{2}$ for the cross-flow configuration [22]. The variation of $\alpha$ with $\tau$ is evaluated experimentally using high-speed images. Therefore, in the above equation, the only unknown is $We_{{\it eff}}$, which is determined through curve fitting of the experimental data for different values of $h/D$. This approach allows us to incorporate the combined effects of size and shape deformation under various experimental conditions. We found that substituting ${\it We}_{{\it eff}}=12$ in Eq. (12) yields the best match with the experimental values of $\alpha$ for the retracting-bag breakup mode. Similarly, ${\it We}_{{\it eff}}=17.5$ aligns well with the experimental values for the bag breakup mode, and ${\it We}_{{\it eff}}=31$ provides the best agreement for the bag-stamen breakup mode. These values of ${\it We}_{{\it eff}}$ effectively incorporate the combined effects of size and shape deformation under different experimental conditions, allowing the theoretical model to better capture the dynamics observed in the experiments.

To gain deeper insights into the analysis of droplet breakup dynamics, we present the results of effective velocity ($V_{{\it eff}}$) and droplet residence time ($t_{R}$) for different values of the normalized height ($h/D$). These results help elucidate the connection between effective velocity and the resident time of the droplet in the airstream. Figure 10(a) illustrates the variation of effective velocity ($V_{{\it eff}}$) with $h/D$. As evident in figure 10(a), $V_{{\it eff}}$ first increases and then decreases. The $V_{{\it eff}}$ represents the relative velocity between the airflow and the droplet at the point of maximum radial deformation. The increase in $V_{{\it eff}}$ up to $h/D=5.6$ is due to higher droplet inertia, causing the drop to experience progressively higher velocity from the outer region to the potential core. Beyond $h/D=5.6$, droplet inertia increases further, leading to a trend reversal, and $V_{{\it eff}}$ begins to decrease as the drop crosses the potential core, interacting with the lower velocity aerodynamic fields. It is to be noted that $We_{{\it eff}}$ is not directly evaluated from $V_{{\it eff}}$ and therefore does not exhibit similar trends. This complex interaction of the droplet with the aerodynamic field leads to different breakup modes. Figure 10(b) illustrates the variation of droplet residence time ($t_{R}$) in milliseconds with $h/D$. The $t_{R}$ represents the time the droplet spends in the aerodynamic field. As shown in figure 10(b), $t_{R}$ decreases with an increase in $h/D$. This decreasing trend is due to the increase in the droplet’s falling inertia (and hence the falling velocity) with height. A higher $h/D$ leads to a shorter residence time because the droplet falls faster through the aerodynamic field. Figure 11 illustrates the variation of ${\it We}_{{\it eff}}$ with $h/D$. The best-fit curve, represented as $We_{{\it eff}}=11+1.8({h/D})+0.4({h/D})^{2}$, qualitatively captures the observed variations for the range of parameters considered in the present study. We utilize the values of the effective Weber number $({\it We}_{{\it eff}})$ to estimate the droplet size distributions that arise from different breakup modes corresponding to various release heights.

In this section, we present the size distribution of child droplets resulting from the retracting bag, bag, and bag-stamen breakup phenomena. We also compare the experimental results with the analytical model [21] by employing the ${\it We}_{{\it eff}}$ instead of ${\it We}$. The volume probability density ($P_{v}$) is defined as the ratio of the volume occupied by all child droplets of a specific diameter to the total volume occupied by droplets of all diameters. To provide a comprehensive analysis, we incorporate both gamma and log-normal distributions. This dual approach allows for a more robust and detailed comparison with experimental data, thereby providing a deeper understanding of the underlying physical processes.

The volume probability density using the gamma distribution function is given by

where $P_{n}={\zeta^{\alpha-1}e^{-\zeta/\beta}/\beta^{\alpha}\Gamma(\alpha)}$; $\zeta\left(=d/d_{0}\right)$; $\Gamma(\alpha)$ represents the Gamma function; $\alpha=(\bar{\zeta}/\sigma_{s})^{2}$ and $\beta=\sigma_{s}^{2}/\bar{\zeta}$ are the shape and rate parameters, respectively; $\bar{\zeta}$ and $\sigma_{s}$ are the mean and standard deviation of the distribution, which are estimated based on the characteristic breakup sizes corresponding to each mode.

The volume probability density ($P_{v}$) for log-normal distribution is given by

where, $\zeta\left(=d/d_{0}\right)$ and $P_{n}$ is the number probability density function for log-normal distribution and it can be determined using following equation as

Here, $\mu$ and $\sigma_{l}$ are the logarithmic mean and logarithmic standard deviation of the distribution, which are estimated based on the characteristic breakup sizes corresponding to each mode.

In the retracting bag, bag and bag-stamen breakup phenomena, the overall size distribution of child droplets is influenced by three distinct modes, namely the node, rim, and bag fragmentation. Hence, it is crucial to account for their respective contributions to the size distribution by performing a weighted summation of each mode. Thus, the total volume probability density ($P_{v}$) can be determined as

where $w_{N}=V_{N}/V_{0}$, $w_{R}=V_{R}/V_{0}$ and $w_{B}=V_{B}/V_{0}$ represent the contributions of volume weights from the node, rim and bag, respectively. Here, $V_{N}$, $V_{R}$, $V_{B}$ and $V_{0}$ are the node, rim, bag and the initial droplet volumes, respectively. The volume probability density for the node, rim and bag breakup modes are denoted as $P_{v,N}$, $P_{v,R}$ and $P_{v,B}$, respectively. The expressions for the characteristic breakup sizes and the weight contributions for each mode can be found in Appendix A. In the following, we present the size distribution of child droplets obtained from three repeated measurements for each set of parameters.

Figure 12(a-c) illustrates the comparison between theoretically predicted characteristic breakup sizes and experimental data for retracting bag breakup ($h/D=0.8$ and ${\it We}_{{\it eff}}=12$). In figure 12(a), it can be seen that for the bag fragmentation mode, the rupture of the bag results in four specific breakup sizes: $d_{RP,B}$ due to Rayleigh-Plateau instability, $d_{sat,B}$ due to non-linear instability of liquid ligaments, $d_{B}$ due to minimum bag thickness, and $d_{rr,B}$ due to receding rim instability (see, appendix §A.3). These sizes represent the smallest characteristic dimensions governing the analytical size distribution in the bag rupture mode. Figure 12(b) shows the characteristic breakup sizes due to rim fragmentation mode. The rim breakup process results in four characteristic sizes: $d_{R}$ due to capillary instability, $d_{rr}$ due to receding rim instability, and $d_{sat,R}$ and $d_{sat,rr}$ due to non-linear instability of liquid ligaments near the pinch-off point (see, appendix §A.2). These characteristic sizes are larger than those produced during the bag ruptures and ultimately govern the analytical size distribution in the rim breakup process. Figure 12(c) displays the three characteristic sizes, $d_{N}$, corresponding to $n=0.2$, $n=0.4$, and $n=1$, obtained after the node breakup process. The value of $n$ represents the volume fraction of the node relative to the rim (see, appendix §A.1). These are the largest characteristic sizes in the entire fragmentation process and govern the size distribution in the node breakup. Thus, figure 12(a-c) confirms that the theoretically predicted characteristic sizes align well with the experimental results. Specifically, the peak of the experimental distribution corresponds closely to the mean value of these characteristic sizes.

In figure 13(a), we compare the analytical prediction of the overall size distribution obtained using the gamma distribution function with our experimental results for $h/D=0.8$ at $\tau=4.78$ (an instant after completion of the breakup phenomenon). This case is associated with the retracting bag breakup corresponding to ${\it We}_{{\it eff}}=12$. It can be seen that there are three distinct modes with one marginal and two significant contributions to the size distribution. The solid line indicates the sum of all the modes (bag+rim+nodes) of the size distribution, while the dotted, dashed, and dash-dotted lines represent the contributions from the individual modes. In this breakup scenario, as the droplet stays in the shear layer, the bag fragmentation has a marginal contribution to the overall size distribution. Therefore, the rim and node fragmentation mainly contribute to the overall size distribution. Due to the Rayleigh-Plateau capillary instability, the breakup of the rim generates child droplets with a distribution peak at $d/d_{0}\approx 0.22$, while the node breakup due to Rayleigh-Taylor instability contributes to the droplet size distribution with a peak at $d/d_{0}=0.34$. The theoretical model employing the effective Weber number provides reasonable predictions for rim and node breakup processes. However, it over-predicts the size distribution for the bag breakup process. This is due to the fact that the current model does not account for the retraction of the bag and its subsequent collapse into the rim. Figure 13(b) illustrates the corresponding comparison with the analytical prediction obtained using the log-normal distribution. It is evident that the log-normal distribution accurately predicts the volume contribution of the retracting bag, showing a distribution peak around $d/d_{0}\approx 0.08$. This accuracy is attributed to the long exponential tail characteristic of the log-normal distribution. Furthermore, the log-normal distribution also effectively predicts the volume contributions resulting from rim and node fragmentation.

Figure 14(a-c) depicts the comparison of theoretically predicted characteristic breakup sizes with experimental data for bag breakup ($h/D=1.9$ and ${\it We}_{{\it eff}}=17.5$). Figure 14(a) represents the characteristic sizes of the bag fragmentation mode. In this case, as the droplet interacts with the shear and potential core region, it exhibits the classical bag breakup process. Figure 14(a) illustrates the four characteristic sizes resulting from Rayleigh-Plateau instability, non-linear instability of liquid ligaments, minimum bag thickness, and receding rim instability in the bag rupture process. These are the smallest characteristic sizes that govern the size distribution for bag fragments. Figure 14(b) depicts the four characteristic sizes resulting from capillary instability, receding rim instability, and non-linear instability of liquid ligaments near the pinch-off point in the rim breakup process. These moderate characteristic sizes determine the size distribution for rim fragments. Figure 14(c) shows the three characteristic sizes resulting from Rayleigh-Taylor instability in the node fragmentation process. These are the largest characteristic sizes that govern the size distribution in the node breakup mode.

The size distribution of the child droplets at $\tau=3.37$ for $h/D=1.9$ associated with the bag breakup phenomenon for ${\it We}_{{\it eff}}=17.5$ is shown in figure 15(a) and 15(b). Figure 15(a) and 15(b) are associated with the gamma and log-normal distributions, respectively. The theoretical predictions of the size distribution for individual modes and the overall size distribution are also depicted. It can be seen that the bag breakup exhibits three distinct modes in the size distribution of the child droplets. In this case, as the droplet interacts with both the shear and potential core region, the droplet undergoes the normal bag breakup process, which involves the breakup of the inflated bag, rim, and nodes. As a result, all these processes contribute to the overall size distribution. In figure 15(a), the first peak at $d/d_{0}\approx 0.07$ is attributed to the bag rupture. The second peak observed at $d/d_{0}\approx 0.19$ is due to the fragmentation of the rim, and the third peak at $d/d_{0}\approx 0.42$ is due to the node breakup. Inspection of figure 15(a) also reveals that the analytically estimated contributions for bag and rim and node breakups provide reasonable predictions for the experimental size distribution associated with these modes. In addition, the theoretically predicted characteristic sizes are in reasonable agreement with the experimental results. The percentage deviation between the experimental and analytical distributions for bag, rim and node breakup processes are $20\%$, $8.7\%$, and $1.4\%$, respectively. We found that the volume fraction of bag, rim, and nodes are $15\%$, $45\%$, and $40\%$, respectively. Inspection of Figure 15(b) reveals that the log-normal distribution significantly underpredicts the volume contribution of the bag. On the other hand, the analytical distribution reasonably predicts the volume contributions for rim and node fragmentation, with slight discrepancies of $8.5\%$ and $16\%$, respectively. A careful inspection of figure 15(a) and 15(b) also reveals that the gamma distribution accurately predicts the volume contribution for smaller child droplets due to its short exponential tail, whereas the log-normal distribution predicts the volume contribution for larger child droplets reasonably well due to its long exponential tail.

Figure 16(a-c) compares theoretically predicted characteristic breakup sizes with experimental data for bag-stamen breakup ($h/D=5.6$ and ${\it We}_{{\it eff}}=31$). In this case, the droplet interacts with the potential core of the flow field. Figure 16(a) represents the four characteristic sizes resulting from Rayleigh-Plateau instability, non-linear instability of liquid ligaments, minimum bag thickness, and receding rim instability during the bag rupture process. These sizes are the smallest and dictate the size distribution for bag fragments. Figure 16(b) shows the four characteristic sizes resulting from capillary instability, receding rim instability, and non-linear instability of liquid ligaments near the pinch-off point in the rim breakup process. These sizes are moderate and determine the size distribution for rim fragments. Figure 16(c) presents the three characteristic sizes resulting from Rayleigh-Taylor instability in the node fragmentation process. These are the largest sizes of child droplets and mainly contribute to the size distribution of the node breakup mode.

Figure 17(a) shows the analytical prediction of the overall distribution obtained using the gamma distribution function with the experimental results for $h/D=5.6$ associated with the bag-stamen breakup at ${\it We}_{{\it eff}}=31$. Similar to the bag breakup ($h/D=1.9$), the bag-stamen breakup ($h/D=5.6$) undergoes three different modes in the size distribution. For $h/D=5.6$, the droplet completely migrates into the potential core region of the airstream and undergoes the bag-stamen breakup process, which involves the breakup of the inflated bag, rim stamen, and nodes. As a result, all these processes contribute to the overall size distribution. In figure 17(a), the first peak at $d/d_{0}\approx 0.058$ corresponds to the fragmentation of the bag. The second peak observed at $d/d_{0}\approx 0.18$ is due to the fragmentation of the rim, and the third peak at $d/d_{0}\approx 0.47$ is due to the node breakup. The comparison with the analytical prediction reveals that the contributions of the bag and node to the volume-weight size distributions are over-estimated and under-estimated, respectively. On the other hand, the contribution of rim fragmentation is slightly under-predicted. This discrepancy arises because the model does not account for the fragmentation of the stamen, which is similar in size to the rim. Furthermore, since the volume of the bag contributes to the formation of the stamen, the model also falls short in predicting the extent of child droplets produced by the bag breakup. However, the theoretically predicted characteristic sizes reasonably agree with the experimental results. We found that the volume fraction of bag, rim, and nodes are $20\%$, $40\%$ and $40\%$, respectively, for $h/D=5.6$. In figure 17(b), we observe that log-normal distribution underestimates the volume contribution of the bag by $42.8\%$, but the analytical distribution accurately predicts the volume contributions for rim and node fragmentation, with minor discrepancies of $4.5\%$ and $18\%$, respectively. It can be seen in figure 17(a) and (b) that the gamma distribution accurately predicts the volume contribution for smaller child droplets due to its short exponential tail, while the log-normal distribution reasonably predicts the volume contribution for larger child droplets due to its long exponential tail.

In summary, while the model initially developed by Jackiw and Ashgriz [21] for bag breakup in a cross-flow configuration involving droplets released with negligible inertia into the airstream, the present study enhances the analytical model by incorporating the effective Weber number. This addition allows the model to account for the inertia effects of droplets released from varying heights, enabling it to predict size distributions for a range of breakup phenomena, including retracting bag, bag and bag-stamen breakups. The size distribution of the retracting bag is primarily influenced by rim and node fragmentation, resulting in a bimodal distribution. In contrast, the bag and bag-stamen breakups yield a tri-modal size distribution due to the collective contributions of the three breakup modes: bag, rim, and node.