3.1.1. Bag and bag-stamen breakup

The bag breakup mode is one of the main modes of droplet breakup. Figure 3(a) shows the bag breakup process of a particle-laden droplet. For comparison, the bag breakup process of a silicone oil droplet is shown in figure 3(b), which has the same effective viscosity as the particle-laden droplet. The breakup process can be divided into an initial deformation stage (0–13 ms in figure 3), and a bag development and breakup stage (after 13 ms in figure 3). During the initial deformation stage, the particle-laden droplet deforms almost the same as the silicone oil droplet. This indicates that during the initial deformation stage, the particle-laden droplet can be regarded as a homogeneous fluid; and compared with the interstitial liquid, the particles lead to a higher effective viscosity of the droplet. However, during the bag development and breakup stage, the heterogeneous effect of particles appears. The particles agglomerate on the liquid film and accelerate the breakup of the liquid film (23.2–30 ms in figure 3a). In contrast, the liquid film of the silicone oil droplet is significantly stretched and deformed, and the breakup starts later (35 ms in figure 3b). The development and breakup of the suspension bag film are similar to the fragmentation of particle-laden liquid sheets generated by a droplet impacting a cylindrical target described by Raux et al. (Reference Raux, Troger, Jop and Sauret2020). However, compared with the particle-laden liquid sheets described by Raux et al. (Reference Raux, Troger, Jop and Sauret2020), the particle distribution on the suspension bag film is more uneven due to the accelerated stretching of the bag film caused by the continuous aerodynamic force. Further quantitative comparisons for the stretch and breakup of the liquid film will be given in § 3.3.

Figure 3.

Image sequences of the bag breakup mode at $We_g = 20.3$, $Oh = 0.32$: (a) suspension with $d_p = 90$ $\mathrm {\mu }$m and $\phi = 0.25$; (b) silicone oil with $\eta = 100$ mPa s. The corresponding video can be found in the supplementary material and movie 1.

As $We_g$ increases, the bag-stamen breakup mode occurs. Different from the bag breakup, the droplet morphology in bag-stamen breakup mode consists of the peripheral bag and the middle stamen. The comparison of the breakup process of the particle-laden droplet and the silicone oil droplet is shown in figure 4. The initial deformation of the particle-laden droplet (0–10 ms in figure 4a) is similar to that of the silicone oil droplet under the same conditions (figure 4b). But after the initial deformation, compared with that of the silicone oil droplet, the peripheral bag of the particle-laden droplet breaks up quickly (20 ms in figure 4a), and the elongation of the peripheral ring is weaker (25 ms in figure 4a).

Figure 4.

Synchronised images of the bag-stamen breakup at $We_g = 28.7$, $Oh = 0.32$: (a) suspension with $d_p = 90$ $\mathrm {\mu }$m and $\phi = 0.25$, (b) silicone oil with $\eta = 100$ mPa s. The corresponding video can be found in the supplementary material and movie 2.

3.1.2. Multimode breakup

The multimode breakup mode is another main mode of droplet breakup. Compared with bag breakup, the multimode breakup occurs at a higher Weber number and has new features. Xu et al. (Reference Xu, Wang and Che2023) classified the multimode breakup into low-order and high-order modes according to the piercing times in the middle of the droplet. A typical process of the low-order multimode breakup is shown in figure 5, while the morphology of the high-order multimode breakup is shown in figure 6.

Figure 5.

Synchronised images of the low-order multimode breakup at $We_g = 71.7$, $Oh = 0.32$: (a) suspension with $d_p = 90$ $\mathrm {\mu }$m and $\phi = 0.25$; (b) silicone oil with $\eta = 100$ mPa s. The corresponding video can be found in the supplementary material and movie 3.

Figure 6.

Synchronised images of the high-order multimode breakup at $We_g = 103.7$, $Oh = 0.32$: (a) suspension with $d_p = 90$ $\mathrm {\mu }$m and $\phi = 0.25$, (b) silicone oil with $\eta = 100$ mPa s. The corresponding video can be found in the supplementary material and movie 4.

For comparison, figure 5(a) shows the breakup process of a particle-laden droplet and figure 5(b) shows the breakup process of a silicone oil droplet under the same conditions. The process of the low-order multimode breakup can be divided into the initial deformation stage (0–4 ms in figure 5), the peripheral retraction stage (4–6.5 ms in figure 5(a) and 4–8 ms in figure 5b) and the subsequent piercing and breakup stage. During the initial deformation stage, the deformation is similar for both fluids, and the particle-laden droplet can be regarded as a homogeneous fluid. During the peripheral retraction stage, the silicone oil droplet tends to form a peripheral sheet, and the peripheral sheet deflects to the leeward of the droplet before the sheet destabilisation develops, i.e. the sheet retraction (8 ms in figure 5b). In contrast, for the particle-laden droplet under the same conditions, the azimuthal destabilisation of the droplet periphery develops and forms fingerings at the periphery (6.5 ms in figure 5a). Hence, the formation of fingers indicates the emergence of particle heterogeneity as the peripheral sheet becomes thinner. In addition, for the piercing in the middle of the droplet, the particle-laden droplet tends to form local piercing with a faster speed and less lateral expansion (9.8 ms in figure 5a) than the silicone oil droplet (11 ms in figure 5b).

Different from the low-order multimode breakup where the droplet middle is pierced once, the droplet middle is pierced multiple times under the high-order multimode breakup. The comparison of the particle-laden droplet and the silicone oil droplet is shown in figure 6. The time interval between two piercings of the particle-laden droplet (7–9.6 ms in figure 6a) is shorter than that of the silicone oil droplet (7–11 ms in figure 6b). In addition to the major piercings, the particle-laden droplet forms some minor piercings (9.6 ms in figure 6a). Similar minor piercings also occur in the case of the low-order multimode breakup with large particles (9.2 ms in figure 7). These piercings in the middle have important effects on the breakup morphology and fragment size, which will be discussed further in § 3.4.

Figure 7.

Synchronised images of the low-order multimode breakup at $We_g = 69.9$, $Oh = 0.32$. The fluid used is a suspension with $d_p = 180$ $\mathrm {\mu }$m and $\phi = 0.25$. The corresponding video can be found in the supplementary material and movie 5.

3.1.3. Shear-stripping breakup

The shear-stripping breakup occurs at a much higher Weber number and is the terminal regime considered by some studies (Theofanous Reference Theofanous2011; Theofanous et al. Reference Theofanous, Mitkin, Ng, Chang, Deng and Sushchikh2012). Figure 8(a) shows the shear-stripping breakup process of a particle-laden droplet, and figure 8(b) shows the breakup process of a silicone oil droplet under the same conditions for comparison. Being different from the bag and multimode breakup modes, here the heterogeneous effect of particles appears at the beginning of the breakup process. In the early stage of the breakup of the particle-laden droplet (0–3 ms in figure 8a), the interstitial liquid is rapidly pushed downstream by the airflow, thus particles protrude from the windward side in the middle of the droplet forming a rough surface, while some particles are stripped directly from the periphery of the droplet. In contrast, for the silicone oil droplet, the windward side in the middle is relatively smoother due to the tension generated by the outward stripping (2 ms in figure 8b), while some ripples on the edge of the windward surface of the silicone oil droplet are formed. After the flattening of the droplet middle, massive piercings occur in the middle of the droplet. The particle-laden droplet forms some large particle clusters (4.5 ms in figure 8a), while the silicone oil droplet forms some liquid clumps (4.5 ms in figure 8b). Finally, the particle clusters or the liquid clumps may be further pierced by the airflow. The particles and the interstitial liquid in the particle-laden droplet are almost completely separated (6 ms in figure 8a), while the silicone oil droplet forms mists of tiny droplets (6 ms in figure 8b). Overall, compared with the continuous breakup of the silicone oil droplet, the breakup of the particle-laden droplet is more discrete, like the shattering of a solid.

Figure 8.

Synchronised images of the shear-stripping breakup at $We_g = 459.5$, $Oh = 0.32$: (a) suspension with $d_p = 180$ $\mathrm {\mu }$m and $\phi = 0.25$; (b) silicone oil with $\eta = 100$ mPa s. The insets show enlarged images of the windward surface of the droplets. The corresponding video can be found in the supplementary material and movie 6.

3.5.1. Fragments in bag breakup mode

To quantify the fragment size, we use the probability density of fragment size weighted by volume, which represents the ratio of the volume of fragments with a certain diameter to the total volume. As shown in figure 13, these distributions of volume probability density in the bag breakup mode contain three distinct peaks. Combined with the morphological evolution of bag breakup, we can know that the three peaks, from small to large, correspond to the fragmentation of different parts of the droplet, including the bag, ring and nodes. Compared with silicone oil droplets, the fragment size distributions of the particle-laden droplets show distinct differences. The positions of the two peaks for small droplets shift to the right, while the position of the peak for large droplets remains almost unchanged. In addition, the heights of different peaks also change, corresponding to the variation of the volume fractions of different parts.

Figure 13.

Volume probability density distribution of the fragments in the bag breakup. The corresponding cases are the stars along the dashed line with $We_{c,0} = 12.2$ in figure 9: (a) silicone oil with $\eta = 35$ mPa s, $We_g = 15.7$, $Oh = 0.11$; (b) suspension with $\phi = 0.10$, $We_g = 16.6$, $Oh = 0.17$; (c) suspension with $\phi = 0.25$, $We_g = 20.3$, $Oh = 0.32$; (d) silicone oil with $\eta = 100$ mPa s, $We_g = 20.3$, $Oh = 0.32$. $d_0 = 3.35$ mm. The dashed lines are the compound gamma distributions with the best weight fitting. Each plot shows the statistical results of approximately 60 repeated experiments under the same conditions.

To further analyse the effect of particles on the fragment size distribution, we need to distinguish the effects of particles on different parts of the droplet (i.e. the bag, ring and nodes). For each part, the fragments essentially come from the rupture of threads or ligaments, so the fragment size distribution corresponding to each part of the droplet can be fitted with a gamma distribution (Villermaux Reference Villermaux2020; Xu, Wang & Che Reference Xu, Wang and Che2022). The parameters of the gamma distribution corresponding to different parts depend on the average fragment size and the corrugation of the ligaments from which the fragments originate (Villermaux, Marmottant & Duplat Reference Villermaux, Marmottant and Duplat2004; Villermaux Reference Villermaux2020). Hence, the parameters of the gamma distribution are different for different parts. The different gamma distributions for different parts can be combined to form a compound gamma distribution. The overall size distribution is fitted by this compound gamma distribution, and different peaks of the overall size distribution correspond to the different sub-gamma distributions (Jackiw & Ashgriz Reference Jackiw and Ashgriz2022; Ade, Chandrala & Sahu Reference Ade, Chandrala and Sahu2023). In this regard, we use a compound gamma distribution to fit the overall size distribution

(3.7)\begin{equation} g( x ) = \sum_{i = 1}^3 {{w_i}{g_{i,v}}( x)}, \end{equation}

where $x=d/{{d}_{0}}$ is the dimensionless fragment size and $d_0$ is the initial diameter of the droplet, ${{w}_{i}}$ is the weight of each part and $w_1 + w_2 + w_3=1$, ${g_{i,v}}( x ) = {\beta _{i}}^{{\alpha _i}}{x^{{\alpha _i} - 1}}{\,{\rm e}^{ - {\beta _i}x}}/\varGamma ( {{\alpha _i}} )$ is the volume probability density function for each part, $\varGamma$ is the gamma function, ${{\alpha }_{i}}$ and ${{\beta }_{i}}$ are the shape parameter and the rate parameter of ${g_{i,v}}$, respectively. Since the compound gamma distribution contains many parameters, a direct fitting using the gamma distribution may yield different results and thus make quantitative comparisons difficult. Here, for the fragment size distribution of the bag breakup, the interaction between the peaks is small due to the large difference in the size of the fragments produced by different parts. So we can determine ${\alpha _i}$ and ${\beta _i}$ of each sub-gamma distribution based on the shape and position of each peak, and then obtain the weight ${w_i}$ of each peak by fitting. In this way, the compound gamma distributions are in good agreement with the measured results, as shown by the dashed lines in figure 13. Based on the parameters of each volume probability density function, we can obtain the shape parameter and the rate parameters of each number probability density function as ${\alpha _i} - 3$ and ${\beta _i}$ (Jackiw & Ashgriz Reference Jackiw and Ashgriz2022), and further estimate the average fragment size of the corresponding part as

(3.8)\begin{equation} {d_i} = \frac{{{\alpha _i} - 3}}{{{\beta _i}}}, \quad i = 1,2,3, \end{equation}

where ${{d}_{1}}$, ${{d}_{2}}$ and ${{d}_{3}}$ are the average fragment sizes corresponding to the bag (${{d}_{b}}$), ring (${{d}_{r}}$) and node (${{d}_{n}}$), respectively. Moreover, the weight ${{w}_{i}}$ of each sub-gamma distribution is the volume fraction of that part of the fragments.

The average fragment size corresponding to different parts of the droplet obtained by (3.8) is shown in figure 14. The fragment size of particle-laden droplets is generally larger than that of silicone oil droplets, and the scaling relations between the correlation length and the fragment size are different for different parts. For the fragmentation of the bag, the liquid film retracts towards particle clusters to form the fragments, as illustrated in the inset of figure 14(a). The sizes of the particle clusters can be characterised by the correlation length. This is because the particle clusters essentially correspond to the concentration fluctuation (i.e. the particle agglomeration or separation), and the coherence length reflects the range of concentration fluctuation caused by the heterogeneous effect of particles. Hence, the fragment size of the bag ($d_b$) can be estimated as

(3.9)\begin{equation} \frac{{{d}_{b}}}{{{d}_{0}}}\sim \frac{\xi }{{{d}_{0}}}. \end{equation}

The relation is in good agreement with the data in figure 14(a), where the scaling in (3.9) is shown as the solid line with a prefactor of $C_1 = 1.12$.

Figure 14.

Average fragment sizes of (a) the bag, (b) the ring and (c) the node under different particle volume fractions and particle sizes. The scatters are the fragment size of the particle-laden droplets obtained from (3.8). The dashed and dotted lines correspond to the fragment sizes of silicone oil droplets with $\eta = 35$ and 100 mPa s, respectively. Here, $\xi /{{d}_{0}}$ is calculated according to (3.6), $d_0 = 3.35$ mm. The corresponding cases are the stars along the dashed line with $We_{c,0} = 12.2$ in figure 9. Each data point shows the statistical results of approximately 60 repeated experiments under the same condition. The insets illustrate the fragment mechanisms for the corresponding parts.

For the fragmentation of the ring, each fragment corresponds to a segment of the liquid ring, as illustrated in the inset of figure 14(b). The initial diameter of the liquid ring, which is approximately equal to the thickness of the droplet ($h$) after the initial flattening, is larger than the correlation length ($\xi$). But as the liquid ring is stretched and thinned, the diameter of the liquid ring ($h_1$) is smaller than the correlation length. At this condition, the concentration fluctuation caused by the particle heterogeneity can lead to the destabilisation of the ring and then produce fragments. Therefore, the correlation length reflecting the range of concentration fluctuation can characterise the wavelength of the destabilisation, which is the length of the segment of the liquid ring where a fragment is produced. In addition, the diameter of the liquid ring ($h_1$) is mainly controlled by the rim expansion ratio, which is affected by the Weber number (Jackiw & Ashgriz Reference Jackiw and Ashgriz2021). Hence, $h_1$ can be considered as a constant due to the same inviscid Weber number in our experiments. Therefore, the fragment size of the ring can be estimated based on the mass conservation as

(3.10)\begin{equation} \frac{{{d_r}}}{{{d_0}}} \sim {\left( {\frac{\xi }{{{d_0}}}} \right)^{1/3}}. \end{equation}

The relation is in good agreement with the data in figure 14(b), where the scaling in (3.10) is shown as the solid line with a prefactor of $C_2 = 0.78$.

Finally, for the fragmentation of the nodes, the fragment size is almost constant unless the correlation length $\xi$ is large ($\xi /{{d}_{0}}>0.1$), as shown in figure 14(c). This is because the effect of particle heterogeneity occurs when the characteristic scale is smaller than the correlation length. For the nodes with a large scale, the size of the node fragments will be affected only when the correlation length is large, that is, a high volume fraction or a large particle size. For particle-laden droplets with $\xi /{{d}_{0}}<0.1$, we find that the fragment size of the node obeys

(3.11)\begin{equation} \frac{{{d}_{n}}}{{{d}_{0}}}={{C}_{3}}\frac{h}{{{d}_{0}}}, \end{equation}

where the thickness of the droplet after the initial flattening $h$ is a constant for the cases in this study, and $C_3$ is a constant equal to 1.55. In figure 14(c), the dashed line for silicone oil droplets with $\eta$ = 35 mPa s corresponds to $C_3 = 1.53$ ($d_n / d_0 = 0.481$), and the dotted line for silicone oil droplets with $\eta = 100$ mPa s corresponds to $C_3 = 1.47$ ($d_n / d_0 = 0.460$). This law is consistent with the end-pinching mechanism proposed by Schulkes (Reference Schulkes1996) for free liquid jets (with $C_3 = 1.55$), that is, the droplets are shed via end pinching caused by the retraction of ligament tips, as illustrated in the inset of figure 14(c). A similar law was also found in the breakup of Worthington jets (Gordillo & Gekle Reference Gordillo and Gekle2010) and unsteady sheets (Wang & Bourouiba Reference Wang and Bourouiba2018), which have $C_3 = 1.5\unicode{x2013}1.6$. Here, our results show that the fragmentation of the node also follows this scaling law for the end-pinching mechanism.

In addition to the average fragment size, the volume fraction of each part is also important for understanding the fragment size distribution. The volume fraction of each part obtained by the weight ${{w}_{i}}$ of each sub-gamma distribution is shown in figure 15(a,b). To analyse the reason of the volume fraction of each part, we need to consider the mass distribution in the droplet after the initial flattening. This is because the evolution of the droplet shape and the subsequent breakup process determine the volume fraction of each part. After the initial flattening, the droplet is deformed into a disk with a thickness of $h\approx 1.05$ mm and a width of ${{d}_{f}}\approx 5.6$ mm, as shown in figure 15(c). Correspondingly, the volume fractions at the rim and middle of the droplet are roughly 56 % and 44 %, respectively (based on volume calculation from figure 15c). Then the middle of the droplet develops into the bag, and the rim develops into the ring and nodes. During this process, liquid flows from the bag to the ring and from the ring to the nodes (Jackiw & Ashgriz Reference Jackiw and Ashgriz2022), as illustrated in figure 15(d). These processes ultimately determine the volume fraction of each part. For the silicone oil droplet ($\phi$ = 0 in figure 15a), the flow from the bag to the ring reduces the volume fraction of the middle bag from the initial 44 % to 25 % after the breakup. The volume fraction of the nodes calculated from the fragment size (3.11) and number (usually approximately 3) is 28 %, which is in agreement with the experimental data (27 %) in figure 15(a). In addition, Jackiw & Ashgriz (Reference Jackiw and Ashgriz2022) predicted the fragment size distribution of water droplets, and found the volume fraction of the bag, ring and nodes were 12 %, 45 % and 38 %, respectively. Compared with their prediction, the volume fraction of the bag in our results is larger, while the volume fraction of the node is smaller. Besides the effect of a slight difference in $We_{c,0}$, this difference may be due to the higher viscosity of the droplet in our experiments, which inhibits the liquid flow between the parts. These comparisons also indicate that the volume fractions obtained from the compound gamma distribution are reasonable.

Figure 15.

Variation of the volume fraction of each part with (a) the particle volume fraction and (b) the particle size. The corresponding cases are the stars along the dashed line with $We_{c,0} = 12.2$ in figure 9. Each data point shows the statistical results of approximately 60 repeated experiments under the same conditions. (c) Illustration of the droplet morphology after the initial flattening. (d) Illustration of the liquid flow in the bag development process and the effect of particles on flow inhibition.

Compared with the silicone oil droplet, there are several differences in the volume fractions of different parts of the particle-laden droplet. First, the volume fraction of the bag of the particle-laden droplet is much larger, i.e. approximately 40 %. This is because, during the bag development process, the particles tend to be retained at the junction of the bag film and the ring. This results in a higher local particle concentration and correspondingly a higher viscosity, which inhibits the liquid flow from the bag to the ring, as illustrated in figure 15(d). In addition, the particle clusters on the liquid film can trap some liquid. Therefore, the bag volume fraction of the particle-laden droplet is maintained at about 40 %, i.e. the initial volume fraction in the middle of the droplet. Second, as the particle volume fraction increases, the volume fraction of the nodes decreases first and then increases, while the volume fraction of the ring increases first and then decreases, as shown in figure 15(a). This is because the higher particle volume fraction inhibits the liquid flow from the ring to the nodes. However, when the particle volume fraction is very large ($\phi$ = 0.35 in figure 15a), the presence of the particles makes the fragment size of the node increase significantly (as shown in figure 14c), and the corresponding volume fraction of the node also increases. Third, as the particle size increases, the volume fraction of the nodes increases gradually, and the volume fractions of the bag and ring decrease, as shown in figure 15(b). This is because the larger particle tends to reside on the larger-scale nodes rather than the smaller-scale bag and ring.

In addition to considering the effects of particles on different parts, the fragment size distribution can also be described by some representative diameter to quantify the overall atomisation effect. A representative diameter is the Sauter mean diameter (SMD), which is the diameter of a droplet whose ratio of volume to surface area is the same as that of the entire fragments (Lefebvre & McDonell Reference Lefebvre and McDonell2017). The SMD can be calculated as

(3.12)\begin{equation} {SMD}=\frac{\displaystyle \sum{{{n}_{i}}d_{i}^{3}}}{\displaystyle \sum{{{n}_{i}}d_{i}^{2}}}= \frac{1}{\displaystyle \sum{{{{w}_{i}}}/{{{d}_{i}}}}}, \end{equation}

where $n_i$ is the number of fragments corresponding to a size of $d_i$ and equal to ${{{w}_{i}}d_{0}^{3}}/{d_{i}^{3}}$. Combining the fragment sizes (obtained by (3.9), (3.10) and (3.11)) and the corresponding volume fractions of different parts, we can estimate the SMD as

(3.13)\begin{align} \frac{SMD}{{{d_0}}} \sim \frac{{1/{d_0}}}{{{w_1}/{d_b} + {w_2}/{d_r} + {w_3}/{d_n}}} = {\left[ {\frac{{{w_1}}}{{{C_1}}}{{\left( {\frac{\xi }{{{d_0}}}} \right)}^{ - 1}} + \frac{{{w_2}}}{{{C_2}}}{{\left( {\frac{\xi }{{{d_0}}}} \right)}^{ - 1/3}} + \frac{{{w_3}}}{{{C_3}}}{{\left( {\frac{h}{{{d_0}}}} \right)}^{ - 1}}} \right]^{ - 1}}. \end{align}

By substituting the prefactors $C_1 = 1.12$, $C_2 = 0.78$, $C_3 = 1.55$ (obtained from (3.9), (3.10) and (3.11)) and the weights $w_1 = 0.405$, $w_2 = 0.345$, $w_3 = 0.25$ (obtained by averaging all the cases considered in this study) into (3.13), we can see that the relationship is in good agreement with the experimental data, as shown in figure 16, where (3.13) is shown as the solid line with a prefactor of 1.4.

Figure 16.

Sauter mean diameter of fragments at different particle volume fractions and sizes in the bag breakup mode. The solid line is (3.13) with a prefactor of 1.4. The corresponding cases are the stars along the dashed line with $We_{c,0} = 12.2$ in figure 9. Here, $\xi /{{d}_{0}}$ is calculated according to (3.6), $d_0 = 3.35$ mm. Each data point shows the statistical results of approximately 60 repeated experiments under the same conditions.

3.5.2. Fragments in multimode breakup

The volume probability density distribution of the fragments in the low-order multimode breakup is shown in figure 17. The fragments are also formed by the fragmentation of three parts, including the middle node, the peripheral nodes and the peripheral ring, as illustrated in figure 17(c). The fragmentation of the middle node forms a distinct peak with the largest fragment size. But the fragment sizes formed by the fragmentation of the peripheral nodes and ring are close, and the corresponding peaks are indistinguishable. Therefore, we still use the compound of three gamma distributions to fit the data, but do not distinguish the effects of particles on different parts. We only use the fitting curves to identify the changes in the overall size distributions. As shown in figures 17(a,b), the fragment size distribution shifts to the right as the particle volume fraction and particle size increase, that is, the overall fragment size of the particle-laden droplets becomes larger.

Figure 17.

Variation of the volume probability density distribution in the low-order multimode breakup with (a) the particle volume fraction and (b) the particle size. The corresponding cases are the stars along the dashed line with $We_{c,0} = 36$ in figure 9, i.e. fluids used are silicone oil with $\eta = 35$ mPa s, $We_g = 51.6$, $Oh = 0.11$, and suspensions with $\phi = 0.10$, $We_g = 56.3$, $Oh = 0.17$ and $\phi = 0.25$, $We_g = 71.7$, $Oh = 0.32$. Here, $d_0 = 3.35$ mm. The solid lines are the compound gamma distributions with the best weight fitting. Each plot shows the statistical results of approximately 90 repeated experiments under the same conditions. (c) Droplet morphology at 14.5 ms in low-order multimode breakup mode, corresponding to the whole breakup process shown in figure 5.

The SMDs of the fragment size distributions are shown in figure 18. For the multimode breakup, the droplet is pierced by RT instability waves to form fragments with a typical size ($d_s$). The breakup is extremely unstable, and the heterogeneity of the particle-laden fluid will intensify the breakup randomness, as described in § 3.4. For this condition, we can refer to the full-wave approach to predict the SMD. The full-wave approach assumes that there is a spectrum of wavelengths which could be excited to appreciable amplitudes and then produce fragments of corresponding size (Mayer Reference Mayer1961; Qin, Yi & Yang Reference Qin, Yi and Yang2018; Yang et al. Reference Yang, Gao, Li and Fu2020). Following this idea, in the condition of the multimode breakup, the fragments originate from the piercing caused by the RT instability wave, so the sizes of the fragments ($d_s$) can be estimated as

(3.14)\begin{equation} {{d}_{s}}={{C}_{s}}{{\lambda }_{RT}}, \end{equation}

where $C_s$ is a constant and ${{\lambda }_{RT}}$ is the RT instability wavelength. In addition, the number probability of fragments with size $d_s$ is required to calculate SMD. However, due to the close size of the fragments originating from different parts, the exact size distribution of fragments is difficult to be quantified by the compound gamma distributions. Since we mainly focus on the effect of particles, we, for simplicity, neglect the variation of the volume probability density with the fragment size $d_s$, that is, the volume of the initial droplet is divided uniformly into $N$ parts and each of which is broken into fragments with size $d_s$. Therefore, the number probability ($n$) of fragments with size $d_s$ is estimated as

(3.15)\begin{equation} n \sim \frac{{d_0^3}}{{Nd_s^3}}. \end{equation}

By integrating over the spectrum of wavelengths and combining (3.14) and (3.15), the SMD can be written as

(3.16)\begin{equation} {SMD}=\frac{\displaystyle \int_{{{\lambda }_{min }}}^{{{\lambda }_{max }}}{nd_{s}^{3}\, {\rm d} \lambda }}{\displaystyle \int_{{{\lambda }_{min }}}^{{{\lambda }_{max }}}{nd_{s}^{2}\, {\rm d} \lambda }}={{C}_{s}}\frac{{{\lambda }_{max }}-{{\lambda }_{min }}}{\ln {{\lambda }_{max }}-\ln {{\lambda }_{min }}}, \end{equation}

where ${{\lambda }_{max }}$ and ${{\lambda }_{min }}$ are the upper and lower limits of the spectrum of wavelengths. For droplet breakup, the wavelength must be smaller than the droplet diameter, i.e. ${{\lambda }_{max }}={{d}_{0}}$. The correlation length of the particle-laden droplet is the typical length over which a fluctuation will be damped, so short-wavelength instabilities will be suppressed by the particles, i.e. ${{\lambda }_{min }}=\xi$. Therefore, the SMD in (3.16) can be written in dimensionless form as

(3.17)\begin{equation} \frac{SMD}{{{d}_{0}}}={{C}_{s}}\frac{1-\xi /{{d}_{0}}}{-\ln ( \xi /{{d}_{0}} )}. \end{equation}

Through the full-wave approach with a lower limit $\xi$, the relation in (3.17) reflects the effect of the particle in intensifying the breakup randomness and suppressing the short-wavelength instabilities, and is in good agreement with the data in figure 18, where (3.17) is the solid line with $C_s = 0.764$.

Figure 18.

Sauter mean diameter of fragments at different particle volume fractions and particle sizes in the low-order multimode breakup. The corresponding cases are the stars along the dashed line with $We_{c,0} = 36$ in figure 9. Here, $\xi /{{d}_{0}}$ is calculated according to (3.6), $d_0 = 3.35$ mm. The solid line is (3.17) with $C_s = 0.764$. Each point shows the statistical results of approximately 90 repeated experiments under the same condition.