3.1. Characterisation of shear flow

A typical velocity field of the jet shear layer obtained by the PIV experiment is shown in figure 2. The velocity field is the average of 400 instantaneous velocity fields, and it has been confirmed in our previous experiment that the average velocity field is stable when the number of instantaneous velocity fields reaches 200 (Xu et al. Reference Xu, Wang and Che2020). From the average velocity field, it can be seen that there is a shear layer with a large velocity gradient between the jet core and the outer region, and this shear layer is used to study the shear effect on droplet breakup.

Figure 2.

Average velocity field of the shear flow without droplet breakup and contours of the droplet during the breakup process at jet velocity $u_g=41.7$ m s$^{-1}$. Here, ${{We}}_g = 230$, ${{We}}_d = 19.9$, $\chi = 0.295$, ethanol droplet diameter $d_0 = 2.4$ mm. The values on the axes are the spatial coordinates defined in figure 1(a).

By controlling the airflow rate, we can change the jet velocity and the shear strength of the shear layer. In our experiments, the range of the jet velocity ($u_g$) is 27.8–83.3 m s$^{-1}$. Correspondingly, the Reynolds number of the jet is $4.4\times 10^4$–$1.3\times 10^5$, in which the hydraulic diameter (24 mm) of the rectangular nozzle is used as the characteristic length for the Reynolds number calculation. The jet flow is turbulent, and according to the studies by Jiao et al. (Reference Jiao, Jiao, Zhang and Du2019) and Zhao et al. (Reference Zhao, Nguyen, Duke, Edgington-Mitchell, Soria, Liu and Honnery2019), the velocity fluctuation increases the asymmetry of droplet deformation and the randomness of droplet breakup. Therefore, besides qualitative discussions on the influence of shear, statistical data are obtained from repeated experiments to eliminate the impact of turbulence randomness on the data analysis.

To further quantify the velocity fields at different jet velocities, we extract the velocity thickness ($\delta _1$) of the shear layer, which is defined as the distance in the $y$-direction from 5 % jet velocity to 95 % jet velocity:

(3.1)\begin{equation} {{\delta }_{1}}(x)=\int_{{{y}_{1}}\left| u=0.05{{u}_{g}} \right.}^{{{y}_{2}} \left| u=0.95{{u}_{g}} \right.}{1}\,{{\rm d}y}. \end{equation}

Since the effect of the airflow on the droplet is essentially a momentum transfer process, we also extract the momentum thickness ($\delta _2$) of the shear layer, which is defined as the thickness of a hypothetical inviscid fluid with a uniform velocity that has the same momentum flow rate as the momentum loss rate due to fluid viscosity in the actual shear layer:

(3.2)\begin{equation} {{\delta }_{2}}(x)=\int_{{{y}_{1}}}^{y_{2}}\frac{u(x,y)}{{{u}_{g}}(x)} \left[1-\frac{u(x,y)}{{{u}_{g}}(x)}\right]{{\rm d}y}, \end{equation}

where the range of integration $y_1< y < y_2$ is some points far from the shear layer. The variations of the velocity thickness and the momentum thickness of the shear layer along the flow direction are shown in figure 3, which shows that both the velocity thickness and the momentum thickness can be considered to increase linearly with $x$, and the thickness of the shear layer at different jet velocities can be regarded as constant.

Figure 3.

Velocity thickness ($\delta _1$) and momentum thickness ($\delta _2$) of the shear layer along the flow direction $x$. The range of the jet velocity ($u_g$) is 27.8–83.3 m s$^{-1}$. The lines are linear fitting.

To understand the relative position of the droplet in the flow field during the droplet breakup process, we extract the droplet contours of a typical breakup process and plot them in the velocity field, as shown in figure 2. It can be seen that in the process of the droplet deformation and breakup, the bottom of the droplet is stretched quickly along the jet direction under the strong shear, while the top of the droplet remains at the original position until it finally breaks. By changing the falling height of the droplet, we can change the downward speed of the droplet entering into the flow field to control the degree of the droplet invasion into the shear flow field, and further change the relative position of the droplet in the flow field.

3.2. Breakup process

A typical breakup process of a droplet in a strong shear flow is shown in figure 4. We let the droplet enter the jet at a low speed, which makes the droplet break up under the strong shear of the shear layer. As the droplet enters the shear layer, the lower left part of the droplet is first flattened and tilted under the effect of the shear airflow (1 ms in figure 4a). Then the direction of the airflow is deflected on the inclined windward side of the droplet, and a high-pressure area is formed. Similar to the pressure surface (i.e. the lower surface) of a wing, the high-pressure area on the inclined windward side produces an upward lift on the droplet. If we divide the droplet into two parts (the front and back parts, as shown in figure 4b), then we can find further that the shear flow has different effects on the front and back parts. At the front part of the droplet, the flattened windward side is almost parallel to the shear layer. Under the combined effect of the downward inertia of the droplet and the upward lifting force of the airflow, a liquid cap appears at the front of the droplet, similar to the impact of a droplet on a wall. At the back part of the droplet, the windward side of the droplet is tilted in the shear layer at a certain angle, and a liquid sheet is stretched out from the tail of the droplet under the strong shear effect.

Figure 4.

Image sequences of droplet breakup in shear flow. The direction of the airflow is from left to right: (a) synchronised images from the bottom view (first row) and the side view (second row); (b) enlarged images for the front and back parts of the droplet at $t = 1$ ms. Here, ${{We}}_g = 230$, ${{We}}_d = 19.9$, $\chi = 0.295$, ethanol droplet diameter $d_0 = 2.4$ mm. The orange lines indicate the velocity boundaries of the shear layer, corresponding to 5 %–95 % of the jet velocity. The corresponding movie can be found in the supplementary material (movie 1) available at https://doi.org/10.1017/jfm.2022.326.

As the lower part of the droplet is flattened and the upper part of the droplet moves downwards, the main body of the droplet deforms continuously (1.7–2.2 ms in figure 4a). In the uniform flow of a high Weber number, the main body of the droplet is flattened as a consequence of the non-uniform pressure distribution around the droplet surface (high pressure at the front and rear stagnation points, as well as low pressure at the equator due to the acceleration of the gas) (Meng & Colonius Reference Meng and Colonius2017). In contrast, the deformation mechanism of the droplet in shear flow is different, which includes not only the flattening caused by the lifting force of the shear layer and the downward inertia of the droplet, but also the stretching by the shearing force along the flow direction. The combined effects make the main body of the droplet deform into a fan shape (2.2 ms in figure 4a).

During the deformation of the droplet's main body, the liquid cap in the front part of the droplet develops into a stable rim and maintains its position in the flow field. The liquid sheet stretched out from the droplet tail in the early stage breaks continuously on a small scale. When the deformation of the main body is over, the droplet is fully stretched. Since the back part of the droplet has a larger windward area and contains more liquid at this time than the early tail sheet breakup, the deformed droplet will break up intensively from the droplet's back part (2.6 ms in figure 4a). The intensive breakup at the droplet's back part will cause the droplet to swing upwards remarkably (as highlighted by the arrow at 2.6–3.3 ms in figure 4a). Finally, depending on the strength of the swing, the front rim of the droplet moves upwards out of the flow field driven by the swing, or the rim falls into the high-speed region of airflow by gravity and breaks up because the lifting force decreases with the disintegration of the liquid film. In addition to figure 4, more image sequences at different conditions are provided in Appendix A.

To analyse further the breakup process of the droplet under the strong shear effect, we divide the breakup process into three stages, according to the interaction between the droplet and airflow: the sheet breakup stage (i.e. the liquid sheet is stretched out from the droplet tail), the swing breakup stage (i.e. the main body of the droplet breaks up), and the rim breakup stage (i.e. the front rim of the droplet collapses or shrinks).

Sheet breakup stage: When the droplet just enters the shear flow, the liquid sheet is stretched out from the lower surface of the droplet, which is the main feature of the sheet breakup stage. The stretching of the sheet is similar to the early deformation of the droplet breakup process in a uniform airflow. But in the shear flow, the sheet stretching is asymmetric and occurs mainly at the droplet tail where the shear effect is strong, while it is symmetrical and occurs along the periphery of the droplet in a uniform flow.

Most of the previous studies (Theofanous Reference Theofanous2011; Theofanous et al. Reference Theofanous, Mitkin, Ng, Chang, Deng and Sushchikh2012; Jalaal & Mehravaran Reference Jalaal and Mehravaran2014; Sharma et al. Reference Sharma, Singh, Rao, Kumar and Basu2021), especially those based on shock tube experiments, indicated that the early sheet originated from the KH instability wave on the surface of the droplet. The KH instability wave caused by the shear layer on the droplet surface was further stretched to form the sheet under the airflow action. In a recent study, Jackiw & Ashgriz (Reference Jackiw and Ashgriz2021) attributed the mechanism of the sheet formation to the local normal pressure of the airflow; i.e. an initial rim around the droplet formed due to the outward flow of the windward face, then the rim was pushed downstream, and a thin sheet was produced to connect the rim to the main body of the droplet. The difference between the two interpretations depends essentially on whether more emphasis is on the shear effect or the normal pressure of the airflow, which is affected by the experimental conditions. Moreover, both interpretations are based on the analysis of the interplay between the aerodynamic force and the surface tension (Villermaux Reference Villermaux1998,  Reference Villermaux2020; Jackiw & Ashgriz Reference Jackiw and Ashgriz2021). Therefore, the two interpretations are closely related. From our experimental phenomenon, the early stretching of the sheet is more consistent with the interpretation of Jackiw & Ashgriz (Reference Jackiw and Ashgriz2021). An initial rim is formed on the periphery of the flattened windward side, then the rim is drafted downstream to form a stretched sheet. The sheet, as it is stretched, becomes thinner and increasingly sensitive to instabilities until it breaks up.

Based on the pressure balance at the initial rim between the Laplace pressure (${2\sigma }/{{{h}_{{init}}}}$) and the airflow pressure (${{{\rho }_{g}}u_{g}^{2}}/{2}$), we can estimate the thickness of the rim ${{{h}_{{init}}}}/{{{d}_{0}}}\sim {4}/{ {{{{We}}}_{g}}}$. As ${{We}}_g$ increases, the thickness of the initial rim decreases due to the stronger aerodynamic force of the air. Hence the thickness of the stretched sheet is thinner at higher ${{We}}_g$. Figure 5 shows the images of the tail sheet before its first breakup under different ${{We}}_g$ with the same ${{We}}_d$. Although different ${{We}}_g$ and $\chi$ lead to different deformation of the droplet body (discussed in detail in § 3.3), it can be seen that as ${{We}}_g$ increases, the thinner sheet in the early stage becomes more unstable and is prone to break up quickly under the stronger shear. Thus the spreading area of the sheet becomes smaller as ${{We}}_g$ increases.

Figure 5.

Bottom-view images of the tail sheet before its first breakup: (a) ${{We}}_g = 102$, $\chi = 0.442$; (b) ${{We}}_g = 230$, $\chi = 0.295$; (c) ${{We}}_g = 408$, $\chi = 0.221$; (d) ${{We}}_g = 637$, $\chi = 0.177$; (e) ${{We}}_g = 917$, $\chi = 0.147$. Here, ${{We}}_d = 19.9$, ethanol droplet diameter $d_0 = 2.4$ mm.

Swing breakup stage: The second stage of the breakup features the intensive swing of the droplet body. With the stretching and breakup of the tail sheet, the main body of the droplet is flattened continuously. As the droplet body deforms, the windward surface area of the droplet increases and the thickness of the droplet decreases. Hence the liquid tongue has a larger acceleration. In addition, as the droplet falls and deforms, the windward surface at the tail of the droplet body becomes more perpendicular to the incoming flow. These factors promote the development of the RT instability and the formation of large fluctuation with transverse corrugations on the liquid tongue at the tail of the droplet body, as shown in figure 6(b). Even at a much higher Weber number (${{We}}_g = 917$), the sheet breakup of the tail edge of the droplet in the early stage is more intensive and thorough, but the liquid ridge formed by the lateral RT instability can still be seen on the edge of the droplet body, as shown from the bottom-view image in figure 6(c). Further modelling analysis of the RT instability will be given in § 3.3. Since this large liquid tongue contains much liquid and the droplet body is thin, the droplet will swing remarkably, driven by the breakup of the liquid tongue.

Figure 6.

(a) Emergence, propagation and merging of the surface wave at the droplet windward side in the swing breakup stage in the side-view images. (b) Wave structures in the bottom-view images at ${{We}}_g = 230$, ${{We}}_d = 19.9$, $\chi = 0.295$, ethanol droplet diameter $d_0 = 2.4$ mm. (c) Wave structures in the bottom-view images at ${{We}}_g = 917$, ${{We}}_d =36.5$, $\chi = 0.199$, ethanol droplet diameter $d_0 = 2.4$ mm (movie 2 in the supplementary material).

Rim breakup stage: After the swing breakup stage, the droplet breakup process enters the rim breakup stage, which features mainly the collapse or shrinkage of the front rim of the droplet. In the first two stages, when sheet breakup and swing breakup occur at the back part of the droplet, the front rim of the droplet is still in the region with low air velocity and maintains a stable position due to the uplift effect of the shear layer. But after the swing breakup stage, there are two scenarios for the rim. In the first scenario, the swing breakup stage has consumed only a little liquid and much liquid is left on the rim, and the strength of the swing effect cannot pull the thick rim out of the shear layer. With the reduction of the lifting force of the shear layer caused by the disintegration of the liquid film, the rim enters the high-speed region of the airflow under gravity, and then is penetrated by the RT instability waves. The front of the droplet appears jagged and eventually collapses, as shown in figure 7(a). In contrast, in the second scenario, the swing breakup stage has consumed much liquid and a little liquid is left on the rim, and the front rim will be driven out of the shear layer by the strong swing effect. After leaving the air jet, ligaments appear on the rim, and the rim shrinks and forms large fragments, as shown in figure 7(b).

Figure 7.

(a) Collapse of the droplet front rim, ${{We}}_g = 230$, ${{We}}_d = 19.9$, $\chi = 0.295$, ethanol droplet diameter $d_0 = 2.4$ mm. (b) Shrinkage of the droplet front rim, ${{We}}_g = 230$, ${{We}}_d =36.5$, $\chi = 0.399$, ethanol droplet diameter $d_0 = 2.4$ mm. The corresponding movies can be found in the supplementary material (movie 3 for rim collapse, and movie 4 for rim shrinkage).

It should be noted that although the droplet breakup under the strong shear effect involves the above three subsequential stages, these three stages of breakup compete with each other due to the conservation of mass. If one stage consumes much liquid and its breakup is intensive, then the other two stages will be weak. For example, when the shear effect is very strong, liquid sheets will be peeled off from the tail of the droplet repeatedly. In this situation, the strength of the swinging breakup stage will be reduced.

3.3. Analysis of swing breakup

The swing breakup process is similar to the flapping breakup of coaxial liquid jets, during which the liquid jet with a coaxial airflow becomes unstable and deviates from its axis to flap (Delon, Cartellier & Matas Reference Delon, Cartellier and Matas2018; Ricard et al. Reference Ricard, Machicoane, Osuna-Orozco, Huck and Aliseda2021). However, different from the flapping breakup in coaxial liquid jets, which originates from the shear instability, the swing breakup in this study is induced by the flattening deformation of the droplet body (the detailed comparison is presented in Appendix B). The droplet body is flattened under the combined effects of the lifting force of the shear layer and the downward inertia of the droplet. Similar to inertia-dominated droplet collisions for low-viscosity liquids (Clanet et al. Reference Clanet, Béguin, Richard and Quéré2004; Roisman Reference Roisman2009; Roisman, Berberović & Tropea Reference Roisman, Berberović and Tropea2009; Opfer et al. Reference Opfer, Roisman, Venzmer, Klostermann and Tropea2014), the flattened thickness of the droplet body ($h$) is

(3.3)\begin{equation} \frac{h}{{{d}_{0}}}\sim {{We}}_{f}^{{-}1/2}, \end{equation}

where ${{We}}_f$ is a Weber number for the flattening of the droplet, indicating the ratio of the flattening force to the surface tension. The flattening force consists of the downward inertia of the droplet (${{F}_{d}}\sim {{\rho }_{d}}u_{d}^{2}d_{0}^{2}$) and the lifting force of the shear layer ($F_l$). Since the windward surface of the droplet is tilted in the airflow like a wing, the lifting force can be estimated according to the wing lift theory (Kundu & Cohen Reference Kundu and Cohen2008) as

(3.4)\begin{equation} {{F}_{l}}\sim {{\rho }_{g}}u_{g}^{2}d_{0}^{2}\sin \alpha, \end{equation}

where $\alpha$ is the angle of attack of a wing, which corresponds to the angle of droplet inclination, as shown in figure 8(a). The angle of droplet inclination $\alpha$ can be obtained from the comparison of the vertical movement of the droplet (scaled by $u_d$) and the horizontal movement of the droplet (scaled by ${{u}_{g}}\sqrt {{{{\rho }_{g}}}/{{{\rho }_{d}}}}$). For small $\alpha$, we have

(3.5)\begin{equation} \sin \alpha \approx \tan \alpha =\frac{{{u}_{d}}}{{{u}_{g}}\sqrt{{{\rho }_{g}}/{{\rho }_{d}}}}=\chi, \end{equation}

which indicates that the angle of droplet inclination can be represented by $\chi$. Using (3.4) and (3.5), we can obtain the flattening Weber number

(3.6)\begin{equation} {{ {{We}}}_{f}}=\frac{\left( {{\rho }_{d}}u_{d}^{2}+{{u}_{g}}{{u}_{d}} \sqrt{{{\rho }_{g}}{{\rho }_{d}}} \right){{d}_{0}}}{\sigma }= {{ {{We}}}_{g}}( {{\chi }^{2}}+\chi ). \end{equation}

By comparing (3.6) with the original definition of the Weber number, we can obtain the droplet flattening velocity ($u_f$) as

(3.7)\begin{equation} {{u}_{f}}=\sqrt{u_{d}^{2}+{{u}_{g}}{{u}_{d}}\sqrt{{{{\rho }_{g}}}/{{{\rho }_{d}}}}}. \end{equation}

Using this droplet flattening velocity, we can obtain the flattening Reynolds number (${{{{Re}}}_{f}}={{\rho }_{d}}{{d}_{0}}{{u}_{f}}/{{\mu }_{d}}$) and a scaling parameter $P={{{{We}}}_{f}}\,{{Re}}_{f}^{-4/5}$, where $P$ is a dimensionless parameter to quantify the relative importance of the inertia used in droplet collision. For the low-viscosity fluid (the test liquid for the results presented in §§ 3.2–3.6 is ethanol), ${{{{We}}}_{f}}<430$, ${{{{Re}}}_{f}}>1253$ and $P<0.6$, so the inertia dominates the droplet deformation, and (3.3) is valid (Clanet et al. Reference Clanet, Béguin, Richard and Quéré2004; Roisman Reference Roisman2009; Roisman et al. Reference Roisman, Berberović and Tropea2009). From (3.3) and (3.6), we can predict the dimensionless thickness of the droplet body:

(3.8)\begin{equation} H=\frac{h}{{{d}_{0}}}={{C}_{1}}\,{{We}}_{g}^{{-}1/2}{{( {{\chi }^{2}}+\chi )}^{{-}1/2}}, \end{equation}

where $C_1$ is a constant.

Figure 8.

(a) Acceleration of the liquid tongue. (b) Width of the liquid tongue ($b_w$) before the instability development, and number ($n$) of bags or transverse corrugations after the instability development. Here, ${{We}}_g = 230$, ${{We}}_d = 11.8$, $\chi = 0.227$, ethanol droplet diameter $d_0 = 2.4$ mm.

As the RT instability develops, the flattening deformation of the droplet body leads further to the swing breakup. For inviscid fluids, the dispersion equation of the RT instability wave (Chandrasekhar Reference Chandrasekhar2013) is

(3.9)\begin{equation} n_{{RT}}^{2}=ak\left[ \frac{{{\rho }_{d}}-{{\rho }_{g}}}{{{\rho }_{d}}+{{\rho }_{g}}}- \frac{{{k}^{2}}\sigma }{a({{\rho }_{d}}+{{\rho }_{g}})} \right], \end{equation}

where $n_{RT}$ and $k$ are the growth rate and the wavenumber of the RT instability wave, and $a$ is the acceleration of the liquid. The growth rate ($n_{RT}$) has a maximum value, and the corresponding most-amplified wavelength is

(3.10)\begin{equation} {{\lambda }_{RT}}=\frac{2{\rm \pi} }{{{k}_{m}}}=2{\rm \pi} \sqrt{\frac{3\sigma }{a({{\rho }_{d}}-{{\rho }_{g}})}}, \end{equation}

where $k_m$ is the most-amplified wavenumber. In the early stage of the swing breakup (as shown in figure 8a), the acceleration of the liquid tongue can be considered as

(3.11)\begin{equation} a=\frac{{{F}_{D}}}{m}=\frac{{{F}_{D}}}{{{\rho }_{d}}h{{A}_{e}}}, \end{equation}

where $A_e$ is the projected area, $h$ is the thickness of the liquid tongue of the droplet, and $F_D$ is the drag force by the gas on the liquid tongue:

(3.12)\begin{equation} {{F}_{D}}=\tfrac{1}{2}{{C}_{D}}{{\rho }_{g}}{{({{u}_{g}}-{{u}_{c}})}^{2}}{{A}_{e}}, \end{equation}

where $C_D \approx 2$ (White Reference White2003), and $u_c$ is the backward velocity of the liquid tongue, $u_c \ll u_g$.

By substituting (3.8), (3.11) and (3.12) into (3.10), we can deduce the most-amplified wavelength of the RT instability:

(3.13)\begin{equation} \frac{{{\lambda }_{RT}}}{{{d}_{0}}}=2{\rm \pi} \sqrt{3}C_{1}^{1/2}\, {{We}}_{g}^{{-}3/4}{{( {{\chi }^{2}}+\chi )}^{{-}1/4}}. \end{equation}

From (3.13), we can obtain the theoretical wavelength of the RT instability wave. For comparison, the experimental wavelength of the RT instability wave can be obtained from the high-speed images as

(3.14)\begin{equation} {{\lambda }_{ex}}=\frac{{{b}_{w}}}{n}, \end{equation}

where $b_w$ is the width of the liquid tongue before the development of the instability wave, and $n$ is the number of bags or transverse corrugations after the development of the instability wave (as illustrated in figure 8b). A comparison between the theoretical and experimental wavelengths of the RT instability is made in figure 9, which shows that (3.13) can predict the experimental wavelength well.

Figure 9.

Comparison between theoretical and experimental instability wavelengths, ${{\lambda }_{RT}}$ and ${{\lambda }_{ex}}$. The experimental instability wavelength is obtained from experimental images, and the theoretical instability wavelength is obtained from (3.13) with a fitted constant $C_1 = 1.41$. The dashed line indicates ${{\lambda }_{RT}}={{\lambda }_{ex}}$. As the wavelength increases, the five black squares correspond to ${{We}}_g = 637$, 408, 312, 159, 102, respectively, $\chi = 0.136$, 0.170, 0.195, 0.273, 0.341, respectively, and ${{We}}_d = 11.8$, ethanol droplet diameter $d_0 = 2.4$ mm. The five red circles correspond to ${{We}}_g = 408$, 312, 230, 159, 102, respectively, $\chi = 0.221$, 0.253, 0.295, 0.354, 0.442, respectively, and ${{We}}_d = 19.9$, ethanol droplet diameter $d_0 = 2.4$ mm. The blue star corresponds to the case of figure 8 with ${{We}}_g = 230$, $\chi = 0.227$, ${{We}}_d = 11.8$, ethanol droplet diameter $d_0 = 2.4$ mm.

Although the above instability analysis is based on the swing breakup of droplets in a strong shear airflow, it can also provide an understanding of droplet breakup in uniform airflow due to the similar physical mechanisms involved. In the study of droplet breakup in uniform airflow, following the theory of Marmottant & Villermaux (Reference Marmottant and Villermaux2004) on coaxial jets, Jalaal & Mehravaran (Reference Jalaal and Mehravaran2014) believed that the liquid tongue that generates the RT instability was a KH instability wave formed by the development of shear instability (i.e. $h\sim {{\lambda }_{KH}}$). They obtained the acceleration of the liquid tongue through KH instability analysis, but their theoretical data and simulated data did not match well. From our experimental phenomenon, except for the breakup of the early stretched liquid film, in the further breakup, such as the recurrent breakup proposed by Dorschner et al. (Reference Dorschner, Biasiori-Poulanges, Schmidmayer, El-Rabii and Colonius2020) and the swing breakup in this study, the thickness and the acceleration of the liquid tongue that generates the RT instability is weakly correlated with the KH instability due to the deformation of the droplet body and the merging of the KH instability waves. However, in uniform airflow, it is difficult to obtain directly the thickness of the droplet tongue that generates the RT instability since the stretched liquid film or the droplet mist at the edge of the droplet often blocks the view. In our experiment, the thickness of the liquid tongue is predicted from analogising the flattening deformation of the droplet body in the shear flow to inertia-dominated droplet collisions and referring to the wing lift theory. In addition, we can measure the experimental wavelength through bottom-view images due to the unidirectional stretch effect of the shear layer. Therefore, quantitative results of the RT instability can be obtained, and good agreement between the experimental data and the theoretical analysis is achieved.

3.4. Regime map of droplet breakup

To analyse further the shear effect of the airflow on droplet breakup, a regime map is produced as shown in figure 10. According to the morphology of the droplet and the mechanism of breakup, the breakup of droplets can be categorised into three regimes, namely the swing breakup with front rim collapse (figures 4 and 18), the swing breakup with front rim shrinkage (figure 17), and the shear-stripping breakup (figure 19). The shear-stripping breakup is the breakup regime of droplets in almost uniform airflow, and liquid sheets are stripped out continuously from the droplet body. In contrast, in a strong shear flow of air, the droplet body swings with a front rim, and the breakup can be divided into two regimes according to the collapse or the shrinkage of the front rim, as discussed in § 3.2.

Figure 10.

Regime map of droplet breakup in airflow with a strong shear effect. Transition 1 is based on (3.15) and hence $\chi = 0.618$. Transition 2 is based on (3.16) and ${{ {{We}}}_{g}}-100= {{{{We}}}_{g}} ( {{\chi }^2}+\chi )$.

The regime map is plotted in the parameter space of $\chi$ and ${{We}}_g$. Here, $\chi$ reflects the relative effect of the droplet falling inertia, and together with ${{We}}_g$, $\chi$ further affects the deformation of the droplet body during the interaction of the droplet with the shear layer, as discussed in § 3.3. It should be noted that the jet velocity and the shear strength of the jet shear layer are related to each other through the thickness of the shear layer (i.e. ${{u}_{g}}=\dot {\gamma }\delta$). Since the thickness of the shear layer $\delta$ almost does not vary with the jet velocity (as shown in figure 3), the gas Weber number (${{We}}_g$) can represent not only the strength of aerodynamic force but also the strength of the shear effect. The change in the temporal evolution of the droplet with different parameters can be attributed to the variation in the relative strength at different stages. With the increase in $\chi$ under the same ${{We}}_g$ (figures 4, 17 and 19), the thickness of the early stretched sheet at the tail of the droplet is almost unchanged due to the same ${{We}}_g$, but the droplet body deforms faster. Hence the swing breakup after the deformation of the droplet body consumes more liquid and its strength is greater, and the front rim contains less liquid. The breakup regime changes gradually from the swing breakup with rim shrinkage to the swing breakup with rim collapse, further to the shear-stripping breakup in which the front rim disappears. With the increase in ${{We}}_g$ under the same $\chi$ (figures 4 and 18), the droplet body deforms faster, but the droplet breaks up faster at different stages. At large ${{We}}_g$, small-scale breakup occurs continuously from the droplet tail, and the distinction between different stages becomes less clear.

To distinguish droplet breakup in shear flow from that in uniform flow, we compare the characteristic time for the droplet passing through the shear layer ($t_y={( {d_0}+\delta )}/{{{u}_{d}}}$) with the time scale of droplet deformation (${{t}^{*}}={{{d}_{0}}}/({{u_g}\sqrt {{{{\rho }_g}}/{{{\rho }_d}}}})$), and find that ${t_y}/{t^{*}}\propto \chi$. When $\chi$ is small, the residence time scale (${{{t}_{y}}}/{{{t}^{*}}}$) of the droplet in the shear layer during the process of droplet deformation is small, and the droplet can pass quickly through the shear flow. In this condition, the droplet does not form a rim in the front of the droplet, but is stretched directly to form a liquid sheet around the edge of the windward surface, which is the same as the shear-stripping breakup in uniform airflow. In contrast, when $\chi$ is large, the droplet deforms mainly in the shear flow. The droplet will form a front rim, and the main body of the droplet will exhibit swing breakup under the strong shear effect. Therefore, the transition from the swing breakup to the shear-stripping breakup occurs when $\chi$ is larger than a threshold. The transition condition can be estimated from

(3.15)\begin{equation} {{ {{We}}}_{f}}={{{{We}}}_{g}}, \end{equation}

which corresponds to the transition condition of the droplet body deformation from being controlled by the shear layer uplift and the droplet falling inertia (i.e. the swing breakup) to being controlled by the aerodynamic force of a uniform airflow (i.e. the shear-stripping breakup). From (3.6) and (3.15), we can obtain the value of $\chi$ as $\chi = 0.618$. This transition condition is shown as Transition 1 in figure 10, which is in good agreement with the experimental data.

For the droplet breakup under the strong shear effect, whether the front rim shrinks or collapses depends on the strength of the swing breakup, that is, the relative amount of liquid contained in the front rim and the swinging liquid film. The amount of liquid consumed in the swing breakup is less when the swing breakup develops quickly, and the rim is thicker when the droplet body deforms slowly. Here, we compare the relative speed of the swing breakup (i.e. ${{ {{We}}}_{g}}- {{ {{We}}}_{g,c}}$, where ${{We}}_{g,c}$ is the critical Weber number for sheet stretching and ${{We}}_{g,c} = 100$) (Guildenbecher et al. Reference Guildenbecher, Lopez-Rivera and Sojka2009; Jackiw & Ashgriz Reference Jackiw and Ashgriz2021) and the deformation of the droplet body (i.e. ${{We}}_f$, which can be obtained from (3.6)) after the tail sheet is stretched out, so we can obtain the transition condition of the front rim from collapse to shrinkage:

(3.16)\begin{equation} {{ {{We}}}_{g}}- {{ {{We}}}_{g,c}}={{{{We}}}_{f}}. \end{equation}

As shown by Transition 2 in figure 10, the transition condition of the front rim from collapse to shrinkage based on (3.16) agrees well with the experimental data.

3.5. Stretching of liquid film

Different from the rapid breakup of the liquid film stretched from the droplet edge in a uniform airflow, the stretching of the liquid film in shear airflow is significant, and has an important influence on the uplift of the droplet front rim and the size of the fragments. We extract the boundary of the liquid film from the bottom-view images via image analysis, and obtain the temporal evolution of the liquid film area during the droplet deformation and breakup process.

Figure 11 shows the evolution of the liquid film area from the instant when the droplet edge just enters the shear flow to the end of the swing breakup stage. From the development of the liquid film in a single experiment, it can be seen that the liquid film stretches out from the tail of the droplet at first, and its area increases gradually until it breaks up. Every sudden decrease in the area of the liquid film corresponds to the occurrence of breakup. At the instant of the film collapsing during the swing breakup stage, the area of the liquid film suddenly decreases from its maximum and does not increase any more. Since the development of instability on the liquid film is extremely quick and the liquid film after stretching is affected easily by turbulence, there is a strong uncertainty about the moment of the liquid film breakup. To compare the results under different parameters, we repeat a large number of experiments to get a statistical result as shown in figure 11.

Figure 11.

Area evolution of the liquid film from the instant when the droplet edge just enters the shear flow to the end of the swing breakup stage. Here, ${{We}}_g = 230$, ${{We}}_d = 19.9$, $\chi = 0.295$, ethanol droplet diameter $d_0 = 2.4$ mm. The red curve shows the results of a single experiment, and the black curve shows the statistic results from repeated experiments. The error bars indicate the standard deviations of 20 repeated experiments.

The temporal evolution of the liquid film area at different $\chi$ is shown in figure 12(a). In the case of ${{We}}_g = 230$, as $\chi$ increases from 0.123 to 0.695 (corresponding to ${{We}}_d$ increasing from 3.45 to 111), the stretch speed of the early liquid film tends to be the same, the average time to complete swing breakup becomes shorter, and the maximum liquid film area first increases and then decreases. When $\chi$ is very small ($\chi = 0.123$), the early liquid sheet at the tail of the droplet deforms slowly and contains much liquid in the low-speed region of the airflow, which leads to a slow development speed and a long development time of the film area at the early stage. After the sheet breakup stage, the liquid left for the swing breakup is little, and the swing breakup is weak, which results in a small maximum area of the liquid film. As $\chi$ increases from 0.123 to 0.399, the downward inertia of the droplet increases, and more liquid quickly invades the shear flow. The early sheet is stretched quickly in the high-speed region of the airflow, and the stretch speed tends to be the same because ${{We}}_g$ is constant. Combined with the uplift effect of the shear layer, the increase of $\chi$ also leads to a faster deformation of the droplet main body, an increased windward area of the droplet, more liquid contained in the liquid tongue before the swing breakup, and finally more intensive swing breakup. Therefore, the maximum area of the liquid film increases as $\chi$ increases. If $\chi$ increases further ($\chi > 0.399$), then the stretch speed of the early sheet remains unchanged, but the maximum area of the liquid film decreases with increasing $\chi$. This is because the further increase in $\chi$ results in too little liquid in the front rim of the droplet. Since the liquid in the front rim is insufficient, when the swing breakup occurs, the front rim cannot maintain its original position but moves downstream, which weakens the stretching of the liquid film. In addition, as $\chi$ increases from 0.123 to 0.695, the average time to the end of swing breakup becomes shorter because the droplet enters the shear flow faster to deform.

Figure 12.

Temporal evolution of the area of the liquid film under the influence of (a) $\chi$ and (b) ${{We}}_g$. Each curve shows the average of 20 repeated experiments.

Further, combined with the regime map of figure 10, the area evolution of the liquid film with different $\chi$ (i.e. varied ${{We}}_d$ with the fixed ${{We}}_g$) in figure 12(a) can be linked with the breakup regimes. As $\chi$ increases from 0.123 to 0.399, the maximum area of the liquid film increases. This is because the swing breakup strength at the tail of the droplet body increases while the front rim remains relatively stable, which strengthens the stretching of the liquid film. This means that the swing breakup at the tail cannot make the front rim move, and after the swing breakup, the front rim enters the high-speed region of the airflow and then collapses, corresponding to the regime of the swing breakup with rim collapse. As $\chi$ increases from 0.399 to 0.695, the swing breakup strength at the tail of the droplet body is further enhanced, but the front rim is affected by the swing breakup and cannot maintain its position, which weakens the stretching of the liquid film and finally causes a reduction in the maximum area of the liquid film. In this condition, the rim will be taken away from the main stream by the swing breakup and then shrinkage, corresponding to the regime of the swing breakup with rim shrink. Finally, when $\chi = 0.695$, the front rim disappears, and the droplet body is pierced after the early stripping along the droplet periphery, which causes the liquid film to disappear quickly after the early sheet stretching, corresponding to the regime of shear-stripping breakup in the uniform airflow. These results reflect the connection between the swing breakup strength and the movement of the front rim, which determines the regime of breakup.

The temporal evolution of the area of the liquid film at different ${{We}}_g$ is shown in figure 12(b). In the case of ${{We}}_d = 19.9$, as ${{We}}_g$ increases from 159 to 917 and $\chi$ decreases from 0.354 to 0.147, the stretching speed of the early liquid film becomes faster, the average time to complete the swing backup becomes shorter, and the maximum liquid film area becomes smaller. This is because an increase in ${{We}}_g$ means a stronger shear effect, and the droplet will stretch and break up faster. And due to the rapid and continuous occurrence of sheet breakup caused by the stronger shear effect, the swing breakup becomes weaker, and the maximum area of the liquid film becomes smaller.

3.6. Fragment size distribution

The fragment size after droplet breakup is the main concern in many practical applications. Under our experimental conditions with high ${{We}}_g$ accompanied by strong shear, the fragment size is analysed from the high-speed images. Due to the long duration of the whole breakup process, the fragments produced in different stages after the completion of the breakup are already dispersed to a large region. In the traditional method of extracting the size of fragments from a single image, it will require a large shooting area to cover all the fragments, which will mean that most small fragments cannot be counted under the limitation of spatial resolution. To circumvent the tradeoff between the shooting area and the spatial resolution, we develop a customised Matlab program based on fragment tracking on image sequences. By tracking each fragment, we can obtain the size of all fragments after the completion of the breakup using only a small shooting area. By tuning the photography setting and image-processing procedure, we can obtain all the fragments sized larger than 100 $\mathrm {\mu }$m. A detailed description of the procedure for measuring the fragment size is given in the supplementary material.

Figure 13 shows the probability density of fragment size weighted by volume, which represents the ratio of the total volume of fragments with a certain diameter to the fragments’ total volume. Under the same ${{We}}_g$ (${{We}}_g = 230$), the different $\chi$ in figure 13 correspond to the three breakup regimes, namely the swing breakup with rim collapse ($\chi = 0.123$), the swing breakup with rim shrinkage ($\chi = 0.399$), and the shear-stripping breakup ($\chi = 1.535$). Due to the differences in the breakup mechanism and the breakup morphology, the fragment size distribution is also different. For the shear-stripping breakup ($\chi = 1.535$), the distribution appears unimodal with a peak at $d_p \sim 0.17$ mm, which is similar to the result of Guildenbecher et al. (Reference Guildenbecher, Gao, Chen and Sojka2017) in uniform airflow. For the swing breakup with rim collapse ($\chi = 0.123$), the size distribution also appears unimodal. But compared with the shear-stripping breakup, the swing breakup with rim collapse generally occurs in a region where the airflow velocity is lower. In addition, the droplet body is almost parallel to the airflow when the front rim collapses. Hence the windward area is small, and less energy is transferred from the airflow to the droplet during the breakup process. The lower local air velocity and less energy transfer leads to larger fragments during the swing breakup with rim collapse than the shear-stripping breakup. For swing breakup with rim shrinkage ($\chi = 0.399$), the front rim is driven out of the shear layer by the strong swing effect and then shrinks, producing larger fragments. We calculate the characteristic diameters of the fragments after droplet breakup in table 2. At ${{We}}_g = 230$ and as $\chi$ increases from 0.123 to 1.535, the size distribution first widens and then narrows. Meanwhile, the characteristic diameters, including the Sauter mean diameter (SMD), the mass median diameter (MMD) and the peak diameter ($d_p$), increase first and then decrease.

Figure 13.

Volume probability density distribution of the fragments after completion of the droplet breakup. (a) The solid lines are the best-fit log-normal distributions, and the dashed lines are the best-fit infinitely divisible distributions. (b) The solid lines are the best-fit compound gamma distributions, and the dashed lines are the best-fit gamma distributions. Each scatterplot shows the statistical results of about 30 repeated experiments under the same conditions. Here, ${{We}}_g = 230$.

Table 2.

Characteristic diameters of the fragments after droplet breakup.

In the spray community, there are many functions to describe the fragment size distribution. Among them, the log-normal distribution is used widely, which is derived from considering the overall breakup process as sequential cascades where liquid volumes are aggregated or split at random (Kolmogorov Reference Kolmogorov1941; Gorokhovski & Saveliev Reference Gorokhovski and Saveliev2003; Hadj-Achour et al. Reference Hadj-Achour, Rimbert, Gradeck and Meignen2021). The log-normal distribution is

(3.17)\begin{equation} {{f}_{\log }}\left( d \right)=\frac{1}{\sqrt{2{\rm \pi} }d{{\sigma }_{\log }}}\exp \left[ -\frac{1}{2\sigma _{\log }^{2}}\,{{( \ln d-\ln {{d}_{m}} )}^{2}} \right], \end{equation}

where $d_m$ is the geometric mass median diameter, and $\sigma _{\log }$ is a scaling parameter.

Based on sequential cascades of breakups, there are many variants of the log-normal distribution (Novikov & Dommermuth Reference Novikov and Dommermuth1997; Rimbert & Sero-Guillaume Reference Rimbert and Sero-Guillaume2004; Saveliev & Gorokhovski Reference Saveliev and Gorokhovski2012). More recently, the infinitely divisible distribution proposed by Novikov & Dommermuth (Reference Novikov and Dommermuth1997) was used by Vallon, Abid & Anselmet (Reference Vallon, Abid and Anselmet2021), which assumed that the liquid underwent a sequential cascading breakup in turbulent sprays similar to the intermittent turbulent dissipation. The infinitely divisible distribution is

(3.18)\begin{equation} {{f}_{\inf }}( d )=\frac{d_{\inf }^{3/2}}{\sqrt{2{\rm \pi} }\,{{\sigma }_{\inf }}\,{{d}^{3/2}}}\exp \left[ -\frac{{{d}_{\inf }}}{2\sigma _{\inf }^{2}}\,{{\left( {{d}_{\inf }}{{d}^{{-}1/2}}-{{d}^{1/2}} \right)}^{2}} \right],\quad d\ge 0, \end{equation}

where $d_{\inf }$ and $\sigma _{\inf }$ are the average and the standard deviation, respectively.

The fitting results of (3.17) and (3.18) for different breakup modes are shown in figure 13(a). The log-normal distribution and the infinitely divisible distribution are very close, and both of them can describe the fragment size distribution reasonably well, especially for the shear-stripping breakup. Due to the high ${{Re}}$ of airflow and the high ${{We}}_g$, the strong turbulence in the shear layer of airflow masks the effect of different breakup mechanisms on the final fragment size distribution to some extent. Therefore, the size distribution is similar to the result of a random cascade breakup, and the different breakup regimes affect only the mean and variance of the distribution. But for the droplet breakup in the shear layer, there are some deviations at diameter $\sim$0.6 mm, especially for the swing breakup with rim shrinkage.

In addition, the gamma distribution is known to be a good approximation of the log-normal distribution (Hadj-Achour et al. Reference Hadj-Achour, Rimbert, Gradeck and Meignen2021) and was used recently by Vallon et al. (Reference Vallon, Abid and Anselmet2021) and Huck et al. (Reference Huck, Osuna-Orozco, Machicoane and Aliseda2022). Different from the log-normal distribution and infinitely divisible distribution, the gamma distribution is based on the assumption that the fragments come from the rupture of objects in the form of threads or ligaments, i.e. the elementary distribution of the fragment size originates from the ligament breakup (Villermaux, Marmottant & Duplat Reference Villermaux, Marmottant and Duplat2004; Villermaux & Bossa Reference Villermaux and Bossa2011; Villermaux Reference Villermaux2020). The gamma distribution is

(3.19)\begin{equation} g( d )=\frac{{{d}^{\alpha -1}}\,{{\rm e}^{-{d}/{\beta }}}}{{{\beta }^{\alpha }}\,\varGamma ( \alpha )}, \end{equation}

where $\alpha$ is the shape parameter and refers to the roughness of the ligament, and $\beta$ is the scale parameter. The fitting results of (3.19) for different breakup modes are shown in figure 13(b). The gamma distribution displays an exponential fall-off at the tail and a relatively narrow distribution. For the droplet breakup in the shear layer of airflow, the gamma fitting underestimates the experimental data at the large fragment sizes.

Since the fragments are formed mainly in two different stages, and the single gamma distribution displays an apparent discrepancy at the large fragment sizes, we further use the compound gamma distribution proposed by Villermaux (Reference Villermaux2020) and Villermaux & Bossa (Reference Villermaux and Bossa2011) to describe the fragment size distribution. The compound gamma distribution is based on the overall fragments originating from two distinct classes of ligaments. The linear superposition of the two elementary gamma distributions leads to a compound distribution

(3.20)\begin{equation} {{g}_{c}}( d )={{A}_{1}}\,{{g}_{1}}( d )+{{A}_{2}}\,{{g}_{2}}( d ), \end{equation}

where $g_1(d)$ and $g_2(d)$ are the elementary gamma distributions with different ligament roughnesses, and $A_1$ and $A_2$ are the weights of $g_1(d)$ and $g_2(d)$, with $A_1+A_2=1$. The fitting results of (3.20) for different breakup modes are shown in figure 13(b). The compound gamma distribution shows clearly a better description of the experimental data only for the swing breakup with rim shrinkage. This is due to the apparent difference in the sizes of fragments produced by the swing breakup of the droplet body and the shrinkage of the front rim. But for the swing breakup with rim collapse, because the sizes of fragments produced in different stages are close, the elementary gamma distributions are difficult to distinguish from each other. Hence the parameters in the compound gamma distribution are not stable, and the compound distribution has no obvious superiority in performance.

A comparison between the log-normal distribution and the compound gamma distribution for the swing breakup with rim shrinkage is shown in figure 14. The log-normal distribution seems to be slightly better across the entire size range, but the compound gamma distribution is superior at the intermediate size and reflects the effects of the two different fragmentation mechanisms on the size distribution.

Figure 14.

Volume probability density distribution of the fragments after completion of the droplet breakup. The blue dashed-dotted line is the best-fit log-normal distribution, the red solid line is the best-fit compound gamma distribution, and the red dashed lines are the sub-gamma distributions. Each scatterplot shows the statistical results of about 30 repeated experiments under the same conditions. Here, ${{We}}_g = 230$.

Finally, table 3 presents the mean and variance obtained by fitting different distributions, and compares them with the experimental results; the calculation of the mean and variance is given in Appendix C. The log-normal distribution and infinitely divisible distribution can fit the mean of the experimental data better, while the gamma and compound gamma distributions can fit the variance better. This is consistent with the view of Villermaux (Reference Villermaux2007,  Reference Villermaux2020) that the gamma distribution can model the experimental data well with prior knowledge of the mean size.

Table 3.

Mean and variance of experimental results and the fitting distributions.

3.7. Effect of the Ohnesorge number

The Ohnesorge number ${{Oh}}$ indicates the ratio of droplet viscous forces to surface tension forces. In the process of droplet breakup, the droplet viscosity induces energy dissipation and hinders droplet deformation. Hence ${{Oh}}$ is an important control parameter for the droplet breakup. Instead of using ethanol in §§ 3.2–3.6, we use silicone oil of different viscosities to study the effect of ${{Oh}}$ on droplet breakup in strong shear airflow. Figure 15(a) shows images of the tail sheet before its first breakup at different ${{Oh}}$ with almost the same $\chi$ and ${{We}}_g$. (The slight variations in $\chi$ and ${{We}}_g$ are due to the variations in the fluid density and the surface tension as the viscosity changes, shown in table 1.) With increasing ${{Oh}}$, the stretched sheet at the tail is more stable, and the stretched area increases slightly, similar to the decrease in ${{We}}_g$ as discussed in § 3.2 and figure 5. When ${{Oh}}$ increases to 0.47, the stretched sheet transforms to a tail bag structure due to the stabilisation effect of the high viscosity, as shown in figure 15(b). This process is similar to the transition from shear-stripping breakup to multimode breakup in uniform airflow when increasing ${{Oh}}$ (Guildenbecher et al. Reference Guildenbecher, Lopez-Rivera and Sojka2009; Radhakrishna et al. Reference Radhakrishna, Shang, Yao, Chen and Sojka2021). In this condition, because the rapid development of the tail bag consumes much liquid, the swing breakup of the droplet body does not occur, and the front rim is lifted and then shrinks. Although the increase in ${{Oh}}$ inhibits the droplet deformation and even alters the breakup mode, the swing breakup of the droplet body still occurs if ${{We}}_g$ is large enough. For example, even when ${{Oh}}$ reaches 0.96 and ${{We}}_g$ increases to 853, the droplet body swings and breaks up, as shown in figure 15(c).

Figure 15.

(a) Bottom-view images of the tail sheet before its first breakup at different ${{Oh}}$. The values of ${{Oh}}$ are, respectively, 0.0054, 0.048, 0.23, 0.47, with corresponding ${{We}}_g = 230$, 239, 230, 219, $\chi = 0.294$, 0.347, 0.353, 0.354, and $d_0 = 2.4$, 2.3, 2.3, 2.2 mm. (b) Synchronised images of the bottom and side views after tail bag development at ${{Oh}} = 0.47$, ${{We}}_g = 219$, $\chi = 0.354$, silicone oil with $d_0 = 2.2$ mm (movie 5 in the supplementary material). (c) Synchronised images of the bottom and side views for the swing breakup of the droplet body at ${{Oh}} = 0.95$, ${{We}}_g = 873$, $\chi = 0.177$ (movie 6 in the supplementary material).

For viscous droplets, the swing breakup also originates from the flattening deformation of the droplet body, but the flattened thickness of the droplet body and the development of the RT instability are both affected by the droplet viscosity. For the cases presented in this section, $100< {{ {{We}}}_{f}}<490$, ${{ {{Re}}}_{f}}<100$ and $P>4.9$, which indicates the viscous case following the droplet collision analogue (Roisman Reference Roisman2009). In this condition, the flattened thickness of the droplet body is determined mainly by the flattening Reynolds number (${{Re}}_f$):

(3.21)\begin{equation} \frac{{{h}_{v}}}{{{d}_{0}}}\sim {{Re}}_{f}^{{-}2/5}, \end{equation}

where the subscript $v$ in $h_v$ indicates the viscous fluid, and ${{Re}}_f$ indicates the ratio of the flattening force to the viscous force of the droplet, ${{ {{Re}}}_{f}}={{\rho }_{d}}{{d}_{0}}{{u}_{f}}/{{\mu }_{d}}$; $u_f$ can be obtained from (3.7). Hence ${{Re}}_f$ can be transformed further into

(3.22)\begin{equation} {{ {{Re}}}_{f}}={{We}}_{g}^{1/2}({\chi ^2+\chi })^{1/2}/{{{Oh}}}. \end{equation}

By substituting (3.22) into (3.21), we can predict the dimensionless thickness of the droplet body ($H_v$) as

(3.23)\begin{equation} {{H}_{v}}=\frac{{{h}_{v}}}{{{d}_{0}}}={{C}_{2}}\, {{We}}_{g}^{-1/5}\,{{{{Oh}}}^{2/5}}{{( {{\chi }^{2}}+\chi )}^{{-}1/5}}. \end{equation}

Considering the RT instability of viscous fluids, when $\rho _l \gg \rho _g$, $\mu _l \gg \mu _g$, the dispersion equation of the instability wave (Joseph, Belanger & Beavers Reference Joseph, Belanger and Beavers1999; Joseph, Beavers & Funada Reference Joseph, Beavers and Funada2002) is

(3.24)\begin{equation} {{n}_{RT,v}}={-}\frac{{{k}^{2}}{{\mu }_{d}}}{{{\rho }_{d}}}\pm \sqrt{\frac{{{k}^{4}}{{\mu }_{d}}^{2}}{{{\rho }_{d}}^{2}}+ak-\frac{{{k}^{3}}\sigma }{{{\rho }_{d}}}}. \end{equation}

By assuming that the viscous and surface tension effects are additive to the leading order, Aliseda et al. (Reference Aliseda, Hopfinger, Lasheras, Kremer, Berchielli and Connolly2008) obtained the form of the most-amplified wavelength as

(3.25)\begin{equation} {{\lambda }_{RT,v}}=2{\rm \pi} \left[ {{\left( \frac{3\sigma }{{{\rho }_{d}}a} \right)}^{1/2}}+ {{\left( \frac{\mu _{d}^{2}}{\rho _{d}^{2}a} \right)}^{1/3}} \right], \end{equation}

where $a$ is obtained by (3.11), but replacing the thickness of the droplet body $h$ by $h_v$ in (3.23). Therefore, we can predict the most-amplified wavelength of the RT instability of viscous fluids as

(3.26)\begin{equation} \frac{{{\lambda }_{RT,v}}}{{{d}_{0}}}=2{\rm \pi} \left( \sqrt{3}\,{{We}}_{g}^{{-}1/2}\,H_{v}^{1/2}+ {{ {{Oh}}}^{2/3}}\,{{We}}_{g}^{{-}1/3}\,H_{v}^{1/3} \right), \end{equation}

where $H_v$ is given in (3.23).

The experimental wavelengths of the RT instability wave are also obtained from the high-speed images according to (3.14). A comparison between the theoretical and experimental wavelengths of the RT instability wave is made in figure 16, which shows that the experimental wavelengths can be well predicted by (3.26) for high-viscosity fluids with fitting parameter $C_2 = 0.75$. The above analysis confirms that the droplet viscosity can affect the development of the RT instability in the swing breakup.

Figure 16.

Comparison between theoretical and experimental instability wavelengths, ${{\lambda }_{RT,v}}$ and ${{\lambda }_{ex}}$, for viscous droplets. The experimental instability wavelength is obtained from experimental images, and the theoretical instability wavelength is obtained from (3.26) with fitted constant $C_2 = 0.75$. The dashed line indicates ${{\lambda }_{RT,v}}={{\lambda }_{ex}}$. As the wavelength increases, the three black squares correspond to ${{We}}_g = 921$, 410, 230, respectively, $\chi = 0.163$, 0.244, 0.326, respectively, and ${{Oh}} = 0.23$, silicone oil with $d_0 = 2.3$ mm. The two red circles correspond to ${{We}}_g = 877$, 390, respectively, $\chi = 0.163$, 0.245, respectively, and ${{Oh}} = 0.47$, silicone oil with $d_0 = 2.2$ mm. The two blue triangles correspond to ${{We}}_g = 873$, 388, respectively, $\chi = 0.164$, 0.245, respectively, and ${{Oh}} = 0.95$, silicone oil with $d_0 = 2.2$ mm.