3.1. The overall drag modification

In this section, we study the overall drag modification by changing the turbulent intensity and the viscosity of the dispersed droplet phase. The skin friction drag of the TC system is evaluated by the friction coefficient, $c_f$ (Van Gils et al. Reference Van Gils, Huisman, Bruggert, Sun and Lohse2011; Wang et al. Reference Wang, Yi, Jiang and Sun2022), which is defined as

(3.1)\begin{equation} c_{f,\phi} = \frac{(1-r_i/r_o)^2}{{\rm \pi}(r_i/r_o)^2}\frac{G}{Re^2}. \end{equation}

Based on the definitions, it can be noticed that $G\sim \mu ^{-2}$ and $Re\sim \mu ^{-1}$. Therefore, the friction coefficient defined as $c_f\sim G/Re^2$ is not explicitly dependent on the viscosity, but only implicitly depends on viscosity through the torque embedded in $G$. The friction coefficient will not be affected if the flow properties (such as $Re$ and $G$) are computed using a general form of viscosity (e.g. $\mu = (1-\phi )\mu _c+\phi \mu _d$). To emphasize the magnitude of the drag modification induced by dispersed droplets, the friction coefficient of the emulsions is normalized by that of the corresponding single-phase flow, i.e. the ethanol–water solution. The normalized friction coefficient, $c_{f,\phi }/c_{f,\phi =0}$, is shown in figure 3.

Figure 3.

The normalized friction coefficients of the TC system, $c_{f,\phi }/c_{f,\phi =0}$, for different $\zeta$. The value of $\zeta$ is shown in each panel, and panel (f) shows the data for rigid spherical particles taken from Wang et al. (Reference Wang, Yi, Jiang and Sun2022). The accuracy of the experiments is indicated by the error bars, which are typically less than 1 %.

We first look into the effect of turbulent intensity, $\dot {\gamma }$ (or, $Re$), on the drag modulation, $c_{f,\phi }/c_{f,\phi =0}$, induced by droplets with fixed $\zeta$. As shown in figure 3(a,b), $c_{f,\phi }/c_{f,\phi =0}$ of droplets with $\zeta <1$ monotonically decreases with increasing $\dot {\gamma }$ (or, $Re$) in a general way, and shows a weak $\dot {\gamma }$ (or, $Re$) dependence when compared with the rigid particles shown in figure 3(f). As the droplet viscosity increases to $\zeta >1$ (figure 3c,d), it can be seen that $c_{f,\phi }/c_{f,\phi =0}$ is somehow damped for the case of the smallest $\dot {\gamma }$ (or, $Re$). This could be the result of the under-developed turbulent flow in the system at this moderate turbulent intensity (Grossmann et al. Reference Grossmann, Lohse and Sun2016; Wang et al. Reference Wang, Yi, Jiang and Sun2022). For the rest of the turbulent intensities (i.e. higher $\dot {\gamma }$ or $Re$) in figure 3(c,d), $c_{f,\phi }/c_{f,\phi =0}$ monotonically decreases with increasing $\dot {\gamma }$ (or, $Re$), but still shows weaker trends than the case of rigid particles. For even higher $\zeta$ (figure 3e), $c_{f,\phi }/c_{f,\phi =0}$ shows a plateau for various $\dot {\gamma }$ (or, $Re$) and seems to depend only on the volume fraction of the droplets. This may be due to the turbulence suppression caused by highly viscous dispersed droplets. In this case, the magnitude of the drag modulation could be mainly affected by the contribution of the surface tension, which is proportional to the volume fraction of the droplets (De Vita et al. Reference De Vita, Rosti, Caserta and Brandt2019). The weak $\dot {\gamma }$ (or, $Re$) dependence could be interpreted by looking into the finite-size effect of the droplet, which can be connected to the particle Reynolds number, $Re_p$, and Stokes number, $St$,

(3.2)\begin{gather} Re_p = \frac{\rho_c u_bd_p}{\mu_c} = \frac{\rho_c d_p}{\mu_c}\frac{0.2(r_o^2-r_i^2)\dot{\gamma}}{r_i}, \end{gather}

(3.3)\begin{gather}St={\frac{\rho_pd_p^2}{18\rho_f[(r_o^2-r_i^2)r_i(r_o-r_i)]^{1/2}}(GRe)^{1/2}}. \end{gather}

Figure 4 presents $Re_p$ and $St$ along with $d_p$ normalized by the dissipation length scale, $\eta _K = (\nu ^3/\epsilon )^{1/4}$, where $\nu$ is the kinematic viscosity of the continuous phase, and $\epsilon$ is the total energy dissipation rate of the flow defined as $\epsilon = \tau \omega _i/({\rm \pi} \rho L(r_o^2-r_i^2))$ (van den Berg et al. Reference van den Berg, van Gils, Lathrop and Lohse2007). For the droplets, one can see that $d_p/\eta _K\simeq 2$ for various $\zeta$, resulting in both a $Re_p$ and $St$ much smaller than that of the rigid finite-size particle in our previous study. Given that $Re_p$ and $St$ in this work are around $50$ and $0.3$ (see figure 4(b,c), respectively), the droplets behave like fluid tracers (Toschi & Bodenschatz Reference Toschi and Bodenschatz2009; Grossmann et al. Reference Grossmann, Lohse and Sun2016; Mathai, Lohse & Sun Reference Mathai, Lohse and Sun2020; Brandt & Coletti Reference Brandt and Coletti2022), and modify the turbulence mainly by altering its volume fraction (Voth & Soldati Reference Voth and Soldati2017), accounting for the $\phi$-dependence of $c_{f,\phi }/c_{f,\phi =0}$. As $Re$ increases, the droplet size decreases as $d_p\sim Re^{-6/5}$ (Yi et al. Reference Yi, Toschi and Sun2021), giving $Re_p\sim Re^{-0.2}$ and $St\sim Re^{-1.9}$. Therefore, the finite-size effect of droplets becomes negligible at high $Re$ in the current turbulent TC system, in accordance with the weak $Re$-dependence of $c_{f,\phi }/c_{f,\phi =0}$ for various $\zeta$.

Figure 4.

(a) The dependence of (a) $d_p/\eta _K$, (b) $Re_p$ and (c) $St$ on the $\zeta$ at $\dot {\gamma }=125$ s$^{-1}$. The results for rigid particles (black hollow symbols) taken from Wang et al. (Reference Wang, Yi, Jiang and Sun2022) are also shown for comparison.

It is worth mentioning that, for different $\zeta$, whether it is smaller than or larger than 1, $c_{f,\phi }/c_{f,\phi =0}$ is always larger than unity, indicating that, in the current system, dispersed oil droplets always increase the overall drag of the system regardless of their viscosity. This persistent drag increase may be counter-intuitive, as one might expect the dispersed phase with low viscosity to induce a drag reduction by reducing the effective viscosity of the system. However, it has been shown in De Vita et al. (Reference De Vita, Rosti, Caserta and Brandt2019) that, by introducing the extra interfacial tension and the viscous stress, the presence of droplets could provide a comparable, or even larger, contribution to the total stress than the reduced stress in the continuous phase. As for the dispersed droplets with high viscosity, the overall drag of the system is enhanced although the turbulent intensity might be suppressed. This could be understood since the turbulence attenuation is not necessary to correspond to an overall drag reduction and vice versa, as suggested in our previous work (Wang et al. Reference Wang, Yi, Jiang and Sun2022). Here, we note that the stress increases of turbulent emulsions mainly come from the interfacial tension and the viscous stress of the dispersed droplets, whereas that for the rigid particle suspension is the frictional contact between solid surfaces to compensate for the reduction in fluid turbulent stress. Therefore, for turbulent emulsions in TC turbulence within the current parameter regimes, the global transport efficiency, i.e. the overall drag of the system, is always increased by introducing dispersed droplets into the system. Future studies are needed to explore in broader regimes the behaviour of the drag modification induced by dispersed droplets.

We next investigate how the drag modulation induced by droplets changes as $\zeta$ increases for fixed turbulent intensity. In figure 5 we show the information of figure 3 in the form of $c_{f,\phi }/c_{f,\phi =0} \sim \zeta$. For a fixed $\dot {\gamma }$, $c_{f,\phi }/c_{f,\phi =0}$ first increases with increasing $\zeta$, which could be attributed to the attenuated coalescence at higher $\zeta$, and consequently results in an increase of the surface tension contribution, and therefore, a higher effective viscosity (De Vita et al. Reference De Vita, Rosti, Caserta and Brandt2019). However, once $\zeta$ exceeds a critical value (in the current work, $1.7\leq \zeta _c\leq 3.8$), $c_{f,\phi }/c_{f,\phi =0}$ of the droplets approaches a plateau value. One could notice that this plateau value of $c_{f,\phi }/c_{f,\phi =0}$ for droplets is smaller than that for the case with rigid finite-size particles at small $Re$ or $\dot {\gamma }$ (figure 5a), while when the flow is highly turbulent ($Re\sim O(10^4)$), as shown in figure 5(c–e), the difference between the turbulent emulsion and the particle suspension becomes negligible. This could be due to the decreasing significance of the contribution of the particulate phase to the total stress at high $Re$ (Wang et al. Reference Wang, Yi, Jiang and Sun2022). Note that, when the $Re$ or $\dot {\gamma }$ is relatively small, the turbulence may be suppressed with dispersing droplets of high viscosity (Crialesi-Esposito et al. Reference Crialesi-Esposito, Rosti, Chibbaro and Brandt2022), thus accounting for the drag reduction that occurs at moderate $\zeta$ (figure 5a,b). Nevertheless, the similarity between the drag of emulsions and rigid particle suspensions increases with increasing $\zeta$, which could be, as shown in later sections, due to the suppressed droplet deformability at higher $\zeta$. Based on the current results, one reasonable speculation could be that the difference in the global transport (drag) of a turbulent emulsion system and a rigid particle suspension would become even smaller on further increasing $\zeta$ in a turbulent TC system. Of course, more studies in wider parameter regimes will be needed to draw a concrete conclusion on this issue.

Figure 5.

The dependence of $c_{f,\phi }/c_{f,\phi =0}$ on $\zeta$ for different shear rates. The value of $\dot {\gamma }$ is shown in each panel. The results for rigid particles (hollow symbols) taken from Wang et al. (Reference Wang, Yi, Jiang and Sun2022) are also shown for comparison.

3.2. Droplet size distribution

As discussed above, the drag of the turbulent emulsion shows a $\zeta$-dependence which could be connected to the droplet deformability. To dive into the $\zeta$-dependence of the droplet microstructure and its relationship with the $c_{f,\phi }/c_{f,\phi =0}$, we look into the DSDs obtained from image analysis. Here, we perform image analysis at the smallest $Re$ (or $\dot {\gamma }$) for $\phi =2\,\%$ and $4\,\%$. One may have noticed that the data quality of drag modulation is better when the flow is highly turbulent (e.g. figure 5d,e) as the turbulence may be not well developed at small $Re$ (or $\dot {\gamma }$). Intuitively, performing image analysis at high $Re$ (or $\dot {\gamma }$) could be better to reveal the physics, which unfortunately cannot be attained in the current work. Since the droplet size decreases with respect to $Re$ in $d_p\sim Re^{-6/5}$ (Yi et al. Reference Yi, Toschi and Sun2021), this means that at high $Re$ (or $\dot {\gamma }$) the droplet borders will soon become undetectable to the cameras. In the current experiments, at the smallest $Re$ (or $\dot {\gamma }$) the droplet borders can be well captured, while at higher $Re$ (or $\dot {\gamma }$), the droplet morphology from the front view becomes unclear and the droplet locations cannot be distinguished from the side view. Additionally, the trend of drag modulation at the smallest $Re$ (or $\dot {\gamma }$) and $\phi =2\,\%, 4\,\%$ (figure 5a) shows a similar behaviour to that of higher $Re$ (or $\dot {\gamma }$), i.e. a two-stage evolution process. Therefore, we also note that the image analysis obtained at small $Re$ (or $\dot {\gamma }$) could provide valuable information for the understanding of the flow at higher $Re$ (or $\dot {\gamma }$). The relatively poor data quality of drag modulation at higher $\phi$ in figure 5(a) could be attributed to the turbulence suppression induced by the highly viscous dispersed phase of high volume fraction.

Figure 6(a) shows that the mean droplet diameter weakly depends on $\zeta$ and $\phi$, while the size range strongly increases as the $\zeta$ increases. Based on the mean droplet diameter and its standard deviations, $We$ and $Ca$ defined by (2.5) and (2.6) are obtained, and the results are shown in figure 6(b,c). In principle, $We$ and $Ca$ decrease with increasing $\zeta$, suggesting that the deformability is suppressed for droplets with high viscosity. As one can see from figure 6(a), as $\zeta$ increases, the maximum droplet size increases, and the probability of finding large droplets becomes higher, which is confirmed by the snapshots presented in figure 1(b) and in accordance with the conclusions in Crialesi-Esposito et al. (Reference Crialesi-Esposito, Rosti, Chibbaro and Brandt2022).

Figure 6.

The dependence of (a) the mean droplet diameter, (b) $We$ and (c) $Ca$ on $\zeta$ at $\dot {\gamma }=125$ s$^{-1}$. The error bars in (a) indicate the interval of standard deviation of the DSD (i.e. $d_p\pm \sigma$). The upper (lower) bound of the error bars shown in (b,c) indicate the values of $We$ and $Ca$ corresponding to the upper (lower) bound of the droplet size shown in (a).

To find out how the viscosity ratio affects the droplet breakup and coalescence, in figure 7 we also show the best fit $d^{-3/2}$ and $d^{-10/3}$ scaling relations of each panel next to the experimental data. It has been reported that, in both bubbly flow (Rivière et al. Reference Rivière, Mostert, Perrard and Deike2021) and emulsion flow (Crialesi-Esposito et al. Reference Crialesi-Esposito, Rosti, Chibbaro and Brandt2022) in HIT conditions, the DSD shows a $d^{-3/2}$ scaling when $d< d_H$, which could be due to the increasing importance of the surface tension effect for small bubbles/droplets (Deane & Stokes Reference Deane and Stokes2002); while when $d>d_H$, the DSDs are well described by the $d^{-10/3}$ scaling, which was first proposed by Garrett, Li & Farmer (Reference Garrett, Li and Farmer2000) and later verified by Deane & Stokes (Reference Deane and Stokes2002). Here, $d_H$ is the Hinze scale defined as the maximum droplet diameter that can stably exist in turbulent emulsions. The $d^{-10/3}$ scaling could be derived with the assumption that the large bubble ($d>d_H$) experiences a purely initial breakup process at a rate depending on the dissipation rate $\epsilon$ and the rate of supply of the dispersed phase.

Figure 7.

The DSDs for different viscosity ratios at $\dot {\gamma } = 125$ s$^{-1}$. The value of $\zeta$ is shown in each panel. The droplet sizes are normalized by the dissipation length scale $\eta _K$ of each single-phase flow. The coloured vertical dashed lines denote the mean droplet size for $\phi =2\,\%$ and $\phi =4\,\%$, respectively (same colour code as the DSDs). The black vertical dotted line indicates the Hinze scale for each case. The statistical errors are indicated by the error bars. The scaling relation $d^{-3/2}$ (green) and $d^{-10/3}$ (blue) are shown for comparison. Note that the expressions of the scaling relation are different in each panel.

As shown in figure 7(a), the DSDs rapidly deviate from the $d^{-10/3}$ scaling at the large size range due to the non-negligible coalescence effects at this small $\zeta$. For small droplet size, the $d^{-3/2}$ scaling could only capture the distribution to some extent in a narrow range, hinting that the interfacial tension effect might be activated for only a limited population of small droplets; the DSDs rapidly fall off when the droplet size further decreases, which might be due to the finite precision of image analysis and failure to capture the droplets with extremely small sizes. This could be the reason that the DSDs shown in figure 7(a) agree with Mukherjee et al. (Reference Mukherjee, Safdari, Shardt, Kenjereš and Van den Akker2019) in the range of small sizes. In the studies of Mukherjee et al. (Reference Mukherjee, Safdari, Shardt, Kenjereš and Van den Akker2019), a $d^{-10/3}$ scaling for large droplets ($d>d_H$) is reported, although the droplets of small size ($d< d_H$) are prone to dissolution and could not be resolved due to the limitation of the employment of diffuse-interface methods. As suggested by the simulation results from Crialesi-Esposito et al. (Reference Crialesi-Esposito, Rosti, Chibbaro and Brandt2022), the $d^{-3/2}$ scaling might still be valid for the significantly small droplets, which unfortunately could not be resolved in the current experiments.

On the other hand, with increasing $\zeta$, the experimental data exhibit stronger coincidences with both scalings. In particular, for the cases of the two largest viscosity ratios (figure 7c,d), the DSDs could be generally described by these two scaling relations with a transition point at around $d/\eta _K \simeq 5\sim 6$. The pronounced consistency between the DSDs and the $d^{-10/3}$ scaling at larger $\zeta$ suggests that the assumption proposed by Garrett et al. (Reference Garrett, Li and Farmer2000) might be fulfilled in turbulent emulsions when the droplets are highly viscous, i.e. the large bubbles/droplets may experience a pure initial breakup process. Note that, at this relatively low shear rate ($\dot {\gamma }=125$ s$^{-1}$), the droplets with high viscosity could be hard to break into pieces and may remain in the initial states, therefore accounting for the secondary peak appearing at the large size in figure 7(d). As for the small droplets at larger $\zeta$ (see figure 7c,d), the DSDs principally follow the behaviour described by the $d^{-3/2}$ scaling. This could be understood considering that, in the present work, the droplet with large $\zeta$ possesses large surface tension, which could prevent the small droplets from merging into larger ones, thus maintaining the $d^{-3/2}$ scaling in this dynamic equilibrium system.

Given the definition of the Hinze scale ($d_H$) in (1.1), we can easily obtain the Hinze scale for various cases, by taking $\epsilon$ as the total energy dissipation rate of the flow, as mentioned above. As shown in figure 7, unlike the cases in HIT conditions (Deane & Stokes Reference Deane and Stokes2002; Mukherjee et al. Reference Mukherjee, Safdari, Shardt, Kenjereš and Van den Akker2019; Soligo et al. Reference Soligo, Roccon and Soldati2019; Rivière et al. Reference Rivière, Mostert, Perrard and Deike2021; Crialesi-Esposito et al. Reference Crialesi-Esposito, Rosti, Chibbaro and Brandt2022), the scale of the transition point is not in line with $d_H$ (the black dotted line), and in fact is smaller than $d_H$. As suggested by Mukherjee et al. (Reference Mukherjee, Safdari, Shardt, Kenjereš and Van den Akker2019) in HIT conditions, the length scale at which the DSDs transition into the $d^{-10/3}$ scaling could be predicted by that at which inertia and surface tension become comparable. Therefore, it sounds reasonable that the length scale of transition in HIT conditions can be approximated by the Hinze scale, which is defined by balancing the surface tension force and turbulent energy fluctuation.

However, the situation is different for TC turbulence where strong inhomogeneity and anisotropy exist. It has been recently reported that, in TC flows (Yi et al. Reference Yi, Wang, van Vuren, Lohse, Risso, Toschi and Sun2022), the fluctuation in the bulk is too weak to generate droplets of the observed size. Instead, it was proposed that the droplet fragmentation occurs within the boundary layers and is controlled by the dynamic pressure caused by the gradient of the mean flow, which can be accounted for using the Levich theory for inhomogeneous turbulent flow (Levich Reference Levich1962). This could partially be the reason that, in the present work, the scales of the transition point are smaller than the Hinze scale. For HIT flow, before the DSDs turn to $d^{-10/3}$ scaling behaviour, the $d^{-3/2}$ scaling for the small droplets is supposed to extend to the scale of $d_H$, where the surface tension can no longer sustain the large droplet under fragmentation by the turbulent fluctuations, while for TC flow, droplet fragmentation could occur within the boundary layers, where the fluctuations are much stronger than in the bulk region. In the spirit of Hinze theory, i.e. $d \sim \sigma ^{3/5}\epsilon ^{-2/5}$ (see also (1.1)), the increase in the turbulent fluctuations (and therefore the increase in the dissipation rate $\epsilon$) would lead to the decrease in the critical length scale of stability for droplets of given surface tension. Hence, the droplets in TC flow could become unstable at a smaller size than that in HIT flow, resulting in an earlier transition to $d^{-10/3}$ scaling before the scale of $d_H$. It should be noted that the experimental data reported here follow both scalings only in narrow ranges, typically less than one decade. As we mentioned above, the extension of $d^{-3/2}$ scaling to the extremely small size in this work is mainly limited by the experimental techniques to resolve extremely small droplets. While for the $d^{-10/3}$ scaling, its indistinct emergence could be attributed to the small size of the limited domain in the present work, which prohibits the emergence of large droplets. Additionally, $Re$ in this work is still in a moderate regime, which might result in a relatively large dissipation length scale and thereby suppress the emergence of $d^{-10/3}$ scaling in the range of large scale. It is expected to further verify these two scalings in the future in broader ranges of size by improving the set-up or increasing the turbulent intensity.

So far, one could picture the physical process of the breakup and coalescence of droplets depending on the droplet deformability. (i) Droplets with small $\zeta$ possess higher deformability, promoting large deformation occurring under the turbulent shear. In this case, the coalescence effect is of comparable importance to the breakup effect and therefore is non-negligible. (ii) Droplets with large $\zeta$ are hard to shear to deformation by the turbulence. Once the large droplets exposed to the large shear stress ($\tau \sim \rho {u^\prime }^2d$) are broken, part of the daughter droplets could be stabilized by the strong interfacial tension. This lack of unstable small droplets results in a reduction of the significantly small and large droplets, as confirmed by the ends of the DSDs shown in figure 7(c,d). The shortage of large droplet formation indicates that the coalescence effect becomes negligible, or in other words, the large droplets experience an initial breakup process as assumed by Garrett et al. (Reference Garrett, Li and Farmer2000). At the end of this section, we return to the subject of uncovering the connection between DSDs and emulsion drag ($c_{f,\phi }/c_{f,\phi =0}$). The vanishing of the coalescence effect at higher $\zeta$, as demonstrated by the DSDs analysis, gives rise to an increase of the total surface area. Therefore, the increasing trend of $c_{f,\phi }/c_{f,\phi =0}$ concerning $\zeta$ results from the increase of the interfacial tension contribution to the total stress, as discussed in the previous section. When the interfacial tension effect becomes strong enough to maintain the droplet shape as $\zeta$ further increases, the droplets are hard to deform and behave in a way like a rigid body, accounting for the approaching of $c_{f,\phi }/c_{f,\phi =0}$ of emulsions to that of rigid particle suspensions at large $\zeta$.

3.3. Droplet distribution along the radial direction of the system

It has been found that the magnitude of drag modification is connected to the preferential clustering in rigid particle suspensions (Calzavarini et al. Reference Calzavarini, Cencini, Lohse and Toschi2008; Wang et al. Reference Wang, Yi, Jiang and Sun2022) and bubbly flows (Colin, Fabre & Kamp Reference Colin, Fabre and Kamp2012; van Gils et al. Reference van Gils, Guzman, Sun and Lohse2013; Maryami et al. Reference Maryami, Farahat, Javad poor and Mayam2014; Alméras et al. Reference Alméras, Mathai, Lohse and Sun2017; Mathai et al. Reference Mathai, Huisman, Sun, Lohse and Bourgoin2018). Using a camera mounted on the side of the system, the droplet distribution along the radial direction of the system can be obtained and analysed by analogy with a rigid particle suspension. Since the droplet volume fractions are high, the droplet locations are manually detected at the smallest $Re$ (or $\dot {\gamma }$), where the droplet can be well detected, as mentioned before. The results are shown in figure 8. Although the droplets are almost spherical (see figure 1b), they show a preferential clustering in the bulk region, which has been reported in previous studies (Hudson Reference Hudson2003; Roccon et al. Reference Roccon, De Paoli, Zonta and Soldati2017; Crialesi-Esposito et al. Reference Crialesi-Esposito, Rosti, Chibbaro and Brandt2022) and can be attributed to the combined effect of wall migration and shear-induced diffusion of droplets.

Figure 8.

The droplet distribution along the radial direction of the system for (a) $\phi =2\,\%$ and (b) $\phi =4\,\%$ at $\dot {\gamma }=125$ s$^{-1}$. The distribution of rigid spherical particles taken from Wang et al. (Reference Wang, Yi, Jiang and Sun2022) is depicted for comparison.

Compared with rigid spherical particles, the droplets distribute even more towards the centre of the system and are in line with the results in previous numerical studies (De Vita et al. Reference De Vita, Rosti, Caserta and Brandt2019; Cannon et al. Reference Cannon, Izbassarov, Tammisola, Brandt and Rosti2021), where they report a migration towards the wall by prohibiting the droplet coalescence by applying a collision force. As $\zeta$ increases the droplet distribution migrates towards the walls as the coalescence effect is increasingly suppressed. Instead of near the walls, the droplets show stronger clustering in the bulk than the rigid particles, partly being the reason that $c_{f,\phi }/c_{f,\phi =0}$ of the emulsion flow is smaller than that of the suspension of rigid particles since the larger drag is connected to the near-wall clustering (Wang et al. Reference Wang, Yi, Jiang and Sun2022).