3.1. Statistical properties of droplets at a low volume fraction

The size distribution of dispersed droplets is an important statistical parameter, as it characterizes the microscopic structure of the turbulent emulsion, which closely links to the macroscopic rheological properties and the global transport properties of the fluid system. At a low volume fraction, the droplet sizes in the turbulent emulsion eventually show a statistically stationary distribution for the current experiments under stationary stirring conditions.

Figure 1(b) shows some typical snapshots of the emulsion for three different Reynolds numbers $Re=\omega _{i}r_id/\nu$. Since in these cases the volume fraction of the oil phase is very low ($\phi =1\,\%$), the viscosity of the emulsion is approximately equal to that of the continuous phase, i.e. $\nu =\nu _{w}$. It is expected that the average droplet size of the emulsion at a higher $Re$ will be smaller than that at a lower $Re$. The reason is that the higher average shear strength, in the cases of larger $Re$, promotes the breakup of oil droplets.

Droplet interfaces are extracted from the recorded images, at various Reynolds numbers, and the diameter of all the detected droplets is calculated and normalized with the average droplet diameter, for each $Re$ case, as $X = D/\left \langle D\right \rangle$. The distribution of the number of droplets of size $X$, as a function of $X$, is computed as the probability density function (PDF) of the droplet size and shown in figure 2(a), for various Reynolds numbers. It is clear (solid lines in figure 2a) that the experimental results at all Reynolds numbers can be well described with the log-normal distribution

(3.1)\begin{equation} P(X) = \frac{a}{X\sigma_0\sqrt{2{\rm \pi}}}\exp\left[-\frac{[\log(X) - \log(X_0)] ^{2}}{2\sigma_0^{2}}\right], \end{equation}

where $a$, $X_0$ and $\sigma _0$ are fitting parameters. The deviation from the standard log-normal distribution for points in the range $X > 1.5$, for two cases of high $Re$ ($Re = 2.08\times 10^{4}$ and $2.60\times 10^{4}$), is due to the relatively fewer statistics for large droplet size. These log-normal distributions suggest that fragmentation is the primary process for droplet generation in the current system. Similar fragmentation processes are also observed in other systems (Villermaux Reference Villermaux2007), including plume formation in Rayleigh–Bénard turbulence (Zhou, Sun & Xia Reference Zhou, Sun and Xia2007; Bosbach, Weiss & Ahlers Reference Bosbach, Weiss and Ahlers2012) among others. In addition, it is found that the fitted value of the standard deviation $\sigma _0$ decreases monotonically with increasing $Re$ (see the inset of figure 2a). This means that the distribution of droplet size becomes narrower at higher $Re$, as clearly shown in figure 2(a). Some additional analyses of the distribution of droplet size are provided in appendix C using the gamma distribution function, which is found to be a good description of the droplet breakup during the atomization process (Villermaux Reference Villermaux2007). The next question is what sets the droplet size in the typical size in the fragmentation process leading to droplet formation.

Figure 2.

(a) The PDF of the droplet diameter, with respect to the average diameter, for various Reynolds numbers, $Re$. The solid lines denote the fitting results with a log-normal distribution function. The statistics are based on ${O}(10^{3})$ droplet samples for each $Re$ value. The statistical error bars are shown for all $Re$ cases. The fitted values of the standard deviation $\sigma _{0}$ as a function of $Re$ are shown in the inset. (b) The average droplet diameter normalized by the gap width as a function of the Reynolds number. The blue circles are the data of the droplet diameter for $\phi =1\,\%$ measured in experiments, and the error bars are based on the errors of the edge detection. The red solid line denotes the power-law dependence based on the Hinze relation using the local energy dissipation rate in the bulk (3.2). The black dashed line represents the weighted fit of the experimental data, the weight being based on the relative error of each data point.

In 1955, Hinze proposed that the maximum stable droplet diameter in a homogeneous and isotropic turbulent flow is given by $D = C(\rho _w/\gamma )^{-3/5}\varepsilon ^{-2/5}$, where $\rho _w$ is the density of the continuous phase (the ethanol–water solution in the present case), $\gamma$ is the surface tension between the two phases, $\varepsilon$ is the energy dissipation rate and the coefficient $C = 0.725$ was obtained by Hinze through fitting with experimental data available at that time (Hinze Reference Hinze1955). The argument of Hinze applies to a dilute distribution of droplets that occasionally coalesce due to collisions and break up due to turbulent stresses. A key element of Hinze's argument consists of assuming that droplets do not produce a significant feedback on the turbulent flow, whose statistics are those of homogeneous and isotropic turbulence. Many studies show that the average droplet size and the maximum size are proportional in turbulent emulsions (Lemenand et al. Reference Lemenand, Della Valle, Zellouf and Peerhossaini2003; Boxall et al. Reference Boxall, Koh, Sloan, Sum and Wu2012). Considering that the maximum droplet diameter in turbulent emulsions is usually unstable due to breakup and occasional coalescence, the average droplet diameter can be used as an indicator of the droplet size in the Hinze relation (Perlekar et al. Reference Perlekar, Biferale, Sbragaglia, Srivastava and Toschi2012).

We notice that the distribution of the energy dissipation rate in Taylor–Couette turbulence is inhomogeneous, i.e. the dissipation in the bulk is much smaller than that in the boundary layers. As the volume of the bulk is much larger than that of the boundary layers in the current parameter regime (Grossmann, Lohse & Sun Reference Grossmann, Lohse and Sun2016), droplets are expected to mainly distribute in the bulk, where the flow is found to be nearly homogeneous and isotropic (Ezeta et al. Reference Ezeta, Huisman, Sun and Lohse2018). The local energy dissipation rate in the bulk can be expressed as $\varepsilon _l\sim u_{_T}^{3}/\ell$, where $u_{T}$ and $\ell$ denote the typical velocity fluctuation and the characteristic length scale of the flow. The typical velocity fluctuation $u_{T}$ can be approximated as $A\omega _{i}r_{i}$ in the current Taylor–Couette turbulent flow (van Gils et al. Reference van Gils, Huisman, Grossmann, Sun and Lohse2012) with an almost constant prefactor $A$ (order of 0.1). As we know that the Reynolds number can be expressed as $Re=\omega _{i}r_{i}d/\nu$, then the typical velocity fluctuation can be expressed as $u_T \sim \omega _{i}r_{i} \sim Re\cdot \nu /d$. Substituting this velocity estimation into the expression for the energy dissipation above, we obtain $\varepsilon _l \sim u_{T}^{3}/\ell \sim Re^{3}\nu ^{3}/d^{4}$ by assuming $\ell \sim d$, and this scaling dependence is also in good agreement with the recent measurement of the local energy dissipation rate in the bulk of Taylor–Couette turbulence (Ezeta et al. Reference Ezeta, Huisman, Sun and Lohse2018). Inserting this local energy dissipation expression into Hinze's relation, one obtains

(3.2)\begin{equation} \left\langle D\right\rangle/d \sim C/d\left(\frac{\rho_w}{\gamma}\right)^{{-}3/5}\varepsilon_l^{{-}2/5} \sim Re^{{-}6/5}, \end{equation}

suggesting that the average droplet diameter has a power-law dependence on $Re$ with an effective power-law exponent of $-$1.20 (equation (3.2)). We compare the dependence of the normalized droplet size on $Re$ from the experiments and the model in figure 2(b). The best fit of the experimental data gives a scaling exponent of $-1.18 \pm 0.05$. We find that the scaling dependence based on the local energy dissipation rate in the bulk (red solid line) agrees well with the experimental data. The results show that the scaling dependence of the droplet size on $Re$ could be connected to turbulent fluctuations in the bulk of the system. The discussion above is a simple analysis based on the scaling law. A more in-depth and quantitative understanding of the droplet formation in a turbulent (Taylor–Couette) emulsion flow deserves further studies in the future.

3.2. Effective viscosity and shear-thinning effects

The torque of the inner cylinder is directly measured by the rheometer sensor for different oil volume fractions, $\phi$, and angular velocities, $\omega _{i}$, as depicted in figure 3(a), which shows that the faster the inner cylinder rotates, the larger is the torque needed to maintain the selected angular velocity. The torque becomes larger when the oil volume fraction is increased at a given angular velocity, indicating that the oil additive will bring an obvious change to the rheological property of the emulsion system. Combined with the flow properties of Taylor–Couette turbulence at various Reynolds numbers (van Gils et al. Reference van Gils, Bruggert, Lathrop, Sun and Lohse2011a,Reference Van Gils, Huisman, Bruggert, Sun and Lohseb; Huisman et al. Reference Huisman, Van Der Veen, Sun and Lohse2014; Ostilla-Mónico et al. Reference Ostilla-Mónico, Van Der Poel, Verzicco, Grossmann and Lohse2014; Grossmann et al. Reference Grossmann, Lohse and Sun2016), we can calculate the effective viscosity of emulsions in these dynamical equilibrium states. We use the same method as that recently proposed for viscosity measurements in a very high-Reynolds-number Taylor–Couette flow (Bakhuis et al. Reference Bakhuis, Ezeta, Bullen, Marin, Lohse, Sun and Huisman2020).

Figure 3.

(a) Torque measurements. The torque that is required to maintain the inner cylinder at a constant angular velocity, $\omega _{i}$, is measured for the emulsion systems with oil volume fractions corresponding to $\phi =0\,\%$, 1 %, 5 %, 10 %, 20 %, 30 % and $40\,\%$. As an estimate for the error on the torque measurements, we use the standard deviation and find that it is smaller than $0.8\,\%$ for all volume fractions (see appendix A for details). The error results are therefore smaller than the symbol size. (b) The dependence between the dimensionless torque, $G$, and the Reynolds number, $Re$, using the effective viscosity. All these sets of data at the various oil volume fractions collapse into a master curve, and the error is less than $1\,\%$. The inset shows the dimensionless torque compensated with $Re^{-1.58}$.

An effective power-law dependence between $G$ and $Re$ can be obtained as $G \propto Re^{\beta }$ for the Taylor–Couette turbulent flow, and the power-law exponent $\beta$ depends on the Reynolds number regime (Grossmann et al. Reference Grossmann, Lohse and Sun2016). Here we assume that the power-law dependence $G \propto Re^{\beta }$ can still be applied to the two immiscible liquids in our Taylor–Couette turbulent flow. As a reference case, this relation can be determined by using the results of the pure ethanol–water mixture ($\phi =0\,\%$) with a known viscosity. When we plot together all data for the various oil fractions in a $G$–$Re$ plot, and collapse them on a master curve with an effective exponent of $\beta = 1.58$ (figure 3b), the effective viscosity is a fitting parameter for each case. To demonstrate the quality of the overlap of the different data sets, all data are compensated by $Re^{1.58}$ (inset of figure 3b), which clearly shows that the effective power-law dependence works very well. Remarkably, the power-law dependence $G \propto Re^{1.58}$ for single-phase Taylor–Couette flows still works well for the present two-phase emulsion flows. By using the effective power-law exponent of $\beta = 1.58$ between $G$ and $Re$, we can calculate the effective viscosity of the emulsion at various $\omega _{i}$ and $\phi$ with an expression of $\nu _{eff}=\nu _{w}(\tau /\tau _{w})^{2.38}$; here $\nu _{w}$, $\tau _{w}$ and $\tau$ denote the viscosity of the ethanol–water solution and the measured torques of the ethanol–water solution and the emulsion, respectively. It should be noted that $\nu _{w}$ is known, but $\nu _{eff}$, $\tau _{w}$ and $\tau$ are dependent on the experimental settings, i.e. $\phi$ and $\omega _{i}$. The detailed calculation of the effective viscosity is documented in appendix B.

To understand the effect of oil addition on the rheology of the emulsion in turbulent shear flows, we systematically vary two parameters of the system, i.e. the oil volume fraction, $\phi$, and the angular velocity of the inner cylinder, $\omega _{i}$. The effective viscosity of emulsions, as a function of $\phi$ and for various $\omega _{i}$, is reported in figure 4(a), where all data are normalized by the viscosity of the ethanol–water solution, $\nu _{w}$. Obviously, the effective viscosity of the emulsion increases with increasing oil volume fraction, $\phi$, for all $\omega _{i}$ cases. While the effective viscosity has a weak dependence on $\phi$ in the dilute regime (e.g. for $\phi <5\,\%$), it displays a stronger dependence for larger $\phi$. The hydrodynamic or contact interactions between oil droplets for larger $\phi$ are expected to yield an increasing viscous contribution, somehow similar to what is observed for the case of dispersions of hard spheres in suspensions (Guazzelli & Pouliquen Reference Guazzelli and Pouliquen2018). The relation between the effective viscosity and the volume fraction of dispersed solid particles in particle–fluid suspensions is also plotted in figure 4(a) for comparison. Strictly speaking, we find that the effective viscosities of the emulsions, at all $\omega _{i}$, are smaller than that of the dependence proposed by Krieger & Dougherty (Reference Krieger and Dougherty1959). Here it needs to be emphasized that this model was developed for suspensions of monodispersed hard spheres in fluids in the viscous regime. This disagreement in the viscosity can be due to the different nature of the dispersed phases: the dispersed solid particles have a fixed, undeformable shape, while the dispersed oil droplets can deform; the solid particles have a fixed size, while the droplets can dynamically coalesce and break up under the flow. The dynamics of the dispersed droplets in emulsions is therefore much richer than that of the solid particles in suspensions.

Figure 4.

(a) The effective viscosity normalized by the viscosity of the ethanol–water mixture, $\nu _{eff}/\nu _{w}$, as a function of the volume fraction of the dispersed oil phase $\phi$ at various angular velocity $\omega _{i}$. The solid line denotes the effective viscosity model for solid particle suspensions of Krieger & Dougherty (Reference Krieger and Dougherty1959). The calculation of the effective viscosity is based on the torque measurements. The relative standard deviation is less than $2.5\,\%$, so the errors bars are smaller than the symbol size. (b) The effective viscosity of the emulsion versus the characteristic shear rate $\dot {\gamma }$ of the flow. The data for the different oil volume fractions, $\phi$, are denoted by the open symbols with various colours in the legend. The solid lines show the fitting results (fits 1–7) using the Herschel–Bulkley model (Herschel & Bulkley Reference Herschel and Bulkley1926) for various volume fractions. The inset shows the power-law index $n$ as a function of the volume fraction $\phi$.

Furthermore, the effective viscosity is found to decrease with increasing $\omega _{i}$ for a given $\phi$ as indicated in figure 4(a). In other words, the turbulent emulsion shows a continued shear-thinning behaviour. To reveal this effect better, we plot the effective viscosity as a function of shear rate in figure 4(b), where the shear rate is defined as $\dot {\gamma }=\omega _{i}r_{i}/d$. Though the Taylor–Couette flow is not a planar shear flow, the shear rate $\dot {\gamma }$ can still represent well the effective shear strength of the system. When the volume fraction of oil is $\phi =0\,\%$ (i.e. pure ethanol–water), the system is a single-phase flow state and, as expected, the effective viscosity does not change with the shear rate $\dot {\gamma }$. With the addition of the oil phase, the effective viscosity of the emulsion decreases with increasing $\dot {\gamma }$, and this effect is more pronounced for high volume fractions, as shown in figure 4(b). This shear-thinning behaviour is similar to what was found in a suspension of deformable microgel particles under steady shear flow (Adams et al. Reference Adams, Frith and Stokes2004).

To quantify the shear-thinning effect of the turbulent emulsion, we compare our data with the Herschel–Bulkley model (Herschel & Bulkley Reference Herschel and Bulkley1926):

(3.3)\begin{equation} \mu_{eff}=k_{0}\dot{\gamma}^{n-1}+\tau_{0}\dot{\gamma}^{{-}1}, \end{equation}

where $\mu _{eff}$ is the effective dynamic viscosity, $k_{0}$ and $n$ represent the consistency and the flow index, respectively, and $\tau _{0}$ is the yield shear stress. As the system is far from the jamming state, the yield shear stress is expected to be zero in the current case ($\tau _{0}=0$). Consequently, the Herschel–Bulkley model can then be simplified as $\nu _{eff}/\nu _{w}=K\dot {\gamma }^{n-1}$. The fitting results using the Herschel–Bulkley model for various volume fractions are also shown in figure 4(b). As expected, the flow index is around 1 at very low volume fractions, suggesting that the fluid behaves like a Newtonian fluid. The flow index, $n$, monotonically decreases with increasing volume fraction of dispersed phase (inset of figure 4b), indicating a more pronounced shear-thinning effect for the emulsions with high oil volume fractions. The agreement between the experimental data and the Herschel–Bulkley model indicates that the shear-thinning effect can be well described by this classical non-Newtonian model, opening an important avenue for the description of the effective viscosity of turbulent emulsion systems.