3.1. Flow regimes of Janus drops in shear flow

Figure 2 shows numerical results of Janus drops and a pure drop in shear flows at $Re=0.2$, with respect to velocity vectors and interface shapes. Two flow regimes can be identified, namely rotation and breakup, according to the morphology of the Janus drops. In the rotation regime, the drop rotates periodically, accompanied with continuous drop deformation (cf. $Ca=0.07$ and $0.35$ in figures 2a and 2b, respectively). At relatively small $Ca$ ($=0.07$), a vortex persists inside the rotating Janus drop, with rotation period $t_R\approx 15.8 \dot {\gamma }^{-1}$; at a relatively larger $Ca$ ($=0.35$), the Janus drop experiences more significant deformation and a longer rotation period ($t_R\approx 42.8 \dot {\gamma }^{-1}$). The flow features are also different: only one vortex when the drop is squeezed the most, which also corresponds to the moment of fast rotation; moreover, we can see one vortex contained in each component droplet at the maximum elongation, which implies that the Janus drop rotates very slowly at this moment. As shown in the supplementary movies available at https://doi.org/10.1017/jfm.2023.963, the rotation dynamics of the Janus drop at $Ca=0.35$ is more closely coupled with its deformation than that at $Ca=0.07$. When $Ca$ is increased further (e.g. $Ca=0.40$), the Janus drop could elongate continuously and end up with the separation of the two component droplets (see figure 2c), i.e. the breakup regime. It is noteworthy that the behaviour of Janus drops in shear is remarkably different from that of pure drops. For example, a pure drop (of the same size as the Janus drop, $2{\rm \pi} R^2$) at $Ca=0.07$ is elongated by shear flow, and moreover, its eventual shape does not change with time (see figure 2d).

Figure 2.

Dynamics of Janus drops and a pure drop in shear flow at $Re=0.2$. (a) Rotating Janus drop at ${Ca=0.07}$. The snapshots from (a i–a iv) correspond to times $\dot {\gamma }t = 5.4, 10.3, 13.3, 18.2$, respectively.(b) Rotating Janus drop at $Ca=0.35$. The snapshots are $\dot {\gamma }t=11.3, 24.7, 32.7, 46.1$ for (b i–b iv), respectively. (c) Breakup of Janus drops at $Ca=0.40$. The snapshots correspond to $\dot {\gamma }t=8.0$ and $30.9$, respectively.(d) Deformed pure drop (of size $2{\rm \pi} R^2$) in shear flow at $Ca=0.07$. The reference velocity is shown in (a i).

Figure 3(a) shows the dynamics of a rotating Janus drop at $Re=0.2$ and $Ca=0.07$, with respect to the evolution of $L$ and $B$. We can see that $L$ and $B$ evolve in an anti-phase manner, which means that the maximum value of $L$ ($B$) and the minimum value of $B$ ($L$) occur at the same time. The deformation dynamics of the Janus drop is shown in figure 3(b), with respect to the averaged values ($\bar {L}$ and $\bar {B}$) of the geometry parameters as a function of $Ca$ for different $Re$. We find $\bar {L} \approx L_0$ and $\bar {B}\approx B_0$ at small $Ca$ (e.g. $Ca<0.1$), and $\bar {L}>L_0$ and $\bar {B}>B_0$ at relatively large $Ca$. Herein, we refer to the former as linear deformation, and the latter as nonlinear deformation.

Figure 3.

(a) Temporal evolution of $L$ and $B$ at $Re=0.2$ and $Ca=0.07$. (b) Numerical results for $\bar {L}$ and $\bar {B}$ for different $Re$: 0.1 ($\square$), 0.2 ($\triangledown$), 0.5 ($\vartriangle$), 1.0 ($\diamond$) and 2.0 ($\triangleright$).

In order to assess quantitatively the deformation and rotation of Janus drops, we define a new deformation parameter,

(3.1)\begin{equation} D=\frac{B_0L-L_0B}{B_0L+L_0B}. \end{equation}

It is straightforward to get $D=0$ for a Janus drop at equilibrium such that the maximum value of $D$, $D_{max}$, occurs at the maximum elongation of the Janus drop and yields

(3.2)\begin{equation} D_{max}=\frac{(\bar{L}+\Delta L)B_0-(\bar{B}-\Delta B)L_0}{(\bar{L}+\Delta L)B_0+(\bar{B}-\Delta B)L_0}. \end{equation}

Figure 4 presents the variation of $D$, the rotation velocity $\omega$ ($=\text {d} \theta /\text {d}t$) and the surface energy $E_s$ of Janus drops in one rotation period with respect to the phase $\phi$ at $Re=0.2$ and different $Ca$, including typical cases of drops with linear deformation ($Ca=0.07$) and nonlinear deformation ($Ca=0.35$). Here, $E_s=\int _{S}\sigma \,{\rm d}l$, where $S$ is the area of the drop surface; because $\sigma _{12}=\sigma _{13}=\sigma _{23}$ in this case, $E_s$ is determined solely by the total area of the drop surface. Several observations can be made about figure 4. First, the variation of $\omega$ for drops with linear deformation is close to a harmonic function, and has a noticeable phase difference from $D$. By contrast, the variation of $\omega$ for drops with nonlinear deformation deviates significantly from harmonic functions, and appears to synchronize with the change of $D$. In particular, $\omega$ reaches its peak value at the smallest $D$, and reaches its lowest value (close to zero) at the largest $D$ (namely $D_{max}$). Second, the occurrence of $D_{max}$ coincides with the apex of $E_s$. However, the smallest $D$ corresponds not to the lowest $E_s$, but to the second peak of $E_s$. Note that the lowest $E_s$ corresponds to $D\approx 0$. From the observations above, we can see that the rotation of Janus drops is coupled with drop deformation, and the coupling is strengthened with increasing $Ca$. Third, the compression process ($D<0$) is longer than the elongation one ($D>0$). This tendency is intensified in the regime of nonlinear deformation.

Figure 4.

Variation of $\omega$, $D$ and $E_s$ as functions of phase $\phi$ ($=\theta -{\rm \pi} /2$) in one rotation period, with $Re=0.2$ and two different $Ca$: (a) $Ca=0.07$ and (b) $Ca=0.35$. The variation of $\omega$ (dotted line) is close to a harmonic function (dashed line). The insets show snapshots of the interfacial shapes of the Janus drops at the extreme points of $E_S$.

3.2. Deformation of rotating Janus drops

Deformation of Janus drops reflects the balance between viscous shear, interfacial tension and drop inertia. Figure 5(a) shows that $\Delta L$ of all rotating Janus drops can be correlated with $Re$ and $Ca$ by

(3.3)\begin{equation} \frac{\Delta L}{R} \propto (1+\beta\,Re)\,Ca, \end{equation}

where $\beta =0.3$ is a fitting parameter. Similarly, $\Delta B/R$ is found to have the same proportionality as in (3.3); see the inset of figure 5(a). Moreover, the value of $\beta$ ($=0.3$) is also the same for $\Delta B$.

Figure 5.

(a) Variation of $\Delta L$ as a function of $(1+\beta \,Re)\,Ca$ on a log-log scale. Variation of $\Delta B$ is shown in the inset. Different symbols indicate various Reynolds numbers $Re$: $0.1$ ($\square$), 0.2 ($\triangledown$), 0.5 ($\triangle$), 1.0 ($\lozenge$) and 2.0 ($\triangleright$). (b) $D_{max}$ as a function of $(1+\beta \,Re)\,Ca$. The solid line represents the (3.5) with $\beta =0.3$.

To understand the correlation between $\Delta L$, $Re$ and $Ca$ in (3.3), we model the deformation of Janus drops by a one-dimensional mass–spring system. In this system, the viscosity and inertia of the drop play the roles of the damping elements and the mass, and the surface tension plays the role of the spring. Given the deviation of a Janus drop from its original shape as $L(t)-\bar {L}$, the scales of inertia, viscous damping and surface tension of the Janus drop can be written as $\rho R^3\ddot {L}$, $\mu R \dot {L}$ and $\sigma (L-\bar {L})$, respectively. Therefore, the mass–spring system can be expressed as

(3.4)\begin{equation} \rho R^3\ddot{L}+\mu R \dot{L} +\sigma (L-\bar{L}) \sim \mu R^2\dot{\gamma}+\rho R^4\dot{\gamma}^2, \end{equation}

where the two terms on the right-hand side represent the effects of viscous stress and pressure exerted by the surrounding fluid, respectively. Note that $\ddot {L}$ has scale $\Delta L/t_R^2$ and $t_R \gg \dot {\gamma }^{-1}$. Therefore, the effect of drop inertia ($\sim \rho R^3\, \Delta L\,t_R^{-2}\ll \rho R^3\,\Delta L\,\dot {\gamma }^2$) is comparatively negligible. In addition, when the Janus drops are most elongated, we have $\dot {L}=0$ and $L-\bar {L}=\Delta L$ (by definition). Consequently, at this particular moment, (3.4) can be simplified in dimensionless form as $\Delta L/R \sim Ca+Re\,Ca$. We note that on the right-hand side of (3.4), the difference in the effective action area between the viscous stress and the pressure is not taken into account. If considering this geometry effect further, then we have $\Delta L/R \propto (1+\beta \,Re)\,Ca$, where $\beta$ is a geometry factor related to the shape of Janus drops. The value of $\beta$ is more or less a constant as indicated by the numerical results (see figure 5a).

Figure 5(b) shows the numerical results of $D_{max}$ as a function of $(1+\beta \,Re)\,Ca$ on a log-log scale, including those of pure drops at $Re=0.1$. The results of stationary pure drops and rotating Janus drops at the maximum elongation collapse onto a single curve. Moreover, $D_{max}$ is proportional to $(1+\beta \,Re)\,Ca$ for $D_{max}\le 0.2$. This can be explained by simplifying the maximum value of $D$ for Janus drops with linear deformation (denoted by $D_{max}^{L}$). Because of $\bar {L}\approx L_0$, $\bar {B}\approx B_0$ and $|B_0\,\Delta L-L_0\,\Delta B|\ll 2L_0B_0$ (which can be derived from $D_{max}\ll 1$), we can obtain $D_{max}^{L}\approx (\Delta L/L_0+\Delta B/B_0)/2$. Given the correlation of $\Delta L$ and $\Delta B$ with $Re$ and $Ca$, we can obtain directly

(3.5)\begin{equation} D_{max}^{L} \propto (1+\beta\,Re)\,Ca. \end{equation}

Rotating Janus drops can thus be classified quantitatively according to the value of $D_{max}$, i.e. $D_{max}\le 0.2$ for linear deformation, and $D_{max}>0.2$ for nonlinear deformation. More precisely, the linear dependence of $D_{max}$ on $Re$ and $Ca$ holds for $(1+0.3\,Re)\,Ca \le 0.1$ in the present study. We note that the linear relation in (3.5) is also found to be valid for Janus drops with different initial shapes (by changing $\sigma _{12}/\sigma _{13}$), in the case of small deformations.

3.3. Coupling between rotation period and drop deformation

Figure 6(a) shows the variation of $t_R$ for Janus drops at $Re=0.2$ with different equilibrium shape (measured by $L_0/B_0$). For Janus drops with linear deformation (e.g. $Ca=0.02$), $t_R$ increases only slowly with $L_0/B_0$, which is very different from the effect of aspect ratio of particles on $t_R$, as given in (1.1a,b). By contrast, a quick increase of $t_R$ with $L_0/B_0$ is observed for those with nonlinear deformation (e.g. $Ca=0.1$ and $L_0/B_0>2$). Therefore, for Janus drops, the change of aspect ratio due to drop deformation determines the rotation period, rather than the equilibrium shape of the drop. Figures 6(b) and 6(c) show the variation of $t_R$ and $D_{max}$ for Janus drops with different viscosity. At low $Ca$ ($<0.1$), $t_R$ and $D_{max}$ maintain more or less the same value, although the viscosity of Janus drops undergoes a change of two orders in magnitude. At relatively high $Ca$ ($>0.1$), the Janus drop with higher viscosity (e.g. $\mu _1/\mu _3=10$) can resist the shear flow with less deformation, leading to an insignificant increase of $D_{max}$ (and $t_R$) with $Ca$. Figure 7(a) shows the variation of $t_R$ with $Ca$ at different $Re$. At the same $Re$, $t_R$ maintains an approximately constant value at low $Ca$, and increases rapidly with $Ca$ at relatively large $Ca$ (i.e. for drops with nonlinear deformation).

Figure 6.

(a) Variation of $t_R$ for Janus drops with different $L_0/B_0$ at $Re=0.2$ and $\mu _1/\mu _3=1$. The empty symbols represent the nonlinear deformation, and the filled symbols represent the linear deformation. Variation of (b) $t_R$ and (c) $D_{max}$ at $Re=0.2$ for $L_0/B_0$ ($=1.732$) and various $\mu _1/\mu _3$: 0.1 ($\square$), 0.5 ($\triangle$), 1 ($\triangledown$), 2 ($\lozenge$) and 10 ($\bigcirc$).

Figure 7.

(a) Rotating period as a function of $Ca$, with $\mu _1=\mu _3$ and various $Re$. (b) The comparison of $t_R$ between numerical results in (a) and theoretical prediction of (3.8) (denoted by the solid line). The inset shows the results for different $\mu _1/\mu _3$ with $Re=0.2$. In (a,b), the filled and empty symbols denote the cases $D_{max}\le 0.2$ and $D_{max}>0.2$, respectively.

The rotation of Janus drops is similar to that of particles, except for flow slip at the interfaces and deformable shape. The slip effect at the drop interface can be approximated by a corrected shear rate $\dot {\gamma }/\xi$, where the correction is $\xi \ge 1$. Based on theoretical analysis of $t_R$ of a fluid sphere and a spherical particle in shear flow at $Re=0$, Bartok & Mason (Reference Bartok and Mason1958) obtained $\xi =2/\sqrt {3}$ for the fluid sphere with matched viscosity, which is adopted in the present study. To take the effect of finite inertia further into account, $\xi$ can be approximated to the first order of $Re$ by

(3.6)\begin{equation} \xi\approx\frac{2}{\sqrt{3}}+\frac{\text{d}\xi}{\text{d}Re}\,Re. \end{equation}

The deformations of rotating Janus drops are characterised by the maximum elongation. An effective aspect ratio of Janus drops, $\lambda _J$, is thus defined by the geometry parameters of Janus drops at that moment:

(3.7)\begin{equation} \lambda_J=\frac{(\bar{B}-\Delta B)L_0}{(\bar{L}+\Delta L)B_0}=\frac{1-D_{max}}{1+D_{max}}. \end{equation}

In this aspect ratio model, it is straightforward to have $\lambda _J=1$ for Janus drops at equilibrium (i.e. $D_{max}=0$) regardless of their differences in shape, which reflects the dominant effect of drop deformation on the rotation period.

Jeffery's theory (1.1a,b) was shown to be applicable to elliptical cylinders (Ding & Aidun Reference Ding and Aidun2000; Zettner & Yoda Reference Zettner and Yoda2001). Following the spirit of Jeffery's theory, the rotation period of 2-D Janus drops can be obtained by introducing a corrected $\dot {\gamma }$ and an effective aspect ratio $\lambda _J$. That is,

(3.8)\begin{equation} t_R\dot{\gamma}=2{\rm \pi}\xi(\lambda_J+\lambda_J^{{-}1})=4{\rm \pi}\xi\,\frac{1+D_{max}^2}{1-D_{max}^2}. \end{equation}

Figure 7(b) shows the variation of $t_R\dot {\gamma }/\xi$ as a function of $D_{max}$. Numerical results for Janus drops agree well with the theoretical prediction with a fitting of $\text {d}\xi /\text {d}Re=0.3$. Because (3.8) is consistent with the calculation of $t_R$ in (1.1a,b), the rotation period of particles and Janus drops can be predicted in a unified manner.

Similarly, we can obtain the theoretical prediction of $\theta$:

(3.9)\begin{equation} \theta-\theta_0=\arctan\left(\frac{1+D_{max}}{1-D_{max}}\tan\left(\frac{\dot{\gamma}(t-t_0)}{2\xi}\,\frac{1-D_{max}^2}{1+D_{max}^2}\right)\right), \end{equation}

where an initial value $\theta _0$ ($={\rm \pi} /2$) and the corresponding time $t_0$ are introduced to avoid the start-up effect in numerical simulations. Figure 8 shows the time variation of $\theta$ of rotating Janus drops at $Re=0.2$ and different $Ca$. The numerical results agree well with the theoretical prediction for drops with linear deformation ($Ca=0.07$), but deviate from it slightly for drops with nonlinear deformation ($Ca=0.15$ and $0.25$), owing to the synchronization of rotation velocity $\omega$ with drop deformation.

Figure 8.

Comparison between the results of simulation and (3.9) of $\theta -\theta _0$ as a function of $\dot {\gamma }(t-t_0)$ at $Re=0.2$ and various $Ca$.