3.1. Droplet formation and characterisation

We first discuss trends related to droplet generation including how the pinch-off process influences the final droplet size. Figure 2(a) depicts the thinning of the ligament through a long exposure image (created by projecting several frames pre-pinch-off) where the region near the neck has longer streaks compared with other regions, indicating a larger velocity. Figure 2(b) illustrates the forces that act on a droplet at various stages and the directional tendency of interface movement. In the dripping regime (in which the experiments are conducted), the dispersed phase is initially convected with the continuous phase fluid, resulting in its stretching. A blunt nose is formed at the tip of the dispersed fluid (see figure 1 b), which is subjected to shear drag and pressure build-up from the channel restrictions (Kalli et al. Reference Kalli, Pico, Chagot, Kahouadji, Shin, Chergui, Juric, Matar and Angeli2023). As the droplet expands, the drag force on the droplet increases (Roumpea et al. Reference Roumpea, Kovalchuk, Chinaud, Nowak, Simmons and Angeli2019), and a reversal of flow is observed at the necking zone (figure 2 a,b), marking the beginning of necking. Then, the interfacial tension begins to pull the interface backward (figure 2 b). Finally, the cylindrical thread is subjected to the capillary (Rayleigh–Plateau) instability, resulting in pinch-off (Baroud et al. Reference Baroud, Gallaire and Dangla2010).

Figure 3.

(a) Time evolution of $\omega ^*$ for $\varphi = 4$ and $\varphi = 10$ . The scale bar represents 50 $\rm \unicode{x03BC} m$ . (b) Decay mechanism of the post-pinch-off vortex during the transient state when the droplet shape is unstable. The purple coloured arrows on the interface depict the tendency of interface motion and the grey zone represents the region being deformed. The yellow vector represents the velocity of the trailing end of the drop interface $(U_{\textit{int}})$ and the blue vector represent the peak velocity $(U_{\textit{peak}})$ inside the droplet, which is typically achieved near the trailing end.

Figure 3. Long description

Panel A: A series of heat maps showing the time evolution of vortices in droplets with phi values of 4 and 10. The color scale ranges from -0.55 (red, CW) to 0.55 (blue, CCW), indicating the direction and magnitude of vorticity. The x-y plane is shown for different time points (T* = 0, 0.56, 1, 1.74, 2.6). The arrows indicate the direction of vortex movement. Panel B: A schematic diagram illustrating the decay mechanism of the post-pinch-off vortex during the transient state when the droplet shape is unstable. The purple arrows depict the interface motion, and the grey zone represents the deformed region. The yellow vector represents the velocity of the trailing end of the drop interface, and the blue vector represents the peak velocity inside the droplet.

Figure 2(c) shows the variation in the dimensionless droplet diameter ( $D^{*}=D_{{d}}/{\mathrm{W}}_{{n}}$ ) and $t_{\textit{pinch}}$ with increasing $Ca_{{c}}$ and $\varphi$ . As has been found in previous studies (Nie et al. Reference Nie, Seo, Xu, Lewis, Mok, Kumacheva, Whitesides, Garstecki and Stone2008), increasing $Ca_{{c}}$ results in a decrease in diameter. For the present device and conditions, $D^{*}$ is seen to decrease as $Ca_c^{-1/3}$ (similar to what was observed by Lee et al. Reference Lee, Walker and Anna2009). This decrease in droplet diameter is due to the fact that, with an increase in $Q_{{c}}$ or $\varphi$ , the shearing effects increase, leading to more rapid thinning of the neck. A faster thinning implies less volume accumulation and hence smaller droplets. The decrease in $t_{\textit{pinch}}$ (figure 2 c,d) with increasing $\varphi$ is consistent with this explanation. It should also be noted that the dimensionless droplet length $L^*_{\textit{drop}}(=L_{\textit{drop}}/W_{{n}})$ in figure 2(d) is plotted against $t^*$ , that is defined from the point where the dimensionless ligament width $W_{\textit{lig}}^*$ $\approx 0.4$ (figure 2 e). This reference ( $W_{\textit{lig}}^*\approx 0.4$ ) comes from an experimental limitation i.e. smaller FOV when recording at 25 000 Hz to capture the ligament dynamics, as mentioned in § 2.

Before pinch-off, $L^*_{\textit{drop}}$ for $\varphi =$ 4 and 6 exhibits a similar slope whereas the higher $\varphi$ cases (8 and 10) display a sharper slope that increases monotonically with $\varphi$ . Post-pinch-off, the droplet length decreases rapidly toward a fixed length in the stable state. This is primarily because of the surface tension of the deformed pinch-off droplet. Further, it can be seen from figure 2(e) that $W_{\textit{lig}}^*$ initially decreases linearly with different slopes for smaller $\varphi$ (4 and 6) and larger $\varphi$ (8 and 10). A change in the slope is then observed, followed by the final decreasing slope that is identical for all $\varphi$ , suggesting a capillary dominated break-up, independent of the value of $Q_{{c}}$ . Hence, the pinch-off can be understood as comprising two stages, the initial stage is dependent on $\varphi$ i.e. the drag, and the second stage is capillary dominated. Interestingly, even in turbulent conditions it is seen that the final breakup process is capillary dominated (Eastwood et al. Reference Eastwood, Armi and Lasheras2004; Zhong & Ni Reference Zhong and Ni2024). In the literature, ligament thinning is presented in terms of the minimum neck width that exhibits a power-law relation in time for various regimes (Castrejón-Pita et al. Reference Castrejón-Pita, Castrejón-Pita, Thete, Sambath, Hutchings, Hinch, Lister and Basaran2015). However, we do not provide such arguments in order to align with the current focus, i.e. investigating the generation of vorticity due to pinch-off rather than the pinch-off itself.

3.2. Vortex dynamics: post-pinch-off vortex

The necking phenomenon results in a bi-directional acceleration of the fluid being evacuated out of the thinning ligament (Castrejón-Pita et al. Reference Castrejón-Pita, Castrejón-Pita, Thete, Sambath, Hutchings, Hinch, Lister and Basaran2015), as shown in figure 2(a). The fluid accelerating in the forward direction flows into the droplet, and the maximum velocity of the fluid is reached at the time of pinch-off. This fluid accelerates into a comparatively quiescent droplet, resulting in vortex formation, as is schematically depicted in figure 2(b) and experimentally observed in figure 3(a). This strong fluid acceleration is due to rapid neck contraction, leading to pinch-off and the very low radius of curvature at the remnant neck thereafter, which results in a high Laplace pressure (figure 2 b) imparting a strong internal flow into the droplet. The resulting vortex remains confined at the trailing tip of the droplet during pinch-off and dies out as the droplet approaches an equilibrium shape. We now focus on this post pinch-off vortex and the transient vortex dynamics of its decay.

The pinch-off of the droplet simultaneously disrupts the supply of vorticity. Subsequent to pinch-off, the pointed tail (i.e. the trailing end) of the droplet rapidly retracts, leading to a sharp decrease in droplet length (see figure 2 d) and a more circular shape, approaching that of the final steady-state form. The vortex created in the tail of the droplet during the pinch-off process starts to decay as soon as the pinch-off occurs, as can be seen in figure 3(a). This decay occurs over a time scale $ T^*\sim 2.3$ , which is much shorter than the time taken for the droplet to minimise its surface energy and become stable in a circular shape. The length of the droplet is much larger for $\varphi = 4$ (as can be deduced from figure 2 d) for which the vorticity zone is spread over a smaller area compared with $\varphi = 10$ , where the area covered is more than 50 % of the droplet size. It is worth noting that most studies dealing with mixing inside micro-droplets ignore this vortex and study only the steady-state vortex (as shown in figure 1 b), which exhibits vorticity in the opposite sense after the decay of the post-pinch-off vortex.

Figure 4(a) quantitatively shows the evolution of vorticity in the droplet before and after pinch-off. For all values of $\varphi$ , $\omega ^*(=\omega t_{\textit{cap}})$ is seen to increase until the pinch-off time, after which it decays with time as $\omega ^* \sim (T^*)^{-1}$ , with an accommodated time lag of $T_{{o}}^* (\sim O(10^{-4}\,\rm s))$ . The vorticity is decreased by the continuous phase fluid acting in the opposite direction at the interface. This causes the vorticity to decay in magnitude, as depicted in figure 4(b). We observe minimal variation in the vorticity values on changing $\varphi$ (figure 4 b). This is because the vorticity generation is primarily a result of the final thinning of the ligament that exhibits a similar rate of contraction (figure 2 e), thereby making the vorticity values independent of the variation in $\varphi$ . This is governed by the capillary time scale, which is a constant for the chosen set of parameters. Further discussion on this mechanism is given later in this section.

Figure 4.

(a) Evolution of $\omega ^*$ during the necking phase. (b) Evolution of $\omega ^*$ post pinch-off. The dotted line represents a fitted curve. (c) Evolution of $U^*_{\textit{int}}=U_{\textit{int}}(t_{\textit{pinch}}/D_{{d}})$ post pinch-off. (d) Control volume for the model presented in (3.4). (e) Evolution of $\varGamma ^*$ during the necking phase. The dashed line represents the peak value of $\varGamma ^*$ presented as the average of each case using (3.4). Here, $U_{\textit{peak}}$ represents the maximum value of the peak velocities for each $\varphi$ post pinch-off. (f) Evolution of $\varGamma ^*$ post pinch-off. The dot-dashed line represents the average of all the cases. (g) The trailing end interface velocity ( $U_{\textit{int}}^*$ ) plotted against the peak velocity ( $U_{\textit{peak}}^*=U_{\textit{peak}}(t_{\textit{pinch}}/D_{{d}})$ ) inside the pinched-off droplet. The green marked zone inside the droplet depicts the area over which the corresponding parameters are evaluated. $T^* = 0$ marks the pinch-off point.

The vortex formed in the post-pinch-off droplet is symmetric about the x-axis (figure 3). Further, owing to the geometry of the channel, we assume the three-dimensional effects to be minimum; hence, stretching of the vortex structure along its vorticity vector is expected to be insignificant. If we scale the core vorticity with the velocity gradients as

(3.1) \begin{equation} \omega \,\sim \frac {U_{{r}}}{L_{{r}}}, \end{equation}

where r as subscript refers to the reference scale, then $U_r$ should scale with the velocity of the trailing end velocity ( $U_{\textit{int}}$ ) and the length scale $L_{{r}}$ will be the viscous length scale i.e. $L_{{r}} \sim t^{1/2}$ . This is because, at the interface, a slip condition is expected, leading to a finite tangential interface velocity. The imposition of the shear from the continuous fluid creates a region with localised velocity gradients near the interface and a jump in velocity gradient is expected across the interface which depends on the viscosity ratio (Ma et al. Reference Ma, Sherwood, Huck and Balabani2014). From the shear matching condition at the interface we have

(3.2) \begin{equation} \left. \mu _{{c}}\frac {\partial u}{\partial y}\right|_{\textit{out}}= \left. \mu _{{d}}\frac {\partial u}{\partial y}\right|_{\textit{in}} , \end{equation}

where the subscript ‘out’ refers to continuous fluid and ‘in’ refers to the dispersed fluid. From equation (3.2), it can be deduced that the velocity gradient inside the dispersed phase droplet will be much larger than in the continuous phase gradient. The magnitude of the inner velocity gradient decreases with distance from the interface (depicted in figure 3 b) and is further decreased by the vortex rotating in the opposite sense. Moreover, the pronounced deformation of the interface at the trailing end post pinch-off will create vorticity because of interface motion. This vorticity must then diffuse into the interior of the droplet (Song & Tryggvason Reference Song and Tryggvason1999). While the specific role of vorticity generated by the interfacial motion could not be definitively identified, observations during the decay phase show the vortex moving away from the trailing end, diffusing inside the droplet, as schematically shown in figure 3(b). Although capturing velocity gradients near the interface is experimentally challenging, their presence is indisputable and they have a considerable impact on the vortex dynamics. The interaction of these gradients near the inner interface with the generated vortex results in its decay. It is further seen (figure 4 c) that the decay of the trailing end interfacial velocity ( $U_{\textit{int}}$ ) follows the scaling $U^*_{\textit{int}}\sim (T^*)^{-1/2}$ . It has been previously shown (Shikhmurzaev Reference Shikhmurzaev2000) for capillary pinch-off of a liquid neck that the retraction velocity of the cusp post pinch-off follows a similar scaling as observed here. The shape of the interface during the pinch-off process, being self-similar, imposes that the tip position scale as $t^{1/2}$ (Eggers Reference Eggers1997), resulting in the retraction velocity decreasing as a power of $1/2$ . Physically, just after pinch-off, the trailing end of the detached droplet forms a highly curved cusp with large capillary pressure. This drives local interface deformation, smoothing the cusp and causing capillary-dominated retraction. As the interface flattens over time, curvature and driving force decrease. Consequently, the tip velocity $(U_{\textit{int}})$ gradually declines, following the stated power law. In our case, $U^*_{\textit{int}}$ appears to be the most relevant velocity scale post pinch-off, since the capillary forces are high at the trailing end at this period, which is expected to affect $\omega ^*$ inside the droplet. Using the relation (3.1) yields the observed scaling for the core vorticity in dimensional form, i.e.

(3.3) \begin{equation} \omega \,\sim \frac {U_{{r}}}{L_{{r}}}\sim \frac {U_{\textit{int}}}{{(\nu t)^{1/2}}}\sim \frac {t^{-1/2}}{t^{1/2}}\sim t^{-1}. \end{equation}

Moreover, $U^*_{\textit{int}}$ has been rendered non-dimensional with a velocity scale $(D_{{d}}/t_{\textit{pinch}})$ that is dependent on $\varphi$ (figure S3, in supplementary data) resulting in a collapse of all the measured data (figure 4 c). The scalings obtained experimentally for $\omega ^*$ indicate that there is a time lag in the scaling of $\omega ^*$ relative to the $U^*_{int}$ scaling that is used to explain the former. To justify this, we define a viscous time scale $\tau _{{vis}}={\mu _{{d}}}/{\rho _{{d}} U_{\textit{int}}^2}$ (Ni Reference Ni2024) for the time period of retraction just after pinch-off. In this phase, we expect a high viscosity, high velocity condition, since the length scales are extremely small with very high capillary and viscous forces. The order of the time lag seen for $\omega ^*$ is $O(10^{-4}\,\rm s)$ , which is of the same order as $\tau _{{vis}}$ with $U_{\textit{int}}\sim (10^{-1} \,\rm m\,s^{-1})$ . Hence, it can be inferred that the interface velocity is imposed on the vorticity decay with a time lag that responds to the relaxation of the interface just post pinch-off. Moreover, the time taken for the vortex core to diffuse in the experiments was observed to be $O(10^{-4})$ s, which is of the same order as the diffusive time scale defined as $L^2/\nu$ considering the radius of the droplet as the characteristic length scale. This justifies the choice of diffusive length scale in 3.3. Furthermore, it is also seen that $t_{\textit{cap}}$ is of the same order as the diffusive time scale i.e. $O(10^{-4})$ . Hence, a ‘cause and effect’ dynamics can be established between them. This retraction of the trailing end acts like an impulsive force, the effect of which is seen on the core vorticity after the characteristic time described above.

Furthermore, the evolution of core circulation ( $\varGamma ^*$ ) is depicted in figure 4(e, f). As expected, $\varGamma ^*$ increases up to the pinch-off time ( $T^*=0$ ). Post pinch-off (figure 4 f), the circulation is seen to decrease weakly. The parameter $\varGamma ^*$ being the area integral of vorticity, indicates that the encompassed area of the vortex is increasing, since the vorticity level is decreasing (figure 4 b). We predict the scale of the maximum circulation ( $\varGamma ^*_{\textit{peak}}$ ) using (3.4) and represent the average of each case in figure 4(f). We consider a control volume (figure 4 d) around the neck of the droplet and the constant decrease of its radius during the necking phase, as observed in figure 2(e), to determine the volume flux of vorticity in the axial direction. We assume that the length of the control volume follows the imposed characteristic length $W_{\textit{lig}}$ . Then the following formulation with the scaling constant as unity for the vorticity flux through a slug approximation (Dabiri & Gharib Reference Dabiri and Gharib2005) yields ( $\varGamma ^*_{\textit{peak}}$ ):

(3.4) \begin{equation} \frac {{\rm d}\varGamma _{{s}}}{{\rm d}t} \sim \frac {1}{2}U_{\textit{ejec}}^2(t)\sim \frac {1}{2}\left ( \frac {\varOmega (t)}{A(t)}\right )^2\sim \frac {1}{2}\left (\frac {\varOmega ^2(t)}{(\pi W_{\textit{lig}}^2/4)^2(t)}\right )\!, \end{equation}

where $\varGamma _{{s}}$ represents the circulation values predicted using the slug flow approximation, $U_{\textit{ejec}}$ is the velocity with which the accelerated fluid comes out of the necking zone in the droplet, $\varOmega$ is the volumetric flux obtained using $W_{\textit{lig}}$ and A is the cross-sectional area of the thinning ligament. The above formulation is applied only during the necking phase, where the rate of thinning is found to be constant, and feeding to the droplet is cut off from the upstream part. The data obtained from figure 2(e) are used to find $\varOmega$ and A(t). A time integral then yields the variation of $\varGamma ^*$ in time and the order of $\varGamma ^*_{\textit{peak}}$ . It can be seen in figure 4(e, f) that the average of the maximum predicted value lies close to the maximum obtained by averaging the experimental data for each $\varphi$ . It is worth noting that the slug model, that is generally an inviscid approach, predicts the maximum value to the correct order and the viscosity will appear as an additional correction term, which can be accounted for in a more exact representation of the circulation production (Dabiri & Gharib Reference Dabiri and Gharib2004; Krieg & Mohseni Reference Krieg and Mohseni2013). On defining a local Reynolds number at the instance we apply the slug flow, we get

(3.5) \begin{equation} \textit{Re}_{\textit{slug}} = \frac {\rho _{{d}}U_{\textit{ejec}}W_{\textit{lig}}}{\mu _{{d}}} \approx 3.4. \end{equation}

The length scale is taken at the point where we start considering the values of $W_{lig}$ and $U_{\textit{ejec}}$ is the velocity of a particle near the thinning zone in the ligament found using the particle tracking methodology (as discussed in § 2), which was found to be of same order until the pinch-off. As can be seen, we get significant fluid inertia during this period, which has also been reported before (Castrejón-Pita et al. Reference Castrejón-Pita, Castrejón-Pita, Thete, Sambath, Hutchings, Hinch, Lister and Basaran2015), suggesting the domination of inertial forces over viscosity during the small time period. The value of $u$ across the centreline is shown in figure S4 (in supplementary) and also indicates high acceleration. This is why (3.4) predicts the circulation values to the correct order. However, although the viscosity is not dominant, it is important since the phenomenon takes place on a time scale similar to the order of $\tau _{{vis}}$ . Hence, to come to an exact solution we need to incorporate viscous effects into the equation, that remains an open problem.

We further observe a linear correlation (figure 4 g) between $U_{\textit{int}}^*$ and the peak velocity ( $U_{\textit{peak}}^*$ inside the droplet, as depicted in figure 3 b) especially when these normalised velocities are $\leqslant 1.1$ , as highlighted through the orange zone in figure 4(g), which represents later stages. This proves $U_{\textit{int}}^*$ to be a suitable choice for the velocity scale, imposed earlier to explain the vortex dynamics. Similar to $U_{\textit{int}}^*$ , the peak velocity i.e. $U_{\textit{peak}}^*$ , is also rendered non-dimensional using the ratio $(D_{{d}}/t_{\textit{pinch}})$ , where $t_{\textit{pinch}}$ is the pinch-off time for the respective $\varphi$ . This ratio has an increasing trend (figure S3 in supplementary data) with increasing $\varphi$ . The data show excellent collapse among various values of $\varphi$ for lower values of velocity (or in the later stages post pinch-off). The correlation becomes weaker for higher values, with $U_{\textit{int}}^*$ being larger than $U_{\textit{peak}}^*$ . The deviation of the correlation from the $45^\circ$ line is because post pinch-off it is expected to have a residual of the peak velocity within the bulk and the newly created surface motion will have a different velocity in the early stages since the interface relaxation is dominated by capillary effects. However, in the later stages the influence of the interfacial motion is diffused within the droplet bulk, therefore it is expected that the two velocities are correlated.

The interface motion is a resultant of the complex force balance arising from capillary-, shear- (viscous) and pressure-driven effects, and the displaced fluid (due to interface motion) within the droplet will diverge in the y-direction as well, the intensity of which depends on the amount of confinement or the droplet size, among other effects. Since these forces and the droplet sizes are dependent on $\varphi$ , the slope of the curves in figure 4(g) are seen to slightly vary with $\varphi$ . But this variation is not significant in the present conditions. It is evident that the high surface tension force post pinch-off at the trailing end governs the vorticity and the peak velocity decay.

Figure 5.

(a) Evolution of $\omega ^*$ for the retracting ligament vortex. The marked zone inside the droplet depicts the state of the droplet for which data are taken. (b) Evolution of $\varGamma ^*$ for the retracting ligament vortex. Here, $u_{{avg,peak}}$ represents the maximum of the average value of the velocity in the retracting ligament fluid, $T^* = 0$ marks the pinch-off point.

Figure 5. Long description

Two scatter plots are presented side by side. Panel A on the left shows the relationship between T* and ω* = ωt_cap, while Panel B on the right shows the relationship between T* and Γ* = Γ/u_avg,peakD_d. Both plots have T* on the horizontal axis, ranging from 0.8 to 2.8. Panel A has ω* = ωt_cap on the vertical axis, ranging from 0 to 0.6. Panel B has Γ* = Γ/u_avg,peakD_d on the vertical axis, ranging from 0 to 7. Each plot contains data points for different values of φ, represented by different symbols and colors: blue circles for φ = 4, blue squares for φ = 6, red triangles for φ = 8, and purple inverted triangles for φ = 10. Both plots show a general trend of decreasing values as T* increases. Panel A includes a dashed line indicating a (T*)^(-1) relationship, and both panels have shaded regions near the lower left corner.

3.3. Vortex dynamics: retracting and advancing ligament vortex

The bi-directional acceleration of the fluid during pinch-off results in a vortex in the retracting fluid as well. This vortex decays with the same scaling as observed for the post-pinch-off vortex (figure 5 a). This decay spans a period of time ( $\sim T^*=2.7$ ) beyond which the values begin to stagnate. The corresponding circulation ( $\varGamma ^*$ = $\varGamma$ / $u_{{avg,peak}} D_{{d}}$ , where $u_{{avg,peak}}$ is the peak of the average velocity values) decays linearly with time, exhibiting a reduction in slope beyond a point, as shown in figure 5(b). An average value of velocity is used to render the circulation dimensionless for this case. We observe a collapse of all the values when normalising using the peak of average value and the steady-state droplet diameter that varies with $\varphi$ .

Figure 6(a) depicts the vorticity contours for the transient advancing ligament, with $T^* = 0$ being the point of pinch-off for $\varphi = 8$ . Figure 6(b) shows $\omega ^*$ for the corresponding contour plots shown in figure 6(a). These vorticity values are determined only for the central recirculating zone and not for the gradients near the interface. We discuss the phenomenon only for the region above the symmetry line, as demarcated in the first contour plot. As can be seen, beyond pinch-off, $\omega ^*$ decreases slightly until $\sim$ $T^* = 13$ (this is continued from the retracted ligament vortex decay), beyond which there is a steep rise. Interestingly, during the rise, we observe an opposite vorticity gradient developing near the interface region (prominent at $T^* = 41, 59$ ). The interface changes its curvature due to pressure from the channel neck and continuous fluid (see the difference in curvature in figure 6(a): $T^*=1.8$ and $T^*=59$ ). The interface motion in a downward direction induces a CCW vorticity near the interface, consistent with what is discussed in the existing literature (Song & Tryggvason Reference Song and Tryggvason1999; Terrington et al. Reference Terrington, Hourigan and Thompson2020). This CCW vorticity increases in strength from the first instance until the curvature changes sharply. However, this is impeded by the CW rotating vortex generated during the pinch-off period. Notably, this retracting ligament vortex at the central region is still being fed by the fluid travelling along the interface (driven by shear) within the dispersed fluid and curling back from the leading end, resulting in its vorticity enhancement. This outer CCW vorticity near the interface is seen to decrease in strength once the dispersed phase completely fills the neck of the channel, i.e. when drag forces dominate over the capillary forces. We see a prominent circulating zone again ( $T^* = 84$ ) in the bulging part of the dispersed phase fluid, i.e. the advancing ligament vortex. This is the same retracted vortex that exhibited a decay and then an increase in its vorticity. During this time the contraction phase begins, and the post-pinch-off droplet vortex starts to build near the neck region (highlighted with a dotted box at $T^* = 84$ ). Subsequently, the advancing ligament vortex in the bulk of the droplet decays and the post-pinch-off vortex comes into the picture.

3.4. Stresses due to post-pinch-off vortex

In this section, we examine the contours of the in-plane shear rate ( $\epsilon$ ) and plot two different quantities: the area integral of stresses $\int \epsilon ,\eta _{{x}},\eta _{{y}} {\rm d}A$ , and the maximum value of stresses, i.e. $\epsilon _{{max}}$ , $\eta _{{x,max}}$ $\eta _{{y,max}}$ .

Figure 6.

(a) The $\omega ^*$ contours at different times for the advancing ligament vortex for $\varphi = 8$ . (b) The six $\omega ^*$ values correspond to the vorticity contour plots in (a). The scale bar represents 50 $\rm \unicode{x03BC} m$ , $T^* = 0$ marks the pinch-off point.

Figure 7.

Contour plots of $\epsilon ^*$ at different times. The scale bar represents 50 $\rm \unicode{x03BC} m$ .

Figure 7. Long description

Panel A: Heat map showing in-plane shear rate distribution within a droplet at time T* = 0 and volume fraction φ = 4. The color scale ranges from -0.3 (red) to 0.3 (blue), indicating the shear rate magnitude. The map shows regions of high shear rate in blue and low shear rate in red, with a noticeable gradient from the top-left to the bottom-right. Panel B: Heat map showing in-plane shear rate distribution within a droplet at time T* = 0.56 and volume fraction φ = 4. The color scale ranges from -0.3 (red) to 0.3 (blue), indicating the shear rate magnitude. The map shows regions of high shear rate in blue and low shear rate in red, with a noticeable gradient from the top-left to the bottom-right. Panel C: Heat map showing in-plane shear rate distribution within a droplet at time T* = 1 and ratio of flow rates φ = 4. The color scale ranges from -0.3 (red) to 0.3 (blue), indicating the shear rate magnitude. The map shows regions of high shear rate in blue and low shear rate in red, with a noticeable gradient from the top-left to the bottom-right. Panel D: Heat map showing in-plane shear rate distribution within a droplet at time T* = 1.74 and volume fraction φ = 4. The color scale ranges from -0.3 (red) to 0.3 (blue), indicating the shear rate magnitude. The map shows regions of high shear rate in blue and low shear rate in red, with a noticeable gradient from the top-left to the bottom-right. Panel E: Heat map showing in-plane shear rate distribution within a droplet at time T* = 2.6 and volume fraction φ = 4. The color scale ranges from -0.3 (red) to 0.3 (blue), indicating the shear rate magnitude. The map shows regions of high shear rate in blue and low shear rate in red, with a noticeable gradient from the top-left to the bottom-right. Panel F: Heat map showing in-plane shear rate distribution within a droplet at time T* = 0 and volume fraction φ = 10. The color scale ranges from -0.3 (red) to 0.3 (blue), indicating the shear rate magnitude. The map shows regions of high shear rate in blue and low shear rate in red, with a noticeable gradient from the top-left to the bottom-right. Panel G: Heat map showing in-plane shear rate distribution within a droplet at time T* = 0.56 and volume fraction φ = 10. The color scale ranges from -0.3 (red) to 0.3 (blue), indicating the shear rate magnitude. The map shows regions of high shear rate in blue and low shear rate in red, with a noticeable gradient from the top-left to the bottom-right. Panel H: Heat map showing in-plane shear rate distribution within a droplet at time T* = 1 and volume fraction φ = 10. The color scale ranges from -0.3 (red) to 0.3 (blue), indicating the shear rate magnitude. The map shows regions of high shear rate in blue and low shear rate in red, with a noticeable gradient from the top-left to the bottom-right. Panel I: Heat map showing in-plane shear rate distribution within a droplet at time T* = 1.74 and volume fraction φ = 10. The color scale ranges from -0.3 (red) to 0.3 (blue), indicating the shear rate magnitude. The map shows regions of high shear rate in blue and low shear rate in red, with a noticeable gradient from the top-left to the bottom-right. Panel J: Heat map showing in-plane shear rate distribution within a droplet at time T* = 2.6 and volume fraction φ = 10. The color scale ranges from -0.3 (red) to 0.3 (blue), indicating the shear rate magnitude. The map shows regions of high shear rate in blue and low shear rate in red, with a noticeable gradient from the top-left to the bottom-right.

Figure 8.

(a) Variation of of $\epsilon ^*_{{max}}$ for $\varphi = 4-10$ . (b) Area integral normalised using $U_{\textit{peak}} D_d$ and (c) maximum values of shear and extension rates for $\varphi = 6$ , $T^* = 0$ marks the pinch-off point.

Figure 7 shows the contour plot for $\epsilon ^*(=\epsilon t_{\textit{cap}})$ with $T^*$ for $\varphi = 4$ and 10. Similar to the $\omega ^*$ contours in figure 3(a), we see that there is a decrease of $\epsilon ^*$ with time and it covers a larger area for smaller droplets compared with larger droplets. The area under the influence of $\epsilon ^*$ demarcates the hot spots where a particle or a living organism will be subjected to intense shearing. As discussed in Tan et al. (Reference Tan, Saleski and Lin2022), the role of droplet size is crucial in bio-chemical processes taking place inside micro-droplets. In droplets with smaller size ( $\varphi$ = 10), a very significant portion is under the influence of shearing, which indicates a less favourable condition for entrapped living organisms. It is to be noted that the time of exposure to higher levels of $\epsilon ^*$ will be less in this regime (post-pinch-off vortex decay period); however, the unbalanced surface tension post-pinch off rapidly accelerates both sides of the fluids that would act like an impact force (Peregrine et al. Reference Peregrine, Shoker and Symon1990) on any organism trapped in high shear zones and it has been shown that impact loadings can alter bacterial behaviour significantly (Hariharan et al. Reference Hariharan, Chowdhury, Rao S, Chakravortty and Basu2023). Furthermore, figure 8(a) shows the variation of $\epsilon ^*_{max}$ for each of the $\varphi$ values. It is seen that the values are slightly larger for smaller $\varphi$ compared with higher $\varphi$ . The reason for this may be the size of the droplet, which is larger for lower $\varphi$ , leading to a higher $D_{{d}}/h$ ratio, indicating maximum activity in the observation plane. The other quantities shown in figures 8(b) and 8(c) for a single value of $\varphi = 6$ follow a similar decay and for all quantities it can be seen that $\epsilon ^*$ is much higher compared with $\eta ^*_{{x}}(=\eta _{{x}} t_{\textit{cap}})$ and $\eta ^*_{{y}}(=\eta _{{y}} t_{\textit{cap}})$ . Between $\eta ^*_{{x}}$ and $\eta ^*_{{y}}$ , the linear shear in the x direction dominates. This is because the flow inside the droplet is driven by a shearing mechanism in the direction of the flow in the x direction. Also, the pinch-off stretches the droplet in the x direction, generating its gradient in the x direction. The initial gradients are large enough to generate high impulsive loading inside the droplets.

Figure 9.

Variation of $\epsilon ^*$ , $\eta ^*_{{x}}$ and $\eta ^*_{{y}}$ for retracted and advancing ligament phases. (a) Area integral normalised by $U_{\textit{peak}} D_d$ ; (b) maximum values. The first part of each plot is for the retracted ligament. The other part of each plot represents the advancing ligament and corresponds to the same time as shown for vorticity plots in figure 6(a), $T^* = 0$ marks the pinch-off point.

3.5. Stresses due to retracted and advancing ligament vortex

Corresponding to figure 6(a), where we depict the vorticity contours, we plot the different stress quantities in figures 9(a) and 9(b) for $\varphi =8$ . Again, $T^*=0$ represents the pinch-off point and the first part of the plots before the axis break represent the stresses during the decay of the retracting ligament vortex (the vorticity contours for the first part of the x axis are not presented whereas the other part can be correlated with the $T^*$ values in figure 6 a). The initial decay in the retracted vortex is very similar to that in the pinch-off droplet, as was seen in figure 8(b,c). However, the extended part of the plot, where we show these quantities for longer $T^*$ , indicates that the area integral of stress (figure 9 a) increases nearly to the same level as during the pinch-off. This increase is due to the fact that different zones with velocity gradients are created with significantly high $\omega ^*$ values in the advancing dispersed fluid. The most interesting aspect is the sustained presence of the retracted vortex with a decrease and subsequent increase in its $\omega ^*$ value (approaching the advancing ligament vortex), which results in large stresses. The final dip seen in figure 9(a,b) is due to the decrease in the velocity gradients and stresses that occur near to the pinch-off ( $T^*\sim 94$ ) when the vortex formed in the bulk of the droplet (as marked in figure 6, $T^* = 84$ ) starts to decay due to a reduction in its feeding as a result of ligament thinning. We found no previous reference to the stresses developed in the advancing ligament, which nevertheless are very important, since this stress remains for a longer period compared with the post pinch-off stress in the droplet.

From a practical point of view, the stresses must be reduced inside the dispersed phase to provide a habitable environment for the encapsulated cells (Ma et al. Reference Ma, Sherwood, Huck and Balabani2014). It should be understood that the particles or organisms remain under stresses right from the beginning of the cycle i.e. inside the remnant ligament, its advancement along the flow direction, during the pinch-off and finally in the steady-state condition (as described in figure 1 b). Although not discussed here, the steady state of the droplet exhibits two counter-rotating vortices of opposite sense (Ma et al. Reference Ma, Sherwood, Huck and Balabani2014) compared with the post pinch-off vortex due to the action of the outer continuous fluid (figure 1 b). The longer the organisms are entrapped, the longer they are subjected to stress. However, once the droplet attains a steady state, it is usually seen that the vorticity existing inside the droplet has relatively low magnitude depending on the $Q_{{c}}$ value. Therefore, a trade-off between the stress duration and stress magnitude occurs at different regimes of droplet formation process.