4.1. Effect of viscosity ratio and drop size

First, we investigate the effect of changes in drop-to-bulk viscosity ratio on drop velocity. Figure 6(a) shows the transient evolution of dimensionless drop axial velocity ( $U_x/U_{av})$ with distance moved by drop centroid $(x_c/H)$ along the channel. The drop velocity first decreases due to the initial deformation, which results in increased resistance to flow, until a minimum is reached, then it starts increasing as the drop becomes more streamlined until it reaches a steady state. The steady velocity is reached after the drop has travelled a few drop diameters, in most cases. We also see from figure 6(a) that the drop steady-state velocity decreases with increasing viscosity ratio, as a higher viscosity ratio means higher viscous drag, higher shear stress in the fluid film between the drop and the walls and reduced internal flow, and hence a lower drop velocity. A more viscous drop also travels a longer distance before reaching a steady state; a drop with $Ca=0.5$ and $\lambda =5$ does not reach a steady state within the span of figure 6(a). This particular drop travelled approximately 40 channel heights before reaching a steady state. The high viscosity ratio coupled with a high capillary number ensures a dominant viscous force and slow deformation. Note that the solution by Boussinesq (Reference Boussinesq1868) gives $U_{max}/U_{av} = 2.096$ for the undisturbed velocity at the centre of a square channel, which is close to the steady-state velocity of the least viscous drop ( $\lambda =0.05)$ with $D/H=0.7$ at the higher capillary number ( $Ca=0.5)$ . Figure 6(b) shows a side view of the transient shape of a 3-D, deforming drop as it travels along the channel for different viscosity ratios. A higher viscosity ratio results in higher deformation at steady state, although it takes longer for a high-viscosity drop to reach a steady state due to increased hydrodynamic resistance.

Figure 7. (a) Cross-stream migration of drop centroid position (solid lines) with distance travelled along the channel for $\lambda =0.05,\,0.5,\,1,\,2,\,5,\,10,\,25,\,30,\,40$ , $D/H=0.6$ , $Ca=0.5$ and initial lateral position $y_c/H=0.6$ . (b) Drop shape at different positions along the channel length while migrating laterally for $\lambda =0.05\text { and } 5$ from (a).

A deformable drop placed away from the centreline of a channel undergoes cross-streamline migration due to its deformed, non-spherical shape. Figure 7(a) shows the centroid position of a drop of size $D/H=0.6$ that was placed off centre initially at $y_c/H=0.6$ for different values of viscosity ratio at a fixed capillary number. The drop migrates towards the channel centreline in all cases. A drop of lower viscosity ratio migrates cross-stream faster than one of higher viscosity ratio in the range of $\lambda \in [0.05,1]$ , due to the slower deformation and larger hydrodynamic resistance of the latter. A similar trend is observed for large viscosity ratios ( $\lambda \gt 25$ ) due to reduced internal flow. However, a non-monotonic behaviour is observed in cross-stream migration for $\lambda \in [2,25]$ . Figure 7(b) shows the deformed drop shapes while undergoing lateral migration for a low and high $\lambda$ . Initially, the drop starts elongating along the flow profile and migrating away from the top wall. Tail formation is evident due to the drop interaction with the nearby top wall. As the drop travels along the channel length and approaches the centreline, its shape becomes more compact and symmetric, eventually reaching a steady shape. Generally, a higher-viscosity drop attains a steady state with higher deformation, and since the phenomenon of cross-stream migration is attributed to drop deformation, it might be counter-intuitive to observe that a high-viscosity drop has a slower cross-stream migration velocity, as can be seen in figure 8(a) for short times. Even though the final steady-state deformation is lower in a low-viscosity drop, the drop deforms faster in the initial stages due to reduced hydrodynamic resistance. This phenomenon is shown in figure 8(b), where the transient tip-to-tip maximum drop length non-dimensionalised by drop diameter ( $l/D$ ) is plotted for different viscosity ratios. This maximum tip-to-tip drop length $(l/D)$ is the maximum distance between two points on the drop surface (in any direction) and is a better metric for cross-stream migration where both the lateral drop length ( $\Delta y)$ and the axial drop length $(\Delta x)$ are important. The low-viscosity drops reach their maximum deformation faster, which allows them to attain high cross-stream velocities rapidly, immediately after release. However, high-viscosity drops experience greater deformation in the long run, helping them maintain a higher cross-stream migration velocity longer. Due to this competition, we see a non-monotonic behaviour in the intermediate viscosity ratio values ( $\lambda \in [2,25]$ ). A similar trend was observed by Griggs et al. (Reference Griggs, Zinchenko and Davis2007) in their work involving the motion of a drop between parallel walls.

Figure 8. (a) Transient cross-stream migration velocity ( $U_y$ ) and (b) deformation in the axial direction of a drop for $\lambda =0.05,\,0.5,\,1,\,2,\,5,\,10,\,25,\,30$ , $D/H=0.6$ , $Ca=0.5$ and initial lateral position $y_c/H=0.6$ .

Figure 9. Change in steady-state (a) drop velocity and (b) deformation with increasing drop size and $\lambda = 0.05,\,0.2,\,0.4,\,0.8,\,1,\,5$ (top to bottom for (a) and bottom to top at large $D/H$ for (b)) for $Ca=0.3$ (solid), and $Ca=0.6$ (dot-dash). The magenta portion of the curves were initially started as a prolate ellipsoid instead of a sphere. The dotted line in (a) and (b) are for drops with $Ca=0.1$ , $\lambda =10$ and the filled circle points ( $\bullet$ ) are numerical data from Wang & Dimitrakopoulos (Reference Wang and Dimitrakopoulos2012) at the same conditions.

Figures 9(a) and 9(b) show curves for the change in drop steady-state velocity and deformation with drop diameter for different values of drop-to-bulk viscosity ratio $(\lambda )$ , respectively. The special case $Ca=0.1$ and $\lambda =10$ is included to show good agreement between the BI simulations done in this work and the simulations done by Wang & Dimitrakopoulos (Reference Wang and Dimitrakopoulos2012) using a spectral-boundary-element method. The velocity decreases with increasing drop size in all cases except $\lambda =0.05$ with $Ca=0.6$ (discussed later). A larger drop has increased interaction with the channel walls, and spans across both fast and slow streamlines which make the drop slower. As in figure 6(a), we see a decreasing steady-state drop velocity with increasing viscosity ratio $(\lambda )$ in figure 9(a). Figure 9(b) shows that, for large drop diameters, a more viscous drop undergoes greater deformation since they experience a larger shear stress. For a fixed capillary number, the drop deformation for small drop sizes ( $D/H\lt 0.6$ ) is not significant, and almost independent of the drop-to-bulk viscosity ratio ( $\lambda$ ). Size effects are more prominent for larger drops ( $D/H\gt 0.7)$ at higher capillary number ( $Ca$ ) values.

Drops with relatively high $\lambda$ or high $Ca$ in figure 9 start experiencing unstable tail formation beyond a certain drop diameter ( $D/H\gtrsim 0.8$ ) when initially started as a sphere. As noted previously, a drop with a higher drop-to-bulk viscosity ratio undergoes larger deformation before reaching a steady state. A higher viscosity ratio coupled with a high capillary number, and greater wall interactions for larger drops, yields large deformation and formation of a four-tailed structure for sufficiently large drops, as shown in figures 10(a) and 10(b). Such drops seldom reach a steady state, and their tails eventually become narrow, leading to pinch off of very small satellite drops. Interestingly, we found that, if a drop is started as a prolate ellipsoid instead of a sphere, such that its minor axis is smaller than the sphere’s radius and therefore is further away from the walls initially, then the simulations could be continued successfully until a steady state is reached without tail formation (figure 10 c), and this part of the curve is coloured magenta in figure 9. Lac & Sherwood (Reference Lac and Sherwood2009) also reported a critical capillary number in their work on drop motion in a cylindrical microchannel, above which a drop does not reach a steady state but instead undergoes continued deformation. Since the focus of the current study is steady-state drop velocities, we did not include pinch-off nor continued simulations after the formation of satellite drops.

Figure 10. Highly deformed, unstable four-tailed drops prior to satellite drop pinch-off at a $Ca=0.6$ , $\lambda =1$ and $D/H=0.92$ . Panel (a) shows a side view of the drop and in (b) we see a perspective from which all the four tails are visible. (c) Steady-state shape of the same drop when simulated with an initial prolate ellipsoid shape.

Tail pinch-off is not limited to high viscosity ratios. We observed similar behaviour in large drops for low viscosity ratios at high capillary numbers. Figure 11 shows one such simulation where a drop with $D/H=0.92$ , $\lambda =0.1$ and $Ca=0.8$ displays satellite drop pinch-off of its tails. The drop deforms gradually and initially forms a dimple in its posterior, which turns into a re-entering jet-like intrusion. Eventually, its tails become increasingly narrow, leading to pinch-off. Lac & Sherwood (Reference Lac and Sherwood2009) observed a similar re-entering jet in their simulations with a low-viscosity drop in a cylindrical tube at a high capillary number. Since cylindrical tubes are radially symmetric, tail formation is not seen in that channel geometry, rather Lac & Sherwood (Reference Lac and Sherwood2009) observed breakup of low-viscosity drops when the re-entrant jet reached the tip of the drop. This phenomenon was also observed experimentally in a cylindrical tube by Olbricht & Kung (Reference Olbricht and Kung1992). More recently, Nath et al. (Reference Nath, Biswas, Dalal and Sahu2017) performed an extensive numerical study on droplet breakup in a cylindrical channel using a coupled level-set and VOF method focusing on the effect of drop size, capillary number and viscosity ratio. They observed several outcomes based on these parameters, where a drop either forms a long tail or a re-entering jet and breaks up into small daughter or satellite droplets, which sometimes coalesced back to a single drop further down the channel.

Figure 11. Cross-section of transient shapes of a drop of undeformed diameter $D/H=0.92$ with $\lambda =0.1$ and $Ca=0.8$ , developing a dimple that eventually becomes a re-entrant jet and finally leads to tail pinch-off (dashed circles) at $t^*=1.925$ .

One of the most interesting results of this work is the increase in steady-state velocity with increasing drop diameter for large $Ca$ and small $\lambda$ , such as observed in figure 9(a) for $Ca=0.6$ and $\lambda =0.05$ and (to a lesser extent) $\lambda =0.2$ , as typically we would expect a larger drop to be slowed down due to wall interactions. When $\lambda \lt 1$ , the drop fluid has a lower viscosity than the suspending fluid, which, generally, makes it easier to drive the flow. A possible explanation is provided by considering the roles of form drag and friction drag. Form drag is due to pressure and increases with the increasing cross-sectional area of the drop perpendicular to the flow direction, whereas friction drag is due to viscous shear stresses and increases with surface area parallel to the flow direction. These two contributions are comparable for spherical particles and viscous drops, but the form drag will be reduced and the friction drag will be increased if a drop becomes elongated. This deformation typically leads to a net increase in drag or a reduction in drop velocity. However, for low-viscosity drops, form drag dominates and so a net decrease in drag or increase in drop velocity can occur if the drop deformation is substantial (large $Ca$ ). This phenomenon is discussed further in § 4.2.

Figure 12. Comparison between numerical (curves) and experimental (points) results for steady-state drop velocity with increasing drop diameter for $\lambda =0.001$ (aqueous glycerol drop), and $0.838$ (PDMS drops) at $Ca=0.74$ . The squares () and circles ( $\bullet$ ) are experimental data for $\lambda = 0.001$ and $\lambda =0.838$ , respectively, with error bars representing 90 % confidence intervals from repeated experiments.

Figure 12 shows a comparison between experimental and simulation results for steady-state drop velocity versus drop diameter at a fixed capillary number ( $Ca=0.74$ ) and two different viscosity ratios, $\lambda =0.001$ and $\lambda =0.838$ . The lower and the higher viscosity ratios correspond to 8 % aqueous glycerol drops and silicon oil drops, respectively, suspended in castor oil (table 1). There is good agreement between the experiments and simulations, where the theory lies within the 90 % confidence intervals of the experimental results for all cases except at $D/H=0.27$ for the lower curve at $\lambda =0.838, Ca=0.74$ . Previous studies have shown that small drops below a certain critical $D/H$ (depending on $\lambda$ and $Ca$ ) and with a viscosity ratio within a specified range ( $\lambda \in [0.5,10]$ for drops in an unbounded flow (Chan & Leal Reference Chan and Leal1979) and $\lambda \in [0.7,10]$ for small drops in a tube (Cappello et al. Reference Cappello, Rivero-Rodríguez, Vitry, Dewandre, Sobac and Scheid2023)) have a tendency to undergo lateral migration towards the walls if started off centre initially. We suspect that our smallest drop in the curve with $\lambda =0.838$ and $Ca=0.74$ might have been placed slightly off centre initially in the experiments and slowly migrated further from the centre as it travelled along the channel length, hence resulting in a slower steady-state velocity observed in experiments. In fact, Cappello et al. (Reference Cappello, Rivero-Rodríguez, Vitry, Dewandre, Sobac and Scheid2023), in their study on the stable, equilibrium, lateral position of drops in a cylindrical microchannel showed that, in the inertialess limit, the critical drop size for lateral migration towards the walls around $\lambda =0.8$ is approximately $D/H=0.27$ , which might be slightly different for a channel of rectangular cross-section but attests to our suspicion. These experiments verify the predicted trend of an increase in steady-state drop velocity with increasing drop diameter for a low viscosity ratio ( $\lambda =0.001$ ) and high capillary number ( $Ca=0.74$ ), whereas the drop velocity decreases with increasing drop size at the larger viscosity ratio.

The maximum flow velocity at the channel centreline for a square channel from the Boussinesq (Reference Boussinesq1868) solution is, $U_{max}/U_{av}\sim 2.096$ . Another interesting observation in figures 9(a) and 12 is that the drop steady-state velocity increases beyond the maximum flow velocity with drop diameter. Lac & Sherwood (Reference Lac and Sherwood2009) explained this behaviour for a cylindrical channel with an asymptotic analysis assuming the flow of a long, slender drop in a capillary. They considered a long slender drop to be similar to an annular flow of two immiscible fluids, where the inner fluid of viscosity $\lambda \mu$ occupies a cylinder of radius $R\delta$ in a capillary of radius $R$ , and the outer fluid of viscosity $\mu$ occupies the annular film of thickness $(1-\delta )R$ . Assuming the inner cylinder to be a long drop with axial velocity $V$ , Lac & Sherwood (Reference Lac and Sherwood2009) obtained the following relation between the drop velocity and the average flow velocity in the channel:

(4.2) \begin{equation} \frac {V}{U_{av}}=\frac {2\lambda +(1-2\lambda )\delta ^2}{\lambda +(1-\lambda )\delta ^4}. \end{equation}

For a pressure-driven flow in a cylindrical channel, $U_{max}/U_{av}=2$ , where $U_{max}$ is the maximum undisturbed flow velocity at the channel centreline. Equation (4.2) reveals that it is possible to attain $V/U_{av}\gt 2$ at $\lambda \lesssim 0.5$ for a range of $\delta$ . Thus, it is possible for a low-viscosity drop with large deformation to obtain a steady-sate velocity that exceeds the maximum velocity of the undisturbed flow. This conclusion is also expected for a rectangular channel, although the analysis would be more complex due to the corners.

4.2. Effect of capillary number

In this section, we further examine the effect of capillary number on the velocity and deformation of a drop. The capillary number is a measure of the relative strength of deforming viscous forces to the restoring interfacial tension forces, so a higher capillary number results in greater deformation. Figure 13 shows simulations of the transient evolution of drop velocity with distance travelled along the channel length. As in figure 6, the drop velocity decreases at first and reaches a minimum, but then starts increasing until it reaches a steady state. Since a higher capillary number is associated with higher deformation, the drop travels a longer distance before it reaches a steady state for higher $Ca$ . The drop steady-state velocity increases with increasing capillary number $(Ca)$ . A drop with higher deformation has more of its volume close to the maximum flow velocity at the channel centreline and farther from the walls, which results in a higher drop velocity with increasing $Ca$ . Figure 13 also shows that more viscous drops have lower velocities and a weaker dependence on $Ca$ .

Figure 13. Simulations of the evolution of drop velocity in the axial direction with distance traversed for a droplet of size $D/H=0.6$ . The dot-dashed curves are for $\lambda =0.05$ with $Ca=0.2,\,0.4,\,0.6,\,0.8$ (bottom to top) and the solid curves are for $\lambda =0.5$ with $Ca=0.2,\,0.4,\,0.6,\,0.8$ (bottom to top).

Figure 14. (a) Cross-stream migration of drop centroid position (solid lines) with distance travelled along the channel for $Ca=0.2,\,0.3,\,0.5,\,0.6$ ; $D/H=0.6$ ; $\lambda =1$ ; and initial lateral position $y_c/H=0.6$ . The dashed line is for a drop with $Ca=0.5$ , $\lambda =1$ , $D/H=0.6$ , and an initial position of $y_c/H=0.6$ in a channel of infinite depth using the algorithm of Navarro et al. (Reference Navarro, Zinchenko and Davis2020). (b) Drop shape at different positions along the channel length while migrating laterally for $Ca=0.5$ corresponding to (a).

As mentioned in § 4.1, a deformable drop placed off centre in a channel for the conditions examined here experiences cross-stream migration towards the centreline. Figure 14(a) shows the transient centroid position of a drop of size $D/H=0.6$ that was initially placed off centre at $y_c/H=0.6$ for different capillary numbers at a fixed drop-to-bulk viscosity ratio. A drop with a higher capillary number deforms more and hence has a more streamlined shape with a higher velocity expected both parallel and perpendicular to the wall. In figure 14(a), we see that a drop with $Ca=0.5$ migrates faster than the ones with $Ca=0.2,\,0.3$ , and the drop with $Ca=0.6$ although it is initially slower than $Ca=0.5$ , eventually crosses over at $x_c/H\approx 30$ and reaches the centreline earliest among the four. This behaviour is because the drop with $Ca=0.6$ has the highest deformability and hence forms the longest tail (which makes it slower than the other cases initially due to increased interactions with its closest wall), but remains highly deformed for a longer time than the low $Ca$ drops, helping it maintain a high lateral velocity for longer and hence reaching the centreline earliest. The dashed line in figure 14(a) is for the transient centroid position of a drop with $Ca=0.5$ , $\lambda =1$ , $D/H=0.6$ and initial lateral position $y_c/H=0.6$ in a channel of infinite depth simulated adopting the numerical framework of Navarro et al. (Reference Navarro, Zinchenko and Davis2020). The drop migrates slightly faster than the one in a square channel, which can be attributed to the absence of added sidewall interactions in an infinite-depth channel as opposed to a square channel. Figure 14(b) shows the transient deformed drop shapes corresponding to the curve for $Ca=0.5$ in the channel of square cross-section in figure 14(a). As in figure 7(b), the drop starts deforming and migrating towards the centreline, forms a tail due to interaction with its closest wall, and then eventually retracts its tail and becomes compact; finally, it reaches a symmetric steady shape at the channel centreline.

Figure 15(a) shows the steady-state shapes of drops of different diameters with increasing capillary number at $\lambda =0.001$ as seen in the experiments overlaid by steady-state drop outlines obtained from simulations at the same conditions, showing good agreement. A more elongated drop can be seen with increasing $Ca$ , due to higher viscous deformation forces. Figure 15(b) shows the steady-state shapes of a drop of size $D/H=0.76$ with increasing capillary number for different viscosity ratios. With increasing $Ca$ , the drop develops a dimple on its backside due to the flow indenting its posterior. The increase in deformation with capillary number is greater for a higher viscosity ratio, since a large $\lambda$ is also associated with greater viscous forces on the interface. Noticeably, we see a wide rear and a tapered front for the drop at $\lambda =1$ and $Ca=0.8$ , but for $\lambda =0.05$ this shape is no longer exhibited at high capillary numbers. The dimple at the posterior end of the drop with $\lambda =0.05$ and $Ca=0.8$ evolves into a re-entering jet at higher capillary numbers when the shape-preserving interfacial tension forces are overpowered, as seen in figure 11.

Figure 15. (a) Steady-state shapes of a drop with increasing drop diameter $D/H=0.52, 0.76, 0.84$ for $Ca=0.28,0.47,0.74$ and $\lambda =0.001$ obtained from experiments overlaid by outlines for steady-state drop shapes obtained from BI simulations at same conditions. (b) Steady-state shapes of a drop of $D/H=0.76$ for $\lambda =0.05,\,0.5,\,1$ with increasing capillary number $Ca=0.2,\,0.5,\,0.8$ from simulations.

Figure 16. Steady-state (a) drop velocity and (b) drop deformation with increasing drop diameter and $Ca=0.1,\,0.2,\,0.3,\,0.4,\,0.5,\,0.6,\,0.7,\,0.8$ (bottom to top) at $\lambda =0.5$ .

Figure 17. Steady-state drop velocity with increasing drop diameter and $Ca=0.1,\,0.2,\,0.3,\, 0.4,\,0.5,\,0.6, 0.7,\,0.8$ (bottom to top) at (a) $\lambda =0.05$ (b) $\lambda =0.2$ . (c) Steady-state shapes of drops with $\lambda =0.05$ , $Ca=0.7$ and $D/H=1,\,1.1,\,1.2,\,1.3,\,1.4$ (left to right).

Figure 16 shows the change in steady-state drop velocity and its deformation with drop diameter and capillary number for a bulk-to-drop viscosity ratio of 0.5. As discussed earlier, the steady-state velocity typically decreases with increasing drop size due to increased interaction with the channel walls and lower fluid velocity near the walls, except for cases with very low drop-to-bulk viscosity ratios. In figure 16, both the steady-state velocity and the deformation of the drop increase with increasing capillary number for drops of all sizes. In figure 17(a), for a very low viscosity ratio, $\lambda =0.05$ , the steady-state velocity decreases with increasing drop radius for lower capillary numbers, whereas it has an opposite trend for higher capillary numbers. This observation is due to the fact that drops with very low $\lambda$ but high $Ca$ experience much less skin friction and hence are able to attain higher velocities, even greater than $U_{max}$ , as explained in § 4.1, whereas for higher viscosity ratios the steady-state velocity gradually approaches $U_{max}$ with increasing capillary number, as seen in figures 16(a) and 17(b). Figure 17(b) shows the change in steady-state velocity with drop diameter for increasing $Ca$ at $\lambda =0.2$ . Here, we see that the drops reach a steady state without breaking, with their velocity almost equal to $U_{max}$ at $Ca=0.8$ , showing little dependence on drop diameter. For a viscosity ratio lower than 0.2, drops at $Ca=0.8$ reach a steady state at a higher velocity with increasing drop diameter, attaining faster speeds than $U_{max}$ . A similar behaviour was found in numerical studies on the motion of a bubble in a cylindrical channel by Rivero-Rodriguez & Scheid (Reference Rivero-Rodriguez and Scheid2018) and Atasi et al. (Reference Atasi, Haut, Pedrono, Scheid and Legendre2018), where they observed a transition in the change in steady-state bubble velocity from decreasing with increasing bubble size to increasing with increasing bubble size between a $Ca$ of 0.25 and 0.5. They also observed a steady-state velocity greater than $U_{max}$ for large bubbles above a critical $Ca$ in this regime. A larger drop is always found to have a higher deformation for any viscosity ratio due to an increase in the ratio of deforming to restoring forces and higher interaction with the walls. Note that a few curves in figures 16 and 17 for the large capillary numbers are discontinued after a certain drop size because drop breakup was observed due to unstable tail or re-entrant jet formation. Figure 17(c) shows the steady-state shapes of some drops with increasing $D/H$ for $\lambda =0.05$ and $Ca=0.7$ .

Figure 18(a) shows the critical capillary number ( $Ca_{cr}$ ) at which the drop steady-state velocity becomes larger than the maximum velocity of the background flow at the channel centreline ( $U_{max})$ and how it changes with viscosity ratio for different drop sizes, as obtained from BI simulations. For a square channel, $U_{max}\sim 2.096$ . In agreement with our previous results, $Ca_{cr}$ increases with increasing viscosity ratio up to $\lambda \approx 0.25$ . On the other hand, $Ca_{cr}$ decreases with increasing drop size owing to the fact that larger drops travel at greater speeds at these low viscosity ratios. A capillary number with drop steady-state velocity greater than $U_{max}$ was not found beyond $\lambda =0.25$ for any of the drop sizes shown in figure18(a). In these cases, the drops experience breakup beyond a certain $Ca$ , with values shown using red squares in figure18(a). A re-entrant jet was observed in all cases, causing breakup in one of the three ways including (i) necking of the origin of the re-entrant jet (figure 18 b), (ii) penetration by the jet up to the tip of the drop (figure 18 c) or (iii) tail pinch-off before the re-entrant jet is able to penetrate a substantial amount (figure 11).

Figure 18. (a) Critical $Ca$ at which either the steady-state velocity becomes greater than $U_{max}$ ( $\circ$ ), or the drop breaks () with increasing viscosity ratio $(\lambda )$ for drops of sizes $D/H=0.6,\,0.7,\,0.8,\,0.92$ (top to bottom). Drop breakup when (b) the origin of the re-entrant jet undergoes necking (dashed circle) for a drop with $D/H=0.92,\,\lambda =0.4,\,Ca=0.9$ or (c) the jet reaches the tip of a drop with $D/H=0.8,\,\lambda =0.25,\,Ca=1.1$ .

Figure 19 shows a comparison between experimental and simulation results of steady-state drop velocity and deformation versus drop diameter for a fixed drop-to-bulk viscosity ratio ( $\lambda =0.001$ , 8 % aqueous glycerol drops in castor oil) and three different capillary numbers, $Ca=0.28, 0.47$ and $0.74$ . The capillary number in the experiments was varied by changing the flow rate of the suspending fluid. As shown in table 1, there is an uncertainty associated with the interfacial tension $(\sigma )$ values measured using the pendant-drop method. Naturally, this uncertainty propagates to the calculated values of capillary number, where $Ca=\mu _e U_{av}/\sigma$ . The confidence bands of the theory curves (solid lines) in figure 19 show the corresponding variation in the simulations. There is a good agreement for $Ca=0.74$ and $Ca=0.47$ with considerable overlap between theory and experiments in both figures 19(a) and 19(b), but for $Ca=0.28$ the measured velocities are 10 %–20 % below the values predicted by the simulations. We suspect it could be an effect of surfactant in our experiments arising from impurities in the castor oil or the blue inorganic dye used for our 8 % aqueous glycerol drops. The presence of surfactants slows down a drop suspended in a plane Poiseuille flow (Pak et al. Reference Pak, Feng and Stone2014) or a rectangular channel (Luo et al. Reference Luo, Shang and Bai2018) due to the Marangoni effect. A lower capillary number corresponds to a greater importance of interfacial tension forces and surfactants in the experiments. Moreover, we suspect that, at a low viscosity ratio, there is a more significant effect of surfactants due to reduced interfacial mobility, although further investigation involving the modelling of a surfactant-covered drop in microchannels is required. Atasi et al. (Reference Atasi, Haut, Pedrono, Scheid and Legendre2018) observed an approximately 10 % decrease in steady-state velocity of nearly spherical bubbles (low $Ca$ ) due to a small amount of surfactant in their numerical study involving the bubble dynamics in a tube. This result affirms the possible explanation for the disparity observed between experiments and simulations in figure 19(a). Nevertheless, our experimental results display similar behaviour as the simulation results in figure 17, where both the steady-state drop velocity and deformation increase with increasing capillary number, decrease with increasing drop diameter for a low capillary number, but increase with increasing drop diameter for a high $Ca$ , and the steady-state drop deformation always increases with capillary number. Figures 17(a) and 19(a) also suggest there is a critical capillary number value near $Ca=0.4$ , at which this switch in behaviour from the steady-state velocity decreasing with increasing $D/H$ to increasing with increasing $D/H$ occurs for $\lambda \ll 1$ .

Figure 19. Comparison between numerical (curves) and experimental (symbols) results for (a) steady-state drop velocity and (b) drop deformation with increasing drop diameter. The comparison was done for $\lambda =0.001$ (aqueous glycerol drop) at $Ca=0.74\pm 0.06,\,0.47\pm 0.03,\,0.28\pm 0.02$ , with the confidence bands representing the uncertainty in capillary number ( $Ca$ ) calculations arising from the standard deviation of interfacial tension $(\sigma )$ measurements. Experimental results are provided at $Ca=0.74$ (), $Ca=0.47$ ( $\bullet$ ) and $Ca=0.28$ ( $\blacktriangle$ ), with the error bars representing 90 % confidence intervals.

Figure 20 shows the steady-state deformed drop height ( $\Delta y/H$ ) with increasing undeformed drop diameter ( $D/H$ ) for different capillary numbers at $\lambda =0.05 \text { and } 0.2$ , obtained from BI simulations. The steady-state deformed drop height is defined here as the transverse length of a deformed drop that has reached steady state non-dimensionalised by channel height. As discussed previously, slender drops can attain a velocity greater than the maximum flow velocity in the channel at low viscosity ratios. There is a threshold value of steady-state drop height ( $\delta$ ) obtained from (4.2) for tubes, below which a drop can become faster than $U_{max}$ , and this $\delta$ increases with decreasing drop-to-bulk viscosity ratio. Therefore, at a fixed viscosity ratio, a drop that is able to survive up to a high enough capillary number to experience enough deformation and reach a steady state with a $\Delta y/H$ less than this critical value of $\delta$ obtained from (4.2) will be able to attain a velocity greater than $U_{max}$ . This critical drop height was found to be $\delta =0.707$ at $\lambda =0.05$ , and we see from figure 20(a) that drops with higher $Ca$ have their curves below the dashed line at $\Delta y/H=0.707$ . In qualitative agreement with this result, we observe in figure 17(a) that large drops with high $Ca$ attain a steady-state velocity higher than $U_{max}$ , and the steady-state velocity increases with increasing drop diameter. The critical drop height below which a drop can go faster than $U_{max}$ at $\lambda =0.2$ was found to be $\delta =0.6$ from (4.2). In figure 20(b), we see that the deformed drop height ( $\Delta y/H)$ obtained from BI simulations at $\lambda =0.2$ for different capillary numbers are mostly above $\Delta y/H =0.6$ . This again is in qualitative agreement with our observation in figure 17(b), where we see that the steady-state velocity increases with capillary number, slowly approaches $U_{max}$ and becomes almost equal to $U_{max}$ at $Ca=0.8$ . The authors would like to reiterate that the above comparison is qualitative, since (4.2) was derived by Lac & Sherwood (Reference Lac and Sherwood2009) for long drops in a cylindrical channel whereas the present study involves a channel of rectangular cross-section and drops of size comparable to the channel height.

Figure 20. Fraction of channel height that the deformed steady-state drop occupies with increasing drop diameter for $Ca=0.1,\,0.2,\,0.3,\,0.4,\,0.5,\,0.6,\,0.7,\,0.8$ (top to bottom) at (a) $\lambda =0.05$ (solid lines) (b) $\lambda =0.2$ (solid lines). The dashed line is at $\Delta y/H=0.707$ in (a) and at $\Delta y/H=0.6$ in (b), signifying the maximum $\delta$ at which a slender drop can have a velocity greater than $U_{max}$ for $\lambda =0.05 \text { and } 2$ , respectively, using (4.2) for tubes. The solid lines are simulations for square channels.

4.3. Effect of aspect ratio

The aspect ratio of a rectangular channel has a significant effect on the steady-state velocity of the drop. In the present finite-depth channel simulations, we specify the volumetric flow rates $Q$ and use the corresponding Boussinesq solution for the flow in a rectangular duct as the undisturbed flow in the absence of a drop. The most straightforward way to compare finite-depth simulations of rectangular channels with different aspect ratios is to keep a fixed $Q/W$ , where the channel width, $W$ , is varied to change the aspect ratio. In this manner, we can match the average flow velocities, $U_{av}$ , between simulations with different aspect ratios.

Figure 21 shows the effect of aspect ratio on the steady-state velocity of a drop non-dimensionalised by the average velocity of the background flow. In figures 21(a) and 21(b), simulation results for drop steady-state velocity with increasing aspect ratio are shown for different values of capillary number and drop-to-bulk viscosity ratio, respectively. In both cases, the drop steady-state velocity first increases slightly up to an aspect ratio of approximately 1.3, and then decreases monotonically with increasing aspect ratio. For the Boussinesq (Reference Boussinesq1868) solution, the channel maximum velocity (centreline velocity) of the background flow decreases with increasing $W/H$ and fixed $Q/W$ , from $U_{max}/U_{av}=2.096$ for a square cross-section ( $W/H=1$ ) and asymptotes to 1.5 as $W/H\to \infty$ , which is the case of Poiseuille flow between plane parallel walls. Due to this decrease in the channel maximum velocity with increasing aspect ratio, the steady-state drop velocity also decreases. However, for increasing aspect ratios in the approximate range $1\lt W/H\lt 2$ , there is a competition between the effect of decreasing interaction with the walls, which helps a drop attain higher steady-state velocity, and a lower channel maximum velocity, which results in lower steady-state velocity. The net result is a maximum in each curve near $W/H=1.3$ , except for a viscosity ratio of 0.05 (figure 21 b, top curve). As discussed in §§ 4.1 and 4.2, a higher steady-state velocity is observed in figure 21(b) with a decreasing viscosity ratio $(\lambda )$ due to lower viscous drag, and a lower steady-state velocity is observed with decreasing capillary number $(Ca)$ in figure 21(a) due to less deformation of the drops leading to a lower percentage of its volume experiencing the maximum flow velocity near channel centreline. However, the dependence on capillary number is weaker than the dependence on viscosity ratio.

Figure 21. (a) Steady-state drop velocity (solid lines) with increasing aspect ratio for $Ca=0.1,\,0.3,\,0.5, 0.7,\,1.0$ (bottom to top), $\lambda =0.5$ and $D/H=0.6$ . (b) Steady-state drop velocity (solid lines) with increasing aspect ratio for $\lambda =0.05,\,0.2,\,0.4,\,0.8,\,1,\,2,\,5,\,10$ (top to bottom), $Ca=0.4$ and $D/H=0.6$ . The dashed line shows the values for $U_{max}/U_{av}$ from the Boussinesq (Reference Boussinesq1868) solution for a rectangular channel.

Figure 22. (a) Steady-state drop deformation with increasing aspect ratio for $Ca=0.1,\,0.3,\,0.5,\,0.7,\,1.0$ (bottom to top), $\lambda =0.5$ and $D/H=0.6$ . (b) Steady-state drop deformation with increasing aspect ratio for $\lambda =0.05,\,0.2,\,0.4,\,1$ (top to bottom among the ones not specifically labelled, i.e. $\lambda =2,\,5,\,10$ ), $Ca=0.4$ and $D/H=0.6$ . The solid lines are for deformation along channel height, $D_{T,y}=\Delta x/\Delta y$ , and the short-dashed lines for deformation along channel width, $D_{T,z}=\Delta x/\Delta z$ .

In figure 22, the variation in steady-state drop deformation with increasing aspect ratio is shown for different capillary numbers (figure 22 a) and drop-to-bulk viscosity ratios (figure 22 b). Two different parameters have been used to describe the change in deformation with aspect ratio: (i) deformation along channel height, $D_{T,y}=\Delta x/\Delta y$ , and (ii) deformation along channel width, $D_{T,z}=\Delta x/\Delta z$ . The aspect ratio of the channel is changed by keeping the height ( $H$ ) constant and increasing the channel width ( $W$ ). Figures 22(a) and 22(b) show that both $D_{T,y}$ and $D_{T,z}$ decrease with increasing aspect ratio due to decreased wall interaction. Again, $D_{T,z}$ decreases more than does $D_{T,y}$ due to the channel width increasing with the aspect ratio. There is a modest effect of viscosity ratio in the curves for steady-state drop deformation versus aspect ratio, as seen in figure 22(b), which is weaker than the effect of change in $Ca$ (figure 22 a). This difference is because the effect of change in $\lambda$ on drop deformation is significant only for large drops at high capillary numbers, as seen previously in figure 9(b). It is interesting to note that there is a rapid decrease in drop deformation, both $D_{T,y}$ and $D_{T,z}$ , with a slight increase in aspect ratio from a square channel. This decrease shows that the effect of wall interactions is more important when the drop surface is in close vicinity to the wall and decays rapidly with distance. Furthermore, this sudden decrease in wall interactions must be responsible for the slight increase in drop steady-state velocity with a small deviation from a square cross-section, such that the reduced wall interaction has a dominant effect over the lower maximum centreline velocity of the channel, as observed in figures 21(a) and 21(b).

The simulations performed with varying aspect ratios for figures 21 and 22 are normalised by the channel average velocity $(U_{av})$ for different aspect ratios but naturally have a varying maximum background velocity at the channel centreline $(U_{max})$ . Therefore, in figure 23, the effect of aspect ratio on drop steady-state velocity normalised by the maximum background velocity $(U_{max})$ at the channel centreline from the Boussinesq (Reference Boussinesq1868) solution is shown. Upon normalising by $U_{max}$ , all the drop velocities for these conditions become less than unity and the variation in $U_s/U_{max}$ is small (less than 10 %). The effect of decreasing $U_{max}$ with increasing aspect ratio on the drop velocity is eliminated by this normalisation, thus isolating the effect of wall interactions. Therefore, we see in figure 23 that $U_s/U_{max}$ first increases with increasing aspect ratio up to approximately $W/H=2.5$ due to a reduction in viscous interaction with the walls and then approaches the asymptote since the sidewalls are then far enough for their effect to be insignificant. The initial increase in $U_s/U_{max}$ with changing aspect ratio can be linked with the sudden decrease in deformation upon deviation from a square cross-section observed in figure 22, confirming that the increase in $U_s/U_{max}$ is due to reduced wall interactions.

Figure 23. Steady-state drop velocity non-dimensionalised by maximum channel velocity with increasing aspect ratio for (a) $Ca=0.1,\,0.3,\,0.5,\,0.7,\,1.0$ (bottom to top), $\lambda =0.5$ and $D/H=0.6$ and (b) $\lambda =0.05,\,0.2,\,0.4,\,0.8,\,1,\,2,\,5,\,10$ (top to bottom), $Ca=0.4$ and $D/H=0.6$ .

Figure 24(a) compares numerical and experimental results for the effect of aspect ratio on the drop steady-state velocity. Experiments were conducted for a square channel and a rectangular channel with an aspect ratio of 2 : 1 with drops of $D/H\lt 1$ . In the case of a square channel, an excellent match is observed, where the theory and experiments agree within 2 %–3 % for all values of $D/H$ , whereas for the 2 : 1 rectangular channel, there is a qualitative similarity between experiments and simulations, but the experimental data points are approximately 3 %–8 % below the simulation results for all but the smallest drop size. Simulation curves are also provided for larger aspect ratios up to a channel of infinite depth. The results for the channel of infinite depth were generated using the algorithm of Navarro et al. (Reference Navarro, Zinchenko and Davis2020). Convergence to the infinite-depth case with increasing $W/H$ is slow, due to the stagnant zones near the corners. The steady-state velocities for drops suspended in an infinite-depth channel are very close to the limit $U_{max}/U_{av}=1.5$ for $D/H\lesssim 0.5$ and then gradually increase with drop size (due to very low $\lambda$ and high $Ca$ ). Interestingly, it is observed that the steady-state drop velocities reach a plateau with increasing drop size for a square channel ( $W/H=1$ ), but in the cases for channels with rectangular cross-section we see a monotonic increase in steady-state velocities with increasing drop size.

Figure 24. (a) Comparison between numerical and experimental results for steady-state drop velocity with increasing drop diameter at $\lambda =0.001$ (aqueous glycerol drop), $Ca=0.74$ and two aspect ratios. The solid theory curves are for increasing aspect ratios, with $W/H=1,\,2,\,5,\,10,\,20$ (top to bottom). The dot-dashed line is at $U_s/U_{av}=1.5$ to show the ratio $U_{max}/U_{av}=1.5$ for a plane Poiseuille flow. The short-dashed line shows the results for an infinite-depth straight channel using the algorithm of Navarro et al. (Reference Navarro, Zinchenko and Davis2020). The circle ( $\bullet$ ) and square () points are experimental results for a square channel and a rectangular channel of 2 : 1 aspect ratio respectively, with the error bars representing 90 % confidence intervals. (b) The solid lines show steady-state drop velocity non-dimensionalised by maximum channel velocity with increasing drop radius for $W/H=1,\,2,\,5,\,10,\,20$ (bottom to top on the right end of the curves), $Ca=0.74$ and $\lambda =0.001$ . The dashed line is for a channel of infinite depth using the algorithm of Navarro et al. (Reference Navarro, Zinchenko and Davis2020) for the same conditions, and the dotted line shows $U_{max}/U_{av}=1.0$ . The dot-dashed line is for $\lambda =50,\,Ca=0.01 \text { and } 0.1 \text { (bottom to top}),\,W/H=10$ and the points () are simulations for a solid sphere between plane parallel walls from Staben et al. (Reference Staben, Zinchenko and Davis2003). (c) Steady-state drop shapes obtained from BI simulations at $\lambda =0.001,\,Ca=0.74,$ and $D/H=0.92$ for $W/H=1,\,5,\,\infty$ . At $W/H=2$ drop breaks off tails (dashed circles) before reaching steady state when started as a sphere.

In figure 24(b), the same simulation results as figure 24(a) are shown but normalised by the maximum velocity of the background flow $(U_{max})$ instead of the average velocity $(U_{av})$ . This simple modification nearly collapses all the curves, showing weak dependence of $U_s/U_{max}$ on aspect ratio for drops with $D/H\lesssim 1$ . In the case of a square channel ( $W/H=1$ ) and a rectangular channel with $W/H=2$ , we observe a substantial difference in steady-state velocities for large drops ( $D/H\gtrsim 1$ ) when compared with the channels with larger aspect ratios. This is because wall interactions have a greater impact on steady-state drop velocities for large drops in narrower confinements. Therefore, even though the effect of $U_{max}$ is nullified upon normalisation, the wall effects are highlighted at $D/H\gtrsim 1$ , showing a lower steady-state velocity for narrower confinements. For the higher aspect ratios, we see a similar behaviour for $U_s/U_{max}$ as in figure 23, where the steady-state velocities asymptote to a fixed value for $W/H \gtrsim 2$ . Figure 24(b) also shows good agreement between the steady-state velocity of a drop with large viscosity ( $\lambda =50$ ) and very small capillary numbers ( $Ca=0.01,\,0.1$ ) in a channel of $W/H=10$ obtained using the numerical framework of the current paper and the velocities of a solid sphere in a Poiseuille flow between plane parallel walls obtained by Staben et al. (Reference Staben, Zinchenko and Davis2003) using BI simulations that have also been previously validated with experiments by Staben & Davis (Reference Staben and Davis2005).

For $W/H=2$ in figures 24(a) and 24(b), a drop of size $D/H=0.92$ was found to form two small satellite drops, leading to pinch-off, as shown in figure 24(c) along with the steady-state drop shapes for a drop of size $D/H=0.92$ at $Ca=0.74,\,\lambda =0.001$ and $W/H=1,5,\text { and }\infty$ . However, when the drop ( $W/H=2,\,D/H=0.92$ ) was started as a prolate ellipsoid instead of a sphere, the simulation could be continued until the drop reached a steady state. Note that in figures 24(a) and 24(b) the curves for $W/H\gt 1$ are discontinued at $D/H=1.3$ . This is because we observed drop breakup due to a small satellite droplet formation (figure 25) at $D/H=1.4$ . Drops with $D/H\geqslant 1$ are initially started as a prolate ellipsoid with a minor axis of $0.4H$ , and the satellite drop breakup at $D/H=1.4$ occurs while the drop tries to relax from its initial highly deformed prolate ellipsoid shape. A prolate ellipsoid was used to keep a consistent initial shape with the narrower confinements ( $W/H=1,2$ ) where an oblate ellipsoid would not fit. It is possible that using an oblate ellipsoid could help avert such drop breakup.

Figure 25. Satellite drop breakup at $D/H=1.4$ for $W/H=2,\,\lambda =0.001,\,Ca=0.74$ .

Figure 26. Streamlines showing the flow patterns in the $x$ - $y$ plane containing the channel centreline, both inside and outside a drop of $D/H=0.8$ at $\lambda =1,\,0.5,\,0.05$ (top to bottom) and $Ca=0.2,\,0.6$ (left to right) translating at steady state inside a straight channel of square cross-section.

4.4. Velocity fields

Besides influencing droplet velocity and deformation, changes in capillary number and viscosity ratio may also lead to differences in the flow behaviour inside droplets, which plays an important role in applications such as microfluidic mixers (Stone & Stone Reference Stone and Stone2005; Wang et al. Reference Wang, Wang, Feng and Lin2015; Fu et al. Reference Fu, Bai, Zhao, Zhang, Jin and Cheng2018; Li et al. Reference Li, Boulafentis, Stathoulopoulos, Liu and Balabani2023). To illustrate the effect of these parameters on the flow inside and outside the droplet, figure 26 shows streamlines for a steady-state drop in a square channel for different viscosity ratios and capillary numbers in a cross-sectional $\boldsymbol {x}$ - $\boldsymbol {y}$ plane at ${z}=0$ . The velocities outside the droplet were obtained from the BI representation of the velocity field outside the droplet:

(4.3) \begin{align} \boldsymbol {u}(\boldsymbol {y})=&\boldsymbol {u}_\infty (\boldsymbol {y}) + 2\int _{S_\infty } \boldsymbol {n}(\boldsymbol {x})\boldsymbol\cdot \boldsymbol {\tau }(\boldsymbol {x}-\boldsymbol {y})\boldsymbol\cdot \boldsymbol {q}(\boldsymbol {x})\, {\rm d}S_x \notag\\ &+ \boldsymbol {F}(\boldsymbol {y})+(\lambda -1)\int _{S_d} \boldsymbol {n}(\boldsymbol {x})\boldsymbol\cdot \boldsymbol {\tau }(\boldsymbol {x}-\boldsymbol {y})\boldsymbol\cdot \boldsymbol {u}(\boldsymbol {x})\, {\rm d}S_x, \end{align}

where $\boldsymbol {F}(\boldsymbol {y})$ is given by (2.3). A detailed derivation can be found in Roure et al. (Reference Roure, Zinchenko and Davis2023). In contrast, to calculate the flow velocity inside the drop, we solved a generalised BI equation, similar to previous works by Gissinger et al. (Reference Gissinger, Zinchenko and Davis2021a ) and Roure et al. (Reference Roure, Zinchenko and Davis2023), using the velocity boundary conditions obtained from the numerical solution of (2.1)–(2.3). As in previous works (Martinez & Udell Reference Martinez and Udell1990; Lac & Sherwood Reference Lac and Sherwood2009; Jing et al. Reference Jing, Lu, Farutin, Guo, Wang, Wang, Misbah and Sui2024), the flow fields are calculated in the droplet frame of reference.

For $\lambda =1$ , and $\lambda = 0.5$ , our results show flow patterns with small recirculation zones in the front portion of the drop, whereas larger recirculation zones are observed in the middle portion of the drop, similar to previous works (Martinez & Udell Reference Martinez and Udell1990; Lac & Sherwood Reference Lac and Sherwood2009; Jing et al. Reference Jing, Lu, Farutin, Guo, Wang, Wang, Misbah and Sui2024). The presence of the double-vortex structure in the external flow also indicates the existence of small recirculation regions in the back portion of the droplet, as reported by Martinez & Udell (Reference Martinez and Udell1990) and Lac & Sherwood (Reference Lac and Sherwood2009), which are not captured in this resolution. As the droplet velocity increases (i.e. for higher $Ca$ and smaller $\lambda$ ), the external vortices, as well as the small vortex structures inside the droplet, decrease in size. For $\lambda =0.05$ and $Ca=0.6$ , as the drop steady-state velocity surpasses the maximum velocity of the background flow, the external vortices are no longer present and the smaller recirculation zones inside the droplet completely vanish.

The fluid flow in a rectangular channel is not axisymmetric. When a drop is pushed through a rectangular channel by a pressure-driven flow, there is larger space between the drop surface and the channel walls in the corner region than near the middle of the walls, so that the surrounding fluid is expected to flow with less resistance near the channel corners. To investigate this corner flow, we plotted the velocity fields of the surrounding liquid flow in a $\boldsymbol {y}$ - $\boldsymbol {z}$ cross-sectional plane of a square channel ( $y\in [0,1]$ and $z\in [-0.5,0.5]$ ) at a position halfway between the drop nose and its centroid in figures 27 $(a)$ – $(c)$ for $\lambda =0.05,\, Ca=0.1,$ and different drop sizes. The observations indicate that the flow is directed into the channel corner from the middle of the walls, verifying this expectation. This flow pattern differs from what is typically observed in axisymmetric channels and is most prominent in drops at low $Ca$ and low $\lambda$ . Figure 27 $(d)$ shows the velocity vectors of the surrounding fluid flow for a drop of $D/H=0.6,\,\lambda =0.05,$ and $Ca=0.1$ in a $\boldsymbol {y}$ - $\boldsymbol {z}$ cross-sectional plane at the nose of the drop. We have previously observed the presence of recirculation vortices in the bulk fluid flow pattern outside the drop in figure 26. These recirculation vortices are usually larger for drops of bigger size. Therefore, we see that the velocity fields in a cross-sectional $\boldsymbol {y}$ - $\boldsymbol {z}$ plane at the nose of the drop changes direction close to the channel centreline in figure 27 $(d)$ . Away from the centreline, there is a weak flow to the corner regions. On the rear half of the drop (not shown), the flow in the $\boldsymbol y$ - $\boldsymbol z$ cross-sectional plane is away from the corners.

Figure 28 shows the velocity vectors of the surrounding fluid flow for a drop of $D/H=0.8,\,\lambda =0.05,$ and $Ca=0.1$ in $(a)$ the $\boldsymbol {x}$ - $\boldsymbol {y}$ plane and $(b)$ the $\boldsymbol {x}$ - $\boldsymbol{y}^{\prime}$ plane, where $\boldsymbol{y}^{\prime}$ is the diagonal of the square cross-section of the microchannel, starting at one corner. The flow pattern outside the drop in the plane containing this diagonal (figure 28 b) is qualitatively similar to that in the $\boldsymbol {x}$ - $\boldsymbol {y}$ plane (figure 28 a), except that there is more space between the drop surface and the walls. Hence, the flow is stronger in the corner regions and the recirculation vortices are smaller in the diagonal plane than the $\boldsymbol {x}$ - $\boldsymbol {y}$ plane.

Figure 27. Velocity fields of the surrounding fluid flow in a $\boldsymbol {y}$ - $\boldsymbol {z}$ cross-sectional plane midway between the drop nose and drop centroid for $\lambda =0.05,\, Ca=0.1,$ and $(a)$ $D/H=0.4$ , $(b)$ $D/H=0.6$ and $(c)$ $D/H=0.8$ ; $(d)$ $\boldsymbol {y}$ - $\boldsymbol {z}$ cross-sectional plane at the nose of a drop for same conditions as $(b)$ . The solid curves show the drop contours in $(a)$ , $(b)$ and $(c)$ .

Figure 28. Velocity fields of the surrounding fluid flow for a drop of $\lambda =0.05,\,Ca=0.1,$ and $D/H=0.8$ in $(a)$ the $\boldsymbol {x}$ - $\boldsymbol {y}$ plane and $(b)$ $\boldsymbol {x}$ - $\boldsymbol{y}^{\prime}$ plane, where $\boldsymbol{y}^{\prime}$ is the diagonal of the square cross-section of the channel. The solid curves show the drop contours.