4.1. Single particle phenomena

The results shown in figure 2 have multiple layers of scientific information, which are rather difficult to comprehend at one go. Thus, in order to elucidate the origin of such migrations of the charged particles inside a microparticle-laden fluid flow, a series of experiments have been performed involving either a single particle or a pair of particles. We initiate the discussions with the motions of single particles under an electric field with conducting and non-conducting surfaces. The experimental time sequence snapshots of a silver coated (Ag-coated) and an uncoated amberlite resin particle of $\sim$500 $\mathrm {\mu }$m radius each, under application of a 9 kV cm$^{-1}$ average electric field, are demonstrated in figures 3(a) and 3(b), respectively. The oscillatory motion of the conductive Ag-coated particle is expected and studied in earlier literature (Drews et al. Reference Drews, Kowalik and Bishop2014, Reference Drews, Cartier and Bishop2015). However, the uncoated amberlite particle also shows similar oscillatory behaviour, as shown by figures 3(b) and 3(e). This observation is particularly interesting, given the fact that the conductivity of the uncoated amberlite particle is rather limited (${\sim } 10^{-10}$ S m$^{-1}$). Figure 3(c) demonstrates the simulated time sequence snapshots of a 500 $\mathrm {\mu }$m radius conductive particle under application of a 9 kV cm$^{-1}$ average electric field.

Figure 3. Time sequence snapshots of (a) a Ag-coated and (b) an uncoated, amberlite resin particle of $\sim$500 $\mathrm {\mu }$m radius each under application of a 9 kV cm$^{-1}$ average electric field. (c) Simulated time sequence snapshots of a 500 $\mathrm {\mu }$m particle under application of a 9 kV cm$^{-1}$ average electric field. The time indicated above each micrograph has unit of seconds (s). The plot (d) shows the variations of the charging time ($t_{ch}$) of a $\sim$500 $\mathrm {\mu }$m radius amberlite resin particle coated with different materials with the average applied electric field. The error bars represent the maximum standard deviations obtained from five sets of experiments. The plot (e) shows the variations of the positions of the centres ($h$) of the particles from the bottom electrode at $z=0$, with time ($t$) corresponding to (a), (b) and (c). The broken (solid) lines correspond to the experimental (simulated) values, respectively. The experiments were visualized under a microscope at 2.5$\times$ magnification. The image panels shown in (a) and (b) correspond to the top view of the particles.

This experiment raises an important question on the mechanism which causes the reversal of the direction of the particles immediately after contact. In this regard, a collisional rebound may be thought of as one of the possibilities. However, the Stokes number (ratio of particle inertia to viscous forces), $St(=\frac {1}{9}({{\rho _s}/{\rho _f}})Re_s)$ (where $Re_s={\rho _s D_s U_i}/{\mu _f}$ and $D_s$ and $U_i$ denote particle diameter and impact velocity, respectively), for the experiments reported here, is found to be $O(10^{-3})$. Prior-art (Joseph et al. Reference Joseph, Zenit, Hunt and Rosenwinkel2001; Birwa et al. Reference Birwa, Rajalakshmi, Govindarajan and Menon2018; Ruiz-Angulo, Roshankhah & Hunt Reference Ruiz-Angulo, Roshankhah and Hunt2019) suggests that below a critical value of $St\hspace {1mm}(\sim 10)$ the particles may not rebound after collision. Hence, any possibility of rebound due to an electrode–particle elastic collision can be safely ignored. The more probable reason can be that the non-conductive amberlite particles too undergo some charge transfer during contact with the electrodes, in a fashion similar to the Ag-coated conductive particles.

Figure 3(d) demonstrates the experimentally evaluated values of average charging time ($t_{ch}$) of an amberlite particle ($r_s$, $\sim$500 $\mathrm {\mu }$m) during contact with the electrodes. The results are reported for the particles coated with different materials and at different electric fields ($E_0$). Two additional types of particles, a conductive Ni-coated (nickel-coated) and a non-conductive Fe-coated (iron-coated) particle, were experimented with to get an approximate knowledge of the oscillatory trends shown by different materials. The Fe-coated particles were kept for two days at room temperature to reduce their conductivity due to oxidation. Figure 3(d) shows that both the conductive Ag-coated and Ni-coated particles show shorter charging times ($t_{ch}$) at the electrodes compared with the non-conductive Fe-coated and uncoated particles. In these experiments, $t_{ch}$ was evaluated by noting the difference in the time of zero approach velocity and the same for a marginal rebound velocity. The experiments suggest that the electrical conductivity of the particle at the surface has a significant influence on the charging time of the particles at the electrodes. Figure 3(e) shows the experimental trajectories of the Ag-coated and uncoated particles corresponding to figures 3(a) and 3(b), respectively.

Figure 3(e) also shows the numerically simulated trajectories in the geometry shown in figure 1(b), which is very similar to the experimental set-up shown in figure 1(a). It may be noted that in the simulations the particles were modelled as conductive. The dimensions shown in the schematic diagram of the computational domain in figure 1(b) were used for all the simulations, unless otherwise stated. The surrounding liquid and the particles were assigned physical properties similar to those mentioned for the experiments in § 2, unless otherwise stated. The particles were assigned the simulated theoretical values of charge given in figure 4(a), which depicts the charge acquired by a conductive particle in contact with the electrode, unless otherwise stated.

Figure 4. (a) The variation of the net charge ($q$) acquired by amberlite particles of $\sim$500 $\mathrm {\mu }$m radius, coated with a variety of materials, with the average applied electric field. The solid (hollow) symbols denote the experimental (numerical) values, respectively. The error bar represents the the maximum standard deviations obtained from five sets of experiments. (b) The variation of the simulated values of dimensionless drag coefficient ($\lambda _d$) with the dimensionless position ($h/r_s$) of a particle ($r_s=500$ $\mathrm {\mu }$m) moving between two electrodes 5 mm apart. Here, $h$ is the position of the centre of the particle measured from the lower electrode at $z=0$.

It can be inferred from figure 3(e) that the motion of the uncoated particle is slightly sluggish compared with the Ag-coated particle. The numerically simulated trajectory again predicts higher particle speeds than both the uncoated and Ag-coated particles. The reason behind this trend can be better understood from figure 4(a), which depicts the average charge ($q$) acquired by the particles at different values of average applied electric field ($E_0$). The method to calculate charge from the experiments is given in § 1.4 of the supplementary material. The drag force on the particles was experimentally estimated using the numerically simulated values of drag coefficient across the channel as depicted in figure 4(b). The hollow symbols in 4(a) correspond to the numerically simulated values of charge, which are approximately equal to the theoretical charge $Q_0$ described above. The Ag-coated and the Ni-coated particles are found to acquire $\sim$73 % while the Fe-coated and uncoated particles are found to contain $\sim$48 % of the theoretical value of charge $Q_0$. As the uncoated particle acquires less charge compared with the Ag-coated particles, and both the particles acquire significantly less charge compared to the theoretical values, they are acted upon by less electric force compared with the simulated particle. Hence, the experimental trajectories show sluggish behaviour compared with the numerical one.

4.2. Two-particle phenomena

Figures 5(a) and 5(b) demonstrate the experimental time sequence snapshots of Ag-coated amberlite resin particle pairs moving inside a liquid medium under application of a 9 kV cm$^{-1}$ average electric field. The videos of these motions have been summarized in supplementary movie 2. The image panel in (a) shows a pair of equal-sized particles of $\sim$550 $\mathrm {\mu }$m radius each, while the smaller particle in (b) is $\sim$400 $\mathrm {\mu }$m. The trajectories in the image panel (a) suggest that the particles, after reversal of charge at the respective electrodes after contact, move towards the electrodes of opposite polarity. During the motion, they appear to undergo an ‘elastic’ collision with each other before reversing their directions of motion. The behaviour is found to be very similar for the set-up with particles having different sizes in the image panel (b), with differences in the location of the point of contact and the path length of oscillations of the individual particles. In such a scenario, the prior-art suggests the possibility of the presence of a thin fluid layer between the particles (Birlasekaran Reference Birlasekaran1991; Joseph et al. Reference Joseph, Zenit, Hunt and Rosenwinkel2001; Khayari & Perez Reference Khayari and Perez2002; Drews et al. Reference Drews, Kowalik and Bishop2014, Reference Drews, Cartier and Bishop2015; Birwa et al. Reference Birwa, Rajalakshmi, Govindarajan and Menon2018; Ruiz-Angulo et al. Reference Ruiz-Angulo, Roshankhah and Hunt2019).

Figure 5. Experimental time sequence snapshots of two Ag-coated particles of (a) equal sizes of $\sim$550 $\mathrm {\mu }$m radius each (b) unequal sizes with radii of the smaller and bigger particles being $\sim$400 and $\sim$550 $\mathrm {\mu }$m, respectively, under application of 9 kV cm$^{-1}$ average electric field. The time indicated above each micrograph has units of milliseconds (ms). Variations of the positions of the centres ($h$) of the particles measured from the lower electrode at $z=0$, with time ($t$) for the image panels shown in (a) and (b). Experimental time sequence snapshots of two uncoated particles of (d) equal sizes of $\sim$550 $\mathrm {\mu }$m radius each (e) unequal sizes with radii of the smaller and bigger particles being $\sim$400 and $\sim$550 $\mathrm {\mu }$m, respectively, under application of 9 kV cm$^{-1}$ electric field. ($f$) Variations of the positions of the centres ($h$) of the particles with time ($t$) for the image panels shown in (d) and (e). The evenly broken (unevenly broken) lines correspond to the case of equal (unequal) sized particles, respectively. The suffix S (B) correspond to the smaller (bigger) particles, respectively. The experiments were visualized under a microscope at 2.5$\times$ magnification. The image panels shown in (a), (b), (d) and (e) correspond to the top view of the particles.

Figure 5(c) demonstrates the trajectories shown by the particle pairs in (a) and (b). The trajectories are reported in terms of positions of the centres ($h$) of the particles measured from the lower electrode at $z=0$, as a function of time ($t$). Figures 5(a) and 5(c) clearly indicate that the equal-sized particle pairs show a rather synchronized oscillatory behaviour. They approach each other with nearly similar speeds until the middle of the channel. After collision, they maintain reasonably the same speed of separation during their reverse motions. Figures 5(b) and 5(c) show that for the unequal-sized particles, the smaller particle accelerates after contact whereas the bigger particle moves rather sluggishly. Interestingly, the motions of the uncoated particles, shown in figures 5(d)–5(f) and supplementary movie 2, are found to be very similar to the Ag-coated particles, as shown in figures 5(a)–5(c).

Further interesting behaviours are observed when particles of two different types are used. For example, the oscillation characteristics of a Ag-coated amberlite particle and an uncoated particle under a 9 kV cm$^{-1}$ electric field are shown in figure 6(a–d). Again, as observed in figure 5, the smaller amberlite particle hastens its speed of return towards the electrode after collision, while the bigger Ag-coated particle moves rather slowly after contact, as shown in figure 6(a) and supplementary movie 3. This behaviour is more clearly indicated in the position versus time plots depicted in figure 6(d). In comparison, in the case of a bigger uncoated particle and a slightly smaller Ag-coated particle, the trajectories of the motion of the particles are found to be rather symmetric, as shown in figures 6(b) (and supplementary movie 3) and 6(d). For the combination of a Ag-coated particle and an uncoated amberlite particle of equal size figures 6(c) and 6(d) depict very marginal acceleration of the amberlite particle and deceleration of the Ag-coated particle after contact.

Figure 6. Experimental time sequence snapshots of a Ag-coated and an uncoated particle of (a) unequal sizes with radii of the smaller uncoated and bigger Ag-coated particles being $\sim$400 and $\sim$550 $\mathrm {\mu }$m, respectively, (b) unequal sizes with radii of the smaller Ag-coated and bigger uncoated particles being $\sim$450 and $\sim$550 $\mathrm {\mu }$m, respectively, and (c) almost equal sizes of $\sim$550 $\mathrm {\mu }$m radius each, under application of a 9 kV cm$^{-1}$ average electric field. The time indicated above each micrograph has units of milliseconds (ms). (d) Variations of the positions of the centres ($h$) of the particles measured from the lower electrode at $z=0$, with time ($t$), for the image panels shown in (a–c). Superscripts A, B and C in the legend indicate the particles shown in (a), (b) and (c), respectively, and Am corresponds to the uncoated particle and Ag corresponds to the Ag-coated particle. The black (yellow) rectangle represents the positive electrode (grounded electrode). The experiments were visualized under a microscope at 2.5$\times$ magnification. The image panels shown in (a), (b) and (c) correspond to the top view of the particles.

The experiments with equal-sized Ag-coated particles shown in figure 5(a) are qualitatively similar to that reported in previous studies (Mersch & Vandewalle Reference Mersch and Vandewalle2011; Bishop et al. Reference Bishop, Drews, Cartier, Pandey and Dou2018). Bishop et al. (Reference Bishop, Drews, Cartier, Pandey and Dou2018) observed that two equal conductive particles undergo electric-field-driven elastic collisions with redistribution of charge on their surfaces, keeping the total charge conserved. In addition to this, the results presented above reveal that (i) even non-conductive and dissimilar (a conductive and a non-conductive) particles and (ii) unequal-sized similar or dissimilar particles also undergo such field-driven collisions with charge reversals. To gain more insight into the nature of such field-driven collisions, quantitative estimations of charge contained by the particles before and after collisions are necessary. Figure 7(a–d) shows the variations of the average velocities ($v_s$) and the fraction of the initial charge retained by the particles after collisions ($q^{*}$) as functions of the average electric field intensity ($E_0$) for different combinations of particles. Figures 7(a) and 7(b) refer to two equal-sized Ag-coated and uncoated particles, each with a radius of $\sim$550 $\mathrm {\mu }$m each. Both the figures suggest that the velocities of approach and separation in the case of equal-sized particles remain reasonably similar as the particles retain approximately the initial amount of charge of opposite polarity after collision. Thus, the equal-sized particles undergo elastic collisions irrespective of the particle type. Figures 7(c) and 7(d) show the variations of unequal-sized Ag-coated and uncoated particles, respectively. As discussed in figures 5 and 6, figures 7(c) and 7(d) show that the velocities of the smaller particles increase and that of the bigger particles decrease after collision. The bigger particle retains almost 30–40 % of its initial charge, while the smaller particle retains approximately 120–150 %. The dotted (dashed) line refers to the separation velocity of the bigger (smaller) particle, predicted by elastic collision theory. It can be seen that $v_s$ of the smaller particle resembles the elastic collision velocities, but $v_s$ of the bigger particle is substantially less compared with the elastic collisions. Thus, it can be inferred that (i) equal-sized particles of similar type undergo ‘elastic’ electric-field-driven collisions and (ii) the collisions between unequal-sized particles are essentially ‘inelastic’.

Figure 7. Variation of the average velocity ($v_s$) of the particles and the fraction of the initial amount of charge retained after collision ($q^{*}$) with average electric field ($E_0$) for (a) equal-sized Ag-coated particles with radius $\sim$550 $\mathrm {\mu }$m each, (b) equal-sized uncoated particles with radius $\sim$550 $\mathrm {\mu }$m each, (c) unequal-sized Ag-coated particles with the radius of the smaller being $\sim$400 $\mathrm {\mu }$m and that of the bigger being $\sim$550 $\mathrm {\mu }$m, and (d) unequal-sized uncoated particles with the radius of the smaller being $\sim$400 $\mathrm {\mu }$m and that of the bigger being $\sim$550 $\mathrm {\mu }$m. In panels (c and d), the notations specify the following: $P_{B{\_}b}$ ($P_{S{\_}b}$), velocities of bigger (smaller) particle before collision; $P_{B{\_}a}$ ($P_{S{\_}a}$), velocities of bigger (smaller) particle after collision; $q_{B{\_}a}$ ($q_{S{\_}a}$), fraction of the initial charge retained by the bigger (smaller) particle after collision; $E_{B{\_}a}$ (;$E_{S{\_}a}$), velocity of the bigger (smaller) particle after elastic collision between them.

We further explore some of the other interesting characteristics of the charge reversal for a set of two-particle systems. For this purpose, we report the magnified time sequence experimental snapshots during the charge reversals of two particles, under application of 5 kV cm$^{-1}$ average electric field, in figure 8(a–c). The image panel in figure 8(a) denotes two equal-sized Ag-coated particles of $\sim$550 $\mathrm {\mu }$m radius each, while figure 8(b) corresponds to two equal-sized uncoated amberlite particles of the same size, as mentioned above. Figure 8(c) captures the dynamics of a Ag-coated (darker shade) and an uncoated amberlite (lighter shade) particle of $\sim$550 $\mathrm {\mu }$m radius each. It can be inferred from figure 8(a–c) and supplementary movie 4, that both the particles in each panel show synchronized motions before and after collisions in all cases. The positively charged upper particle (returning from the positive upper electrode) first shows apparent contact with the negatively charged bottom particle (returning from the grounded electrode), after which both the particles reverse their trajectory, as demonstrated in figures 5 and 6. The time periods for charge reversals of the particles last for $10^{-2}-10^{-1}$ s. However, it may be noted here that such time measurements reported are not exact due to experimental artefacts, such as optical aberrations, while capturing the videos.

Figure 8. Experimental time sequence snapshots show the contact dynamics of (a) two Ag-coated, (b) two uncoated and (c) a Ag-coated and an uncoated particle of $\sim$550 $\mathrm {\mu }$m radius each under application of a 5 kV cm$^{-1}$ average electric field. The time indicated below each micrograph has units of milliseconds (ms). (d) Variations of the time periods for charge reversals ($t_c$) for the particle pairs shown in panels (a–c) with the average applied electric field. The experiments were visualized under a microscope at 10$\times$ magnification. The image panels shown in (a), (b) and (c) correspond to the top view of the particles. The dashed lines in the image panels of (a–c) indicate the initial positions of the particles during their close approach with each other.

Figure 8(d) shows the variations of the time period for charge reversals ($t_c$) for the particle pairs shown in 8(a–c). The plot suggests that for all the three cases shown in 8(a–c), $t_c$ decreases with increasing field intensity, suggesting that the charge reversal kinetics during collision is directly proportional to the applied electric field. A higher value of electric field thus provides a greater driving force for rapid charge transfer between the particles. Experiments suggest that, at high electric fields, the local enhancement of electric field between the particles becomes progressively higher as they approach each other. This provides a relatively higher driving force for charge transfer, leading to shorter $t_c$, as illustrated in the computational study by Flittner & Přibyl (Reference Flittner and Přibyl2017) for the case of an oscillating droplet. Further, figure 8(d) suggests that the uncoated particles show slightly higher contact times than the Ag-coated particles. The Ag-coated particles with greater amounts of surface charge compared with the uncoated particles are expected to be subjected to greater local field enhancements during their collision. This may be a possible reason for the smaller $t_c$ than the corresponding uncoated particles.

The image panels in figure 9(a–d) and corresponding supplementary movie 5 depict the flight of two unequal-sized particles inside a liquid medium under a 5 kV cm$^{-1}$ electric field. The time sequence experimental snapshots in panel (a) correspond to Ag-coated particles of which the radius of the bigger particle is $\sim$550 $\mathrm {\mu }$m and the smaller particle is $\sim$450 $\mathrm {\mu }$m. Panel (b) shows the response for uncoated particles of approximately the same size as those in (a). Panels (c) and (d) depict bigger (smaller) Ag-coated (uncoated) particles. It is interesting to note that the contact dynamics shown by the unequal-sized particles is remarkably different from that observed in the case of equal-sized particles discussed above in figure 8. Figure 9 depicts that, even after the contact, the bigger particle carrying a larger amount of charge continues its motion while in contact with the smaller particle for a brief period before reversing its direction towards the oppositely charged electrode. The smaller particle, however, reverses its direction of motion almost immediately after the charge reversal.

Figure 9. Experimental time sequence snapshots of (a) two unequal-sized Ag-coated, (b) two unequal-sized uncoated, (c) a bigger Ag-coated and smaller uncoated and (d) a smaller Ag-coated and a bigger uncoated amberlite resin particles under application of a 5 kV cm$^{-1}$ average electric field. The time indicated below each micrograph has units of milliseconds (ms). The radii of the big and small particles are $\sim$550 and $\sim$450 $\mathrm {\mu }$m, respectively. The experiments were visualized under a microscope at 10$\times$ magnification. The images correspond to the top view of the particles. The dashed lines in the image panels of (a–d) indicate the initial positions of the bigger particles during their close approach with the smaller ones.

In view of these observations, the ‘inelastic’ collision characteristics of significantly different sized particles can be summarized as (a) the bigger particle, after its engagement with the smaller particle approaching from the opposite direction, briefly continues its motion in the original direction along with the smaller particle, at a much reduced speed (images (ii)–(iv) of each panel); (b) the smaller particle then detaches itself from the union and reverses its direction of motion at a slightly higher speed than that of its approach (images (iv)–(vii); (c) after its disengagement from the smaller particle, the bigger particle still continues its flight towards the smaller particle for a very brief period of time (images (vi) and (vii)) before finally reversing its direction towards the opposite electrode; and (d) the bigger particle exhibits a much smaller speed after colliding with the smaller one. The experiments shown so far suggest that the charge reversal mechanisms between a pair of equal- or unequal-sized particles are very different. While it is observed that the equal-sized particles temporarily freeze for a very brief period of time, during the anticipated charge reversal (figure 8a–c), the unequal-sized particles form a union and move briefly along the direction of the bigger particle, during which the charge reversal is expected to take place.

In fact, the multi-particle PLFF systems shown in figure 2 also show such pair-wise repeated ‘elastic’ and ‘inelastic’ collisions inside the liquid medium, for equal- and unequal-sized particles. With the progress in time, a combination of these types of pair-wise motions and charge reversals during the collision lead to the large-scale assemblage of the particles inside a liquid medium. The charged particles keep showing this pair-wise oscillatory behaviour with repeated reversal of motions after charge reversals until there are some void spaces available during the chain formation.

4.3. Numerical investigations

The experiments on the pairs of equal- and unequal-sized particles, shown in figures 5–9, uncover the following key details: (a) the equal-sized particles acquire charge of nearly equal quantity of opposite sign; (b) there is a disparity of charge acquired by unevenly sized particles with the smaller particle getting overcharged and the bigger particle getting undercharged after a collision. In order to obtain a physical explanation of such observations on the charge reversal and subsequent motions of the particle pairs, a set of numerical simulations were carried out using parameters that largely emulate the experimental conditions. The model employed a system of two particles suspended in a liquid medium to qualitatively uncover the salient features of the oscillatory motions between a pair of electrodes under electric field. The results shown here help in identifying the underlying mechanism of the formation of particle chains, as shown previously in figure 2.

In the first simulation, a pair of uniformly sized particles oscillating between two parallel electrodes is considered, as schematically shown in the figure 1(b). The simulated time sequence snapshots shown in figures 10(a) and 10(b), along with supplementary movie 6, demonstrate the response of two equal-sized particles 1 (negatively charged) and 2 (positively charged) of 500 $\mathrm {\mu }$m radius each, under a 9 kV cm$^{-1}$ electric field for $q_{2t}=1$ and $q_{2t}=0.5$, respectively. Here, $q_{2t}$ is defined as the ratio of surface charge density of the particle 2 to its theoretical value. Figure 10(a) shows the variation of the positions of the centres ($h$) of particles 1 and 2, measured from the electrode at $z=0$, with time, under a 9 kV cm$^{-1}$ electric field for different values of $q_{2t}$. It can be inferred from the plot (a) that for $q_{2t}=1$, when the particles contain equal and opposite charge, the displacements of the particles from their initial positions are similar. The negatively charged particle moves towards the positively charged anode, while the positively charged particle migrates towards the cathode.

Figure 10. Time sequence snapshots of two equal-sized particles 1 and 2 of 500 $\mathrm {\mu }$m radius each, under application of a 9 kV cm$^{-1}$ average electric field for (a) $q_{2t}=1$ and (b) $q_{2t}=0.5$. Here, $q_{2t}$ represents the ratio of charge ($q$) contained by particle 2 to the theoretical value of charge. Particle 1 contains the theoretical value of charge of opposite sign. The time indicated above each micrograph has units of milliseconds (ms). The solid lines in the image panels of (a) and (b) indicate the initial positions of the particles during their close approach with each other. (c) The positions of the centres of two particles ($h$) measured from the bottom electrode at $z=0$, with time for different values of $q_{2t}$. The upper (black) graphs correspond to particle 1 and the lower (red) graphs correspond to particle 2. The inset plot represents the variation of $h$ of particle 2 during the initial time. The simulations were carried out employing an axisymmetric domain.

This synchronized motion of equal-sized particles is qualitatively confirmed from the experimental trajectory plots depicted in figures 5–8. For $q_{2t}=0.5$, wherein particle 2 contains half the amount of charge contained by particle 1, figures 10(b) and 10(c) demonstrate that the trajectories are not similar. Particle 1, containing higher charge, shows expected motion towards the oppositely charged electrode. In contrast, particle 2, containing less charge, follows particle 1 for a brief period of time before migrating towards the oppositely charged lower electrode. Thus, the results in the simulated image set (b) is found to be very similar to the experimental results involving the unevenly sized particles in figures 5, 6 and 9. This observation leads to a reasonable confirmation that, during the process of dynamic charge reversals of the particles of unequal sizes, the charge distribution is rather unequal. The bigger particle shows a relatively slower motion after the ‘inelastic’ collision because it gains relatively less charge of opposite polarity.

Figure 11(a,b) demonstrates the simulated time sequence snapshots of a bigger particle 1 (negatively charged) and a smaller particle 2 (positively charged) of radii 500 and 400 $\mathrm {\mu }$m, respectively, under a 9 kV cm$^{-1}$ electric field for $q_{1t}=1$ and $q_{1t}=0.2$, respectively. Here, $q_{1t}$ represents the ratio of charge contained by particle 1 to the theoretical value of charge. Particle 2 contains the theoretical amount of charge in all the cases. In the first case, wherein the smaller particle 2 contains a lower magnitude of charge compared with particle 1, similar results are obtained as previously shown in figure 10(b). The negatively charged particle 1 rapidly moves towards the positively charged upper electrode, while the positively charged particle 2 follows particle 1 for a very brief period of time, before migrating towards the grounded electrode. However, experiments contradict this observation, thus suggesting that the bigger particle contains a smaller amount of charge after the collision. This result is further validated by the observations shown in figure 11(b) along with supplementary movie 7, where the bigger particle contains 20 % of the theoretical value of charge. In this case, in line with the experimental observations shown in figure 9, the bigger particle follows the smaller one for a brief period of time before reversing its direction.

Figure 11. Simulated time sequence snapshots of two unequal-sized particles 1 and 2 of radii 500 and 400 $\mathrm {\mu }$m, respectively, under application of a 9 kV cm$^{-1}$ average electric field for (a) $q_{1t}=1$ and (b) $q_{1t}=0.2$. Here, $q_{1t}$ represents the ratio of charge ($q$) contained by particle 1 to the theoretical value of charge. Particle 2 contains the theoretical amount of charge of opposite sign. The time indicated above each snapshot has units of milliseconds (ms). The solid lines in the image panels of (a) and (b) indicate the initial positions of the particles during their close approach with each other. (c) The positions of the centres of the two particles ($h$) measured from the bottom electrode at $z=0$, with time for different values of $q_{1t}$. The upper (black) graphs correspond to particle 1 and the lower (blue) graphs correspond to particle 2. The inset plot represents the variation of $h$ of particle 1 during the initial time. The simulations were carried out employing an axisymmetric domain.

Figure 11(c) demonstrates the variations of the positions of the centres of the two particles ($h$) measured from the electrode at $z=0$, with time for different values of $q_{1t}$. The inset plot shows that particle 1 shows motion towards particle 2 only if the net charge on particle 1 is less than that of 2. The plots suggest that the larger the difference between the charges on the particles, the greater is the distance traversed by particle 1 in the direction of particle 2. Another key observation is that as the charge on 1 decreases, the speed of particle 2 increases, although the magnitude of charge on particle 2 remains constant. This observation is qualitatively similar to the experimental observations shown in figures 5 and 6 where the smaller particles show slight increment in speed compared with their original after charge reversal.

Figure 12 compares the simulated velocities of Ag-coated particles with the experimental ones shown in figure 7. Figure 12(a) shows that in the case of the equal-sized particles, the simulated velocities predict the experimental ones with reasonable accuracy. It may be noted here that charge assigned to each particle was $Q_0$. In the case of the unequal-sized particles, based on the observations made in figure 7(c,d), the simulated particles were assigned $Q_0$ charge before collision, and after collision the charge assigned to the bigger particle was $0.35Q_0$ and that to the smaller particle was $1.35Q_0$. With these assigned values of charges, figure 12(b) shows that the simulated values of $v_s$ mimic the experimental values very well. The slight differences between the experimental and simulated values may be attributed to factors such as (i) the effect of low but finite liquid conductivity is not included in the simulations in which the liquid is assumed to be non-conductive, (ii) the effect of gravity is ignored in the simulations, (iii) the particles in the experiments are not perfect conductors as assumed in the simulations and (iv) there are inaccuracies associated with assignment of values of parameters such as liquid viscosity, particle charge and liquid permittivity in the simulations.

Figure 12. Variation of the average velocity ($v_s$) of the particles with average electric field ($E_0$) for (a) equal-sized Ag-coated particles with radius 550 $\mathrm {\mu }$m each and (b) unequal-sized Ag-coated particles with the radius of the smaller being 400 $\mathrm {\mu }$m and that of the bigger being 550 $\mathrm {\mu }$m. In panel (a), the dotted lines refer to simulated velocities. In panel (b), the notations specify the following: $P_{B{\_}b}$ ($P_{S{\_}b}$), velocities of bigger (smaller) particle before collision; $P_{B{\_}a}$ ($P_{S{\_}a}$), velocities of bigger (smaller) particle after collision; $P_{B{\_}sim{\_}b}$ ( $P_{S{\_}sim{\_}b}$), simulated velocities of bigger (smaller) particle before collision; $P_{B{\_}sim{\_}a}$ ($P_{S{\_}sim{\_}a}$), simulated velocities of bigger (smaller) particle after collision.

Figures 13(a) and 13(b) demonstrate the experimental time sequence snapshots of two Ag-coated particles under 8 and 5 kV cm$^{-1}$ electric fields, respectively. The diameters of the big and small particles are $\sim$550 and $\sim$450 $\mathrm {\mu }$m, respectively. The image panels (a) and (b) show that the average displacement of the bigger particle towards the smaller particle under two different electric fields is rather similar. However, in the case of the lower electric field, as shown in (b), the contact time between the particles as well as the time of retraction of the bigger particle is much higher compared with the higher electric field situation demonstrated in (a). This behaviour is intuitive for lower field values as charge reversal kinetics is expected to be slow, leading to an ‘inelastic’ collision with a higher contact time. The sluggish return of the bigger particle can be attributed directly to the lower value of electric force acting on it at lower field intensity. These observations are qualitatively confirmed by the simulated values of the variations of the positions of the centres of a negatively charged (400 $\mathrm {\mu }$m) particle 1 and a positively charged (500 $\mathrm {\mu }$m) particle 2 ($q_{2t}=0.5$) with time at different applied electric field intensities, as demonstrated in figure 13(c). The inset plot in (c) clearly demonstrates that, while the bigger particle 2 undergoes similar displacement towards the smaller particle 1 at all studied field strengths, the time of flight of particle 2 increases with decreasing field.

Figure 13. Experimental time sequence snapshots of Ag-coated amberlite resin particles under application of (a) 8 kV cm$^{-1}$ and (b) 5 kV cm$^{-1}$ average electric fields. The time indicated below each micrograph has units of milliseconds (ms). The diameters of the big and small particles are $\sim$550 and $\sim$450 $\mathrm {\mu }$m, respectively. The experiments were visualized under a microscope at 10$\times$ magnification. The images correspond to the top view of the particles. The dashed lines in the image panels of (a and b) indicate the initial position of the bigger particle during its close approach with the smaller one. (c) Simulated values of the variations of the positions of the centres ($h$) measured from the lower electrode at $z=0$ of a 400 $\mathrm {\mu }$m particle 1 and a 500 $\mathrm {\mu }$m particle 2 with time ($t$), at different applied electric field intensities. The upper (blue) graphs correspond to particle 1 and the lower (red) plots correspond to a particle 2. The inset plot represents the magnified view of the positions of particle 2 during the initial time period. The simulations were carried out employing an axisymmetric domain.

Having discussed the origins of the oscillatory behaviours of the particles alongside their charge reversal dynamics under electric field, we now shift our focus to explore some of the important factors that drive the proposed phenomena. Figures 14(a) and 14(b) demonstrate the simulated distribution of the $z$-component of the Maxwell stress tensor ($\boldsymbol {\tau }_z$) over two equal-sized particles 1 (negatively charged) and 2 (positively charged) of 500 $\mathrm {\mu }$m radius each, under application of a 9 kV cm$^{-1}$ average electric field. Here, particle 1 is assigned the theoretical amount of charge. Figure 14(c) demonstrates that the magnitude of the Coulomb force ($F_C$) acting on the particles is very large compared to the dielectrophoretic force ($F_D$), and thus the former almost entirely contributes to the electrical forces acting on the particles. Figure 14(a) correspond to $q_{2t}=1$, wherein both the particles contain equal charge of opposite polarity. The images suggest that the negatively charged particle 1 experiences an attractive force towards the anode, while the positively charged particle 2 experiences a pull towards the cathode. This eventually leads to the separation of the particles from each other.

Figure 14. Simulated time sequence snapshots of the distribution of the $z$-component of the Maxwell stress tensor ($\boldsymbol {\tau }_z$) over two equal-sized particles 1 and 2 of 500 $\mathrm {\mu }$m diameter under application of a 9 kV cm$^{-1}$ average electric field for (a) $q_{2t}=1$ and (b) $q_{2t}=0.5$. Here, $q_{2t}$ represents the ratio of charge contained by particle 2 to its theoretical value of charge. Particle 1 contains the theoretical values of charge. The time indicated above each snapshot has units of milliseconds (ms). (c) Variations of the Coulomb force ($F_{C}$) and the dielectrophoretic force ($F_{D}$) acting at points $1^{t}$ and $2^{b}$ with time. The simulations were carried out employing an axisymmetric domain.

In order to obtain more in-depth information regarding the distribution of electric force on the particles, we consider points $1^{t}(2^{t})$ at the topmost point of particle 1 (particle 2) and $1^{b}(2^{b})$ at the bottom-most point of particle 1 (particle 2), respectively. It may be noted here that the usage of the words ‘top’ and ‘bottom’ is only directed towards describing the results because the gravitational influence has been neglected in the formulation. Figure 15(a), corresponding to the case shown in figure 14(a), depicts the variation of $\boldsymbol {\tau }_z$ with time at the topmost and bottom-most points of the two particles. It can be seen from the plot that the top pole of particle 1 experiences a strong attractive force towards the anode, while the bottommost point experiences a weak attractive force towards particle 2. This facilitates the net movement of the particle towards the anode. On the other hand the bottom pole of particle 2 experiences a strong attractive force towards the cathode, while the topmost point experiences a weak attractive force towards particle 1. This facilitates the net movement of the particle towards the cathode. In a nutshell, when the particles carrying opposing charges of equal magnitude move in close proximity under an electric field, they experience a mutual attraction as well as an attractive force towards the electrodes of opposite polarity. While the brief association during the charge reversal is caused by the particle–particle attractive force, they move apart towards the electrodes owing to the larger particle-electrode attractive force.

Figure 15. Variations of the $z$-component of the Maxwell stress tensor ($\boldsymbol {\tau }_z$) at the topmost and bottom-most points of two equal-sized particles 1 and 2 of 500 $\mathrm {\mu }$m diameter under a 9 kV cm$^{-1}$ electric field for (a) $q_{2t}=1$ and (b) $q_{2t}=0.5$. Here, $q_{2t}$ represents the ratio of charge contained by particle 2 to its theoretical value of charge. Particle 1 contains the theoretical values of charge. The subscripts $t$ and $b$ in the legends denote the topmost and bottom-most points of particles 1 and 2 corresponding to figure 14. (c) Surface plot showing the variations of the magnitude of the electric field intensity ($E$) and (d) variations of $E$ along a cut line passing through the centres of particles 1 and 2 (axis of symmetry shown in figure 1b) at an average applied field intensity of 9 kV cm$^{-1}$, corresponding to the case shown in figure 14(a). (e) Surface plot showing the variations of $E$. ($f$) Variations of $E$ along a cut line passing through the centres of particles 1 and 2 (axis of symmetry shown in figure 1b) at an average applied field intensity of 9 kV cm$^{-1}$, corresponding to the case shown in figure 14(b). The colour scale in panels (c and e) from blue to red indicates increasing magnitude of $E$. The simulations were carried out employing an axisymmetric domain.

Figure 15(c) depicts the distribution of electric field magnitude ($E$) along a cut line passing through the centres of the particles (axis of symmetry shown in figure 1b). The plots show that the electric field magnitude is highest at the topmost part of particle 1 and the bottom-most part of particle 2, which increases as the particles approach the respective electrodes. Thus, points $1^{t}$ and $2^{b}$ experience larger attractive forces from the electrodes, which progressively increase as the particles move towards them. The electric field in the gap between the particles gradually decrease as the particles move apart, indicating reduction in the mutual attraction between the particles.

Figure 14(b) demonstrates the simulated distribution of the $z$-component of the Maxwell stress tensor ($\boldsymbol {\tau }_z$) over particles 1 and 2 for $q_{2t}=0.5$, wherein particle 2 contains half of the charge contained by particle 1 of opposite polarity. In this situation, the negatively charged particle 1 experiences a pull towards the positive electrode, while the positively charged particle 2 containing less charge briefly experiences a pull towards particle 1 and moves along its direction. The distributions of the stress due to electric field on the topmost and bottom-most points of particles 1 and 2, depicted in figure 15(b), reveal that the positive upward force acting on point $2^{t}$ is significantly larger than the downward force acting on point $2^{b}$ during the initial time period when the particles arrive at near proximity, and eventually decrease with increase in the separation distance. Thus, particle 2 briefly moves towards particle 1, before reversing its motion towards the oppositely charged electrode. The variation of the magnitude of electric field corresponding to this situation is depicted in figure 15(e, f), which suggests a significant increment of electric field between the particles during their close proximity. The electric field in the gap decreases as the particles move apart, leading to the decrease in the upward force experienced by $2^{t}$ compared with the downward force experienced by $2^{b}$ thus causing it to move away from the other particle. In this case, contrary to the case for $q_{2t}=1$, $1^{b}$ experiences a significantly larger downward force compared with the upward force experienced by $1^{t}$, during their close proximity, for a very brief period of time. However, particle 1 still experiences a larger net positive force, which causes it to move towards the upper electrode.

The simulated results shown in figures 14(b) and 15(b) can be summarized as follows: when two particles carrying unequal amounts of charge of the opposite polarity, are placed in close proximity under the influence of an externally applied electric field, the particles experience mutual attraction towards each other and towards the respective electrodes of opposite polarity. When their separation distance is small, the attraction experienced by the particle of less charge towards the particle of higher charge is larger than its attraction towards the electrode of opposite polarity. Thus, the particle of lower charge follows the particle of higher charge briefly, until the separation between them increases to the extent that it experiences more attractive force towards the electrode of opposite polarity, after which the particle starts moving towards the electrode. The net force on the particle of greater charge always acts towards the electrode of opposite polarity and it moves towards the electrode.

To summarize the above discussed cases, we present two model simulations shown in supplementary movies 8 and 9. Movie 8 depicts the motion of particle 1 (initially positively charged) and particle 2 (initially negatively charged) under a 9 kV cm$^{-1}$ electric field. On contact with each other they undergo charge transfer, which is modelled using an arbitrary rate constant to mimic the experimental time scales. Post charge transfer, they retain equal charge of opposite polarity and, under the influence of the externally applied field, they move apart in a synchronized manner. This simulation is qualitatively similar to the experimental motions of equal particles shown in supplementary movie 2. Movie 9 depicts the motion of a smaller particle 1 (initially positively charged) and a bigger particle 2 (initially negatively charged) under the application of a 9 kV cm$^{-1}$ average electric field. In this case post contact and charge transfer, while particle 1 retains the initial charge magnitude of opposite polarity, particle 2 retains only 20 % of initial charge of opposite polarity. Thus, it can be seen travelling in the direction of 1 for a brief period of time, before reversing its direction. This simulation emulates the experimental motions of unequal particles shown in supplementary movies 2 and 3.

4.3.1. Flow patterns and role of fluid properties

Thus far, the major focus of the discussion has been to elucidate the motions of the particles inside the liquid medium under an electric field. In this direction, the surrounding liquid medium is found to play a key role. For example, the dielectric contrast at the liquid–particle interface decides the capacity of the particles to retain charges, which is a crucial factor in determining the extent of Coulombic force experienced by them. Further, formation of interesting flow patterns can also be envisaged during their periodic approach and reversal of the spherical particles between the electrodes. Such flow patterns may not only be important in deciding the speeds of particle migration, however; they may also help in deciding the directions of the particle motion. In such a scenario, the viscosity of the liquid is expected to play a crucial role especially when the particles sizes are different.

Figures 16 and 17 show the velocity and streamline fields in the surrounding liquid medium at the various stages of migration of a pair of equal- and unequal-sized particles under electric field, respectively. The plots suggest that, as the spherical particles move in the confined space between the electrodes, they entrain the liquid surrounding them. Subsequently, high fluid velocities are observed near the front and rear ends of the particles, as shown in the first three frames of the image sets (a) and (c) of figures 16 and 17. The liquid velocity between the particles increases as the particles approach each other with the increase in their speed. In the process, the liquid between drains out and during the collision the ejection of the liquid happens at a relatively higher velocity. Following this, once the particles separate and move towards the electrodes, they again entrain liquid with them until they collide with the electrodes, as shown by the frames after collision of the image sets (a) and (c) of figures 16 and 17.

Figure 16. Simulated time sequence snapshots of (a) fluid velocity and (b) streamline fields for viscosity of the fluid $\mu _{rf}=0.01$ Pa.s. (c) Fluid velocity and (d) streamline fields for viscosity of the fluid $\mu _{rf}=0.1$ Pa.s. The particles considered are of equal size with radius of 500 $\mathrm {\mu }$m each. The other parameters used for the simulations are $E_0=9$ kV cm$^{-1}$ and $\varepsilon _{rf}=3$. The time indicated above the snapshots has units of milliseconds. The velocities indicated by the colour scales have units of ms$^{-1}$.

Figure 17. Simulated time sequence snapshots of (a) fluid velocity and (b) streamline fields for viscosity of the fluid $\mu _{rf}=0.01$ Pa.s. (c) Fluid velocity and (d) streamline fields for viscosity of the fluid $\mu _{rf}=0.1$ Pa.s. The particles considered are of unequal sizes with the radius of the bigger particle being 550 $\mathrm {\mu }$m and the smaller being 400 $\mathrm {\mu }$m. The other parameters used for the simulations are $E_0=9$ kV cm$^{-1}$ and $\varepsilon _{rf}=3$. The time indicated above the snapshots has units of milliseconds. The velocities indicated by the colour scales have units of ms$^{-1}$.

Once the particles attain the steady to and fro oscillatory motion between the electrodes, a pair of counter-rotating circulation loops are found to appear across the particles near their equatorial regions. The arrows on the streamline plots denote the directions of the liquid flows in the zones of recirculations. In such a situation, the local stagnation points appear at the centres of the loops, where the liquid velocities nearly vanish, as shown in the image sets (b) and (d) of figures 16 and 17. As the particles move towards or away from each other, a stagnation point develops at the axial region between the particles, with the liquid around it moving in opposite directions. Examining the streamline plots for $\mu _{rf}=0.01$ Pa.s and $\mu _{rf}=0.1$ in figures 16 and 17, it can be seen that the circulation loops are centred much closer to the particles in the case of $\mu _{rf}=0.01$ Pa.s compared with $\mu _{rf}=0.1$ Pa.s. Interestingly, the recirculation zones are of similar size, for a very synchronized motion of the particles with an ‘elastic’ collision in the middle. However, if an ‘inelastic’ collision takes between unequal-sized particles, the sizes and shapes of the recirculation zones in the surrounding liquid medium are asymmetric, as shown in figure 17(b,d). Further, a comparison between the images of figure 17(b,d) suggests that the recirculations near the electrodes are also suppressed for the high-viscosity case, and the velocity of liquid ejection between the particle and electrode is smaller (figure 17b,d). The results also suggest that the multi-particle assemblages shown in figure 2 are expected to have an array of such flow patterns around the particles while undergoing incessant to and fro motion between the electrodes.

Apart from the flow structures in the surrounding liquid, the viscosity of the fluid also influences the kinetics of the particle migration. For example, figures 16 and 17 show that the velocities of particle oscillations decrease with increase in the liquid viscosity, due to an increase in the hydrodynamic drag force. Interestingly, the point of collision can also be modulated by tuning the viscosity of the liquid. Figure 20(a) shows the variations of the positions ($h$) of two unequal particles with time for two different liquid viscosities. The figure suggests that as the liquid viscosity increases, the point of collision shifts upwards. With the increase in viscosity of the liquid medium, the drag force experienced by each of the particles increases, with the electrical force acting on them remains constant. In the case of the smaller particle, for $\mu _{rf}=0.1$ Pa.s, due to an increase in the viscous drag, the net force acting downwards decreases more compared with the net upward force experienced by the bigger particle. Hence, the smaller particle travels a shorter distance downwards before its collision with the bigger particle. This, in turn, shifts the point of collision slightly upwards for $\mu _{rf}=0.1$ Pa.s. compared with $\mu _{rf}=0.01$ Pa.s.

Vorticity ($\boldsymbol {\omega }=\boldsymbol {\nabla }\times \boldsymbol {v}_f$) fields show the local fluid rotation or spinning near a point. Figure 18 shows the azimuthal vorticity ($\omega _\theta ={\partial v_{fr}}/{\partial z}-{\partial v_{fz}}/{\partial r}$) fields, for equal-sized particles, where $v_{fr}$ and $v_{fz}$ denote the $r$ and $z$ directional fluid velocities, respectively. Figure 19 denotes the vorticity fields for unequal-sized particles. At the surface of the solid, the fluid velocity goes to zero (no-slip condition), which results in very high velocity gradients in the regions surrounding the particles, leading to the generation of vorticity in the fluid. Figures 18 and 19 show that regions of strong positive and negative vorticity develop in the liquid in the immediate vicinity of the particles. Interestingly, the first three frames of the image sets shown in figures 18 and 19 show a common-flow-up configuration (CFUp) of twin vortices accompanying the bottom particle during its motion towards the top particle (Lu & Zhai Reference Lu and Zhai2019). Again in the same set of images, the vortices surrounding the top particle show a common-flow-down arrangement (CFDn) during its motion towards the bottom one. This can be more explicitly understood by the arrows on the first three frames of image sets (b) and (d) of figures 16 and 17. Importantly, the 2nd, 3rd and 4th frames of the image sets (a) and (b) of figures 18 and 19 (and the corresponding images on figures 16 and 17) suggest that after the collision there is a reversal of the direction of the rotations in the vortices. In such a situation, the top particle moving towards the top electrode is accompanied by a pair of CFUp vortices, whereas the bottom one is surrounded by a pair of CFDn vortices until it arrives the bottom electrode. In the case of the unequal-sized particles, after the collision, the smaller particles travels faster towards the top electrode and, for a brief period of time, both the particles travel downwards. Thus, the last images of the image sets shown in figure 19 depict negative vorticity fields around both the particles. It can be seen from both figures 18 and 19 that the magnitudes of vorticity are higher for $\mu _{rf}=0.01$ Pa.s compared with $\mu _{rf}=0.1$ Pa.s. This is because the velocity gradient in the lower viscosity fluid near the solid is greater than the higher viscosity fluid.

Figure 18. Simulated time sequence snapshots of azimuthal vorticity ($\omega _\theta$) fields for (a) viscosity of the fluid $\mu _{rf}=0.01$ Pa.s. and (b) viscosity of the fluid $\mu _{rf}=0.1$ Pa.s. The particles considered are of equal size with radius of 500 $\mathrm {\mu }$m each. The other parameters used for the simulations are $E_0=9$ kV cm$^{-1}$ and $\varepsilon _{rf}=3$. The time indicated above the snapshots has units of milliseconds. The $\omega _\theta$ values indicated by the colour scales are normalized with the maximum values. For $\mu _{rf}=0.01$ Pa.s, the values are normalized by $\omega _\theta =6\times 10^{4}$ s$^{-1}$, and, for $\mu _{rf}=0.1$ Pa.s, the values are normalized by $\omega _\theta =3000$ s$^{-1}$.

Figure 19. Simulated time sequence snapshots of azimuthal vorticity ($\omega _\theta$) fields for (a) viscosity of the fluid $\mu _{rf}=0.01$ Pa.s. and (b) viscosity of the fluid $\mu _{rf}=0.1$ Pa.s. The particles considered are of unequal sizes with the radius of the bigger particle being 550 $\mathrm {\mu }$m and the smaller being 400 $\mathrm {\mu }$m. The other parameters used for the simulations are $E_0=9$ kV cm$^{-1}$ and $\varepsilon _{rf}=3$. The time indicated above the snapshots has units of milliseconds. The $\omega _\theta$ values indicated by the colour scales are normalized with the maximum values. For $\mu _{rf}=0.01$ Pa.s, the values are normalized by $\omega _\theta =8\times 10^{4}$ s$^{-1}$, and, for $\mu _{rf}=0.1$ Pa.s, the values are normalized by $\omega _\theta =5000$ s$^{-1}$.

Another point of difference in the $\omega _\theta$ fields is that they are much diffused in the case of $\mu _{rf}=0.1$ Pa.s compared with $\mu _{rf}=0.01$ Pa.s. The vorticity transport equation can be written as ${\partial \boldsymbol {\omega }}/{\partial t}+(\boldsymbol {v}_{f}\boldsymbol {\cdot }\boldsymbol {\nabla })\boldsymbol {\omega }= (\boldsymbol {\omega }\boldsymbol {\cdot }\boldsymbol {\nabla })\boldsymbol {v}_{f}+\nu \nabla ^2 \boldsymbol {\omega }$, where $\nu$ refers to the kinematic viscosity of the fluid. The second term on the left-hand side of the equation denotes the convection of vorticity, while the second term of the right-hand side of the equation refers to the diffusion of vorticity with diffusivity, $\nu$. Since the fluid velocities are much less, the convection of vorticity can be ignored. Thus, the vorticity is transported from the regions near the surface of the solid mainly by diffusion. The length scale of the diffusion can be defined as $\sim \sqrt {\nu t}$. Thus, for $\mu _{rf}=0.1$ Pa.s, $\nu$ is greater compared to $\mu _{rf}=0.01$ Pa.s. Also, the time for diffusion is greater due to lower velocity in the case of $\mu _{rf}=0.1$ Pa.s. Thus, the vorticity is transported to a larger distance compared with $\mu _{rf}=0.01$ Pa.s, and the $\omega _\theta$ fields in the case of $\mu _{rf}=0.1$ are more diffused compared with $\mu _{rf}=0.01$ Pa.s.

Finally, the relative permittivity ($\varepsilon _{rf}$) of the fluid is another important parameter affecting the oscillation characteristics of the particles. Generally, relative permittivity in conjunction with the fluid conductivity are more important in determining the particle dynamics. But in the present study the fluid medium is considered as non-conductive. Hence, the variation of fluid conductivity is not considered. Figures 20(b) and 20(c) show the positions ($h$) of the particles with time for varying values of $\varepsilon _{rf}$. Figure 20(b) denotes the case of equal-sized particles, and figure 20(c) refers to unequal-sized particles with the smaller particle being at the top. Both the figures indicate that the velocities of the particles increase with increasing fluid permittivity. As $\varepsilon _{rf}$ increases, the electric field around the particle increases due to increasing bound charge density, leading to a greater Coulombic force acting on the particle. Thus, the particle speed is expected to increase in such circumstances.

Figure 20. Simulated variations of the positions of the centres of the particles ($h$) from the lower electrode at $z=0$ with time ($t$) for (a) two unequal-sized particles with the radius of the bigger particle being 550 $\mathrm {\mu }$m and the smaller being 400 $\mathrm {\mu }$m for different values of $\mu _{rf}$, (b) two equal-sized particles with radius of 500 $\mathrm {\mu }$m each and (c) two unequal-sized particles with the radius of the bigger particle being 550 $\mathrm {\mu }$m and the smaller being 400 $\mathrm {\mu }$m for different values of $\varepsilon _{rf}$. The other parameters used for (a) are $E_0=9$ kV cm$^{-1}$ and $\varepsilon _{rf}=3$. The other parameters used for (b) and (c) are $E_0=9$ kV cm$^{-1}$ and $\mu _{rf}=0.25$ Pa.s.