4.2.1 Drop velocity

It is evident from (4.4) and (4.5) that there are two important effects which alter the drop velocity: charge convection (the strength of which is represented by $Re_{E}$ ) and shape deformation (the strength of which is represented by $Ca$ ). In addition to these two dimensionless numbers, there are the following important electrohydrodynamic parameters which affect the drop motion: the relative strength of the components of applied electric field $(E_{x},E_{z})$ , the Mason number $(M)$ , the viscosity ratio $(\unicode[STIX]{x1D706})$ and the electrical property ratios $(R,S)$ . The effect of these parameters on the cross-stream migration velocity $(U_{dx})$ and steady state transverse position of the drop $(x_{d,\infty })$ are of prime importance. Before presenting the effect of charge convection and shape deformation on the drop velocity and drop trajectory, we investigate the physical reasons behind the charge-convection-induced and shape-deformation-induced cross-stream migration of a drop.

In the present study, in the presence of a tilted electric field and background plane Poiseuille flow, we have obtained not only alteration in the axial drop velocity but also a non-intuitive cross-stream migration of the drop due to charge-convection effects, which is denoted by $U_{dx}^{(Re_{E})}$ . The effect of charge convection on the cross-stream migration velocity can be well understood if we systematically investigate the charge distribution at the drop surface and alteration of the same due to fluid flow. Towards this we consider three representative cases: Case I with axial electric field $\boldsymbol{E}_{\infty }=\boldsymbol{e}_{z}$ (or $\unicode[STIX]{x1D703}_{t}=0$ ), Case II with transverse electric field $\boldsymbol{E}_{\infty }=\boldsymbol{e}_{x}$ (or $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/2$ ) and Case III with tilted electric field $\boldsymbol{E}_{\infty }=(\boldsymbol{e}_{x}+\boldsymbol{e}_{z})/\sqrt{2}$ (or $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/4$ ). Next we show the effect of fluid flow on the surface charge distribution. We show the contour plot of the surface charge density, $q_{s}$ . To represent the explicit variation of surface charge density with $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D719}$ , we also plot the variation of $q_{s}(\unicode[STIX]{x1D703})$ for $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/4$ , $3\unicode[STIX]{x03C0}/4$ and the variation of $q_{s}(\unicode[STIX]{x1D719})$ for $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/4$ , $3\unicode[STIX]{x03C0}/4$ . The choice of two $\unicode[STIX]{x1D719}$ values while showing the variation of $q_{s}(\unicode[STIX]{x1D703})$ is made so that two $\unicode[STIX]{x1D719}$ values are symmetrically located with respect to the axial plane $(\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/2)$ . Likewise, the choice of two $\unicode[STIX]{x1D703}$ values while showing the variation of $q_{s}(\unicode[STIX]{x1D719})$ is made so that two $\unicode[STIX]{x1D703}$ values are symmetrically located with respect to the transverse plane $(\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2)$ .

Figure 2.

This representation depicts the charge distribution at the drop surface. Charge distribution in (a–c) axial (Case I), (d–f) transverse (Case II) and (g–i) tilted (Case III) electric fields, respectively. (a,d,g) $q_{s}^{(0)}$ , (b,e,h) $q_{s}^{(0)}+Re_{E}q_{s}^{(Re_{E})}$ at $x_{d}=5$ (centreline), and (c,f,i) $q_{s}^{(0)}+Re_{E}q_{s}^{(Re_{E})}$ at $x_{d}=4$ (off-centreline), respectively. The parameters employed are $R=0.5$ , $S=2$ , $M=2$ , $\unicode[STIX]{x1D706}=0.2$ , $Re_{E}=0.2$ and $H=10$ . Colourbar represents the magnitude of the dimensionless surface charge density.

Case I (axial electric field). In figure 2(a–c) we show the charge distribution on the drop surface in the presence of an axial electric field ( $\boldsymbol{E}_{\infty }=\boldsymbol{e}_{z}$ or $\unicode[STIX]{x1D703}_{t}=0$ ) while the background flow is considered as plane Poiseuille flow. The parameters used to calculate the charge distribution are given in the figure caption. In the absence of charge convection (i.e.  $Re_{E}=0$ ), the contour plot of the surface charge density, $q_{s}^{(0)}$ , is depicted in figure 2(a). Figure 2(a) shows that the surface charge distribution is symmetric about the axial plane and antisymmetric about the transverse plane. Explicit variation of the surface charge density with $\unicode[STIX]{x1D703}$ is depicted in figure 3(a), which shows that when $Re_{E}=0$ the variation of $q_{s}(\unicode[STIX]{x1D703})$ is antisymmetric about the $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ plane. Figure 3(a) also depicts that the variation of $q_{s}(\unicode[STIX]{x1D703})$ is independent of $\unicode[STIX]{x1D719}$ , which signifies the symmetry of surface charge density about the axial plane. The charge distribution coupled with a tangential electric field creates a tangential electric force $(F_{\unicode[STIX]{x1D703}}^{E})$ which is the jump of tangential electric stress across the drop interface and can be obtained as $F_{\unicode[STIX]{x1D703}}^{E}=q_{s}(\boldsymbol{E}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D703}})$ . The distribution of the tangential electric force $(F_{\unicode[STIX]{x1D703}}^{E})$ with $\unicode[STIX]{x1D703}$ is represented in figure 3(b), which shows that $F_{\unicode[STIX]{x1D703}}^{E}$ is also antisymmetric about the transverse plane. Hence, the flow field generated due to this antisymmetric electric force $(F_{\unicode[STIX]{x1D703}}^{E})$ in the presence of $\boldsymbol{E}_{\infty }=\boldsymbol{e}_{z}$ is also antisymmetric, which finally leads to zero electrohydrodynamic force on the drop. This is the physical picture in the absence of charge convection (i.e.  $Re_{E}=0$ ). An interesting thing happens when we consider the charge-convection effect. The surface charge convection in the presence of plane Poiseuille flow markedly alters the charge distribution on the drop surface, as shown in figure 2(b,c). Figure 2(b) depicts the contours of the surface charge density, $q_{s}^{(0)}+Re_{E}q_{s}^{(Re_{E})}$ , when the drop is at the centreline ( $x_{d}=5$ , considering $H=10$ ) of the imposed flow. Comparison between $Re_{E}=0$ and $Re_{E}=0.2$ curves of figure 3(a) reveals that there is a significant reduction in charge density near $\unicode[STIX]{x1D703}=0$ and $\unicode[STIX]{x03C0}$ for $Re_{E}=0.2$ . This reduction in charge density is due to the velocity field at the drop interface. The tangential velocity field in the absence of charge convection is shown as an inset of figure 3(a). This leading-order tangential velocity at the drop surface is a linear combination of tangential velocities in a uniform electric field and plane Poiseuille flow. Combination of the uniform electric field and plane Poiseuille makes $V_{s,\unicode[STIX]{x1D703}}^{(0)}$ asymmetric about the transverse plane. The tangential velocity $(V_{s,\unicode[STIX]{x1D703}}^{(0)})$ is positive near $\unicode[STIX]{x1D703}=0$ and negative near $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}$ . This velocity distribution drives the charges away from $\unicode[STIX]{x1D703}=0$ , $\unicode[STIX]{x03C0}$ and creates an asymmetric charge distribution about the transverse plane. The asymmetry of charge distribution about the transverse plane is more clearly depicted in figure 3(c), in which we show the variation of charge distribution with $\unicode[STIX]{x1D719}$ for $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/4$ and $3\unicode[STIX]{x03C0}/4$ . Here, we plot the magnitude of the charge density, $|q_{s}|$ , so that the variation for both $\unicode[STIX]{x1D703}$ is noticeable. In the absence of charge convection (i.e.  $Re_{E}=0$ ), $q_{s}(\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/4)=-q_{s}(\unicode[STIX]{x1D703}=3\unicode[STIX]{x03C0}/4)$ for any $\unicode[STIX]{x1D719}$ , which signifies the antisymmetric charge distribution about the transverse plane. But when $Re_{E}=0.2$ , the magnitude of charge density at $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/4$ is very much different as compared with the magnitude of charge density at $\unicode[STIX]{x1D703}=3\unicode[STIX]{x03C0}/4$ . This asymmetric structure creates an asymmetric distribution of the tangential electric force, as depicted in figure 3(d). An asymmetric $F_{\unicode[STIX]{x1D703}}^{E}$ creates asymmetric electrohydrodynamic flow, which finally yields a net electrohydrodynamic force on the drop and an alteration in the drop velocity in the axial direction. There is no electrohydrodynamic force in the transverse direction. This is due to the fact that the charge distribution about the axial plane remains symmetric even in the presence of charge convection. This is reflected by the $\unicode[STIX]{x1D719}$ independence of $q_{s}(\unicode[STIX]{x1D703})$ and $F_{\unicode[STIX]{x1D703}}^{E}(\unicode[STIX]{x1D703})$ in figure 3(a) and figure 3(b), respectively.

Figure 3.

(a) Variation of $q_{s}(\unicode[STIX]{x1D703})$ for $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/4$ , $3\unicode[STIX]{x03C0}/4$ considering $x_{d}=5$ . (b) Variation of $F_{\unicode[STIX]{x1D703}}^{E}(\unicode[STIX]{x1D703})$ for $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/4$ , $3\unicode[STIX]{x03C0}/4$ considering $x_{d}=5$ . (c) Variation of $q_{s}(\unicode[STIX]{x1D719})$ for $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/4$ , $3\unicode[STIX]{x03C0}/4$ considering $x_{d}=5$ . (d) Variation of $F_{\unicode[STIX]{x1D703}}^{E}(\unicode[STIX]{x1D719})$ for $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/4$ , $3\unicode[STIX]{x03C0}/4$ considering $x_{d}=5$ . The parameters employed are $R=0.5$ , $S=2$ , $M=2$ , $\unicode[STIX]{x1D706}=0.2$ and $H=10$ .

Figure 4.

(a) Variation of $q_{s}(\unicode[STIX]{x1D703})$ for $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/4$ , $3\unicode[STIX]{x03C0}/4$ considering $x_{d}=4$ . (b) Variation of $F_{\unicode[STIX]{x1D703}}^{E}(\unicode[STIX]{x1D703})$ for $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/4$ , $3\unicode[STIX]{x03C0}/4$ considering $x_{d}=4$ . (c) Variation of $q_{s}(\unicode[STIX]{x1D719})$ for $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/4$ , $3\unicode[STIX]{x03C0}/4$ considering $x_{d}=4$ . (d) Variation of $F_{\unicode[STIX]{x1D703}}^{E}(\unicode[STIX]{x1D719})$ for $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/4$ , $3\unicode[STIX]{x03C0}/4$ considering $x_{d}=4$ . The parameters employed are $R=0.5$ , $S=2$ , $M=2$ , $\unicode[STIX]{x1D706}=0.2$ and $H=10$ .

When the drop is at an off-centreline location $(x_{d}=4)$ , the contours of surface charge density are depicted in figure 2(c). To investigate the asymmetry in the charge distribution and associated electric force, we plot the variation of $q_{s}$ and $F_{\unicode[STIX]{x1D703}}^{E}$ with $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D719}$ in figure 4(a–d). Figure 4(a) depicts that the variations of charge distribution with $\unicode[STIX]{x1D703}$ for $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/4$ and $\unicode[STIX]{x1D719}=3\unicode[STIX]{x03C0}/4$ are different, which signifies asymmetry about the axial plane. Similar asymmetry is also found in the variation of the electric force in figure 4(b). Asymmetry about the axial plane arises for a drop with an off-centreline position because in this situation the drop encounters a different velocity at different $\unicode[STIX]{x1D719}$ positions. On the other hand, figure 4(c) depicts that the variations of charge distribution with $\unicode[STIX]{x1D719}$ for $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/4$ and $\unicode[STIX]{x1D703}=3\unicode[STIX]{x03C0}/4$ are different, which signifies asymmetry about the transverse plane. Similar asymmetry is found in the variation of electric force in figure 4(d). The asymmetry in the charge distribution and electric force about the transverse plane finally creates an asymmetric electrohydrodynamic flow across the transverse plane, which yields a change in drop velocity in the axial direction. Notably, the asymmetry in the charge distribution and electric force about the axial plane also creates an asymmetric electrohydrodynamic flow around the drop interface. But this electrohydrodynamic flow is such that the net electrohydrodynamic force vanishes. However, asymmetry in the charge distribution and electric force creates a non-zero electrical torque of the form

An important thing to note here is that the direction of the electric torque is determined by the $R/S$ ratio and $x_{d}$ . This electric torque is exactly balanced by the torque due to the hydrodynamic stresses, which causes the drop to be torque-free

where the hydrodynamic torque acting on the drop is obtained from (Happel & Brenner Reference Happel and Brenner1981) $\boldsymbol{T}^{H(Re_{E})}=-8\unicode[STIX]{x03C0}\unicode[STIX]{x1D735}(r^{3}\unicode[STIX]{x1D712}_{-2}^{(Re_{E})})$ .

Case II (transverse electric field). In the presence of a transverse electric field ( $\boldsymbol{E}_{\infty }=\boldsymbol{e}_{x}$ or $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/2$ ), the charge distribution on the drop surface is shown in figure 2(d–f). Contrary to Case I, here we do not obtain a change in the drop velocity in the direction of the applied external electric field. The effect of surface charge convection leads to an asymmetric charge distribution across the transverse plane, which leads to alteration in the drop velocity in the axial direction only. The charge distribution across the axial plane remains antisymmetric for $x_{d}=5$ (refer to figure 2 e). However, for the case of $x_{d}=4$ (drop positioned at an off-centreline location), similar to Case I the asymmetry in the charge distribution about the axial plane (refer to figure 2 f) leads to the generation of an electric torque in the $y$ -direction as

which is further balanced by the hydrodynamic torque so as to satisfy the torque-free condition.

Case III (tilted electric field). The charge distribution due to the application of a tilted uniform electric field having both axial and transverse components is shown in figure 2(g–i) by considering $\boldsymbol{E}_{\infty }=(\boldsymbol{e}_{x}+\boldsymbol{e}_{z})/\sqrt{2}$ or $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/4$ . In this situation, the charge distribution becomes asymmetric about both the axial as well as the transverse planes even when the drop is at the centreline $(x_{d}=5)$ (refer to figure 2 h). This leads to cross-stream migration of the drop in the $x$ -direction along with alteration in the axial velocity as well. In this case also the drop experiences an electrical torque of the form

which is further balanced by the hydrodynamic torque. So, the charge convection leads to an alteration in axial drop velocity when the charge distribution about the transverse place is asymmetric. The asymmetric charge distribution about the axial plane always leads to generation of electric torque on the drop. But the asymmetric charge distribution about the axial plane alters the cross-stream velocity of the drop only when the applied electric field has both axial and transverse components. This new mechanism of cross-stream migration of the drop in the presence of background plane Poiseuille flow is very different from dielectrophoresis (in which a uncharged particle moves in a non-uniform electric field) or electrophoresis (in which a charged particle moves in an electric field) because here the drop has zero net charge and the applied electric field is uniform.

Figure 5.

Sign of charge-convection-induced cross-stream migration velocity of the drop $(U_{dx}^{(Re_{E})})$ on $R{-}S$ plane for (a)  $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/4$ and (b)  $\unicode[STIX]{x1D703}_{t}=3\unicode[STIX]{x03C0}/4$ .

An important thing to note from (4.5) is that the direction of the charge-convection-induced cross-stream migration velocity of the drop (which means sign of $U_{dx}^{(Re_{E})}$ ) is determined by the components of the applied electric field $(E_{x},E_{z})$ , or equivalently the tilt angle $(\unicode[STIX]{x1D703}_{t})$ and electrical property ratios $(R,S)$ , which is given by the term $G_{Re_{E}}=E_{x}E_{z}(S-R)(3R-S+3)$ . When $G_{Re_{E}}>0$ , the drop migrates in the positive $x$ -direction, and the reverse happens (drop migrates in the negative $x$ -direction) for $G_{Re_{E}}<0$ . This migration velocity is independent of the transverse location of the drop. To show the effect of $R$ and $S$ on the charge-convection-induced cross-stream velocity of the drop, we construct a regime diagram showing the sign of $U_{dx}^{(Re_{E})}$ (figure 5 a for $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/4$ and figure 5 b for $\unicode[STIX]{x1D703}_{t}=3\unicode[STIX]{x03C0}/4$ ) in the $R{-}S$ plane. From figure 5 it is clear that the two curves $R=S$ and $S=3R+3$ indicate the lines of zero charge-convection-induced cross-stream velocity $(U_{dx}^{(Re_{E})}=0)$ even in the presence of a tilted electric field. So, the mere presence of a tilted electric field is not a sufficient condition for charge-convection-induced cross-stream motion of the drop in plane Poiseuille flow. However, $R=S$ or $S=3R+3$ do not necessarily imply that the shape-deformation-induced drop migration in the cross-stream direction will be zero.

The effect of shape deformation in terms of $Ca$ has been previously shown in several studies (Chan & Leal Reference Chan and Leal1979; Mortazavi & Tryggvason Reference Mortazavi and Tryggvason2000; Griggs, Zinchenko & Davis Reference Griggs, Zinchenko and Davis2007; Wang & Dimitrakopoulos Reference Wang and Dimitrakopoulos2011; Chaudhury, Mandal & Chakraborty Reference Chaudhury, Mandal and Chakraborty2016) for the case of plane Poiseuille flow, but without any consideration of an electric field. In plane Poiseuille flow, if the drop is placed at an off-centre position, the drop shape becomes asymmetric with respect to the channel centreline, which leads to cross-stream migration of a deformable drop (Chan & Leal Reference Chan and Leal1979; Leal Reference Leal1980). Chan & Leal (Reference Chan and Leal1979) have studied the cross-stream migration of a deformable drop in plane Poiseuille flow analytically and found the dependence of $U_{dx}^{(Ca)}$ on $\unicode[STIX]{x1D706}$ as: the drop migrates towards the centreline for $\unicode[STIX]{x1D706}<0.5$ or $\unicode[STIX]{x1D706}>10$ , and the drop migrates away from the centreline for $0.5<\unicode[STIX]{x1D706}<10$ . Here, the presence of an electric field leads to an additional deformation of the drop into an oblate/prolate shape which modulates the drop motion in the axial as well as the cross-stream directions. This alteration of drop velocity in the presence of shape deformation is due to two different reasons: first, the extra deformation caused by the electric field alters the shape (or equivalently the cross-sectional area in the axial and transverse directions) of the drop. This altered cross-sectional area can increase/decrease the hydrodynamic drag force acting on the drop. Second, the extra deformation alters the surface charge distribution. The surface charge distribution coupled with a tangential electric field leads to generation of a tangential electric force which can further drive electrohydrodynamic flow around the drop interface. An asymmetric distribution of surface charge and associated tangential electric force induce an asymmetric electrohydrodynamic flow which further yields an electrohydrodynamic force on the drop. So, similar to the charge convection, shape deformation also induces asymmetry in charge distribution and electric force depending on the direction of the electric field. Towards this, the most important thing to note here is that when the electric field is applied in the axial direction $(\boldsymbol{E}_{\infty }=\boldsymbol{e}_{z})$ or transverse direction $(\boldsymbol{E}_{\infty }=\boldsymbol{e}_{x})$ , the shape-deformation-induced cross-stream migration due to the electric field is zero, which means the electric field does not affect $U_{dx}^{(Ca)}$ . This is due to the fact that when $\boldsymbol{E}_{\infty }=\boldsymbol{e}_{z}$ or $\boldsymbol{E}_{\infty }=\boldsymbol{e}_{x}$ the drop becomes symmetrically stretched/compressed in the axial/transverse direction depending on the sign of the Taylor discriminating function $\unicode[STIX]{x1D6FA}_{T}$ . The electric field contributes to the cross-stream migration in terms of shape deformation only when the electric field is tilted, because only then does the drop shape become asymmetric relative to the channel centreline. Figure 6 depicts that dependence of the shape-deformation-induced cross-stream migration velocity of the drop $(U_{dx}^{(Ca)})$ on $\unicode[STIX]{x1D706}$ is markedly altered due to the presence of the electric field. The electric field not only alters the magnitude of $U_{dx}^{(Ca)}$ but also the direction of drop migration, as is evident from figure 6 for the electrical property ratios $(R,S)=(0.5,5)$ .

Figure 6.

Variation of shape-deformation-induced cross-stream migration velocity of drop $(U_{dx}^{(Ca)})$ with viscosity ratio $(\unicode[STIX]{x1D706})$ . Other parameters have the following values $Ca=0.1$ , $M=0.1$ , $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/4$ , $H=10$ and $x_{d,0}=4$ .

Now, we show the combined effect of charge convection and shape deformation on the drop velocity. To explore the dependence of electrical property ratios, in figure 7 we plot the variation of $U_{dx}/U_{dx,\boldsymbol{E}_{\infty }=\mathbf{0}}$ in the $R{-}S$ plane for the parameters specified in the figure caption. The colourbar represents the magnitude of $U_{dx}/U_{dx,\boldsymbol{E}_{\infty }=\mathbf{0}}$ , where we have normalized $U_{dx}$ by the cross-stream migration velocity of the drop in the absence of the electric field (represented by $U_{dx,\boldsymbol{E}_{\infty }=\mathbf{0}}$ ). Figure 7(a) depicts that application of a uniform electric field dramatically alters the cross-stream velocity of the drop. Depending on the electrical property ratios $(R,S)$ , the cross-stream velocity of the drop can be much greater/less than the case of no electric field. An electric field not only changes the magnitude of the cross-stream migration velocity of the drop, but also can change the direction of migration, as depicted by figure 7(b) (represented by a negative value of $U_{dx}/U_{dx,\boldsymbol{E}_{\infty }=\mathbf{0}}$ ). However, the net effect due to the electric field will be decided by the magnitude of the governing parameters ( $Re_{E}$ , $Ca$ , $M$ , $\unicode[STIX]{x1D706}$ , $R$ , $S$ and $\unicode[STIX]{x1D703}_{t}$ ). An interesting thing to note from figure 7 is that there are $R$ , $S$ values for which $U_{dx}/U_{dx,\boldsymbol{E}_{\infty }=\mathbf{0}}=1$ , which represents a zero net effect of the applied electric field on $U_{dx}$ . Another interesting thing to note from figure 7(b) is that there is also a line of $U_{dx}=0$ (line of zero cross-stream migration), which means the charge-convection and shape-deformation effects are individually zero at those property values or the combined effects of charge convection and shape deformation cancel each other.

Figure 7.

Combined effect of charge convection and shape deformation on the normalized cross-stream migration velocity of the drop $(U_{dx}/U_{dx,\boldsymbol{E}_{\infty }=\mathbf{0}})$ on the $R{-}S$ plane for (a)  $\unicode[STIX]{x1D706}=15$ and (b)  $\unicode[STIX]{x1D706}=0.25$ . Other parameters have the following values $Ca=0.2$ , $Re_{E}=0.1$ , $M=0.1$ , $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/8$ , $H=10$ and $x_{d,0}=4$ .

An important limiting situation can be obtained by considering $R=S$ , which represents a charge-free interface. This situation is similar to the case of a perfect dielectric drop suspended in another perfect dielectric medium. In this case the charge-convection-induced cross-stream migration velocity vanishes (this is also evident from the $R=S$ line shown in figure 5), but the shape-deformation-induced component of the cross-stream migration velocity remains non-zero, as a perfect dielectric drop always deforms to a prolate shape in the presence of a uniform electric field. This prolate deformation causes an asymmetry in shape when the electric field is tilted with respect to the flow direction, leading to cross-stream migration of a perfect dielectric drop.

So far, we have considered the cross-stream migration velocity, but the axial velocity is also of great importance in the context of the trajectory of the drop. Equation (4.4) shows that both charge convection and shape deformation alter the axial drop velocity. The charge-convection-induced alteration in the axial velocity of the drop is due to the presence of an asymmetric charge distribution about the transverse plane, while the deformed shape of the drop leads to a change in the cross-sectional area and charge distribution of the drop, which also leads to change in axial velocity of drop. In figure 8 we show the combined effect of charge convection and shape deformation on $U_{dz}$ . Here we have normalized $U_{dz}$ by the imposed flow velocity at the centre of the drop, which is represented by $|\boldsymbol{V}_{\infty }|$ . In the absence of any externally applied electric field, the drop always lags behind the flow, as the axial velocity of the drop is less than the velocity of the imposed flow at the centre of the drop, as obtained by Chan & Leal (Reference Chan and Leal1979) and also shown in figure 8 (solid blue line). With increasing viscosity ratio, the drop becomes much more viscous as compared to the suspending medium, and moves much more slowly. Figure 8(a) shows that application of an electric field has the ability to move the drop much faster (for  $\unicode[STIX]{x1D703}_{t}=0$ ) or slower (for  $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/4$ , $\unicode[STIX]{x03C0}/2$ ) as compared to the case of no electric field. Another interesting fact to note here is that the presence of an electric field can also lead to faster/slower motion of the drop as compared to the flow velocity at the centre of the drop (represented by $U_{dz}/|\boldsymbol{V}_{\infty }|>1$ ). Not only does the tilt angle alter $U_{dz}$ significantly, but also the electrical property ratios $(R,S)$ have a profound effect, as evident from figure 8(b). So, depending on the tilt angle $(\unicode[STIX]{x1D703}_{t})$ , electrohydrodynamic property ratios $(R,S,\unicode[STIX]{x1D706})$ and dimensionless numbers $(Ca,Re_{E},M)$ , the axial velocity of the drop can increase or decrease due to the presence of an electric field.

Figure 8.

Combined effect of charge convection and shape deformation on the normalized axial velocity of the drop $(U_{dz}/|\boldsymbol{V}_{\infty }|)$ with viscosity ratio $(\unicode[STIX]{x1D706})$ for (a)  $(R,S)=(5,0.5)$ and (b)  $(R,S)=(0.5,5)$ . Other parameters have the following values $Ca=0.2$ , $Re_{E}=0.2$ , $M=1$ , $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/8$ , $H=10$ and $x_{d}=4$ (which gives $|\boldsymbol{V}_{\infty }|=0.96$ ).

Figure 9.

(a) Drop trajectory in plane Poiseuille flow in the absence of an electric field for different initial transverse position of the drop. Other parameters are used as $Ca=0.1$ , $\unicode[STIX]{x1D706}=0.5$ , and $H=10$ . (b) Drop trajectory in plane Poiseuille flow in the presence of an electric field considering the effect of shape deformation and charge convection separately. Other parameters are used as $M=0.1$ , $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/4$ , $R=5$ , $S=0.5$ , $\unicode[STIX]{x1D706}=0.5$ , $H=10$ and $x_{d,0}=4$ . (c) Drop trajectory in plane Poiseuille flow in the presence of an electric field for $x_{d,0}=4,5,6$ . Other parameters are taken as $Ca=0.1$ , $Re_{E}=0.1$ , $M=0.1$ , $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/4$ , $R=5$ , $S=0.5$ , $\unicode[STIX]{x1D706}=0.5$ and $H=10$ .

4.2.2 Drop trajectory

Now we investigate the effects of charge convection and shape deformation in terms of different relevant electrohydrodynamic parameters on the quasi-steady-state drop trajectory (represented by (3.31)) and cross-stream migration time scale (represented by (3.33)). First, we plot drop trajectory in plane Poiseuille flow disregarding the effect of the electric field in figure 9(a), which has previously obtained by Chan & Leal (Reference Chan and Leal1979). For $\unicode[STIX]{x1D706}=0.5$ , if the drop is placed at some off-centreline location ( $x_{d,0}=4$ or 6), the drop migrates towards the centreline of the flow and the final steady state transverse position of the drop will be the centreline $x_{d}=H/2=5$ . If the drop is initially placed at the centreline $(x_{d,0}=5)$ , then the drop will move only in the axial direction. Distinctively different drop trajectories are obtained when a tilted uniform electric field is applied in the presence of plane Poiseuille flow, as shown in figure 9(b,c). In figure 9(b) we show the independent effects of charge convection and shape deformation on the drop trajectory when the drop is placed at an off-centreline position $(x_{d,0}=4)$ . In the absence of shape deformation $(Ca=0)$ , the drop migrates away from the channel centreline solely due to charge convection $(Re_{E}=0.1)$ because $G_{Re_{E}}<0$ (which leads to transverse motion of the drop in the negative $x$ -direction) for $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/4$ and the electrical properties under consideration $(R=5,S=0.5)$ . In this case we have shown the drop trajectory only for short times, because for longer times the drop moves close to the lower wall, where the effect of the wall in terms of the hydrodynamic lift force will be significant, which we have not considered in the present analysis. On the other hand, in the absence of charge convection $(Re_{E}=0)$ , the drop migrates towards the centreline solely due to shape deformation $(Ca=0.1)$ , but surprisingly the drop does not attain a steady state transverse position at the centreline. This is attributed to the altered shape of the drop, which is very much different in the presence of an electric field. The combined effect of charge convection $(Re_{E}=0.1)$ and shape deformation $(Ca=0.1)$ on the drop trajectory will add linearly at this order of approximation, and the drop initially moves towards the centreline, but the final steady state transverse position of the drop is distinctly different from the centreline. The drop settles below the centreline, as shown in figure 9(b). When the electric field is absent, the drop will always settle to the centreline of the flow irrespective of the initial drop position $(x_{d,0})$ for $\unicode[STIX]{x1D706}=0.5$ (as evident from figure 9 a). In the presence of the electric field, the drop settles to a fixed transverse position irrespective of the initial transverse position of the drop $(x_{d,0})$ for fixed electrohydrodynamic parameters (as evident form figure 9 c, which considers $x_{d,0}=4,5,6$ ). But the difference lies in the fact that the presence of the electric field may yield the steady state transverse position of the drop at $x_{d,\infty }\neq H/2$ . Even when the drop is initially placed at the centreline $x_{d,0}=H/2=5$ , the drop migrates away from the centreline to a different transverse position at steady state, as shown in figure 9(c). This apparently surprising behaviour is due to the fact that, in the presence of a tilted electric field, the charge distribution and drop shape are both asymmetric with respect to the channel centreline, which is otherwise not possible in the absence of the electric field.

Figure 10 depicts the effect of the Mason number $(M)$ on the drop trajectory for the electrohydrodynamic parameters specified in the figure caption. The Mason number physically represents the relative strength of the electrical stress as compared with the hydrodynamic stress. As compared to the case of no electric field (represented by $\boldsymbol{E}_{\infty }=\mathbf{0}$ ), the drop migrates to different steady state transverse positions for $M>0$ . For smaller values of $M(=0.05,0.1)$ , the drop migrates towards the centreline, but finally settles below the centreline. However, for larger values of $M(=0.2)$ , the drop migrates further away from the centreline. Therefore, the increase in Mason number leads to settling of the drop further away from the centreline, which is attributed to the increase in asymmetry in charge distribution and shape deformation. However, there is no noticeable change in the time required for the drop to settle at the steady state transverse position. This is due to the fact that the cross-stream migration time scale $(t_{m})$ depends only on the viscosity ratio and capillary number (as evident from (3.33)).

Figure 10.

Drop trajectory for different values of the Mason number $(M)$ . Different parameters are considered as $Ca=0.1$ , $Re_{E}=0.1$ , $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/4$ , $R=5$ , $S=0.5$ , $\unicode[STIX]{x1D706}=0.5$ , $H=10$ and $x_{d,0}=4$ .

The effect of the relative strength of the components of the electric field ( $E_{x}$ and $E_{z}$ ) are represented by the tilt angle $(\unicode[STIX]{x1D703}_{t})$ . In figure 11 we show the drop trajectory for different $\unicode[STIX]{x1D703}_{t}$ for the electrohydrodynamic parameters specified in the figure caption. When $\unicode[STIX]{x1D703}_{t}=0$ , $\unicode[STIX]{x03C0}/2$ , $\unicode[STIX]{x03C0}$ , there is no effect of electric field in terms of alteration in cross-stream migration velocity, as evident from (4.5) and also depicted in figure 11. But in the regime $0<\unicode[STIX]{x1D703}_{t}<\unicode[STIX]{x03C0}/2$ and $\unicode[STIX]{x03C0}/2<\unicode[STIX]{x1D703}_{t}<\unicode[STIX]{x03C0}$ , the application of a uniform electric field dramatically alters the drop trajectory. For the electrohydrodynamic properties under consideration, the drop initially starts to move towards the centreline, but settles below (for $0<\unicode[STIX]{x1D703}_{t}<\unicode[STIX]{x03C0}/2$ as depicted in figure 11 a) or above (for $\unicode[STIX]{x03C0}/2<\unicode[STIX]{x1D703}_{t}<\unicode[STIX]{x03C0}$ as depicted in figure 11 b) the centreline. The effect of tilt angle is most significant near $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/4$ , $3\unicode[STIX]{x03C0}/4$ , as shown in figure 11(a,b), and the effect of tilt angle reduces for all other values of tilt angle $(\unicode[STIX]{x1D703}_{t})$ .

Figure 11.

Drop trajectory for different tilt angles $(\unicode[STIX]{x1D703}_{t})$ . Different parameters used are $Ca=0.1$ , $Re_{E}=0.1$ , $M=0.1$ , $R=5$ , $S=0.5$ , $\unicode[STIX]{x1D706}=0.5$ , $H=10$ and $x_{d,0}=4$ .

4.2.3 Steady state transverse position of the drop

Figure 12 shows the steady state transverse position of the drop $(x_{d,\infty })$ on the $R{-}S$ plane for the parameters mentioned in the figure caption. The colourbar in figure 12 represents the value of the steady state transverse position of the drop $(x_{d,\infty })$ . Figure 12(a) depicts that, for the combination of small $R$ and large $S$ , $x_{d,\infty }<5$ (where we have considered the channel height $H=10$ ), which represents that the drop settles below the centreline. The drop settles above the centreline $(x_{d,\infty }>5)$ for large $R$ and small $S$ . However, this dependence on $(R,S)$ becomes reversed when $\unicode[STIX]{x1D703}_{t}=5\unicode[STIX]{x03C0}/8$ in figure 12(b), in which small $R$ and large $S$ lead to settling of the drop above the centreline. An interesting thing to note from figure 12 is that there is a line of $x_{d,\infty }=5$ which signifies the state of settling of the drop at the centreline. This is associated with the line of zero cross-stream migration velocity $(U_{dx}=0)$ of the drop previously shown in figure 7(b).

Figure 12.

Variation of the steady state transverse position of the drop $(x_{d,\infty })$ , on the $R{-}S$ plane for (a)  $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/8$ and (b)  $\unicode[STIX]{x1D703}_{t}=5\unicode[STIX]{x03C0}/8$ . Other parameters have the following values: $Ca=0.1$ , $Re_{E}=0.1$ , $M=0.1$ , $\unicode[STIX]{x1D706}=1$ , and $H=10$ .

Figure 13.

Variation of the steady state transverse position of the drop $(x_{d,\infty })$ with the tilt angle $(\unicode[STIX]{x1D703}_{t})$ for different values of Mason number $(M)$ . Electrical property ratios are taken as $(R,S)=(5,0.5)$ and $(R,S)=(0.5,1.25)$ to plot (a) and (b) respectively. Other parameters employed are $Ca=0.1$ , $Re_{E}=0.1$ , $H=10$ and $\unicode[STIX]{x1D706}=0.25$ .

The pivotal effect of the tilt angle $(\unicode[STIX]{x1D703}_{t})$ and Mason number $(M)$ on the steady state transverse position of the drop are shown in figure 13 for the parameters specified in the figure caption. We recall (3.32) in the following form

where $\unicode[STIX]{x1D6E4}$ is a known function of $R,S,\unicode[STIX]{x1D706},H,Ca$ and $Re_{E}$ that can be easily obtained by comparing (4.10) and (3.32). It is evident from (4.10) that the necessary conditions for $x_{d,\infty }\neq H/2$ are the following: $M\neq 0$ (which means the presence of an electric field) and $\sin (2\unicode[STIX]{x1D703}_{t})\neq 0$ (which means a tilted electric field). The sign of the term $\sin (2\unicode[STIX]{x1D703}_{t})\unicode[STIX]{x1D6E4}$ is of great importance because this decides whether the drop will settle below or above the centreline. Figure 13(a) depicts that, for a particular non-zero magnitude of $M$ , the variation of the steady state transverse position of the drop $(x_{d,\infty })$ is sinusoidal with the tilt angle $(\unicode[STIX]{x1D703}_{t})$ . The drop settles to the centreline $x_{d,\infty }=5$ (considering $H=10$ ) when $\unicode[STIX]{x1D703}_{t}=0$ , $\unicode[STIX]{x03C0}/2$ , $\unicode[STIX]{x03C0}$ ; whereas, for any other values of the tilt angle, the drop settles either above or below the centreline for the parameters under consideration. For figure 13(a), $\unicode[STIX]{x1D6E4}>0$ , which gives $\sin (2\unicode[STIX]{x1D703}_{t})\unicode[STIX]{x1D6E4}>0$ for $0<\unicode[STIX]{x1D703}_{t}<\unicode[STIX]{x03C0}/2$ and $\sin (2\unicode[STIX]{x1D703}_{t})\unicode[STIX]{x1D6E4}<0$ for $\unicode[STIX]{x03C0}/2<\unicode[STIX]{x1D703}_{t}<\unicode[STIX]{x03C0}$ . So, for given physical properties we can alter $x_{d,\infty }$ by changing $\unicode[STIX]{x1D703}_{t}$ . One interesting thing to note from figure 13(a) is that there is an optimum value of tilt angle ( $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/4$ , $3\unicode[STIX]{x03C0}/4$ ) for which $|x_{d,\infty }-H/2|$ is maximum and the drop settles far away from the centreline. The reason behind the existence of such an optimum $\unicode[STIX]{x1D703}_{t}$ is that the asymmetry created in the charge distribution and shape deformation due to the presence of the electric field are maximum for $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/4$ , $3\unicode[STIX]{x03C0}/4$ . Next, we plot figure 13(b) for $\unicode[STIX]{x1D6E4}<0$ using the properties mentioned in the figure caption. Here we obtain $\sin (2\unicode[STIX]{x1D703}_{t})\unicode[STIX]{x1D6E4}<0$ for $0<\unicode[STIX]{x1D703}_{t}<\unicode[STIX]{x03C0}/2$ and $\sin (2\unicode[STIX]{x1D703}_{t})\unicode[STIX]{x1D6E4}>0$ for $\unicode[STIX]{x03C0}/2<\unicode[STIX]{x1D703}_{t}<\unicode[STIX]{x03C0}$ . The effect of Mason number on the steady state transverse position of the drop is also depicted in figure 13. With increasing $M$ , the drop moves further away from the centreline. This is attributed to the fact that the increase in $M$ results in an increase of the relative magnitude of the electrical stress as compared to viscous stresses, which brings about the change in flow field, and finally the drag force on the drop. Accordingly, $M$ only controls the strength of alteration in drop motion due to an electric field, whereas the tilt angle $\unicode[STIX]{x1D703}_{t}$ has the ability to affect the direction in which the drop settles with respect to the centreline.

4.3.1 Numerical method and simulation set-up

Two-phase electrohydrodynamic simulations are performed using the open-source scientific package Gerris (http://gfs.sf.net) developed by Popinet (Reference Popinet2003, Reference Popinet2009), which employs the volume-of-fluid (VOF) method in a finite-volume framework. The choice of Gerris is made because the Electrohydro module (López-Herrera, Popinet & Herrada Reference López-Herrera, Popinet and Herrada2011) of Gerris solves the incompressible Navier–Stokes equation coupled with an electric potential and electric charge density very accurately (Ferrera et al. Reference Ferrera, López-Herrera, Herrada, Montanero and Acero2013; Cimpeanu, Papageorgiou & Petropoulos Reference Cimpeanu, Papageorgiou and Petropoulos2014; Datta, Das & Das Reference Datta, Das and Das2015; López-Herrera et al. Reference López-Herrera, Gañán-Calvo, Popinet and Herrada2015). Another important factor is that Gerris is equipped with dynamic adaptive mesh refinement, which allows one to use a greater number of computational cells near the desired regions dynamically (Popinet Reference Popinet2003). This is very advantageous in the present simulations because we have to use a very long channel to simulate the drop motion towards steady state. The electric potential $(\unicode[STIX]{x1D713})$ satisfies the Poisson equation and the electric charge density $(q_{v})$ satisfies the charge conservation equation of the following form (López-Herrera et al. Reference López-Herrera, Popinet and Herrada2011)

The right-hand side of the charge transport equation $\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70E}\unicode[STIX]{x1D735}\unicode[STIX]{x1D713})$ is the Ohmic current, which is the equivalent of the electromigration term in electrokinetic theory (Saville Reference Saville1997). This is obtained by noting that the conservation of the volumetric charge density may be written as (Saville Reference Saville1997) $\unicode[STIX]{x2202}q_{v}/\unicode[STIX]{x2202}t+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}q_{v}=\unicode[STIX]{x1D735}\boldsymbol{\cdot }[\sum _{k}\unicode[STIX]{x1D714}^{k}e^{2}z^{k}n^{k}\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}+\unicode[STIX]{x1D714}^{k}k_{B}T\unicode[STIX]{x1D735}q_{v}]$ , $k=1,2,\ldots ,N$ , where $\unicode[STIX]{x1D714}^{k}$ represents the mobility of the $k$ th species, $n^{k}$ is the ionic concentration of the $k$ th species, $e$ is the charge of a proton, $k_{B}$ is the Boltzmann constant and $T$ is the absolute temperature. In the limit where the charge diffusion is negligible, the description may be reduced simply as (4.11). It is important to note here that Gerris solves for the charge density $(q_{v})$ in the bulk fluid. However, as the charge-relaxation time scale is very small as compared to the convective time scale (represented by small value of $Re_{E}$ ), the charges which are present at the bulk fluid will be accumulated at the fluid–fluid interface very quickly (López-Herrera et al. Reference López-Herrera, Popinet and Herrada2011; Ferrera et al. Reference Ferrera, López-Herrera, Herrada, Montanero and Acero2013). So, we can expect that (4.11) will effectively represent the bulk fluid as charge-free, and the charge conservation equation in the bulk will be the same as the charge conservation equation at the interface (represented in boundary condition  $e4$ ). Fluid flow is governed by the continuity equation and the Navier–Stokes equation of the form (López-Herrera et al. Reference López-Herrera, Popinet and Herrada2011)

where $Re=\unicode[STIX]{x1D70C}_{e}V_{c}a/\unicode[STIX]{x1D707}_{e}$ is the Reynolds number. The electric force density $(\boldsymbol{F}_{e})$ and surface tension force $(\boldsymbol{F}_{s})$ are implemented in Gerris as volume forces by López-Herrera et al. (Reference López-Herrera, Popinet and Herrada2011). The electric force density is related to the electric field $(\boldsymbol{E})$ and the charge density $(q_{v})$ as $\boldsymbol{F}_{e}=q_{v}\boldsymbol{E}-(1/2)|\boldsymbol{E}|^{2}\unicode[STIX]{x1D735}\unicode[STIX]{x1D700}$ . The surface tension force is related to the radius of curvature of the interface $(\unicode[STIX]{x1D705})$ as $\boldsymbol{F}_{s}=\unicode[STIX]{x1D705}\unicode[STIX]{x1D6FF}_{s}\boldsymbol{n}$ , where $\unicode[STIX]{x1D6FF}_{s}$ is the Dirac delta function and $\boldsymbol{n}$ is the unit normal vector on the drop interface. Gerris implements the volume-of-fluid (VOF) method to capture the fluid–fluid interface. The volume fraction $(c)$ satisfies an advection equation of the form

The drop phase is represented by $c=1$ , whereas the suspending medium is represented by $c=0$ . Variation of all the electrohydrodynamic properties across the drop interface is expressed in terms of the following weighted arithmetic interpolation (López-Herrera et al. Reference López-Herrera, Popinet and Herrada2011): $\unicode[STIX]{x1D707}=c\unicode[STIX]{x1D706}+(1-c)$ , $\unicode[STIX]{x1D700}=cS+(1-c)$ and $\unicode[STIX]{x1D70E}=cR+(1-c)$ .

Figure 14.

Schematic representation of the simulation set-up. 2D rectangular domian of length $L$ and height $H$ is considered which is bounded by the inlet left boundary $(S_{L})$ , the outlet right boundary $(S_{R})$ , and the top and bottom walls $(S_{T},S_{B})$ . The drop is initially placed at some transverse position $x_{d,0}$ . The steady state transverse position $(x_{d,\infty })$ , is obtained from the simulation data.

To simulate the electrohydrodynamic motion of the drop, we consider a two-dimensional (2D) rectangular domain of dimensionless length $L$ and dimensionless height $H$ (refer to figure 14). We consider $H=4$ and $L=180$ to simulate the drop dynamics. The choice of a relatively small domain height of $H=4$ is based on the fact that a larger channel height requires a very large length of channel to get steady state drop velocity. This large computational domain is associated with a large computational time. As the electric potential varies in both the axial and transverse directions, the periodic boundary condition is not applicable in the present set-up. We need to use a full length domain to simulate the drop dynamics until the steady state is reached. To reduce the excessively large length of the computational domain and reduce the computational time, we have taken $H=4$ . The whole computational domain is composed of 45 square boxes of size $H=4$ . All the boxes are decomposed using square surface elements with cell size in the bulk fluid as $H/8$ . In the present study, we have used a gradient-based adaptive mesh refinement (Popinet Reference Popinet2003) in which the computational cells are divided with quadtree discretization where the gradient in volume fraction is present. The cell size near the drop interface is $H/256$ , which is equivalent to 64 grid points over the drop radius. Recent studies have employed a similar grid size to simulate drop electrohydrodynamics using Gerris (López-Herrera et al. Reference López-Herrera, Popinet and Herrada2011; Ferrera et al. Reference Ferrera, López-Herrera, Herrada, Montanero and Acero2013; Chen et al. Reference Chen, Song, Li and Hu2015) and found excellent agreement with existing results. We apply the following boundary conditions for the electric potential and the velocity field in the respective domain boundaries:

The initial velocity field throughout the domain is considered as plane Poiseuille flow. The drop is placed initially at an axial location $z_{d,0}=3H/2$ so that drop is not affected by the inlet boundary $(S_{L})$ . The initial transverse position of the drop $(x_{d,0})$ is taken very close to the channel centreline to minimize the wall effects. While showing the drop trajectories, the axial location at any time $t$ , $z_{d}(t)$ , is expressed as $z_{d}(t)-z_{d,0}$ . We have validated the numerical code in the following two cases: (i) drop trajectory in Poiseuille flow and (ii) drop deformation in a uniform electric field. We have also performed grid independence study considering the combined effect of a tilted electric field and Poiseuille flow. Details regarding model validation and grid independence are presented in appendix H.

Figure 15.

(a) Numerically obtained drop trajectories in plane Poiseuille flow in the absence of an electric field for different initial transverse positions of the drop. (b) Analytically obtained drop trajectories in plane Poiseuille flow in the absence of an electric field for different initial transverse positions of the drop. Other parameters have the following values: $Ca=0.2$ , $\unicode[STIX]{x1D706}=0.1$ and $H=4$ . For the numerical simulations, we have considered $Re=0.1$ .

4.3.2 Drop trajectory

Now, we show the drop trajectory, $z_{d}(t)-x_{d}(t)$ , obtained from 2D numerical computations and compare that with the analytical solution. Figure 15(a) depicts the drop trajectory in Poiseuille flow in the absence of an electric field for the parameters mentioned in the figure caption. Chan & Leal (Reference Chan and Leal1979) have shown analytically that for $\unicode[STIX]{x1D706}=0.1$ the drop settles to the centreline. The present analytical solution in the limit of $\boldsymbol{E}_{\infty }=\mathbf{0}$ yields the same result as that of Chan & Leal (Reference Chan and Leal1979), which is depicted in figure 15(b). The present numerical computations show that the drop settles to the centreline for different initial transverse positions $(x_{d,0})$ . Though the steady state transverse position from numerical and analytical solutions show the exact same result (i.e.  $x_{d,\infty }=H/2=2$ ), the numerically obtained axial length traversed by the drop differs significantly from the analytical solution. This difference is attributed to the presence of the bounding walls and the associated hydrodynamic lift force acting on the drop (Chan & Leal Reference Chan and Leal1979; Uijttewaal et al. Reference Uijttewaal, Nijhof and Heethaar1993; Uijttewaal & Nijhof Reference Uijttewaal and Nijhof1995). As we have performed our numerical simulation considering the channel height $H=4$ , the wall effects are not negligible.

Figure 16.

(a) Numerically obtained drop trajectory in plane Poiseuille flow in the presence of a tilted electric field $(\unicode[STIX]{x1D703}_{t}=5\unicode[STIX]{x03C0}/16)$ for $R=0.75$ . (b) Analytically obtained drop trajectory in plane Poiseuille flow in the presence of a tilted electric field $(\unicode[STIX]{x1D703}_{t}=5\unicode[STIX]{x03C0}/16)$ for $R=0.75$ . (c) Numerically obtained drop trajectory in plane Poiseuille flow in the presence of a tilted electric field $(\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/4)$ for $R=1.25$ . (d) Analytically obtained drop trajectory in plane Poiseuille flow in the presence of a tilted electric field $(\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/4)$ for $R=1.25$ . Other parameters have the following values: $S=1$ , $Ca=0.2$ , $Re_{E}=0.2$ , $\unicode[STIX]{x1D706}=0.1$ and $H=4$ . Numerical simulations we performed for $Re=0.1$ .

The effect of a tilted electric field is shown in figure 16 for the parameters mentioned in the figure caption. Figures 16(a) and 16(b) are obtained for $R=0.75$ and $\unicode[STIX]{x1D703}_{t}=5\unicode[STIX]{x03C0}/16$ , while figures 16(c) and 16(d) are obtained for $R=1.25$ and $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/4$ . Figures 16(a) and 16(c) are obtained from numerical simulations, while figures 16(b) and 16(d) are obtained from analytical solutions. Two important things to note from figure 16(a) are as follows: first, in the presence of a tilted electric field, the charge convection and shape deformation lead to settling of the drop below the centreline. Second, the off-centreline steady state transverse position $(x_{d,\infty })$ is independent of the initial transverse position $(x_{d,0})$ . For all $x_{d,0}$ values we obtain nearly equal values of $x_{d,\infty }$ . This is true even when the drop is initially placed at the centreline (i.e.  $x_{d,0}=H/2=2$ ). Figure 16(b) depicts the drop trajectory for the same parameters obtained from the analytical solution. Comparison between the analytical and numerical solutions reveals that both methods qualitatively represent similar results. Similar qualitative agreement between the numerical and analytical solutions is also depicted in figures 16(c) and 16(d), in which the drop settles above the centreline.

To investigate the effect of tilt angle on the drop trajectory and steady state transverse position of the drop, we plot the drop trajectory for the following four different values of $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/4$ , $5\unicode[STIX]{x03C0}/16$ , $3\unicode[STIX]{x03C0}/8$ and $7\unicode[STIX]{x03C0}/16$ , while the other parameters are mentioned in the figure caption. Figure 17(a) depicts the numerically obtained drop trajectories. Numerical simulations show that the drop settles away from the flow centreline for $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/4$ . With increasing tilt angle, the drop still settles in an off-centreline transverse position, but the drop settles much closer to the centreline. This is due to the fact that the asymmetry in charge distribution and shape deformation are larger for $\unicode[STIX]{x1D703}_{t}=\unicode[STIX]{x03C0}/4$ . Analytical solutions depicted in figure 17(b) compare qualitatively well with the numerical simulations.

Figure 17.

(a) Numerically obtained drop trajectory in plane Poiseuille flow in the presence of a tilted electric field for different tilt angles. (b) Analytically obtained drop trajectory in plane Poiseuille flow in the presence of a tilted electric field for different tilt angles. Different parameters are taken as $Ca=0.2$ , $Re_{E}=0.2$ , $R=1.25$ , $S=1$ , $\unicode[STIX]{x1D706}=0.1$ and $H=4$ . Numerical simulations were performed for $Re=0.1$ .

Possible reasons for the lack of an exact match between the analytical and numerical solutions are the presence of bounding walls and a 2D domain. Bounding walls in the presence of drop deformation lead to a hydrodynamic lift force on the drop towards the centreline (Chan & Leal Reference Chan and Leal1979; Uijttewaal et al. Reference Uijttewaal, Nijhof and Heethaar1993; Uijttewaal & Nijhof Reference Uijttewaal and Nijhof1995), which facilitates faster migration of the drop. That is why the numerically obtained drop trajectories show relatively faster settling of the drop to the steady state transverse position as compared to the analytically obtained drop trajectories. This effect is present even in the absence of an electric field (refer to figure 15 a). Another way in which bounding wall affects the drop motion is the non-uniformity and asymmetry in an electric field due to the positioning of drop at an off-centreline location (Halim & Esmaeeli Reference Halim and Esmaeeli2013; Esmaeeli Reference Esmaeeli2016). In the present study, we have restricted ourselves to a 2D domain to reduce computational cost. A three-dimensional (3D) numerical simulation, which is expected to compare better with the analytical solution, is considered as a future study.