2.1 Governing equations and boundary conditions

Throughout our analysis, we present the governing equations and the respective boundary conditions in non-dimensional form unless specified otherwise. Under the consideration of the leaky dielectric model (Taylor Reference Taylor1966; Melcher & Taylor Reference Melcher and Taylor1969), the bulk fluids are charge-free and the electric potentials, ${\it\psi}_{i,e}$ , satisfy the Laplace equation of the form

The distribution of the electric potentials $({\it\psi}_{i,e})$ can be obtained by solving (2.1) by using the following boundary conditions (Saville Reference Saville1997; Xu & Homsy Reference Xu and Homsy2006):

1. (e1) specified unperturbed electric field at infinity; at $r\rightarrow \infty ,{\it\psi}_{e}\rightarrow {\it\psi}_{\infty }$ , where $\boldsymbol{E}_{\infty }=-\boldsymbol{{\rm\nabla}}{\it\psi}_{\infty }$ and $\boldsymbol{E}_{\infty }$ is the electric field vector applied at infinity;

2. (e2) bounded electric field inside the drop;

3. (e3) continuity of the electric potential at the drop interface; at $r=r_{s},{\it\psi}_{i}={\it\psi}_{e}$ ;

4. (e4) charge conservation at the drop interface; at $r=r_{s},\boldsymbol{n}\boldsymbol{\cdot }(R\boldsymbol{{\rm\nabla}}{\it\psi}_{i}-\boldsymbol{{\rm\nabla}}{\it\psi}_{e})=-Re_{E}\boldsymbol{{\rm\nabla}}_{s}\boldsymbol{\cdot }(q_{s}\boldsymbol{V}_{s})$ , where $q_{s}=\boldsymbol{n}\boldsymbol{\cdot }(S\boldsymbol{{\rm\nabla}}{\it\psi}_{i}-\boldsymbol{{\rm\nabla}}{\it\psi}_{e})$ is the surface-charge distribution, $\boldsymbol{n}$ represents the outward unit normal vector on the drop interface and $\boldsymbol{V}_{s}$ is the fluid velocity at the drop interface; here $\boldsymbol{{\rm\nabla}}_{s}=\boldsymbol{{\rm\nabla}}-\boldsymbol{n}(\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}})$ is the surface gradient operator; the deformed drop interface is represented by the radial location $r=r_{s}=1+f({\it\theta},{\it\phi})$ , where a non-zero $f({\it\theta},{\it\phi})$ represents the deviation of the drop shape from sphericity; the normal unit vector $\boldsymbol{n}$ can be defined as $\boldsymbol{n}=\boldsymbol{{\rm\nabla}}(r-r_{s})/|\boldsymbol{{\rm\nabla}}(r-r_{s})|$ .

The velocity and pressure fields satisfy the Stokes equation and the condition of incompressibility of the form (Happel & Brenner Reference Happel and Brenner1981)

where $\boldsymbol{u}_{i}$ and $p_{i}$ represent velocity and pressure fields inside the drop, respectively, while subscript $e$ has been used to denote the same quantities outside the drop. One thing to note here is that pressure $p_{i}$ and $p_{e}$ contain the contribution of the gravitational potential.

In a reference frame attached to the drop centre moving with a velocity $\boldsymbol{U}_{d}$ , the flow field satisfies the following boundary conditions:

1. (f1) uniform flow at infinity; $\boldsymbol{u}_{e}(r\rightarrow \infty )=-\boldsymbol{U}_{d}$ ;

2. (f2) velocity and pressure fields inside the drop are bounded;

3. (f3) velocity fields satisfy the following boundary condition at the drop interface $(r=r_{s})$ : $\boldsymbol{u}_{i}=\boldsymbol{u}_{e}$ and $\boldsymbol{u}_{i}\boldsymbol{\cdot }\boldsymbol{n}=\boldsymbol{u}_{e}\boldsymbol{\cdot }\boldsymbol{n}=0$ ;

4. (f4) stress balance at the drop interface $(r=r_{s});{\bf\tau}_{e}\boldsymbol{\cdot }\boldsymbol{n}-{\bf\tau}_{i}\boldsymbol{\cdot }\boldsymbol{n}=(1/Ca)(\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{n})\boldsymbol{n}$ , where ${\bf\tau}_{i}$ and ${\bf\tau}_{e}$ represent the total stress tensors consisting of the hydrodynamic and electrical effects of the internal and external fluid, respectively.

2.2 Expansion in perturbation of $Re_{E}$ and $Ca$

The above equations and boundary conditions represent a two-way coupled electrohydrodynamic problem where electric field affects the flow field via the generation of Maxwell stress at the drop interface, while the flow field affects the electric field by redistributing the surface charges by convection (Xu & Homsy Reference Xu and Homsy2006; He, Salipante & Vlahovska Reference He, Salipante and Vlahovska2013; Lanauze et al. Reference Lanauze, Walker and Khair2015). On top of that, the drop interface deforms and the drop shape function is a priori unknown. For problems dealing with drop motion and deformation, it is typical to perform a regular domain perturbation in orders of capillary number. But here we also would like to consider the fluid-flow-induced redistribution of surface charges and subsequent alteration in drop velocity. Towards this, we go ahead to solve the problem under consideration via a double asymptotic expansion in $Re_{E}$ and $Ca$ by assuming

Now, we expand the electric potential, velocity, pressure, stress fields and the unknown drop velocity in the following regular perturbation form

Here $\boldsymbol{U}_{d}^{(0)}$ is the drop velocity in the absence of interfacial charge convection and shape deformation, $\boldsymbol{U}_{d}^{(Re_{E})}$ denotes the first contribution of charge convection on the drop velocity and $\boldsymbol{U}_{d}^{(Ca)}$ is the first contribution of shape deformation on the drop velocity. One thing to note here is that we have included $p_{e}^{(1/Ca)}$ and ${\bf\tau}_{e}^{(1/Ca)}$ to balance the Laplace pressure across the drop interface (Leal Reference Leal2007).

The deformed shape of the drop is also expanded in the following form

where $f^{(Ca)}$ , $f^{(Ca\,Re_{E})}$ and $f^{(Ca^{2})}$ are the corrections to the deformed drop shape at different orders of approximation and each are $O(1)$ quantity. An important thing to note is that the first effect of charge convection on the drop shape appears at $O(Ca\,Re_{E})$ . The outward unit normal $\boldsymbol{n}$ and the curvature of the deformed drop shape, $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{n}$ , can be calculated using (2.5) as (Wohl & Rubinow Reference Wohl and Rubinow1974; Ajayi Reference Ajayi1978; Chan & Leal Reference Chan and Leal1979)

and

Considering the above, the governing equations for the electric potentials $({\it\psi}_{i,e}^{(0)})$ at the leading order are given as

which are subjected to the following boundary conditions

where ${\it\xi}|_{r=1}$ represents the evaluation of the quantity ${\it\xi}$ at the spherical drop interface $(r=1)$ .

The leading-order velocity and pressure fields $(\boldsymbol{u}_{i,e}^{(0)},p_{i,e}^{(0)})$ satisfy the following governing equations

which are subjected to the following boundary conditions

One thing to note here is that the constant term present in the expression for the curvature (2.7) does not appear in the normal stress balance (last boundary condition of (2.11)) because of the fact that the $2/Ca$ term is balanced by $(1/Ca)p_{e}^{(1/Ca)}$ .

On similar lines, we obtain the following governing equations for the $O(Re_{E})$ electric potentials $({\it\psi}_{i,e}^{(Re_{E})})$

which are subjected to the, now altered, $O(Re_{E})$ boundary conditions (Feng Reference Feng1999; Xu & Homsy Reference Xu and Homsy2006)

It is to be noted that in the boundary condition for the normal electric field, the effect of surface-charge convection now appears through the redistribution of the surface charge.

The $O(Re_{E})$ velocity and pressure fields $(\boldsymbol{u}_{i,e}^{(Re_{E})},p_{i,e}^{(Re_{E})})$ satisfy the following governing equations

which are subjected to the boundary conditions

The governing equations for the electric potentials $({\it\psi}_{i,e}^{(Ca)})$ at the $O(Ca)$ are given by

which are subjected to the, now altered, $O(Ca)$ boundary conditions (Feng Reference Feng1999; Xu & Homsy Reference Xu and Homsy2006)

where $[{\it\xi}|_{r=1+Ca\,f^{(Ca)}}]^{(Ca)}$ represents the $O(Ca)$ contribution from the quantity ${\it\xi}$ evaluated at the deformed drop interface $r=1+Ca\,f^{(Ca)}$ . The $O(Ca)$ contributions can be obtained by performing the Taylor series expansion about $r=1$ and collecting the terms which are $O(Ca)$ , in the following way

The $O(Ca)$ velocity and pressure fields $(\boldsymbol{u}_{i,e}^{(Ca)},p_{i,e}^{(Ca)})$ satisfy the following governing equations

which are subjected to the following boundary conditions

3.1 Description of electric potential and flow fields

At each order of perturbation, the electric potentials $({\it\psi}_{i,e}^{(k)})$ satisfy the Laplace equation, solution of which is given by

where the superscript $k$ represents the order of perturbation. We only consider $k=0$ , $k=Re_{E}$ and $k=Ca$ . Here ${\it\psi}_{\infty }^{(k)}$ is the unperturbed applied potential at far field $(r\rightarrow \infty )$ having non-zero value only at the leading order of perturbation as ${\it\psi}_{\infty }^{(0)}=-r(E_{x}P_{1,1}\cos {\it\phi}+E_{z}P_{1,0})$ . Here $P_{n,m}(\cos {\it\theta})$ is the associated Legendre polynomial of degree $n$ and order $m$ with $\cos {\it\theta}$ as the argument. The unknown coefficients ( $a_{n,m}^{(k)},\hat{a}_{n,m}^{(k)},b_{-n-1,m}^{(k)}$ and $\hat{b}_{-n-1,m}^{(k)}$ ) can be determined at each order of perturbation by using respective boundary conditions.

As the velocity and pressure fields $(\boldsymbol{u}_{i,e}^{(k)},p_{i,e}^{(k)})$ satisfy the Stokes equation and the condition of incompressibility at each order of perturbation, we can represent the velocity and pressure fields inside the drop, $(\boldsymbol{u}_{i}^{(k)},p_{i}^{(k)})$ , by using the Lamb’s general solution in the following form (Happel & Brenner Reference Happel and Brenner1981)

whereas the velocity and pressure fields outside the drop, $(\boldsymbol{u}_{e}^{(k)},p_{e}^{(k)})$ , can be represented using Lamb’s general solution as (Happel & Brenner Reference Happel and Brenner1981)

where $\boldsymbol{v}_{e}^{(k)}$ is the disturbance velocity field outside the drop which vanishes at infinity. In (3.2) and (3.4), $\boldsymbol{r}$ represents the position vector of magnitude $r$ . The expressions of the solid harmonics present in the velocity and pressure fields ( $p_{n}^{(k)},{\it\Phi}_{n}^{(k)},{\it\chi}_{n}^{(k)},p_{-n-1}^{(k)},{\it\Phi}_{-n-1}^{(k)}$ and ${\it\chi}_{-n-1}^{(k)}$ ) are mentioned in appendix A.

3.2 Leading-order solution

At the leading order of perturbation, there is no charge convection, allowing the electric field to be decoupled from the flow field. Thus, the leading-order electric field distribution can be solved independently. The solution of (2.8) which gives the potential distribution inside and outside the drop can be easily obtained by linear combination of the potential distribution due to the component of electric field applied in the longitudinal direction $(E_{z})$ and potential distribution due to the component of electric field applied in the lateral direction $(E_{x})$ in the following form (Lee, Im & Kang Reference Lee, Im and Kang2000; Im & Kang Reference Im and Kang2003):

where ${\it\psi}_{z}^{(0)}$ and ${\it\psi}_{x}^{(0)}$ represent the potential distributions due to $E_{z}$ and $E_{x}$ , respectively. The leading-order surface-charge distribution at the drop surface is also obtained by combining the charge distributions due to longitudinal and laterally applied electric fields in the following form (Lee et al. Reference Lee, Im and Kang2000; Im & Kang Reference Im and Kang2003):

Calculation of the velocity and pressure fields can be easily done by using the linearity of the leading-order problem. The flow field generated due to $\boldsymbol{E}_{\infty }$ is called the Taylor solution $(\boldsymbol{u}^{E(0)},p^{E(0)})$ (Taylor Reference Taylor1966; Saville Reference Saville1997), and the flow field generated due to setting of the drop in gravitational field is the so-called Hadamard–Rybczynski solution $(\boldsymbol{u}^{g(0)},p^{g(0)})$ (Happel & Brenner Reference Happel and Brenner1981; Leal Reference Leal2007). These solutions are independent of each other. Hence, the leading-order solution is the linear combination of these two solutions. As the spherical harmonic representation is useful for further calculations, we represent the velocity and pressure fields in the following form (Lee et al. Reference Lee, Im and Kang2000; Im & Kang Reference Im and Kang2003):

where the complete expressions of the solid harmonics and their coefficients are given in appendix B.

The sedimenting velocity of a drop is obtained by imposing the force-free condition. At steady state, the buoyancy force balances the hydrodynamic force on the drop. In a dimensional form, this can be represented at the leading order as (Hetsroni & Haber Reference Hetsroni and Haber1970; Leal Reference Leal2007)

where $\bar{p}_{-2}^{(0)}$ is the dimensional solid harmonic. Using the previously prescribed non-dimensional scheme, equation (3.13) may be written as

where $\boldsymbol{e}_{z}=\cos {\it\theta}\boldsymbol{e}_{r}-\sin {\it\theta}\boldsymbol{e}_{{\it\theta}}$ and $p_{-2}^{(0)}=1/r^{2}[A_{-2,0}^{(0)}P_{1,0}+A_{-2,1}^{(0)}\cos {\it\phi}P_{1,1}+\hat{A}_{-2,1}^{(0)}\sin {\it\phi}P_{1,1}].$ The coefficients $A_{-2,0}^{(0)}$ , $A_{-2,1}^{(0)}$ and $\hat{A}_{-2,1}^{(0)}$ are given in appendix B. Now substituting the expression of $p_{-2}^{(0)}$ in (3.14), we obtain the drop velocity as

This shows that at the leading order of perturbation the applied uniform electric field does not contribute to the force acting on the drop.

Towards determining the deformation of the drop, we decompose the correction term $f^{(Ca)}({\it\theta},{\it\phi})$ in the following surface harmonics form (Hetsroni & Haber Reference Hetsroni and Haber1970):

where $L_{n,m}^{(Ca)}$ and $\hat{L}_{n,m}^{(Ca)}$ are the unknown coefficients. The non-zero coefficients $(L_{n,m}^{(Ca)},\hat{L}_{n,m}^{(Ca)})$ are obtained by satisfying the normal stress balance equation (last boundary condition of (2.11)) as

where ${\it\Omega}_{T}=R^{2}+1-2S+(3/5)(R-S)((3{\it\lambda}+2)/({\it\lambda}+1))$ is the Taylor discriminating function (Saville Reference Saville1997; Vizika & Saville Reference Vizika and Saville2006).

3.3 The $O(Re_{E})$ solution

Having obtained the leading-order velocity profile, we may now attempt to solve (2.12) and (2.14) which are subjected to the boundary conditions (2.13) and (2.15), respectively. The first effect of charge convection through the leading-order charge distribution and the velocity field is to affect the $O(Re_{E})$ electric field and, consequently, the $O(Re_{E})$ velocity and pressure fields. For convenience, we rewrite the boundary condition here: $\boldsymbol{e}_{r}\boldsymbol{\cdot }(R\boldsymbol{{\rm\nabla}}{\it\psi}_{i}^{(Re_{E})}-\boldsymbol{{\rm\nabla}}{\it\psi}_{e}^{(Re_{E})})|_{r=1}=-\boldsymbol{{\rm\nabla}}_{s}\boldsymbol{\cdot }(q_{s}^{(0)}\boldsymbol{V}_{s}^{(0)})|_{r=1}$ . This differs from the leading-order condition (i.e.  $\boldsymbol{e}_{r}\boldsymbol{\cdot }(R\boldsymbol{{\rm\nabla}}{\it\psi}_{i}^{(0)}-\boldsymbol{{\rm\nabla}}{\it\psi}_{e}^{(0)})|_{r=1}=0$ ). Given that the electric potential fields $({\it\psi}_{i,e}^{(Re_{E})})$ are harmonic functions, it is instructive to first evaluate the right-hand side of the charge convection boundary condition in order to establish the various surface harmonics that it may contain. It can be recast as (Kim & Karrila Reference Kim and Karrila1991)

By substituting the form of the velocity field (given in (3.9)) and the accumulated charge (from (3.8)) in terms of the leading-order electric field, we split the above expression into various surface harmonics by using the orthogonality of the spherical surface harmonics. If we write the expression given in (3.18) as $\sum _{n=0}^{\infty }\sum _{m=0}^{n}(Z_{n,m}\cos (m{\it\phi})+\hat{Z}_{n,m}\sin (m{\it\phi}))P_{n,m},$ we obtain the following non-zero terms: $Z_{1,0},Z_{1,1},Z_{2,0},Z_{2,1},Z_{3,0},Z_{3,1},Z_{3,2}$ and $Z_{3,3}$ . The complete expressions of the $Z$ values are given in appendix C. Thus, it is expected that $O(Re_{E})$ electric potential inside and outside the drop will have the following form:

Now, we use the rest of the boundary conditions from (2.13) and obtain the above unknown coefficients $(a_{n,m}^{(Re_{E})}\,\;\text{and}\;\,b_{-n-1,m}^{(Re_{E})})$ which are given in appendix D. Surface-charge distribution at this order is obtained as

The $O(Re_{E})$ velocity and pressure fields inside and outside the drop are obtained in the following form:

One thing to note here is that the flow field contains non-zero harmonics up to $n=4$ . The solid harmonics present in (3.22)–(3.25) are obtained by application of the boundary conditions (2.15). We do not present those here for the sake of brevity. However, the detailed procedure to obtain these unknown solid harmonics is given in the supplementary material available at http://dx.doi.org/10.1017/jfm.2016.84. To determine the drop velocity, we need the expression of $p_{-2}^{(Re_{E})}$ which is given by

where $A_{-2,0}^{(Re_{E})}$ , $A_{-2,1}^{(Re_{E})}$ and $\hat{A}_{-2,1}^{(Re_{E})}$ are obtained as

In tune with the leading-order drop velocity estimation, we may proceed in a similar fashion to evaluate $O(Re_{E})$ drop velocity

Thus, employing the force-free condition at $O(Re_{E})$ , we obtain drop velocity at $O(Re_{E})$ as

3.4 The $O(Ca)$ solution

On similar lines, the electric potentials at $O(Ca)$ are obtained as

where the detailed expressions of the coefficients ( $a_{n,m}^{(Ca)}$ and $b_{-n-1,m}^{(Ca)}$ ) are given in appendix E.

The velocity and pressure fields at $O(Ca)$ are obtained in a similar form as given in (3.22)–(3.25). In this case also the flow field contains non-zero harmonics up to $n=4$ . The solid harmonics are obtained by application of the boundary conditions (2.20). We do not present those here for the sake of brevity. However, the detailed procedure to obtain these unknown solid harmonics is given in the supplementary material. The solid harmonic $p_{-2}^{(Ca)}$ , necessary to proceed further to calculate the drop velocity, is given by

where $A_{-2,0}^{(Ca)}$ , $A_{-2,1}^{(Ca)}$ and $\hat{A}_{-2,1}^{(Ca)}$ are obtained as

Now, the force-free condition at $O(Ca)$ is given by

which yields the following form of the drop velocity

3.5 Combined effect of charge convection and shape deformation

Combining drop velocities obtained at different orders of perturbation, we can write the drop velocity as

The above equation is the main result of the entire analysis and shows that the charge convection and the shape deformation affect the components of drop velocity in both the longitudinal as well as lateral directions.

Figure 2.

Variation of dimensional drop velocity $\bar{U}_{d}\,(\text{m}~\text{s}^{-1})$ with the strength of applied electric field $E_{ref}^{2}~(\text{V}^{2}~\text{m}^{-2})$ . Our analytical results are compared with the experimental results of Xu & Homsy (Reference Xu and Homsy2006) for two different drop sizes ( $a=2.8$ and $3.3~\text{mm}$ ) over a wide range of the strength of applied electric field. Other parameters as employed by Xu & Homsy (Reference Xu and Homsy2006), have the following values: $R=0.6452$ , $S=0.35$ , ${\it\lambda}=0.357$ , ${\it\varepsilon}_{e}=8{\it\varepsilon}_{0}$ , ${\it\mu}_{e}=1.4~\text{Pa}~\text{s}$ , ${\it\gamma}=5\times 10^{-3}~\text{N}~\text{m}^{-1}$ , ${\it\sigma}_{e}=2.4\times 10^{-12}~\text{S}~\text{m}^{-1}$ , ${\it\rho}_{i}=1000~\text{kg}~\text{ m}^{-3}$ and ${\it\rho}_{e}=957~\text{kg}~\text{m}^{-3}$ .

4.1 Validation

If we consider an applied electric field acting in the $z$ -direction, i.e. $\boldsymbol{E}_{\infty }=(0,1)$ , we obtain the drop velocity as

which corroborates with the drop sedimentation velocity obtained by Xu & Homsy (Reference Xu and Homsy2006). In figure 2 we compare our analytical results with that of previously reported experimental results of Xu & Homsy (Reference Xu and Homsy2006). They have performed experiments on the measurement of drop velocity in the presence of electric field applied in the direction of gravity for two different drop sizes $a=2.8$ and $3.3~\text{mm}$ (physical properties are mentioned in the caption). Figure 2 depicts reasonable agreement between theory and experiment.

Figure 3.

Contours depicting the variation of the (a) $q_{s}^{(0)}$ and (b) $q_{s}^{(0)}+Re_{E}q_{s}^{(Re_{E})}$ for a longitudinal electric field. The parameters employed are $R=0.5,S=2$ , $M=2$ , ${\it\lambda}=0.5$ and $Re_{E}=0.2$ .

4.2 Effect of charge convection and shape deformation on the longitudinal velocity of drop

The drop velocity in the longitudinal direction ( $z$ -direction) is obtained from (3.36) as

where $U_{dz}^{(0)}=1$ represents the dimensionless Hadamard–Rybcznski velocity, $U_{dz}^{(Re_{E})}$ represents the correction in longitudinal drop velocity due to charge convection at the drop surface and $U_{dz}^{(Ca)}$ represents the correction in longitudinal drop velocity due to shape deformation of the drop. Before studying the effect of different controlling parameters on the drop velocity, we investigate the physical reasons behind the charge-convection- and shape-deformation-induced alterations in the longitudinal drop velocity.

Regarding the effect of charge convection, previously Feng (Reference Feng1999) showed that despite charge convection, for a neutrally buoyant drop, the charge distribution remains anti-symmetric with respect to the equatorial plane $({\it\theta}={\rm\pi}/2)$ of the drop (please refer to figure 3). The consequence of this is that there is no extra force acting on the drop and consequently the drop remains stationary. The problem of a drop with a different density than the suspending fluid in the presence of a longitudinal applied electric field was then considered by Xu & Homsy (Reference Xu and Homsy2006) who highlighted the pivotal role of the surface-charge convection in breaking the antisymmetry about the equatorial plane thus giving rise to a net drag on the drop. This can be seen from figure 3 where we plot (3.8) and (3.21) by setting $E_{z}=1$ and $E_{x}=0$ . The choice of the parameters used in figure 3 are mentioned in the figure caption. Figure 3(a) depicts the leading-order charge distribution $(q_{s}^{(0)})$ which is antisymmetric about the equatorial plane. However, it can be clearly seen from figure 3(b) that the surface convection for a sedimenting drop does lead to a breaking in the antisymmetric structure of the charge distribution. It is precisely this altered asymmetric charge distribution that results in a non-zero force and subsequently the longitudinal drop velocity. On the other hand, the deformation-induced alteration in longitudinal velocity of the drop is due to the change in drag force on the drop due to change in projected cross-sectional area in the longitudinal direction. In figure 4 we show the shape of the drop in the presence of uniform electric field acting in the longitudinal direction. Figures 4(a) and 4(b) show the extent of drop deformation as compared with the spherical shape in $xz$ -plane and $xy$ -plane, respectively. From figure 4(b) it is clear that the effective cross-sectional area projected in the direction of sedimentation of the drop due to deformation has increased as compared with the undeformed case, which reduces the drop velocity for the parameters given in the figure caption.

Figure 4.

(a) Drop shape projected on the (a) $xz$ -plane and (b) $xy$ -plane. For the deformed drop shape we have only shown the leading-order deformation. Other parameters employed are $R=0.5$ , $S=2$ , $M=2$ , ${\it\lambda}=0.5$ and $Ca=0.2$ .

Now, we study the effect of different controlling parameters on the longitudinal velocity of the drop. The most important controlling parameters and non-dimensional numbers which govern the drop velocity are $E_{x}$ , $E_{z}$ , ${\it\lambda}$ , $R$ , $S$ , $M$ , $Re_{E}$ and $Ca$ . The components of the applied electric field $E_{x}$ and $E_{z}$ can be represented by an equivalent tilt angle $({\it\theta}_{t})$ which is previously defined as ${\it\theta}_{t}=\tan ^{-1}(E_{x}/E_{z})$ . Towards investigating the effects of charge convection and shape deformation on the settling velocity of the drop, we show following three cases: drop settling velocity in the absence of deformation $(1+Re_{E}U_{dz}^{(Re_{E})})$ , drop settling velocity in the absence of charge convection $(1+CaU_{dz}^{(Ca)})$ and finally the combined effect of deformation and charge convection $(1+Re_{E}U_{dz}^{(Re_{E})}+CaU_{dz}^{(Ca)})$ . To compare these results with that of sedimentation of drop in the absence of any external electric field, we also plot the $U_{dz}=1$ line representing the Hadamard–Rybczynski velocity.

Figure 5 depicts the variation of longitudinal drop velocity with the tilt angle $({\it\theta}_{t})$ in the presence of charge convection effect where we deliberately neglect the effects of deformation by setting $Ca=0$ . Here we consider two different sets of electrical properties $(R,S)$ which are mentioned in the figure caption. It is apparent form figure 5 that charge convection can increase $(U_{dz}>1)$ or decrease $(U_{dz}<1)$ the longitudinal drop velocity as compared to the Hadamard–Rybczynski velocity $(U_{dz}=1)$ . Net effect of charge convection is determined by the sign of the correction term $U_{dz}^{(Re_{E})}$ . From (3.29) it is noticeable that the sign of $U_{dz}^{(Re_{E})}$ is decided solely by the combination of the conductivity and the permittivity ratio: $G_{Re_{E},z}=(S-R)(3R-S+3)$ . If $G_{Re_{E},z}<0$ (which is the case for $R=0.2$ and $S=10$ ), the charge convection resists the drop motion in the longitudinal direction, and the reverse happens for $G_{Re_{E},z}>0$ (which is the case for $R=0.1$ and $S=2.5$ ). Another important aspect that can be noted from figure 5 is the effect of charge convection towards altering (decrease/increase) the drop velocity in the longitudinal direction which is maximum when ${\it\theta}_{t}=0,{\rm\pi}$ and minimum when ${\it\theta}_{t}={\rm\pi}/2$ . This is due to the fact that the asymmetry in charge distribution with respect to the equatorial plane is maximum for ${\it\theta}_{t}=0,{\rm\pi}$ and minimum for ${\it\theta}_{t}={\rm\pi}/2$ .

Figure 5.

Variation of drop velocity in the longitudinal direction $(U_{dz})$ with tilt angle $({\it\theta}_{t})$ for $G_{Re_{E}}>0$ and $G_{Re_{E}}<0$ . Different parameters considered here have the following values: for $G_{Re_{E}}<0$ : $R=0.2$ , $S=10$ and $M=0.5$ , and for $G_{Re_{E}}>0$ : $R=0.1$ , $S=2.5$ and $M=2$ . Other parameters have the following values: ${\it\lambda}=0.1$ and $Re_{E}=0.2$ . The black dashed line represents the $U_{dz}=1$ line which is the Hadamard–Rybczynski velocity.

Figure 6.

(a) Variation of drop velocity in the longitudinal direction $(U_{dz})$ with tilt angle $({\it\theta}_{t})$ in the absence of charge convection $(Re_{E}=0)$ for ${\it\Omega}_{T}<0$ and ${\it\Omega}_{T}>0$ . (b) Variation of cross-sectional area of a drop projected on $xy$ -plane with tilt angle for ${\it\Omega}_{T}<0$ and ${\it\Omega}_{T}>0$ . We have normalized $A_{xy}$ by ${\rm\pi}$ . (c) Cross-sectional area of the deformed drop on the $xy$ -plane for ${\it\Omega}_{T}<0$ . (d) Cross-sectional area of the deformed drop on the $xy$ -plane for ${\it\Omega}_{T}>0$ . Different properties used for ${\it\Omega}_{T}<0$ are $R=0.1$ , $S=2.5$ and $Ca=0.1$ . Similarly, for ${\it\Omega}_{T}>0$ , we have used $R=10$ , $S=0.5$ and $Ca=0.2$ . Other parameters have the following values: $M=2$ and ${\it\lambda}=0.1$ .

To investigate the pivotal role of drop deformation on the longitudinal velocity of the drop, we plot the longitudinal drop velocity $(U_{dz})$ with tilt angle $({\it\theta}_{t})$ in the presence of shape deformation effect where we deliberately neglect the effects of charge convection by setting $Re_{E}=0$ in figure 6(a). We consider two different cases: ${\it\Omega}_{T}<0$ (oblate drop) and ${\it\Omega}_{T}>0$ (prolate drop). When the tilt angle is ${\it\theta}_{t}=0$ , which is equivalent to the case of $\boldsymbol{E}_{\infty }=(0,1),$ the deformation and motion of the drop is exactly axisymmetric about the longitudinal direction which has previously been studied by Xu & Homsy (Reference Xu and Homsy2006). They have reported that a drop in the presence of axial electric field falls slower when ${\it\Omega}_{T}<0$ and faster when ${\it\Omega}_{T}>0$ . Similar observations are depicted in figure 6(a) for ${\it\theta}_{t}=0$ . But with increase in the tilt angle $({\it\theta}_{t})$ there is a remarkable change in the drop velocity. For the case of ${\it\Omega}_{T}<0$ , the drop velocity increases with the tilt angle and reaches maximum at ${\it\theta}_{t}={\rm\pi}/2$ and then decreases. Towards explaining the physical reason behind this behaviour, we appeal to figure 6(b) where we show the effect of tilt angle on the variation of cross-sectional area of the deformed drop projected in the direction of sedimentation $(A_{xy})$ . We have normalized the projected cross-section area by the projected area of spherical drop. Figure 6(b) clearly shows that for ${\it\Omega}_{T}<0$ (oblate shape) and ${\it\theta}_{t}=0$ , $A_{xy}>1$ which means a greater cross-sectional area of the deformed drop in the direction of sedimentation which results in greater drag force and subsequently a reduction in drop velocity. With the increase in tilt angle, $A_{xy}$ reduces and goes below unity until it reaches a minimum at ${\it\theta}_{t}={\rm\pi}/2$ which results in faster sedimentation of the drop as compared with the Hadamard–Rybczynski velocity. For ${\rm\pi}/2\leqslant {\it\theta}_{t}\leqslant {\rm\pi}$ , $A_{xy}$ increases with ${\it\theta}_{t}$ and drop velocity reduces accordingly. The reverse trend is observed for a prolate drop $({\it\Omega}_{T}>0)$ for which $A_{xy}$ increases and drop velocity reduces with increase in the tilt angle and reaches a minima at ${\it\theta}_{t}={\rm\pi}/2$ as shown in figures 6(a) and 6(b). The corresponding drop shapes are shown in figures 6(c) and 6(d) for ${\it\Omega}_{T}<0$ and ${\it\Omega}_{T}>0$ , respectively, for four different tilt angles $({\it\theta}_{t}=0,{\rm\pi}/8,{\rm\pi}/4,{\rm\pi}/2)$ . An important thing to note here is that the sign of $U_{dz}^{(Ca)}$ is decided by the parameter $G_{Ca,z}=(2E_{z}^{2}-E_{x}^{2}){\it\Omega}_{T}$ . The parameter $G_{Ca,z}$ changes sign with tilt angle and there is always two particular values of tilt angle which are ${\it\theta}_{t}=\tan ^{-1}(\sqrt{2})$ and ${\it\theta}_{t}={\rm\pi}-\tan ^{-1}(\sqrt{2})$ for which $U_{dz}^{(Ca)}$ vanishes. The physical reason behind this can be obtained from figure 6(b) which depicts that for ${\it\theta}_{t}=\tan ^{-1}(\sqrt{2})(\text{or }{\it\theta}_{t}={\rm\pi}-\tan ^{-1}(\sqrt{2}))$ , the projected cross-sectional area $(A_{xy})$ is unity which means there is no effective change in $A_{xy}$ as compared with a spherical shape. This is a situation in which the drop deforms but that deformation does not lead to any change in drop velocity in the longitudinal direction.

Figure 7.

Variation of the drop velocity in the longitudinal direction $(U_{dz})$ with tilt angle $({\it\theta}_{t})$ for $R=0.2$ , $S=10$ and $M=0.5$ , and for $R=0.1$ , $S=2.5$ and $M=2$ . Other parameters have the following values: ${\it\lambda}=0.1$ , $Ca=0.1$ and $Re_{E}=0.2$ . The black dashed line represents the $U_{dz}=1$ line which is the Hadamard–Rybczynski velocity.

Figure 7 depicts that depending on the magnitude of the tilt angle $({\it\theta}_{t})$ , the combined effect of charge convection and drop deformation reduce the drop velocity in the longitudinal direction as compared with the Hadamard–Rybczynski velocity for $R=0.2$ , $S=10$ . But for $R=0.1$ , $S=2.5$ , figure 7 shows that depending on the tilt angle, the drop velocity may be greater or less than the Hadamard–Rybczynski velocity. Interestingly, figure 7 also shows two critical values of the tilt angle for which the longitudinal velocity of the drop becomes unity $(U_{dz}=1)$ . The critical tilt angles are obtained as

where the terms $X_{1}=2[Re_{E}\{(R+1)(32S-24R)-8S(S+1)\}+Ca{\it\Omega}_{T}(2R+3)(3{\it\lambda}^{2}-{\it\lambda}+8)]$ and $X_{2}=[Re_{E}\{(R+1)(36R-48S)+12S(S+1)\}+Ca{\it\Omega}_{T}(2R+3)(3{\it\lambda}^{2}-{\it\lambda}+8)].$ It signifies that the longitudinal velocity of the drop remains unaltered for ${\it\theta}_{t}={\it\theta}_{t,cr}^{I},{\it\theta}_{t,cr}^{\mathit{II}}$ even in the presence of finite drop deformation $(Ca,{\it\Omega}_{T}\neq 0)$ and charge convection $(Re_{E}\neq 0)$ . However, this critical tilt angle exists only when $X_{1}X_{2}>0$ (which is the case for $R=0.1$ , $S=2.5$ ) depending on the relative magnitude of the electrohydrodynamic parameters $(R,S\text{ and }{\it\lambda})$ and non-dimensional numbers $(Ca,Re_{E}\text{ and }M)$ .

Figure 8.

Variation of the longitudinal velocity of the drop $(U_{dz})$ with viscosity ratio $({\it\lambda})$ . The parameters have the following values: $R=0.2$ , $S=10$ , ${\it\theta}_{t}={\rm\pi}/8$ , $M=0.5$ , $Ca=0.1$ and $Re_{E}=0.2$ .

In figure 8 we show the variation of the drop velocity in the longitudinal direction, $U_{dz}$ , with viscosity ratio $({\it\lambda})$ . Figure 8 depicts that the effect of charge convection reduces monotonically as the drop becomes more viscous and finally becomes negligible as ${\it\lambda}\gg 1$ . This can be explained by the fact that with the increase in viscosity ratio, the velocity at the drop surface reduces which leads to a diminishing effect of charge convection in terms of smaller magnitude of $U_{dz}^{(Re_{E})}$ . However, the effect of deformation due to electric field persists for high viscosity ratios and contributes to the longitudinal drop velocity. The limit of ${\it\lambda}\rightarrow \infty$ , the longitudinal drop velocity reduces to

So for highly viscous drops, alteration of the drop velocity occurs only due to deformation of the drop as depicted in figure 8 and also by (4.4). Another limiting condition is achieved for ${\it\lambda}\rightarrow 0$ in which both of the effects, charge convection and deformation, contribute to the drop velocity and yield

These two limiting conditions are also shown in figure 8.

We have identified two important parameters $G_{Re_{E},z}=(S-R)(3R-S+3)$ and $G_{Ca,z}=(2E_{z}^{2}-E_{x}^{2}){\it\Omega}_{T}$ which decide the net effect of charge convection and shape deformation, respectively towards increase/decrease in drop velocity in the $z$ -direction. Previously we have shown the importance of the parameter $G_{Ca,z}$ in terms of tilt angle $({\it\theta}_{t})$ on the direction of motion of the drop in the longitudinal motion. Now we investigate the combined effect of electrical properties $(R,S)$ towards determining the longitudinal drop motion. In figures 9(a) and 9(b) we plot the sign of $U_{dz}^{(Re_{E})}$ and $U_{dz}^{(Ca)}$ , respectively on $R$ – $S$ plane. Both figures show that there are two distinct regimes on $R$ – $S$ plane in which the correction in the longitudinal drop velocity becomes positive or negative. In figure 9(a) we identify two regime boundaries given by $R=S$ and $S=3R+3$ , where the former case is tantamount to a perfect dielectric case with zero surface charge. In figure 9(b) two regimes are distinguished by the zero-deformation line $({\it\Omega}_{T}=0)$ . The correction in the longitudinal velocity due to deformation of the drop is positive for a prolate drop and the reverse is true for oblate drop as depicted in figure 9(b). The combined effect of charge convection and deformation towards dictating the longitudinal motion on the $R$ – $S$ plane is shown for two different tilt angles in figures 10(a) and 10(b) for fluid properties mentioned in the figure caption. Figure 10(a) depicts that for ${\it\theta}_{t}=0$ , there is a marked reduction in the longitudinal drop velocity for small conductivity ratios and large permittivity ratios. The drop velocity remains mostly unaffected for large conductivity ratios. For ${\it\theta}_{t}={\rm\pi}/2$ , figure 10(b) depicts that the longitudinal drop velocity is drastically altered as compared with the case of ${\it\theta}_{t}=0$ . In sharp contrast to the case of ${\it\theta}_{t}=0$ , figure 10(b) shows that drop velocity gets affected for small as well as high conductivity ratios. Another interesting facet is the presence of regions in which the drop velocity is greater that unity which means that the combined effect of charge convection and deformation aids the drop settling.

Figure 9.

(a) Different regimes of the correction in longitudinal velocity due to charge convection $(U_{dz}^{(Re_{E})})$ . (b) Different regimes of the correction in longitudinal velocity due to drop deformation $(U_{dz}^{(Ca)})$ . For the later plot, we have considered ${\it\lambda}=0.1$ and ${\it\theta}_{t}={\rm\pi}/4$ .

Figure 10.

Variation of the longitudinal velocity of drop $(U_{dz})$ in $R$ – $S$ plane for (a) ${\it\theta}_{t}=0$ and (b) ${\it\theta}_{t}={\rm\pi}/2$ . Other parameters have the following values: ${\it\lambda}=0.1$ , $M=0.5$ , $Ca=0.1$ and $Re_{E}=0.2$ . The colourbar and contour levels represent the magnitude of $U_{dz}$ .

4.3 Effect of charge convection and shape deformation on the lateral velocity of drop

Now we consider the lateral motion of the drop. We had obtained the drop velocity in the lateral direction ( $x$ -direction) as

This expression depicts that there are two distinct effects contributing to the lateral motion of the drop: charge convection represented by $U_{dx}^{(Re_{E})}$ which persists even for a spherical drop and the drop deformation represented by $U_{dx}^{(Ca)}$ which remains finite even in the absence of surface charges (the case of perfect dielectric drop in perfect dielectric medium). Before investigating the effect of different controlling parameters on the lateral velocity of the drop, we study the physical reason behind the non-intuitive lateral motion of the drop. One important thing to note here is that the lateral migration obtained in the present analysis is neither due to dielectrophoresis in which the presence of a non-uniform electric field is a must (Feng Reference Feng1996), nor due to electrophoresis in which drop must contain a finite amount of charge (Masliyah & Bhattacharjee Reference Masliyah and Bhattacharjee2006). In the present analysis we obtain lateral migration of the uncharged drop in the presence of a uniform electric field. We first study the charge-convection-induced lateral motion of a spherical drop. We consider the following three cases: (i) longitudinal electric field $({\it\theta}_{t}=0)$ , (ii) lateral electric field $({\it\theta}_{t}={\rm\pi}/2)$ and (iii) combined longitudinal and lateral fields or equivalently tilted electric field (with ${\it\theta}_{t}={\rm\pi}/4$ ). The case of a longitudinal electric field has been discussed previously in figure 3. Application of a longitudinal electric field leads to asymmetric charge distribution about the equatorial plane which results in alteration in the drop velocity in the longitudinal direction only. A striking result obtained in (4.6) is that even if there is an applied electric field in lateral direction, the influence of charge convection is solely affecting the longitudinal velocity without causing any motion in the lateral direction. To understand this from the point of view of symmetries in the charge distribution, we appeal to figure 11. It is seen from figure 11(a) that the absence of surface convection leads to an antisymmetric charge distribution about the meridional plane and maintains symmetry about the equatorial plane. In the presence of surface convection due to sedimentation, however, it is observed that the antisymmetric structure is preserved about the meridional plane, but the symmetry is broken about the equatorial plane as depicted in figure 11(b). This leads to a net drag force in the longitudinal direction without affecting the motion in the lateral direction, leading to an alteration in the longitudinal drop velocity. Hence, an applied electric field acting only in the $z$ - or $x$ -direction ( ${\it\theta}_{t}=0$ or ${\it\theta}_{t}={\rm\pi}/2$ ) is insufficient to bring asymmetric charge distribution about the meridional plane due to which there is no velocity of the drop in the lateral direction. Now, we depict the charge distribution due to surface convection in figure 12 where it can be seen that the presence of both components of the applied electric field cause an asymmetry about both the equatorial and meridional plane leading to a force in both the longitudinal and lateral directions, respectively. The findings are also seen in the expressions for the droplet velocity by setting $\boldsymbol{E}_{\infty }=(1/\sqrt{2},1/\sqrt{2})$ or equivalently ${\it\theta}_{t}={\rm\pi}/4$ in (4.6).

Figure 11.

Contours depicting the variation of (a)  $q_{s}^{(0)}$ and (b)  $q_{s}^{(0)}+Re_{E}q_{s}^{(Re_{E})}$ for a lateral electric field. The parameters employed are $R=0.5$ , $S=2$ , $M=2$ , ${\it\lambda}=0.5$ and $Re_{E}=0.2$ .

Figure 12.

Contours depicting the variation of $q_{s}^{(0)}+Re_{E}q_{s}^{(Re_{E})}$ for a tilted electric field $({\it\theta}_{t}={\rm\pi}/4)$ . The parameters employed are $R=0.5$ , $S=2$ , $M=2$ , ${\it\lambda}=0.5$ and $Re_{E}=0.2$ .

Figure 13.

(a) Variation of the drop velocity in the lateral direction $(Re_{E}U_{dx}^{(Re_{E})})$ due to charge convection with tilt angle $({\it\theta}_{t})$ for the parameters mentioned in the legend. (b) Variation of the drop velocity in the lateral direction due to deformation $(CaU_{dx}^{(Ca)})$ with tilt angle $({\it\theta}_{t})$ for the parameters mentioned in the legend. Other parameters have the following values: ${\it\lambda}=0.1$ , $Ca=0.1$ and $Re_{E}=0.2$ .

In figure 13(a), we show the variation of $Re_{E}U_{dx}^{(Re_{E})}$ with tilt angle for the fluid properties as mentioned in the caption. A closer look into (4.6) reveals that the parameter which governs the direction of the charge-convection-induced lateral motion of the drop can be written as $G_{Re_{E},x}=E_{x}E_{z}(S-R)(3R-S+3)=\sin (2{\it\theta}_{t})(S-R)(3R-S+3)/2$ . In contrast to $G_{Re_{E},z}$ which is independent of the direction of the applied electric field, here $G_{Re_{E},x}$ is markedly altered by the tilt angle $({\it\theta}_{t})$ . Depending on the sign of $G_{Re_{E},x}$ , the drop can migrate laterally in the positive $x$ -direction (for $G_{Re_{E},x}>0$ ) or in the negative $x$ -direction (for $G_{Re_{E},x}<0$ ). Figure 13(a) depicts the variation of $U_{dx}^{(Re_{E})}$ when the tilt angle $({\it\theta}_{t})$ is sinusoidal. A few important features that can be noted from figure 13(a) are as follows: (i) charge-convection-induced lateral velocity vanishes for ${\it\theta}_{t}=0$ , ${\rm\pi}/2$ , ${\rm\pi}$ which is due to the fact that for these tilt angles there is no asymmetry in charge distribution with respect to the meridional plane; (ii) charge-convection-induced lateral velocity becomes maximum for ${\it\theta}_{t}={\rm\pi}/4$ , $3{\rm\pi}/4$ which is due to the fact that the asymmetry in charge distribution with respect to the meridional plane is maximum for ${\it\theta}_{t}={\rm\pi}/4$ , $3{\rm\pi}/4$ .

In figure 13(b) we plot variation of $CaU_{dx}^{(Ca)}$ with the tilt angle $({\it\theta}_{t})$ for the properties as mentioned in the caption. Similar to the case of charge convection, the direction of the deformation-induced lateral migration of the drop is determined by the parameter $G_{Ca,x}=E_{x}E_{z}{\it\Omega}_{T}=\sin (2{\it\theta}_{t}){\it\Omega}_{T}/2$ . So, the direction of applied electric field and the Taylor discriminating function $({\it\Omega}_{T})$ govern the direction of lateral migration due to shape deformation. In this case also we observe sinusoidal behaviour of $U_{dx}^{(Ca)}$ with ${\it\theta}_{t}$ . Similar to the case of $U_{dx}^{(Re_{E})}$ , deformation-induced lateral migration of the drop vanishes at ${\it\theta}_{t}=0$ , ${\rm\pi}/2$ , ${\rm\pi}$ and it becomes maximum at ${\it\theta}_{t}={\rm\pi}/4$ , $3{\rm\pi}/4$ as depicted in figure 13(b).

In figure 14 we plot the variation of lateral velocity of the drop with viscosity ratio. Similar to the case of longitudinal velocity, with the increase in viscosity ratio, there is a monotonic decrease in charge-convection-induced lateral velocity of the drop because with the increase in viscosity ratio of the drop, the circulations at the drop surface diminishes which subsequently reduces the convection of charges and subsequently the asymmetry in charge distribution as shown in figure 14. The limit of ${\it\lambda}\rightarrow \infty$ , the lateral drop velocity reduces to

Hence, for highly viscous drops, alteration of drop velocity occurs only due to deformation of the drop as depicted in figure 14 and also by (4.7). Another limiting condition is achieved for ${\it\lambda}\rightarrow 0$ in which both of the effects, charge convection and deformation, contribute to the drop velocity and yield

Figure 14.

Variation of the lateral velocity of the drop $(U_{dx})$ with viscosity ratio $({\it\lambda})$ . The parameters have the following values: $R=0.2$ , $S=10$ , ${\it\theta}_{t}={\rm\pi}/4$ , $M=0.5$ , $Ca=0.1$ and $Re_{E}=0.2$ .

Figure 15.

Variation of the lateral velocity of the drop $(U_{dx})$ in $R$ – $S$ plane for (a)  $(Re_{E},Ca)=(0.2,0)$ , (b)  $(Re_{E},Ca)=(0,0.1)$ and (c)  $(Re_{E},Ca)=(0.2,0.1)$ . The parameters have the following values: ${\it\lambda}=0.1$ , $M=0.5$ , ${\it\theta}_{t}={\rm\pi}/4$ . The colourbar and contour levels represent the magnitude of  $U_{dx}$ .

Now, we consider the effects of electrical properties $(R,S)$ on the lateral velocity of the drop. The effect of $(R,S)$ towards dictating the sign of lateral velocity is the same as that has been shown previously for the longitudinal velocity in figure 9. Here we show the effect of $(R,S)$ for ${\it\theta}_{t}={\rm\pi}/4$ , since the lateral motion is most significant in this case. In figure 15(a), we plot $U_{dx}$ on $R$ – $S$ plane considering charge convection effect but neglecting the shape deformation of the drop $(Ca=0)$ . The effect of drop deformation in the absence of charge convection $(Re_{E}=0)$ is shown in figure 15(b), while the combined effect of the charge convection and the shape deformation is depicted in figure 15(c). Figure 15(a–c) depict that depending on the relative magnitudes of conductivity ratio $(R)$ and permittivity ratio $(S)$ , the lateral velocity of the drop can be in the positive $x$ -direction or in the negative $x$ -direction. The $R=S$ and $S=3R+3$ lines are shown to figure 15(a) which signifies the zero lateral velocity of the drop considering charge convection effects. The $R=S$ line represents the case of a perfectly dielectric drop having zero surface charge, while $S=3R+3$ denotes the zero-lateral-migration line for a leaky dielectric drop. Similar to the case of longitudinal velocity, for the case of ${\it\Omega}_{T}=0$ , the deformation-induced motion vanishes in the lateral direction as depicted in figure 15(b).

4.4 Effect of charge convection and deformation on the direction of drop motion

So far we have discussed the effect of charge convection and deformation on the longitudinal and lateral motion of the drop. Here we investigate the combined effect of charge convection and deformation on the direction of motion of the drop; more specifically, the deviation in direction of drop motion form the direction of gravity ( $z$ -direction). Towards this, we define the deviation angle ${\it\theta}_{d}=\tan ^{-1}(U_{dx}/U_{dz})$ which quantifies the angle made by the drop velocity vector, $\boldsymbol{U}_{d}$ , with the $z$ -direction. When the drop is sedimenting in the positive $z$ -direction, the parameters that can affect the direction of drop motion are $E_{x}$ and $E_{z}$ (or equivalently the tilt angle, ${\it\theta}_{t}$ ), ${\it\lambda}$ , $R$ , $S$ and the non-dimensional numbers $Ca$ , $Re_{E}$ and $M$ . In figure 16(a,b) we plot variation of the deviation angle $({\it\theta}_{d})$ with tilt angle $({\it\theta}_{t})$ for the parameters specified in the caption. We consider three different viscosity ratios ${\it\lambda}=0.1$ , 1 and 10. Figure 16(a,b) depicts that there is an optimum tilt angle for which magnitude of the deviation angle $({\it\theta}_{d})$ is maximum or equivalently the deviation in direction of drop motion from $z$ -direction is maximum. Previously we have come across the fact that for specified values of electrical and hydrodynamic properties, the magnitude of lateral velocity of the drop $(U_{dx})$ become maximum for ${\it\theta}_{t}={\rm\pi}/4$ . However, maximum change in the drop direction may not necessarily take place for ${\it\theta}_{t}={\rm\pi}/4$ because the lateral velocity of the drop $(U_{dz})$ also changes with the tilt angle $({\it\theta}_{t})$ . It is apparent from figure 16(a,b) that viscosity ratio and electrical properties $(R,S)$ markedly alter the direction of drop motion. Drops with high viscosity ratios encounter less change in direction of motion (from the impending axial motion) as compared with drops with low viscosity ratios.

Figure 16.

Variation of ${\it\theta}_{d}$ (in radians) with tilt angle $({\it\theta}_{t})$ for (a)  ${\it\Omega}_{T}>0$ and (b) for ${\it\Omega}_{T}<0$ . The parameters considered for ${\it\Omega}_{T}>0$ : $R=10$ , $S=2$ , $Ca=0.2$ , $Re_{E}=0.2$ , $M=2$ and for ${\it\Omega}_{T}<0$ : $R=0.2$ , $S=10$ , $Ca=0.1$ , $Re_{E}=0.2$ , $M=0.5$ . We have considered three different viscosity ratios ${\it\lambda}=0.1$ , 1 and 10. The horizontal black dashed line represent the zero lateral migration line $({\it\theta}_{d}=0)$ .

Figure 17.

Schematic of the experimental set-up. (a) Depicts the front view while (b) represents the side view of the set-up. In the side view, the electrodes are essentially the projections of the electrodes in the front view.

4.5 Experimental investigations

A schematic representation of the experimental set-up is given in figure 17. The set-up consists of an acrylic cell and electrode assembly. The height and width of the acrylic cell are designed as 13 cm each, and the depth is taken as 10 cm. Two brass electrodes attached at the two faces of the container are shown in figure 17. The two faces to which the electrodes are attached are aligned at 45° with regard to the vertical direction. The dimensions of the electrodes are $5~\text{cm}\times 7~\text{cm}$ . The thicknesses of the electrodes are 8 mm, and the perpendicular distance between the centres of the faces of the two electrodes is 10 cm. The dimensions are chosen so that the wall effects are minimized (Xu & Homsy Reference Xu and Homsy2006). The edges of the electrodes are rounded to minimize the edge effects related to electric field. The cell is filled with castor oil (Sigma Aldrich) while a drop of silicone oil (AR-10, Sigma Aldrich) is injected manually using a syringe (Dispo Van) from the open top end. For the experiment, the top electrode is energized with high-voltage DC power source using an in-house power supply (High Voltage Laboratory, Department of Electrical Engineering, Indian Institute of Technology Kharagpur) with an output range of 0–20 kV while the bottom electrode is grounded. After the silicone oil drop sediments sufficiently and reaches steady state, the electrodes are energized. The electrodes are ramped up within 20–30 s to the desired voltage level. The voltage is then held constant until the drop migrates sufficient distance and experiences wall effects. The motion of the drop is captured using a Nikon D5200 camera with a Nikon AF NIKKOR 24–85 mm macro lens.

Figure 18.

Comparison between experimental and analytical results concerning the droplet deviation angle $({\it\theta}_{d})$ for ${\it\theta}_{t}={\rm\pi}/4$ and $E_{ref}=14.14~\text{kV}$ .

The density and viscosity of the castor oil are ${\it\rho}_{e}=959~\text{kg m}^{-3}$ and ${\it\mu}_{e}=1~\text{Pa}~\text{s}$ while those of silicone oil are ${\it\rho}_{i}=1010~\text{kg m}^{-3}$ and ${\it\mu}_{i}=20~\text{mPa}~\text{s}$ . There is no common consensus on the values of the permittivity and conductivity of both the oils (Vizika & Saville Reference Vizika and Saville2006; Xu & Homsy Reference Xu and Homsy2006). Castor oil has a high propensity to absorb moisture and hence has a wide range of permittivity values, while silicone oil has a very sensitive dependence of the conductivity on the ambient temperature (Xu & Homsy Reference Xu and Homsy2006). The values of the conductivities and relative permittivities for both the oils are considered as (Vizika & Saville Reference Vizika and Saville2006; Xu & Homsy Reference Xu and Homsy2006; Salipante & Vlahovska Reference Salipante and Vlahovska2010): ${\it\sigma}_{e}=10^{-11}~\text{S m}^{-1}$ and ${\it\varepsilon}_{e}=2.6$ for castor oil, and ${\it\sigma}_{i}=10^{-12}~\text{S m}^{-1}$ and ${\it\varepsilon}_{i}=2.9$ for silicone oil. For a 2 mm diameter silicone oil drop we obtain the non-dimensional numbers as: $Ca=0.165$ , $Re_{E}=0.38$ and $M=2.787$ . The combined effect of deformation and charge convection on the drop deviation angle $({\it\theta}_{d})$ is given in figure 18, corresponding to a DC voltage of 14.14 kV. These results can be put in the perspective of the main contention of the analysis presented in the manuscript that is based on the claim that the presence of a tilted uniform electric field is responsible for causing lateral migration of an otherwise sedimenting drop: a phenomenon which is absent in either of the cases of a vertically applied electric field or a horizontally applied electric field. The experimental results clearly reveal that the presence of a tilted electric field (here ${\it\theta}_{t}={\rm\pi}/4$ ) is responsible for the lateral migration of the drop.