2.1 Governing equations and boundary conditions

We consider a physical scenario where a viscous drop of radius $a$ is sedimenting in a viscous medium with an uniform velocity $\widetilde{U}_{d}\boldsymbol{e}_{z}$ and acted upon by a DC electric field in the parallel direction as shown in figure 1. The drop surface is considered to be covered with non-ionic surfactant molecules and thus the electromigration phenomena associated with ionic surfactant transport can be neglected. Physical properties such as viscosity ( $\unicode[STIX]{x1D707}$ ), density ( $\unicode[STIX]{x1D70C}$ ), electrical conductivity ( $\unicode[STIX]{x1D70E}$ ) and dielectric permittivity ( $\unicode[STIX]{x1D716}$ ) are designated with a subscript ‘i’ inside the drop, while a subscript ‘e’ is used for the corresponding properties in the outside medium. In the absence of any flow field the drop interface is uniformly coated with surface concentration $\widetilde{\unicode[STIX]{x1D6E4}}_{eq}$ . The present study focuses on the surfactant effects and hence an electric field direction parallel to the $z$ -axis is chosen. This results in no azimuthal variation of relevant physical variables, thereby allowing us to adopt an axisymmetric platform without sacrificing the important physical behaviours arising from the coupled effects of surfactant and electric field. In the presence of a surface flow, uniformity in surfactant concentration is destroyed and an axisymmetric distribution $\widetilde{\unicode[STIX]{x1D6E4}}(\unicode[STIX]{x1D703})$ is achieved, leading to a non-uniform distribution of surface tension $\widetilde{\unicode[STIX]{x1D6FE}}(\unicode[STIX]{x1D703})$ . The deformed shape is represented as  $\widetilde{r}_{S}(\unicode[STIX]{x1D703})$ .

Figure 1.

Schematic representation of the physical set-up of a Newtonian drop of undeformed radius $a$ settling with a uniform velocity $\widetilde{U}_{d}$ in a surrounding unbounded Newtonian fluid medium under the combined action of gravity and a uniform DC electric field $E\boldsymbol{e}_{z}$ . An axisymmetric coordinate system ( $\widetilde{r},\unicode[STIX]{x1D703}$ ), which is fixed to the centroid of the drop, is used for the present study. The direction of drop motion, i.e. the $\widetilde{z}$ -axis, is chosen to be aligned with the direction of the gravitational field.

2.1.1 Surfactant transport

For an isothermal process the interfacial tension depends on the concentration of surfactant molecules on the drop surface. A closed form relationship between the surface tension and surfactant concentration is given by the Langmuir limit (Leal Reference Leal2007):

(2.1) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D6FE}}(\widetilde{\unicode[STIX]{x1D6E4}})=\widetilde{\unicode[STIX]{x1D6FE}}_{c}+R_{g}T\widetilde{\unicode[STIX]{x1D6E4}}_{\infty }\ln \left(1-{\displaystyle \frac{\widetilde{\unicode[STIX]{x1D6E4}}}{\widetilde{\unicode[STIX]{x1D6E4}}_{\infty }}}\right),\end{eqnarray}$$

where $\widetilde{\unicode[STIX]{x1D6E4}}_{\infty }$ stands for the maximum accommodatable surfactant concentration on the interface. The other parameters: $\widetilde{\unicode[STIX]{x1D6FE}}_{c}$ , $R_{g}$ and $T$ are the interfacial tension of the surfactant-free or clean drop, the ideal gas constant and the absolute temperature, respectively. Next we assume that the surfactant concentration is dilute everywhere along the surface. Thus nonlinearities associated with $\widetilde{\unicode[STIX]{x1D6E4}}/\widetilde{\unicode[STIX]{x1D6E4}}_{\infty }$ can be neglected. This allows us to consider a linear constitutive behaviour between surface tension and the surfactant concentration (Stone & Leal Reference Stone and Leal1990; Li & Pozrikidis Reference Li and Pozrikidis1997; Leal Reference Leal2007; Nganguia et al. Reference Nganguia, Young, Vlahovska, Bławzdziewcz, Zhang and Lin2013; Das et al. Reference Das, Mandal and Chakraborty2017a ). The dimensional form of the same is given as

(2.2) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D6FE}}(\widetilde{\unicode[STIX]{x1D6E4}})=\widetilde{\unicode[STIX]{x1D6FE}}_{c}-R_{g}T\,\widetilde{\unicode[STIX]{x1D6E4}}.\end{eqnarray}$$

At equilibrium, for a uniformly coated drop, the interfacial tension decreases and can be expressed as $\widetilde{\unicode[STIX]{x1D6FE}}_{eq}=\widetilde{\unicode[STIX]{x1D6FE}}_{c}-R_{g}T\,\widetilde{\unicode[STIX]{x1D6E4}}_{eq}$ . Under such a condition, equation (2.2) becomes

(2.3) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D6FE}}(\widetilde{\unicode[STIX]{x1D6E4}})=\widetilde{\unicode[STIX]{x1D6FE}}_{eq}+R_{g}T(\widetilde{\unicode[STIX]{x1D6E4}}_{eq}-\widetilde{\unicode[STIX]{x1D6E4}}).\end{eqnarray}$$

Assuming a bulk-insoluble surfactant (Stone & Leal Reference Stone and Leal1990; Li & Pozrikidis Reference Li and Pozrikidis1997; Leal Reference Leal2007) the surfactant distribution at the drop surface is governed by

(2.4) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D735}}_{S}\boldsymbol{\cdot }(\widetilde{\boldsymbol{u}}_{S}\widetilde{\unicode[STIX]{x1D6E4}})=D_{S}\widetilde{\unicode[STIX]{x1D6FB}}_{S}^{2}\widetilde{\unicode[STIX]{x1D6E4}},\quad \text{at}~\tilde{r}=\tilde{r}_{S}(\unicode[STIX]{x1D703}),\end{eqnarray}$$

where $\widetilde{\boldsymbol{u}}_{S}=\widetilde{\boldsymbol{u}}_{i}|_{\widetilde{r}=\widetilde{r}_{S}}$ is the velocity at the drop surface and $D_{S}$ denotes the surface-diffusion coefficient.

We adopt the following non-dimensional scheme: $\text{length}\sim a$ ; $\text{electric potential}\sim aE$ ; $\text{conductivity}\sim \unicode[STIX]{x1D70E}_{e}$ ; $\text{permittivity}\sim \unicode[STIX]{x1D716}_{e}$ and $\text{viscosity}\sim \unicode[STIX]{x1D707}_{e}$ ; $\text{velocity}\sim \widetilde{U}_{HR}$ ; surfactant concentration ${\sim}\widetilde{\unicode[STIX]{x1D6E4}}_{eq}$ ; $\text{hydrodynamic stress}\sim \unicode[STIX]{x1D707}_{e}\widetilde{U}_{HR}/a$ and electric stress ${\sim}\unicode[STIX]{x1D716}_{e}E^{2}$ . Sedimentation of a drop without electric effects results in a drop velocity expression, widely known as the Hadamard–Rybczynski velocity, $\widetilde{U}_{HR}=(2/9)ga^{2}((\unicode[STIX]{x1D70C}_{i}-\unicode[STIX]{x1D70C}_{e})/\unicode[STIX]{x1D707}_{e})((\unicode[STIX]{x1D706}+1)/(\unicode[STIX]{x1D706}+2/3))$ (Leal Reference Leal2007), where $\unicode[STIX]{x1D706}$ is the viscosity ratio between the drop and surrounding fluid. However, another velocity scale ( $\unicode[STIX]{x1D716}_{e}aE^{2}/\unicode[STIX]{x1D707}_{e}$ ) appears when electric effects are also present in the system. In the present work we choose the Hadamard–Rybczynski velocity ( $\widetilde{U}_{HR}$ ) as the reference velocity scale with a rationale to obtain a correction of settling velocity (Xu & Homsy Reference Xu and Homsy2006; Mandal, Bandopadhyay & Chakraborty Reference Mandal, Bandopadhyay and Chakraborty2017a ; Mandal et al. Reference Mandal, Chakrabarti and Chakraborty2017b ) in the subsequent analysis. Henceforth we will also use the following notations for various parameters to describe the physical situation: viscosity ratio $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D707}_{i}/\unicode[STIX]{x1D707}_{e}$ , conductivity ratio $R=\unicode[STIX]{x1D70E}_{i}/\unicode[STIX]{x1D70E}_{e}$ , permittivity ratio $S=\unicode[STIX]{x1D716}_{i}/\unicode[STIX]{x1D716}_{e}$ , electric Reynolds number $Re_{E}=\unicode[STIX]{x1D716}_{e}\widetilde{U}_{HR}/a\unicode[STIX]{x1D70E}_{e}$ , capillary number $Ca=\unicode[STIX]{x1D707}_{e}\widetilde{U}_{HR}/\unicode[STIX]{x1D6FE}_{eq}$ , Mason number $M=\unicode[STIX]{x1D716}_{e}aE^{2}/\unicode[STIX]{x1D707}_{e}\widetilde{U}_{HR}$ and surface Péclet number $Pe=a\,\widetilde{U}_{HR}/D_{S}$ . Physically the electric Reynolds number $Re_{E}$ stands for the relative importance of the charge relaxation time scale and charge convection time scale in the problem (Xu & Homsy Reference Xu and Homsy2006; Mandal et al. Reference Mandal, Bandopadhyay and Chakraborty2016b ). From here onwards we will drop all the $\widetilde{\,}$ symbols in the dimensional variables to denote the corresponding dimensionless forms for brevity.

2.1.2 Electrostatic problem

According to the leaky dielectric model introduced by Melcher & Taylor (Reference Melcher and Taylor1969) the electric potentials $\unicode[STIX]{x1D711}_{i}$ and $\unicode[STIX]{x1D711}_{e}$ for the inner and outer fluids, respectively, satisfy the Laplace equation as follows (Taylor Reference Taylor1966; Melcher & Taylor Reference Melcher and Taylor1969; Arp, Foister & Mason Reference Arp, Foister and Mason1980; Saville Reference Saville1997):

(2.5) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D711}_{i}=0\\ \unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D711}_{e}=0.\end{array}\right\}\end{eqnarray}$$

The general solution of the above equations in terms of Legendre polynomials is given by

(2.6) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D711}_{i}=\mathop{\sum }_{n=1}^{\infty }(a_{n}r^{n}+\hat{b}_{-n-1}r^{-n})\text{P}_{n}(\unicode[STIX]{x1D702})\\ \displaystyle \unicode[STIX]{x1D711}_{e}=\mathop{\sum }_{n=1}^{\infty }(\hat{a}_{n}r^{n}+b_{-n-1}r^{-n})\text{P}_{n}(\unicode[STIX]{x1D702}),\end{array}\right\}\end{eqnarray}$$

where the constants $a_{n},b_{-n-1},\hat{a}_{n}$ and $\hat{b}_{-n-1}$ are arbitrary. The electric potential is subjected to the following boundary conditions:

1. (i) $\unicode[STIX]{x1D711}_{i}$ is finite-valued everywhere inside the drop. Thus all the decaying harmonics in the expression of $\unicode[STIX]{x1D711}_{i}$ vanish ( $\hat{b}_{-n-1}=0$ ).

2. (ii) $\unicode[STIX]{x1D711}_{e}$ satisfies the far-field condition of uniform electric field along the positive $z$ -direction, as shown in figure 1. In mathematical notation the dimensional form of this condition, as $\widetilde{r}\rightarrow \infty$ , takes the form $-\widetilde{\unicode[STIX]{x1D735}}\widetilde{\unicode[STIX]{x1D711}}=E$ . With the adopted non-dimensionalization scheme, as $r\rightarrow \infty$ , it becomes $\unicode[STIX]{x1D711}_{e}=-r\text{P}_{1}(\unicode[STIX]{x1D702})$ , where $\unicode[STIX]{x1D702}=\cos \unicode[STIX]{x1D703}$ and $\text{P}_{1}$ is the Legendre polynomial of first degree.

3. (iii) At the drop interface the potential is continuous, i.e. at $r=r_{S}(\unicode[STIX]{x1D703})$ , $\unicode[STIX]{x1D711}_{i}=\unicode[STIX]{x1D711}_{e}$ .

4. (iv) The interfacial charge conservation at the steady state is satisfied by the electric potential, i.e.

   (2.7) $$\begin{eqnarray}\text{at}~r=r_{S}(\unicode[STIX]{x1D703}),\quad \boldsymbol{n}\boldsymbol{\cdot }(R\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}_{i}-\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}_{e})=-Re_{E}\unicode[STIX]{x1D735}_{S}\boldsymbol{\cdot }(q_{S}\boldsymbol{u}_{S}),\end{eqnarray}$$
   where $\boldsymbol{n}$ represents the outward unit normal at the drop interface, $\unicode[STIX]{x1D735}_{S}$ is the surface divergence operator and $\boldsymbol{u}_{S}$ is the fluid velocity at the interface. Here $q_{S}$ represents the interfacial charge density, defined as $q_{S}=\boldsymbol{n}\boldsymbol{\cdot }(S\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}_{i}-\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}_{e})|_{r=r_{S}}$ . In (2.7), the left-hand side represents current due to Ohmic conduction while the right-hand side stands for the current due to charge convection at the droplet surface. Here we would like to draw the attention of the reader to the fact that (2.7) does not contain any term which explicitly contains any surfactant-related parameter. However, the surface velocity appearing in the right-hand side will be shown to become severely affected by the interfacial Marangoni stress due to surfactants. Thus the surfactant distribution will have its indirect effect on the electric potential distribution through a coupling between the fluid flow and electric field.

2.1.3 Hydrodynamic problem

In the creeping flow limit, the velocity ( $\boldsymbol{u}_{i},\boldsymbol{u}_{e}$ ) and pressure ( $p_{i},p_{e}$ ) both inside and outside the drop satisfy the Stokes equation:

(2.8) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D735}p_{i}=\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}_{i},\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{i}=0\\ \unicode[STIX]{x1D735}p_{e}=\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}_{e},\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{e}=0.\end{array}\right\}\end{eqnarray}$$

The corresponding velocity components can be expressed in terms of the streamfunction ( $\unicode[STIX]{x1D6F9}$ ) as $u_{r}=-(1/r^{2})(\unicode[STIX]{x2202}\unicode[STIX]{x1D6F9}/\unicode[STIX]{x2202}\unicode[STIX]{x1D702})$ and $u_{\unicode[STIX]{x1D703}}=-(1/(r\sqrt{1-\unicode[STIX]{x1D702}^{2}}))(\unicode[STIX]{x2202}\unicode[STIX]{x1D6F9}/\unicode[STIX]{x2202}r)$ . Under these considerations, equation (2.8) becomes a set of fourth-order partial differential equations, given as

(2.9) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}{\mathcal{L}}^{2}({\mathcal{L}}^{2}\unicode[STIX]{x1D6F9}_{i}=0)\\ {\mathcal{L}}^{2}({\mathcal{L}}^{2}\unicode[STIX]{x1D6F9}_{e}=0),\end{array}\right\}\end{eqnarray}$$

where $\unicode[STIX]{x1D6F9}_{i}$ , $\unicode[STIX]{x1D6F9}_{e}$ are the streamfunctions for the drop phase and the continuous phase, respectively, and ${\mathcal{L}}^{2}$ is a linear operator in the spherical polar coordinate system defined as ${\mathcal{L}}^{2}=\unicode[STIX]{x2202}^{2}/\unicode[STIX]{x2202}r^{2}+((1-\unicode[STIX]{x1D702}^{2})/r^{2})(\unicode[STIX]{x2202}^{2}/\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{2})$ . The general solutions for the streamfunctions in terms of Gegenbauer polynomials $\text{Q}$ (Leal Reference Leal2007) are given by

(2.10) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D6F9}_{i}=\mathop{\sum }_{n=1}^{\infty }(A_{n}r^{n+3}+B_{n}r^{n+1}+{\hat{C}}_{n}r^{2-n}+\hat{D}_{n}r^{-n})\text{Q}_{n}(\unicode[STIX]{x1D702})\\ \displaystyle \unicode[STIX]{x1D6F9}_{e}=\mathop{\sum }_{n=1}^{\infty }(\hat{A}_{n}r^{n+3}+\hat{B}_{n}r^{n+1}+C_{n}r^{2-n}+D_{n}r^{-n})\text{Q}_{n}(\unicode[STIX]{x1D702}),\end{array}\right\}\end{eqnarray}$$

where $A_{n},B_{n},C_{n},D_{n},\hat{A}_{n},\hat{B}_{n},{\hat{C}}_{n},\hat{D}_{n}$ are arbitrary constants.

The velocity and pressure fields of the drop phase ( $\boldsymbol{u}_{i},p_{i}$ ) and that of the surrounding medium ( $\boldsymbol{u}_{i},p_{e}$ ) satisfy the following boundary conditions:

1. (i) $\boldsymbol{u}_{i},p_{i}$ are finite-valued everywhere inside the drop. Thus for $\unicode[STIX]{x1D6F9}_{i}$ all the decaying harmonics have to be zero.

2. (ii) $\boldsymbol{u}_{e},p_{e}$ are finite-valued outside the drop and in the far-field locations the disturbance velocity generated due to drop motion must die out. Now from a reference frame attached to the drop (as shown in figure 1) which is moving with a uniform velocity $U_{d}$ , the surrounding fluid appears to move in the negative $z$ direction. Mathematically this condition at $r\rightarrow \infty$ reduces to $\boldsymbol{u}_{e}=-\boldsymbol{U}_{d}$ .

3. (iii) At the drop interface the velocity is continuous i.e.  $\boldsymbol{u}_{i}=\boldsymbol{u}_{e}$ and at steady state the normal component of the velocity vanishes ( $\boldsymbol{u}_{i}\boldsymbol{\cdot }\boldsymbol{n}=\boldsymbol{u}_{e}\boldsymbol{\cdot }\boldsymbol{n}=0$ ) by virtue of the no-penetration condition at the interface.

4. (iv) The gradient in surface tension generated due to non-uniformity in the surfactant concentration gives rise to an additional component of interfacial stress known as the Marangoni stress ( $\boldsymbol{T}^{Ma}$ ). The dimensionless form of $\boldsymbol{T}^{Ma}$ is given by

   (2.11) $$\begin{eqnarray}\boldsymbol{T}^{Ma}=Ma[(1-\unicode[STIX]{x1D6E4})(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n})\boldsymbol{n}+\unicode[STIX]{x1D735}_{S}\unicode[STIX]{x1D6E4}],\end{eqnarray}$$
   where the Marangoni number ( $Ma$ ) is defined as $Ma=R_{g}T\,\widetilde{\unicode[STIX]{x1D6E4}}_{eq}/\unicode[STIX]{x1D707}_{e}\widetilde{U}_{HR}$ . This embodies the relative strength of Marangoni stress and the viscous stress. The interfacial balance between the hydrodynamic and electrical components of stress, which would exist otherwise, now gets modified by the appearance of $\boldsymbol{T}^{Ma}$ . The resulting stress balance equation at the interface takes the following form
   (2.12) $$\begin{eqnarray}\text{at}~r=r_{S}(\unicode[STIX]{x1D703}),\quad \unicode[STIX]{x27E6}\boldsymbol{T}^{H}\unicode[STIX]{x27E7}+M\unicode[STIX]{x27E6}\boldsymbol{T}^{E}\unicode[STIX]{x27E7}=\frac{1}{Ca}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n})\boldsymbol{n}+Ma[(1-\unicode[STIX]{x1D6E4})(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n})\boldsymbol{n}+\unicode[STIX]{x1D735}_{S}\unicode[STIX]{x1D6E4}].\end{eqnarray}$$
   Here $\unicode[STIX]{x27E6}~\unicode[STIX]{x27E7}$ represents the jump or difference of any quantity (e.g.  $\unicode[STIX]{x27E6}\boldsymbol{T}\unicode[STIX]{x27E7}=\boldsymbol{T}_{e}-\boldsymbol{T}_{i}$ ) across the interface, while $\boldsymbol{T}^{H}=\unicode[STIX]{x1D749}^{H}\boldsymbol{\cdot }\boldsymbol{n}$ and $\boldsymbol{T}^{E}=\unicode[STIX]{x1D749}^{E}\boldsymbol{\cdot }\boldsymbol{n}$ are the viscous and electric parts of the traction vector. The forms of the viscous and electric stress tensors are given by $\unicode[STIX]{x1D749}_{i}^{H}=-p_{i}\unicode[STIX]{x1D644}+\unicode[STIX]{x1D706}[\unicode[STIX]{x1D735}\boldsymbol{u}_{i}+(\unicode[STIX]{x1D735}\boldsymbol{u}_{i})^{\text{T}}]$ ; $\unicode[STIX]{x1D749}_{e}^{H}=-p_{e}\unicode[STIX]{x1D644}+[\unicode[STIX]{x1D735}\boldsymbol{u}_{e}+(\unicode[STIX]{x1D735}\boldsymbol{u}_{e})^{\text{T}}]$ ; $\unicode[STIX]{x1D749}_{i}^{E}=S[\boldsymbol{E}_{i}\boldsymbol{E}_{i}^{\text{T}}-(1/2)|\boldsymbol{E}_{i}|^{2}\unicode[STIX]{x1D644}]$ and $\unicode[STIX]{x1D749}_{e}^{E}=[\boldsymbol{E}_{e}\boldsymbol{E}_{e}^{\text{T}}-(1/2)|\boldsymbol{E}_{e}|^{2}\unicode[STIX]{x1D644}]$ .

For later convenience $Ma$ is related to the physicochemical parameter $\unicode[STIX]{x1D6FD}$ by the relation $Ma=\unicode[STIX]{x1D6FD}/(Ca(1-\unicode[STIX]{x1D6FD}))$ , where $\unicode[STIX]{x1D6FD}$ is known as the surface elasticity number and defined as $\unicode[STIX]{x1D6FD}=-(\text{d}(\widetilde{\unicode[STIX]{x1D6FE}}/\widetilde{\unicode[STIX]{x1D6FE}}_{c}))/(\text{d}(\widetilde{\unicode[STIX]{x1D6E4}}/\widetilde{\unicode[STIX]{x1D6E4}}_{eq}))=(R_{g}T\,\widetilde{\unicode[STIX]{x1D6E4}}_{eq})/\widetilde{\unicode[STIX]{x1D6FE}}_{c}$ . This quantifies the sensitivity of the interfacial tension to the surfactant concentration. Theoretically $\unicode[STIX]{x1D6FD}$ can vary between 0 and 1 (Stone & Leal Reference Stone and Leal1990; Li & Pozrikidis Reference Li and Pozrikidis1997). To visualize the importance of the specific terms in the right-hand side of (2.12), tangential and normal components of stress balance are presented separately as:

(2.13) $$\begin{eqnarray}\text{at}~r=r_{S}(\unicode[STIX]{x1D703}),\quad \unicode[STIX]{x27E6}T_{t}^{H}\unicode[STIX]{x27E7}+M\unicode[STIX]{x27E6}T_{t}^{E}\unicode[STIX]{x27E7}=\underbrace{\frac{\unicode[STIX]{x1D6FD}}{Ca(1-\unicode[STIX]{x1D6FD})}(\unicode[STIX]{x1D735}_{S}\unicode[STIX]{x1D6E4})\boldsymbol{\cdot }\boldsymbol{t}}_{\text{Marangoni}\,\,\text{effect}},\end{eqnarray}$$

and

(2.14) $$\begin{eqnarray}\text{at}~r=r_{S}(\unicode[STIX]{x1D703}),\quad \unicode[STIX]{x27E6}T_{n}^{H}\unicode[STIX]{x27E7}+M\unicode[STIX]{x27E6}T_{n}^{E}\unicode[STIX]{x27E7}=\frac{1}{Ca}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n})+\underbrace{\frac{\unicode[STIX]{x1D6FD}}{Ca(1-\unicode[STIX]{x1D6FD})}[(1-\unicode[STIX]{x1D6E4})(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n})]}_{\text{Marangoni}\,\,\text{effect}},\end{eqnarray}$$

where $\boldsymbol{t}$ is the unit vector in the tangential direction at the drop surface.

The dimensionless form of the surfactant transport equation becomes

(2.15) $$\begin{eqnarray}Pe\unicode[STIX]{x1D735}_{S}\boldsymbol{\cdot }(\boldsymbol{u}_{S}\unicode[STIX]{x1D6E4})=\unicode[STIX]{x1D6FB}_{S}^{2}\unicode[STIX]{x1D6E4}.\end{eqnarray}$$

In addition, the surfactant present on the drop surface must satisfy a mass conservation constraint (Kim & Subramanian Reference Kim and Subramanian1989) given by

(2.16) $$\begin{eqnarray}\int _{0}^{\unicode[STIX]{x03C0}}\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D703})r_{S}^{2}(\unicode[STIX]{x1D703})\sin \unicode[STIX]{x1D703}\,\text{d}\unicode[STIX]{x1D703}=2.\end{eqnarray}$$

We note that the mathematical model becomes nonlinear in nature owing to the presence of convective transport at the drop interface, as depicted by the left-hand side of (2.15). Secondly the flow field and the surfactant distribution at the interface become coupled to each other.

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

A source of complexity in solving the governing equations and boundary conditions arises from the convection–diffusion transport equation governing the distribution of insoluble surfactants (2.15) on the surface and its coupling with the flow problem (2.12). Moreover, the deformed surface is a priori unknown and must be determined through normal stress balance. Adding to these, is the nonlinearity associated with the charge convection equation, thus rendering the physical system at hand impossible to solve analytically for an arbitrary range of parameters. Hence to proceed with an analytical treatment we evaluate various dimensionless parameters based on the dimensional values of the property and geometric dimensions reported elsewhere (Xu & Homsy Reference Xu and Homsy2006; Mhatre & Thaokar Reference Mhatre and Thaokar2013). We then employ a double asymptotic expansion with $Ca$ and $Re_{E}$ as perturbation parameters (Xu & Homsy Reference Xu and Homsy2006; Mandal et al. Reference Mandal, Bandopadhyay and Chakraborty2016c ). Accordingly any generic variable $\unicode[STIX]{x1D709}$ can be expanded in the following form

(2.17) $$\begin{eqnarray}\unicode[STIX]{x1D709}=\unicode[STIX]{x1D709}^{(0)}+Ca\,\unicode[STIX]{x1D709}^{(Ca)}+Re_{E}\,\unicode[STIX]{x1D709}^{(Re_{E})}+O(Re_{E}^{2},Ca^{2},CaRe_{E}).\end{eqnarray}$$

Special care is to be taken while expanding the pressure and stress inside the droplet. In order to balance the capillary pressure in the absence of any flow field, these are expanded as (Chan & Leal Reference Chan and Leal1979; Mandal et al. Reference Mandal, Bandopadhyay and Chakraborty2016c ):

(2.18) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle p_{i}=\frac{1}{Ca}p_{i}^{(1/Ca)}+p_{i}^{(0)}+Ca\,p_{i}^{(Ca)}+Re_{E}\,p_{i}^{(Re_{E})}+\cdots \\ \displaystyle \unicode[STIX]{x1D749}_{i}=\frac{1}{Ca}\unicode[STIX]{x1D749}_{i}^{(1/Ca)}+\unicode[STIX]{x1D749}_{i}^{(0)}+Ca\,\unicode[STIX]{x1D749}_{i}^{(Ca)}+Re_{E}\,\unicode[STIX]{x1D749}_{i}^{(Re_{E})}+\cdots \,.\end{array}\right\}\end{eqnarray}$$

The drop shape is unknown a priori and we look for a solution of the same by expanding it in the form:

(2.19) $$\begin{eqnarray}r_{S}=1+Ca\,f^{(Ca)}+Ca\,Re_{E}\,f^{(Ca\,Re_{E})}+Ca^{2}\,f^{(Ca^{2})}+\cdots .\end{eqnarray}$$

Here $f^{(Ca)}$ , $f^{(Ca\,Re_{E})}$ and $f^{(Ca^{2})}$ are the shape functions, which can be expressed as a sum of surface harmonics:

(2.20) $$\begin{eqnarray}f^{(j)}=\mathop{\sum }_{n=1}^{\infty }L_{n}^{(j)}\text{P}_{n}(\unicode[STIX]{x1D702});\quad j=Ca,Ca\,Re_{E},Ca^{2}.\end{eqnarray}$$

We define the drop deformation parameter as

(2.21) $$\begin{eqnarray}{\mathcal{D}}={\displaystyle \frac{r_{S}|_{\unicode[STIX]{x1D703}=0}-r_{S}|_{\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2}}{r_{S}|_{\unicode[STIX]{x1D703}=0}+r_{S}|_{\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2}}},\end{eqnarray}$$

which will be used later in the interpretation of different deformation-related effects.

The surfactant concentration is also expected to deviate from the equilibrium value owing to deformation of the drop from the initial sphericity. Thus the concentration ( $\unicode[STIX]{x1D6E4}$ ) is also expanded in a similar fashion to that of the shape deformation (Stone & Leal Reference Stone and Leal1990; Ha & Yang Reference Ha and Yang1995; Mandal et al. Reference Mandal, Bandopadhyay and Chakraborty2016b ) and given as

(2.22) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D6E4}^{(0)}+Ca\,\unicode[STIX]{x1D6E4}^{(Ca)}+Ca\,Re_{E}\,\unicode[STIX]{x1D6E4}^{(Ca\,Re_{E})}+Ca^{2}\unicode[STIX]{x1D6E4}^{(Ca^{2})}+\cdots .\end{eqnarray}$$

Here $\unicode[STIX]{x1D6E4}^{(0)}$ turns out to be 1 when the mass conservation constraint (2.16) is incorporated. Similar to the shape function, the remaining surfactant harmonics can be expressed as

(2.23) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}^{(j)}=\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D6E4}_{n}^{(j)}\text{P}_{n}(\unicode[STIX]{x1D702});\quad j=Ca,Ca\,Re_{E},Ca^{2}.\end{eqnarray}$$

In what follows we will proceed to obtain analytical solutions for the diffusion-dominated regime or the low-Péclet limit ( $Pe\ll 1$ ). The practical relevance of this regime is discussed in great detail by Das et al. (Reference Das, Mandal and Chakraborty2017a ) with reference to the experimental situations (Stebe et al. Reference Stebe, Lin and Maldarelli1991). It is to be noted that having chosen $Ca$ as an independent dimensionless parameter, we cannot consider $Pe$ as another independent parameter (Stone & Leal Reference Stone and Leal1990; Mandal, Bandopadhyay & Chakraborty Reference Mandal, Bandopadhyay and Chakraborty2016a ), since both these dimensionless numbers contain the same $\widetilde{U}_{HR}$ as the velocity scale. Instead we use a ratio of the two, $k=Pe/Ca=\widetilde{\unicode[STIX]{x1D6E4}}_{eq}a/\unicode[STIX]{x1D707}_{e}D_{S}$ , which becomes an $O(1)$ dimensionless constant for $Pe\ll 1$ and consists of various material properties of surface-active agents.

The detailed expressions for the governing equations and boundary conditions of the electric as well as flow problem for the case of a clean, surfactant-free drop can be found in Mandal et al. (Reference Mandal, Bandopadhyay and Chakraborty2017a ,Reference Mandal, Chakrabarti and Chakraborty b ). The boundary conditions of the electrostatic problem and the velocity boundary conditions (no slip and kinematic condition), remaining unaltered, will not be repeated here. However, the presence of surface-active molecules on the drop interface triggers Marangoni flow owing to the variability in interfacial tension. This in turn couples the surfactant transport phenomena with the flow problem through the Marangoni stress term in the stress balance equation (2.12). The tangential and normal stress balance equations along with the corresponding equations for surfactant transport, at different perturbation orders, are discussed below.

Leading order:

The boundary conditions, in this order, are evaluated on the undeformed surface (at $r=1$ ) and are as follows:

1. (i) Tangential stress balance:

   (2.24) $$\begin{eqnarray}\text{at}~r=1,\quad \unicode[STIX]{x27E6}T_{\unicode[STIX]{x1D703}}^{H(0)}\unicode[STIX]{x27E7}+M\unicode[STIX]{x27E6}T_{\unicode[STIX]{x1D703}}^{E(0)}\unicode[STIX]{x27E7}=\frac{\unicode[STIX]{x1D6FD}}{1-\unicode[STIX]{x1D6FD}}\frac{\text{d}\unicode[STIX]{x1D6E4}}{\text{d}\unicode[STIX]{x1D703}}^{(Ca)}.\end{eqnarray}$$

2. (ii) Normal stress balance:

   (2.25) $$\begin{eqnarray}\text{at}~r=1,\quad \unicode[STIX]{x27E6}T_{r}^{H(0)}\unicode[STIX]{x27E7}+M\unicode[STIX]{x27E6}T_{r}^{E(0)}\unicode[STIX]{x27E7}=-(2f^{(Ca)}+\unicode[STIX]{x1D6FB}^{2}f^{(Ca)})-\frac{2\unicode[STIX]{x1D6FD}}{1-\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6E4}^{(Ca)}.\end{eqnarray}$$

3. (iii) Surfactant transport:

   (2.26) $$\begin{eqnarray}\text{at}~r=1,\quad k\unicode[STIX]{x1D735}_{S}\boldsymbol{\cdot }(\boldsymbol{u}_{S}^{(0)})=\unicode[STIX]{x1D6FB}_{S}^{2}\unicode[STIX]{x1D6E4}^{(Ca)}.\end{eqnarray}$$

$O(Re_{E})$ : effect of finite charge convection:

Now we proceed to capture the effects of surface charge convection by adopting the electric Reynolds number ( $Re_{E}$ ) as a perturbation parameter without taking account of the shape distortion. Thus the boundary conditions are applied at $r=1$ as done for the leading-order case. The corresponding equations take the forms:

1. (i) Tangential stress balance:

   (2.27) $$\begin{eqnarray}\text{at}~r=1,\quad \unicode[STIX]{x27E6}T_{t}^{H(Re_{E})}\unicode[STIX]{x27E7}+M\unicode[STIX]{x27E6}T_{t}^{E(Re_{E})}\unicode[STIX]{x27E7}=\frac{\unicode[STIX]{x1D6FD}}{1-\unicode[STIX]{x1D6FD}}\frac{\text{d}\unicode[STIX]{x1D6E4}}{\text{d}\unicode[STIX]{x1D703}}^{(Ca\,Re_{E})}.\end{eqnarray}$$

2. (ii) Normal stress balance:

   (2.28) $$\begin{eqnarray}\text{at}~r=1,\quad \unicode[STIX]{x27E6}T_{n}^{H(Re_{E})}\unicode[STIX]{x27E7}+M\unicode[STIX]{x27E6}T_{n}^{E(Re_{E})}\unicode[STIX]{x27E7}=-(2f^{(Ca\,Re_{E})}+\unicode[STIX]{x1D6FB}^{2}f^{(Ca\,Re_{E})})-\frac{2\unicode[STIX]{x1D6FD}}{1-\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6E4}^{(Ca\,Re_{E})}.\end{eqnarray}$$

3. (iii) Surfactant transport:

   (2.29) $$\begin{eqnarray}\text{at}~r=1,\quad k\unicode[STIX]{x1D735}_{S}\boldsymbol{\cdot }(\boldsymbol{u}_{S}^{(Re_{E})})=\unicode[STIX]{x1D6FB}_{S}^{2}\unicode[STIX]{x1D6E4}^{(Ca\,Re_{E})}.\end{eqnarray}$$

$O(Ca)$ : effect of drop-shape deformation:

All the quantities at this order are to be evaluated on the deformed interface $r=r_{S}(\unicode[STIX]{x1D703})=1+Ca\,f^{(Ca)}$ , which is calculated with the help of the leading-order solution for the shape function $f^{(Ca)}$ . However, since the drop shape is not known a priori, following the domain perturbation technique, we evaluate the corresponding quantities at $r=1$ by a Taylor series expansion about $r=1$ (Happel & Brenner Reference Happel and Brenner1983; Bandopadhyay et al. Reference Bandopadhyay, Mandal, Kishore and Chakraborty2016; Mandal et al. Reference Mandal, Bandopadhyay and Chakraborty2016c ). Mathematically, for a generic quantity this can be stated as

(2.30) $$\begin{eqnarray}[\unicode[STIX]{x1D709}|_{at\,r=1+Ca\,f^{(Ca)}}]^{(Ca)}=\unicode[STIX]{x1D709}^{(Ca)}|_{at\,r=1}+f^{(Ca)}\left.\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{(0)}}{\unicode[STIX]{x2202}r}\right|_{r=1}.\end{eqnarray}$$

The relevant stress boundary conditions and surfactant transport equations take the following forms:

1. (i) Tangential stress balance:

   (2.31) $$\begin{eqnarray}\text{at}~r=1,\quad \unicode[STIX]{x27E6}T_{t}^{H(Ca)}\unicode[STIX]{x27E7}+M\unicode[STIX]{x27E6}T_{t}^{E(Ca)}\unicode[STIX]{x27E7}=\frac{\unicode[STIX]{x1D6FD}}{1-\unicode[STIX]{x1D6FD}}\left[\frac{\text{d}\unicode[STIX]{x1D6E4}}{\text{d}\unicode[STIX]{x1D703}}^{(Ca^{2})}+f^{Ca}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(\frac{1}{r}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}^{(Ca^{2})}\right)\right].\end{eqnarray}$$

2. (ii) Normal stress balance:

   (2.32) $$\begin{eqnarray}\displaystyle \text{at}~r & = & \displaystyle 1,\quad \unicode[STIX]{x27E6}T_{n}^{H(Ca)}\unicode[STIX]{x27E7}+M\unicode[STIX]{x27E6}T_{n}^{E(Ca)}\unicode[STIX]{x27E7}=2f^{(Ca)}(f^{(Ca)}+\unicode[STIX]{x1D6FB}^{2}f^{(Ca)})-(2f^{(Ca^{2})}+\unicode[STIX]{x1D6FB}^{2}f^{(Ca^{2})})\nonumber\\ \displaystyle & & \displaystyle -\,\frac{2\unicode[STIX]{x1D6FD}}{1-\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6E4}^{(Ca^{2})}+\frac{\unicode[STIX]{x1D6FD}}{1-\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6E4}^{(Ca)}(2f^{(Ca)}+\unicode[STIX]{x1D6FB}^{2}f^{(Ca)}).\end{eqnarray}$$

3. (iii) Surfactant transport:

   (2.33) $$\begin{eqnarray}\displaystyle \text{at}~r & = & \displaystyle 1,\quad k\unicode[STIX]{x1D735}_{S}\boldsymbol{\cdot }(\boldsymbol{u}_{S}^{(0)}\unicode[STIX]{x1D6E4}^{(Ca)}+\boldsymbol{u}_{S}^{(Ca)})=\unicode[STIX]{x1D6FB}_{S}^{2}\unicode[STIX]{x1D6E4}^{(Ca^{2})}+f^{(Ca)}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}(\unicode[STIX]{x1D6FB}_{S}^{2}\unicode[STIX]{x1D6E4}^{(Ca)})\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}^{(Ca)}\left(\frac{2}{r^{2}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}^{(Ca)}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(\frac{1}{r}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}^{(Ca)}\right)-\frac{1}{r^{2}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}^{(Ca)}\right).\end{eqnarray}$$
   The expressions for hydrodynamic and electric traction vectors as appeared in (2.31) and (2.32) can be found elsewhere (Mandal et al. Reference Mandal, Bandopadhyay and Chakraborty2016b ) and are not repeated here.

2.3 Solution methodology

In this section we describe the solution methodology to obtain the detailed expressions of electric potential, streamfunction, shape deformation and the surfactant distribution at different orders of perturbation. Since at each order the electric potential satisfies the Laplace equation, the general solution of $\unicode[STIX]{x1D711}_{i}$ and $\unicode[STIX]{x1D711}_{e}$ can be expressed as a sum of spherical harmonics, keeping in view the boundedness and far-field conditions:

(2.34) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D711}_{i}^{(j)}=\mathop{\sum }_{n=0}^{\infty }r^{n}a_{n}^{(j)}\text{P}_{n}(\unicode[STIX]{x1D702})\\ \displaystyle \unicode[STIX]{x1D711}_{e}^{(j)}=\unicode[STIX]{x1D711}_{\infty }^{(j)}+\mathop{\sum }_{n=0}^{\infty }r^{-n-1}b_{-n-1}^{(j)}\text{P}_{n}(\unicode[STIX]{x1D702}),\end{array}\right\}\end{eqnarray}$$

where $\unicode[STIX]{x1D711}_{\infty }^{(j)}$ denotes the unperturbed potential to match with the far-field condition of uniform electric field and only exists in the leading order. Thus it takes a form of $\unicode[STIX]{x1D711}_{\infty }^{(0)}=-r\text{P}_{1}(\unicode[STIX]{x1D702})$ . Similar to the electric potential, the general solution of streamfunction which satisfies the fourth-order partial differential equation in both the regions takes the form:

(2.35) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D6F9}_{i}^{(j)}=\mathop{\sum }_{n=1}^{\infty }[A_{n}^{(j)}r^{n+3}+B_{n}^{(j)}r^{n+1}]\text{Q}_{n}(\unicode[STIX]{x1D702})\\ \displaystyle \unicode[STIX]{x1D6F9}_{e}^{(j)}=U_{d}^{(j)}r^{2}\text{Q}_{1}(\unicode[STIX]{x1D702})+\mathop{\sum }_{n=1}^{\infty }[C_{n}^{(j)}r^{2-n}+D_{n}^{(j)}r^{-n}]\text{Q}_{n}(\unicode[STIX]{x1D702}).\end{array}\right\}\end{eqnarray}$$

Here $\text{Q}_{n}(\unicode[STIX]{x1D702})=\int _{-1}^{\unicode[STIX]{x1D702}}\text{P}_{n}(\unicode[STIX]{x1D702})\,\text{d}\unicode[STIX]{x1D702}$ denotes the $n$ th degree of Gegenbauer polynomial (Leal Reference Leal2007). Now the task remains to obtain the coefficients of the electric potential, streamfunction and surfactant concentration using an appropriate set of boundary conditions at each order of perturbation and invoking the proper orthogonality conditions of the Legendre and Gegenbauer polynomials.

In order to obtain the drop migration velocity, at different orders of perturbation, we follow the steps as followed by several authors previously (Xu & Homsy Reference Xu and Homsy2006; Mandal et al. Reference Mandal, Chakrabarti and Chakraborty2017b ). In the first step the constant coefficients appearing in the electric potential distribution ( $a_{n},b_{-n-1}$ ) (2.34), are obtained by solving the boundary conditions as mentioned in § 2.1.2. The solution is then used to determine the electric force on the drop given by $\boldsymbol{F}^{E}=2\unicode[STIX]{x03C0}\int _{\unicode[STIX]{x1D703}=0}^{\unicode[STIX]{x03C0}}(\unicode[STIX]{x1D749}_{e}^{E}\boldsymbol{\cdot }\boldsymbol{n})r_{S}^{2}\sin \unicode[STIX]{x1D703}\,\text{d}\unicode[STIX]{x1D703}$ . The next step is to determine the streamfunction coefficients ( $A_{n},B_{n},C_{n},D_{n}$ ) (2.35) to obtain the hydrodynamic force on the drop by using the expression $\boldsymbol{F}^{H}=2\unicode[STIX]{x03C0}\int _{\unicode[STIX]{x1D703}=0}^{\unicode[STIX]{x03C0}}(\unicode[STIX]{x1D749}_{e}^{H}\boldsymbol{\cdot }\boldsymbol{n})r_{S}^{2}\sin \unicode[STIX]{x1D703}\,\text{d}\unicode[STIX]{x1D703}$ . Consequently, the fact that the drop is force-free yields

(2.36) $$\begin{eqnarray}\boldsymbol{F}^{H}+\boldsymbol{F}^{E}+\boldsymbol{F}^{B}=\mathbf{0},\end{eqnarray}$$

allowing $U_{d}$ to be determined. Here $\boldsymbol{F}^{B}=((2\unicode[STIX]{x03C0}(2\unicode[STIX]{x03C0}+3\unicode[STIX]{x1D706}))/(1+\unicode[STIX]{x1D706}))\boldsymbol{e}_{z}$ is the buoyancy force acting on the drop and appears in the leading-order force balance only. Moreover, the hydrodynamic force on the drop can be conceptually split into two parts: namely the drag force experienced by the falling drop in a viscous fluid ( $\boldsymbol{F}_{drag}^{H}$ ) and the electrohydrodynamic force ( $\boldsymbol{F}_{EHD}^{H}$ ). The electrohydrodynamic force owes its existence to the imbalance of tangential stress at the interface and the consequent asymmetric flow. To bring out the alterations in the flow field and potential distribution, induced by non-uniform distribution of surfactant molecules on the drop surface, the surfactant transport equation has to be solved concurrently with the velocity and stress boundary conditions.

Before proceeding with the explicit expressions of the solution of various order of perturbation, we require the outward unit normal ( $\boldsymbol{n}$ ) and the curvature of the drop surface ( $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}$ ) at any point on the interface, which are calculated as follows (Ramachandran & Leal Reference Ramachandran and Leal2012):

(2.37) $$\begin{eqnarray}\displaystyle \boldsymbol{n}=\frac{\unicode[STIX]{x1D735}(r-r_{S})}{|\unicode[STIX]{x1D735}(r-r_{S})|} & = & \displaystyle \boldsymbol{e}_{r}-Ca\unicode[STIX]{x1D735}f^{(Ca)}-Ca\,Re_{E}\unicode[STIX]{x1D735}f^{(Ca\,Re_{E})}\nonumber\\ \displaystyle & & \displaystyle -\,Ca^{2}\left\{\unicode[STIX]{x1D735}f^{(Ca^{2})}+\frac{1}{2}(\unicode[STIX]{x1D735}f^{(Ca)}\boldsymbol{\cdot }\unicode[STIX]{x1D735}f^{(Ca)})\boldsymbol{e}_{r}\right\}-\cdots\end{eqnarray}$$

(2.38) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n} & = & \displaystyle 2-Ca(2f^{(Ca)}+\unicode[STIX]{x1D6FB}^{2}f^{(Ca)})-Ca\,Re_{E}(2f^{(Ca\,Re_{E})}+\unicode[STIX]{x1D6FB}^{2}f^{(Ca\,Re_{E})})\nonumber\\ \displaystyle & & \displaystyle -\,Ca^{2}\{-2f^{(Ca)}(f^{(Ca)}+\unicode[STIX]{x1D6FB}^{2}f^{(Ca)})+2f^{(Ca^{2})}+\unicode[STIX]{x1D6FB}^{2}f^{(Ca^{2})}\}-\cdots \,.\end{eqnarray}$$

2.3.1 Leading-order solution

The leading-order electric potential distribution does not contain any information about the sedimentation effects, charge convection or the surfactant contribution. It remains the same as the solution found in the pioneering work of Taylor (Reference Taylor1966):

(2.39) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D711}_{i}^{(0)}=-\frac{3r\cos \unicode[STIX]{x1D703}}{R+2}\\ \displaystyle \unicode[STIX]{x1D711}_{e}^{(0)}=-r\cos \unicode[STIX]{x1D703}+\frac{(R-1)\cos \unicode[STIX]{x1D703}}{(R+2)r^{2}}.\end{array}\right\}\end{eqnarray}$$

The corresponding charge distribution is given by

(2.40) $$\begin{eqnarray}q_{S}^{(0)}=\frac{3\cos \unicode[STIX]{x1D703}(R-S)}{R+2}.\end{eqnarray}$$

To obtain the pressure and velocity fields inside and outside the droplet we need the expressions for streamfunction in the corresponding case as given below:

(2.41) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D6F9}_{i}^{(0)}=\mathop{\sum }_{n=1}^{2}(A_{n}^{(0)}r^{n+3}+B_{n}^{(0)}r^{n+1})\text{Q}_{n}(\unicode[STIX]{x1D702})\\ \displaystyle \unicode[STIX]{x1D6F9}_{e}^{(0)}=r^{2}U_{d}^{(0)}\text{Q}_{1}(\unicode[STIX]{x1D702})+\mathop{\sum }_{n=1}^{2}(C_{n}^{(0)}r^{2-n}+D_{n}^{(0)}r^{-n})\text{Q}_{n}(\unicode[STIX]{x1D702}),\end{array}\right\}\end{eqnarray}$$

where the constants $A_{n}^{(0)},B_{n}^{(0)},C_{n}^{(0)},D_{n}^{(0)}$ can be found in the MATLAB file of the supplementary material, available at https://doi.org/10.1017/jfm.2018.415.

The corresponding local surfactant concentration distribution is obtained as:

(2.42) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}^{(Ca)}=\mathop{\sum }_{n=1}^{2}\unicode[STIX]{x1D6E4}_{n}^{(Ca)}\text{P}_{n}(\unicode[STIX]{x1D702}),\end{eqnarray}$$

where

(2.43) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D6E4}_{1}^{(Ca)}=\frac{k(2+3\unicode[STIX]{x1D706})(1-\unicode[STIX]{x1D6FD})}{2(1+\unicode[STIX]{x1D706})((3\unicode[STIX]{x1D706}-k+2)\unicode[STIX]{x1D6FD}-3\unicode[STIX]{x1D706}-2)}\\ \displaystyle \unicode[STIX]{x1D6E4}_{2}^{(Ca)}=\left(\frac{-3kM(R-S)(1-\unicode[STIX]{x1D6FD})}{\begin{array}{@{}r@{}}((-R^{2}-4R-4)k+5\unicode[STIX]{x1D706}R^{2}+5R^{2}+20\unicode[STIX]{x1D706}R+20R+20\unicode[STIX]{x1D706}+20)\unicode[STIX]{x1D6FD}\\ -\,5\unicode[STIX]{x1D706}R^{2}-5R^{2}-20\unicode[STIX]{x1D706}R-20R-20\unicode[STIX]{x1D706}-20\end{array}}\right)\!.\end{array}\right\}\end{eqnarray}$$

Upon using the force-free condition (2.36), the leading-order drop velocity is obtained as

(2.44) $$\begin{eqnarray}U_{d}^{(0)}=1-\underbrace{\frac{1}{3}\frac{k\unicode[STIX]{x1D6FD}}{(1+\unicode[STIX]{x1D706})((2+3\unicode[STIX]{x1D706})(1-\unicode[STIX]{x1D6FD})+k\unicode[STIX]{x1D6FD})}}_{\text{correction}\,\,\text{due}\,\,\text{to}\,\,\text{surfactant}}.\end{eqnarray}$$

Interestingly, although in the present problem both the settling and EHD effects are considered, the leading-order droplet velocity does not contain any electric effects and the leading-order correction to the Hadamard–Rybczynski velocity ( $\widetilde{U}_{HR}$ ) comes solely from surfactant effects. Such a condition is realized since in the leading order the electrohydrodynamic force ( $\boldsymbol{F}_{EHD}^{H}$ ) does not contribute to the hydrodynamic force and a balance is established only between the viscous drag ( $\boldsymbol{F}_{drag}^{H}$ ) and buoyancy ( $\boldsymbol{F}^{B}$ ). In addition, the leading-order drop velocity always gets reduced if the surfactant-induced interfacial tension variation is present, since the correction term turns out always to be positive $((2+3\unicode[STIX]{x1D706})(1-\unicode[STIX]{x1D6FD})+k\unicode[STIX]{x1D6FD})>0$ considering the practical range of the physical parameters $(k\sim O(1),0<\unicode[STIX]{x1D6FD}<1,\unicode[STIX]{x1D706}>0)$ .

Using the information about the surfactant distribution and drop velocity, the normal stress balance equation (2.25) is solved to obtain the harmonics of shape function. The shape function thus obtained takes the form $f^{(Ca)}=L_{2}^{(Ca)}\text{P}_{2}(\unicode[STIX]{x1D702})$ , where

(2.45) $$\begin{eqnarray}L_{2}^{(Ca)}=\frac{3}{4}\frac{M(c_{1}\unicode[STIX]{x1D6FD}k+c_{2}\unicode[STIX]{x1D6FD}+c_{3})}{(d_{1}\unicode[STIX]{x1D6FD}k+d_{2}\unicode[STIX]{x1D6FD}+d_{3})},\end{eqnarray}$$

where $c_{1}=(-R^{2}-2R+4S-1)$ , $c_{2}=(5\unicode[STIX]{x1D706}R^{2}+5R^{2}+9\unicode[STIX]{x1D706}R-19S\unicode[STIX]{x1D706}+6R-16S+5\unicode[STIX]{x1D706}+5)$ , $c_{3}=-c_{2}$ , $d_{1}=-R^{2}-4R-4$ , $d_{2}=5\unicode[STIX]{x1D706}R^{2}+5R^{2}+20\unicode[STIX]{x1D706}R+20R+20\unicode[STIX]{x1D706}+20$ and $d_{3}=-d_{2}$ .

2.3.2 $O(Re_{E})$ solution

Using the solution of leading order as obtained above and following the boundary conditions in § 2.2, the electrostatic potential is calculated as follows:

(2.46) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D711}_{i}^{(Re_{E})}=\mathop{\sum }_{n=1}^{3}r^{n}a_{n}^{(Re_{E})}\text{P}_{n}(\unicode[STIX]{x1D702})\\ \displaystyle \unicode[STIX]{x1D711}_{e}^{(Re_{E})}=\mathop{\sum }_{n=1}^{3}r^{-n-1}b_{-n-1}^{(Re_{E})}\text{P}_{n}(\unicode[STIX]{x1D702}),\end{array}\right\}\end{eqnarray}$$

where the constants $a_{n}^{(Re_{E})}$ , $b_{-n-1}^{(Re_{E})}$ are provided in the MATLAB file of the supplementary material. The corresponding surface charge density turns out to be

(2.47) $$\begin{eqnarray}q_{S}^{(Re_{E})}=\frac{108(1-\unicode[STIX]{x1D6FD})(R-S)}{5d_{4}}[c_{4}\cos ^{3}\unicode[STIX]{x1D703}+c_{5}\cos ^{2}\unicode[STIX]{x1D703}+c_{6}\cos \unicode[STIX]{x1D703}+c_{7}],\end{eqnarray}$$

where $c_{4},c_{5},c_{6},c_{7}$ and $d_{4}$ are functions of different dimensionless parameters. Again the complete expression of $q_{S}^{(Re_{E})}$ can be found in the supplementary MATLAB file.

The streamfunction and surfactant distribution are obtained as:

(2.48) $$\begin{eqnarray}\displaystyle & \left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D6F9}_{i}^{(Re_{E})}=\mathop{\sum }_{n=1}^{4}(A_{n}^{(Re_{E})}r^{n+3}+B_{n}^{(Re_{E})}r^{n+1})\text{Q}_{n}(\unicode[STIX]{x1D702})\\ \displaystyle \unicode[STIX]{x1D6F9}_{e}^{(Re_{E})}=r^{2}U_{d}^{(Re_{E})}\text{Q}_{1}(\unicode[STIX]{x1D702})+\mathop{\sum }_{n=1}^{4}(C_{n}^{(Re_{E})}r^{2-n}+D_{n}^{(Re_{E})}r^{-n})\text{Q}_{n}(\unicode[STIX]{x1D702})\end{array}\right\} & \displaystyle\end{eqnarray}$$

(2.49) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}^{(Ca\,Re_{E})}=\mathop{\sum }_{n=1}^{4}\unicode[STIX]{x1D6E4}_{n}^{(Ca\,Re_{E})}\text{P}_{n}(\unicode[STIX]{x1D702}) & \displaystyle\end{eqnarray}$$

while the drop settling velocity takes the form:

(2.50) $$\begin{eqnarray}U_{d}^{(Re_{E})}=-\frac{6}{5}\frac{M(3R+3-S)(R-S)}{(R+2)^{2}(2R+3)(1+\unicode[STIX]{x1D706})(2+3\unicode[STIX]{x1D706})}\times \underbrace{\left[\frac{(2+3\unicode[STIX]{x1D706})}{(2+3\unicode[STIX]{x1D706})+{\displaystyle \frac{k\unicode[STIX]{x1D6FD}}{1-\unicode[STIX]{x1D6FD}}}}\right]^{2}}_{k_{1}=\text{correction}\,\,\text{due}\,\,\text{to}\,\,\text{surfactant}}.\end{eqnarray}$$

The physical significance of zero correction of the $O(Re_{E})$ drop velocity as appeared in the above equations is discussed later in the results section. Next we calculate the shape function of this order to investigate the surfactant-induced modifications on the deformation behaviour caused by charge convection. Mathematically,

(2.51) $$\begin{eqnarray}f^{(Ca\,Re_{E})}=\mathop{\sum }_{n=2}^{4}L_{n}^{(Ca\,Re_{E})}\text{P}_{n}(\unicode[STIX]{x1D702}).\end{eqnarray}$$

The harmonics $L_{n}^{(Ca\,Re_{E})}$ , being too cumbersome to present, are provided in the supplementary MATLAB file.

2.3.3 $O(Ca)$ solution

Solving the boundary conditions as outlined in § 2.2, we obtain the deformation-induced modifications in the solution of electric problem as follows:

(2.52) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D711}_{i}^{(Ca)}=\mathop{\sum }_{n=1}^{3}r^{n}a_{n}^{(Ca)}\text{P}_{n}(\unicode[STIX]{x1D702}),\\ \displaystyle \unicode[STIX]{x1D711}_{e}^{(Ca)}=\mathop{\sum }_{n=1}^{3}r^{-n-1}b_{-n-1}^{(Ca)}\text{P}_{n}(\unicode[STIX]{x1D702}),\end{array}\right\}\end{eqnarray}$$

where the non-zero harmonics are as follows:

(2.53a-c ) $$\begin{eqnarray}a_{1}^{(Ca)}=-{\displaystyle \frac{18(R-1)}{5(R+2)^{2}}}L_{2}^{(Ca)},\quad b_{-2}^{(Ca)}={\displaystyle \frac{6}{5}}{\displaystyle \frac{(R-1)^{2}}{(R+2)^{2}}}L_{2}^{(Ca)}\quad \text{and}\quad b_{-4}^{(Ca)}={\displaystyle \frac{9}{5}}{\displaystyle \frac{(R-1)}{R+2}}L_{2}^{(Ca)}.\end{eqnarray}$$

Consequently the surface charge density is obtained as

(2.54) $$\begin{eqnarray}q_{S}^{(Ca)}=\frac{6}{5}\frac{(R-1)(15(\cos \unicode[STIX]{x1D703})^{2}R+30(\cos \unicode[STIX]{x1D703})^{2}-7R-3S-20)\cos \unicode[STIX]{x1D703}}{(R+2)^{2}}L_{2}^{(Ca)}.\end{eqnarray}$$

Similarly the streamfunction and surfactant distribution turn out to be:

(2.55) $$\begin{eqnarray}\displaystyle & \displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D6F9}_{i}^{(Ca)}=\mathop{\sum }_{n=1}^{4}(A_{n}^{(Ca)}r^{n+3}+B_{n}^{(Ca)}r^{n+1})\text{Q}_{n}(\unicode[STIX]{x1D702})\\ \displaystyle \unicode[STIX]{x1D6F9}_{e}^{(Ca)}=r^{2}U_{d}^{(Ca)}\text{Q}_{1}(\unicode[STIX]{x1D702})+\mathop{\sum }_{n=1}^{4}(C_{n}^{(Ca)}r^{2-n}+D_{n}^{(Ca)}r^{-n})\text{Q}_{n}(\unicode[STIX]{x1D702})\end{array}\right\} & \displaystyle\end{eqnarray}$$

(2.56) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}^{(Ca^{2})}=\mathop{\sum }_{n=1}^{4}\unicode[STIX]{x1D6E4}_{n}^{(Ca^{2})}\text{P}_{n}(\unicode[STIX]{x1D702}) & \displaystyle\end{eqnarray}$$

while the drop settling velocity takes the form:

(2.57) $$\begin{eqnarray}U_{d}^{(Ca)}={\textstyle \frac{1}{5}}U_{d}^{(0)}(L_{2}^{(Ca)}c_{6}+Mc_{7}),\end{eqnarray}$$

where $c_{6}=c_{6}(R,\unicode[STIX]{x1D6FD},k,\unicode[STIX]{x1D706})$ and $c_{7}=c_{7}(R,\unicode[STIX]{x1D6FD},k,\unicode[STIX]{x1D706},S)$ are complex functions of physical properties and are given in the supplementary MATLAB file.

We now determine the shape function of this order to investigate the surfactant-induced change in the higher-order deformation behaviour. Mathematically,

(2.58) $$\begin{eqnarray}f^{(Ca^{2})}=L_{0}^{(Ca^{2})}+\mathop{\sum }_{n=2}^{4}L_{n}^{(Ca^{2})}\text{P}_{n}(\unicode[STIX]{x1D702}).\end{eqnarray}$$

Again the harmonics $L_{n}^{(Ca^{2})}$ are provided in the supplementary MATLAB file. Apart from these spherical harmonics a constant term ( $L_{0}^{(Ca^{2})}$ ) has to be included in the higher-order deformation shape function to ensure the conserved volume of fluid. After applying the volume conservation constraint (Mandal et al. Reference Mandal, Bandopadhyay and Chakraborty2016b ), we obtain

(2.59) $$\begin{eqnarray}L_{0}^{(Ca^{2})}=-{\textstyle \frac{1}{5}}(L_{2}^{(Ca)})^{2}.\end{eqnarray}$$

The surfactant concentration can be expressed as:

(2.60) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}^{(Ca^{2})}=\unicode[STIX]{x1D6E4}_{0}^{(Ca^{2})}+\mathop{\sum }_{n=1}^{4}\unicode[STIX]{x1D6E4}_{n}^{(Ca^{2})}\text{P}_{n}(\unicode[STIX]{x1D702}).\end{eqnarray}$$

The term $\unicode[STIX]{x1D6E4}_{0}^{(Ca^{2})}$ can be obtained from the mass constraint equation (2.16). However, with the motive of representing the surfactant distribution on a spherical surface, we employ a deformed-to-spherical surface transformation for $\unicode[STIX]{x1D6E4}$ through a projection operator as previously used by Vlahovska, Loewenberg & Blawzdziewicz (Reference Vlahovska, Loewenberg and Blawzdziewicz2005), Das et al. (Reference Das, Mandal and Chakraborty2017a ):

(2.61) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{S}=\unicode[STIX]{x1D6E4}\frac{r_{S}^{2}}{\boldsymbol{e}_{r}\boldsymbol{\cdot }\boldsymbol{n}},\end{eqnarray}$$

where $\unicode[STIX]{x1D6E4}_{S}$ is the projected concentration on the spherical surface. This gives

(2.62) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{0}^{(Ca^{2})}=-{\textstyle \frac{2}{5}}(L_{2}^{(Ca)})^{2}-{\textstyle \frac{2}{5}}L_{2}^{(Ca)}\unicode[STIX]{x1D6E4}_{2}^{(Ca)}.\end{eqnarray}$$

To check the validity of the present analytical calculations, we compared the above solutions with previous theoretical and experimental works for different limiting conditions (please refer to appendix A).

3.1 Effects of surfactant on a spherical drop

We first focus our attention towards the condition where the drop deformation is insignificant and the combination of settling and EHD is affected by charge convection alone.

Figure 2.

Settling velocity $U_{d}$ versus $\unicode[STIX]{x1D6FD}$ for different electric Reynolds number ( $Re_{E}$ ). (a) System-A. (b) System-B. Other parameters are $k=1$ and $M=5$ .

In figure 2 the variation of drop settling velocity $U_{d}=U_{d}^{(0)}+Re_{E}U_{d}^{(Re_{E})}$ with the elasticity parameter $\unicode[STIX]{x1D6FD}$ has been shown for different values of $Re_{E}$ . It is observed that $U_{d}$ decreases continuously with increasing $\unicode[STIX]{x1D6FD}$ for both system-A and system-B, while the effect of $Re_{E}$ is just the opposite for the two cases. It is also interesting to find that charge-convection-modulated velocity deviation becomes extremely small at higher values of  $\unicode[STIX]{x1D6FD}$ . The corresponding behaviour can be understood by noting the expression of $U_{d}^{(Re_{E})}$ in (2.50). The discriminating factor ${\mathcal{F}}=(3R+3-S)(R-S)$ , which determines the sign of change in drop settling velocity modification by charge convection ( ${\mathcal{F}}<0$ for system-A and ${\mathcal{F}}>0$ for system-B), remains unchanged even if the surfactant effects are present. However, the presence of surfactants can alter the amount of correction ( $k_{1}$ ) to $U_{d}^{(Re_{E})}$ , determined by a coupling of viscosity ratio $\unicode[STIX]{x1D706}$ with surfactant parameters $\unicode[STIX]{x1D6FD}$ and  $k$ . For a mathematical justification of the above behaviour, we revisit the correction factor $k_{1}$ appearing in (2.50). Keeping in view the practical range of various parameters, it is found that the $k_{1}\leqslant 1$ condition is satisfied always (please refer to appendix B for a detail proof of the statement) and hence the magnitude of $U_{d}^{(Re_{E})}$ decreases due to surfactants. It is identified from the present calculation that the electric force $\boldsymbol{F}^{E(Re_{E})}$ turns out to be zero and thus the hydrodynamic force $\boldsymbol{F}^{H(Re_{E})}$ is the sole force responsible for the velocity correction due to charge convection.

Figure 3.

Streamline pattern for (a) clean drop (system-A) and (b) for surfactant-coated case of the same drop where $k=1$ and $\unicode[STIX]{x1D6FD}=0.8$ . $Re_{E}=0.2$ , $M=5$ are chosen for both the cases. Flow directions are shown for some representative streamlines with respect to the drop reference frame. (c) Surface tension distribution ( $\unicode[STIX]{x1D6FE}$ ) for different values of  $\unicode[STIX]{x1D6FD}$ . (d) Interfacial velocity distribution for different values of $\unicode[STIX]{x1D6FD}$ , $k$ and $Re_{E}$ . (e) Tangential electric stress jump for different values of  $\unicode[STIX]{x1D6FD}$ . In the inset we have shown the direction of traction at the interface for a representative case of leading order.

To obtain a comprehensive physical picture, we plot the streamline pattern, surface velocity and tangential component of electric stress jump in figure 3. From figure 3(a) it can be observed that an asymmetric flow pattern about the equator builds up due to redistribution of charges by the charge convection phenomenon. As a consequence circulation cells, with a flow direction from the poles to the stagnation points, set in. This being the picture for a surfactant-free drop, the presence of surface-active agents on the interface introduces Marangoni stress in competition to the electric stress. In figure 3(b) the change in flow pattern due to a surfactant is highlighted. To identify the behaviour of the Marangoni stress and determine its traction direction we plot the surface tension and surface concentration distribution for different values of $\unicode[STIX]{x1D6FD}$ in figure 3(c) (for a clear picture of the traction vector direction, an example has been provided in the inset of figure 3 e). It is found from figure 3(c) that an increase in $\unicode[STIX]{x1D6FD}$ causes a significant variation of surface tension from its equilibrium value and the driving force for the Marangoni flow, i.e. the surface tension gradient $(|\unicode[STIX]{x1D6FE}_{max}-\unicode[STIX]{x1D6FE}_{min}|)$ increases. From the inset of the same figure we find that the surface concentration tends to reach an equilibrium concentration as $\unicode[STIX]{x1D6FD}$ increases. This apparently contradictory behaviour of surface concentration with respect to surface tension can be explained with due consideration of the underlying physical mechanisms. With an increase in $\unicode[STIX]{x1D6FD}$ the surface tension gradients become strong enough to restrict convective transport of surfactant molecules towards the stagnation points and the gradient in surface concentration $|\unicode[STIX]{x1D6E4}_{max}-\unicode[STIX]{x1D6E4}_{min}|$ gets reduced. Similar qualitative observations were also made in the case of a viscous drop in linear shear by Li & Pozrikidis (Reference Li and Pozrikidis1997). Now according to the traction directions determined by the surface tension gradient, Marangoni stress tries to trigger a fluid flow towards polar region, thus opposing the electrohydrodynamically induced flow. The change in flow pattern in figure 3(b) can be justified by the shift in stagnation points which can be visualized from the surface velocity plots in figure 3(d). As $\unicode[STIX]{x1D6FD}$ increases, the stagnation points shift more towards the front pole ( $\unicode[STIX]{x1D703}=0$ ). A similar trend in the shift of sign reversal point of the tangential electric stress jump $T_{t}^{E}$ is also found in figure 3(e). To substantiate such observation we provide maps for the position of the stagnation points $\boldsymbol{u}_{S}=0$ and the stress jump $T_{t}^{E}=0$ as a function of a systematic variation of $\unicode[STIX]{x1D6FD}$ and $Re_{E}$ later in figures 14(a) and 14(b), respectively. A displacement in the stagnation point indicates a change in the force balance at the interface. Figures 14(a) and 14(b) show that the displacement of stagnation points does not exactly follow the location of the sign reversal points of only $\unicode[STIX]{x27E6}T_{t}^{E}\unicode[STIX]{x27E7}$ , although the trend is similar. Such locations of stagnation points are determined by a balance of hydrodynamic, electric and Marangoni stresses. As a consequence the structure of the vortices near the front pole is altered. In order to obtain a deeper insight, we plot the maximum velocity magnitude as well as the maximum vorticity magnitude within both kinds of roll in figures 4(a) and 4(b), thus obtaining a quantification of strength of the corresponding vortices (here vorticity is defined as $\unicode[STIX]{x1D74E}=\unicode[STIX]{x1D735}\times \boldsymbol{u}$ ). It is evident from these figures that both $|\unicode[STIX]{x1D74E}_{max}|$ and $|\boldsymbol{u}_{max}|$ are reduced for both kinds of circulation cell with an increase in surfactant parameter  $\unicode[STIX]{x1D6FD}$ . This creates alterations in the resulting hydrodynamic drag force experienced by the drop.

Figure 4.

(a) Variations in maximum magnitudes of velocity ( $|\boldsymbol{u}_{max}|$ , dashed line) and vorticity ( $|\unicode[STIX]{x1D74E}_{max}|$ , solid line) within the primary rolls near the front pole for system-A. (b) Corresponding variations for the smaller rolls near the rear pole. Here parameter values $k=1$ and $Re_{E}=0.2$ are used. (c) Vorticity iso-contours for a surfactant-free drop with system-A properties.

Interestingly, the nature of variation of surfactant concentration (as shown in the inset of figure 3 c) shows a resemblance to the temporal evolution of a thin film when surfactant deposition at the surface takes place, as reported by Jensen & Grotberg (Reference Jensen and Grotberg1992, Reference Jensen and Grotberg1993). Specifically, the steady state film thickness for an axisymmetric spherical drop (Jensen & Grotberg Reference Jensen and Grotberg1992), when varied with respect to a stretched spatio-temporal variable $\unicode[STIX]{x1D709}=x/t^{1/4}$ , follows a similar trend to that of the present inset figure. The physical situation analysed by those authors, however, is quite different from the present analysis, especially in the application of the lubrication theory. With the help of a similarity transformation they derived a coupled set of differential equations for the film-thickness evolution and surfactant distribution. Similar to the present work, they have also chosen a linear constitutive relation between the surface tension and the surfactant concentration, assuming dilute concentrations everywhere. A significant difference exists in terms of the gravity effect. They assumed a negligible Bond number $Bo=\unicode[STIX]{x1D70C}g{H^{\ast }}^{2}/S^{\ast }$ , where $\unicode[STIX]{x1D70C}$ , $H^{\ast }$ and $S^{\ast }$ are the reference scales of fluid density, height of the film and surface tension, respectively. In contrast, the present work takes gravity into account by considering the Hadamard–Rybczynski velocity as the reference velocity (please refer to § 2.1.1).

In some of the representative cases discussed, modulations created by surfactant parameter variations are not visually distinguishable. However, a different combination of parameters (e.g.  $R=0.01$ , $S=1.7$ , $\unicode[STIX]{x1D706}=0.03$ and $M=2.7$ ) following Xu & Homsy (Reference Xu and Homsy2006) (please refer to appendix D) reveals the variations in a more clear and distinguishable manner, as shown in figure 15. Since in that case, the discriminating factor ${\mathcal{F}}$ remains positive, the resultant effect of $\unicode[STIX]{x1D6FD}$ on the charge-convection-induced velocity increase gives the same trend as figure 2(a) (not shown here for brevity). It is interesting to observe that such a parameter choice gives rise to additional two pairs of secondary circulation cells near the poles in the absence of surfactant molecules (please refer to figure 15 a). Now an increase in the elasticity parameter $\unicode[STIX]{x1D6FD}$ makes the secondary rolls become diminishingly small (shown in figure 15 b). The corresponding prominent modulations in $T_{t}^{E}$ are also depicted in figure 15(c). A phase diagram depicting the parametric space where these vortices form or disappear is presented in figure 5. It provides a profound understanding of the combined influence of charge convection and the Marangoni effect on the formation of additional vortices. The surfactant-induced modifications in the internal circulation rolls, as discussed above, can severely alter the mixing characteristics inside the drop.

Figure 5.

Phase diagram showing two distinct regions on the $Re_{E}{-}\unicode[STIX]{x1D6FD}$ plane depending on the formation of secondary rolls near the poles. The parameters are chosen according to Xu & Homsy (Reference Xu and Homsy2006).

Figure 6.

Surface charge distribution for leading order as well as including the charge convection effect ( $q_{S}=q_{S}^{(0)}+Re_{E}q_{S}^{(Re_{E})}$ ) for various values of  $\unicode[STIX]{x1D6FD}$ . Here $k=1$ , $M=5$ and the other parameters are as for system-A.

The intricate behaviour of the flow not only affects the drop velocity but also causes a significant change in charge distribution, as described in figure 6. For a surfactant-free drop the primary rolls near the front pole possess more strength of flow as compared to the smaller rolls adjacent to the rear pole, as can be observed from the vorticity distribution plot in figure 4(c). Thus the charges are more dominantly swept away from the front pole towards the stagnation point as compared to what is achieved by the smaller rolls adjacent to the rear pole. Such behaviour qualitatively agrees with the work of Mandal et al. (Reference Mandal, Bandopadhyay and Chakraborty2016b ) for a surfactant-free, neutrally buoyant drop motion driven by dielectrophoresis. Again referring to figures 4(a) and 4(b) it is found that presence of surfactants weakens the strength of both primary rolls near the front pole and the smaller roles near the rear pole. In addition to this, for a similar increase in $\unicode[STIX]{x1D6FD}$ from 0 to 0.8, the decrease in $|\unicode[STIX]{x1D74E}_{max}|$ or $|\boldsymbol{u}_{max}|$ is more pronounced for the case of primary rolls as compared to the other kind. Thus the hindering of charge convection towards stagnation points by the Marangoni effect is more effective near the front poles as compared to the rear poles.

Figure 7.

(a) Variation of drop settling velocity $U_{d}$ with elasticity parameter $\unicode[STIX]{x1D6FD}$ for various values of property ratio  $k$ . (b,c) Streamline patterns for $\unicode[STIX]{x1D6FD}=0.8$ and for $k=0.1$ and $k=1$ , respectively. $Re_{E}=0.2$ and $M=5$ are chosen for all the three figures. Other parameters are as for system-A. Flow directions are shown for some representative streamlines with respect to the drop reference frame.

Figure 7(a) depicts the effect of the property ratio $k$ on the drop settling velocity. In the extreme limit of $k=0$ the drop is uniformly coated with surfactant. Consequently the correction term in leading-order drop velocity becomes zero while the factor $k_{1}$ in $O(Re_{E})$ drop velocity becomes 1. Thus the drop velocity becomes the same as that of a surfactant-free drop given by

(3.1) $$\begin{eqnarray}\lim _{k\rightarrow 0}U_{d}=1-\frac{6}{5}\frac{M(3R+3-S)(R-S)}{(R+2)^{2}(2R+3)(1+\unicode[STIX]{x1D706})(2+3\unicode[STIX]{x1D706})}.\end{eqnarray}$$

As seen from the figure, with an increase in $k$ the drop velocity decreases as compared to the surfactant-free drop. Here surface advection of surfactant molecules is relatively weak and hence redistribution of charges for non-zero $Re_{E}$ can significantly alter the drop velocity. It can be inferred from the correction term $k_{1}$ in (2.50) that an increase in $k$ will always reduce the magnitude of the charge-convection-induced correction to the drop velocity. The functional nature of the correction factor $k_{1}$ is shown in the inset of figure 7(a). The primary effect of increasing $k$ is to increase the surface Péclet number, and thus it should have increased the charge convection effect due to enhanced EHD-induced convection of fluid. But the secondary competing effect of the increased interfacial tension gradient due to increased  $k$ dominates in this case, and the opposing Marangoni flow intensifies. Such behaviour is similar to that caused by the elasticity parameter  $\unicode[STIX]{x1D6FD}$ . It can be verified from the streamline structures presented in figures 7(b) and 7(c). Similar to the effect of increasing  $\unicode[STIX]{x1D6FD}$ , the size of the secondary vortices increases with increasing  $k$ . Another important observation from figure 7(a) is that, as $\unicode[STIX]{x1D6FD}$ increases, the retardation effect initiated by increasing  $k$ gets intensified. This can be attributed to a combined additive influence of $k$ and $\unicode[STIX]{x1D6FD}$ on the enhancement of Marangoni flow opposite to the EHD flow.

Previously it has been found that even in the absence of charge convection, the viscosity ratio can remarkably alter the surfactant effect on the drop velocity (please refer to figure 2 and the corresponding discussion in Mandal et al. (Reference Mandal, Bandopadhyay and Chakraborty2016a ) or § 4 in Teigen & Munkejord (Reference Teigen and Munkejord2010)). Following a similar trend, the leading-order corrections to drop velocity brought by surfactant become maximum when the drop viscosity is negligible with respect to the outer medium ( $\unicode[STIX]{x1D707}_{i}\ll \unicode[STIX]{x1D707}_{e}$ ). Such a condition closely reduces to the case of a bubble in an otherwise viscous medium. The limit is obtained as a correction to the Hadamard–Rybczynski velocity as:

(3.2) $$\begin{eqnarray}\lim _{\unicode[STIX]{x1D706}\rightarrow 0}U_{d}^{(0)}=1-\underbrace{\frac{1}{3}\frac{k\unicode[STIX]{x1D6FD}}{(2(1-\unicode[STIX]{x1D6FD})+k\unicode[STIX]{x1D6FD})}}_{\text{correction}\,\,\text{due}\,\,\text{to}\,\,\text{surfactant}}.\end{eqnarray}$$

In a similar manner the $O(Re_{E})$ drop velocity ( $U_{d}^{(Re_{E})}$ ) takes a maximum value under such a condition given by:

(3.3) $$\begin{eqnarray}\lim _{\unicode[STIX]{x1D706}\rightarrow 0}U_{d}^{(Re_{E})}=-\frac{3}{5}\frac{(R-S)(3R-S+3)M}{(2R+3)(R+2)^{2}}\times \underbrace{\frac{4(1-\unicode[STIX]{x1D6FD})^{2}}{(2(1-\unicode[STIX]{x1D6FD})+k\unicode[STIX]{x1D6FD})^{2}}}_{k_{1}=\text{correction}\,\,\text{due}\,\,\text{to}\,\,\text{surfactant}}.\end{eqnarray}$$

It is to be noted here that the correction term due to surfactant $k_{1}$ also becomes maximum in that limit. On the other hand when the drop becomes highly viscous ( $\unicode[STIX]{x1D707}_{i}\gg \unicode[STIX]{x1D707}_{e}$ ) the surface velocity vanishes, resembling the case of a solid sphere. In this case the internal circulations of the drop are already retarded so much due to viscous effects that the additional Marangoni effect has hardly any additional retardation effect to bring in. Thus the leading-order surfactant correction becomes zero while the $O(Re_{E})$ drop velocity vanishes altogether.

The Mason number ( $M$ ), which depicts the relative importance of electric stress to that of hydrodynamic stress, enhances the effect of charge convection on the drop velocity and the corresponding alterations by surfactant. This can be understood from the $O(Re_{E})$ velocity perturbation in (2.50).

3.2 Effects of surfactant on a deformable drop

In this section we investigate how the deformability of the droplet, affected primarily by surfactant parameters $\unicode[STIX]{x1D6FD}$ and  $k$ , can influence the drop settling velocity and the resulting drop shape. Here we consider only the $O(Ca)$ correction to distinguish it from the $O(Re_{E})$ modifications as discussed in the preceding section. During drop deformation the transport of surfactants on the interface, driven by the internal flow field and Marangoni-induced surface stresses, leads to a phenomenon called tip stretching. This refers to the mechanism in which accumulation of surfactant molecules, favoured by the flow direction, causes the local surface tension to reduce, thus making such localized areas on the drop surface more vulnerable to deformation. Accommodation of such effects in the present mathematical model comes through the second term on the right-hand side of the normal stress balance equation (2.14). The intricate interplay among these competing physical mechanisms dictates the resulting modifications introduced by the presence of surfactants.

Figure 8.

Variation of velocity $U_{d}$ with elasticity parameter ( $\unicode[STIX]{x1D6FD}$ ) for different values of the capillary number ( $Ca$ ) with $k=1$ and $M=5$ . Other parameters are chosen as for system-A.

Figure 8 describes the effect of the elasticity parameter ( $\unicode[STIX]{x1D6FD}$ ) on drop settling velocity with increasing order of deformability. For a specific value of capillary number ( $Ca$ ), an increase in $\unicode[STIX]{x1D6FD}$ causes the drop to deform more to an oblate spheroid shape. This represents higher and higher flow obstruction with elevated hydrodynamic drag and a corresponding decrease in drop velocity results.

Figure 9.

(a) Leading-order deformation parameter ( ${\mathcal{D}}$ ) versus capillary number ( $Ca$ ) for different values of elasticity parameter  $\unicode[STIX]{x1D6FD}$ . (b) Distribution of surface tension for different values of elasticity parameter  $\unicode[STIX]{x1D6FD}$ . In the inset we show the corresponding concentration variation. Parameters are $k=1$ , $M=5$ and those pertaining to system-A.

In figure 9(a) the variation of leading-order deformation parameter ( ${\mathcal{D}}$ ) with capillary number ( $Ca$ ) has been shown for different  $\unicode[STIX]{x1D6FD}$ . Accumulation of surfactant molecules from the poles towards the stagnation points becomes more dominant in this case as compared the opposing effect of Marangoni flow. As a consequence the surface tension is lowest adjacent to the stagnation points (denoted by marker ‘ $\times$ ’ in figure 9 b). In order to balance the local normal stresses generated due to electrohydrodynamics, the drop surface becomes more prone to deformation in these regions. This is reflected in figure 9(a). It is also observed from the same figure that the surfactant effect on deformation gets enhanced as the capillary number ( $Ca$ ) increases. This is due to the fact that an increase in $Ca$ indicates a corresponding increase in surface Péclet number ( $Pe=k\,Ca$ ) which pushed more surfactant molecules to these zones, causing the interfacial tension to reduce further. Now for system-B the drop takes a prolate spheroid shape. Since in this case the surfactant molecules are swept towards polar regions by the dominant action of electrohydrodynamic stress, the prolate behaviour is intensified. Thus the drag experienced by the drop is further reduced and finally it settles faster than the surfactant-free case (shown in figure 16).

Figure 10.

Asymmetric deformation versus $Re_{E}$ for (a) various $\unicode[STIX]{x1D6FD}$ and (b) various  $k$ .

An important aspect of the drop deformation characteristics is that the drop shape becomes asymmetric about the equator. Such an asymmetry is typical for a drop settling with electrohydrodynamic modulation even if the surfactant effects are not present (Spertell & Saville Reference Spertell, Saville and Plesset1976; Xu & Homsy Reference Xu and Homsy2006). Such behaviour originates from the antisymmetry of the $L_{3}^{(Ca\,Re_{E})}\text{P}_{3}(\cos \unicode[STIX]{x1D703})$ term present in the expression of $f^{(Ca\,Re_{E})}$ as (2.51). To capture such asymmetry we define a new deformation parameter as (Xu & Homsy Reference Xu and Homsy2006)

(3.4) $$\begin{eqnarray}{\mathcal{D}}_{A}={\displaystyle \frac{r_{S}|_{\unicode[STIX]{x1D703}=0}-r_{S}|_{\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}}}{r_{S}|_{\unicode[STIX]{x1D703}=0}+r_{S}|_{\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}}}}.\end{eqnarray}$$

In figure 10 we show the effect of surfactant on the asymmetric deformation about the equator triggered by charge convection for a deformable drop. The present study reveals that the asymmetric behaviour of shape deformation gets suppressed for increasing values of both the elasticity parameter $\unicode[STIX]{x1D6FD}$ (figure 10 a) and the property ratio $k$ (figure 10 b).

Based on the electrical conductivity ratio ( $R$ ) and permittivity ratio ( $S$ ) some interesting limiting conditions can be deduced in this case. When the $R\rightarrow \infty$ limit is reached the drop behaves as a perfectly conducting one. Such a condition closely resembles various experimental situations as in Mhatre & Thaokar (Reference Mhatre and Thaokar2013) and Ervik et al. (Reference Ervik, Penne, Hellesø, Munkejord and Müller2018), where aqueous drops are considered in an oil medium. In such conditions the electric field acts perpendicularly to the drop surface and the tangential electric field components becomes identically zero, having no effect on $O(Re_{E})$ velocity perturbation. Hence such a limit can alter only the $O(Ca)$ correction to drop velocity, which is obtained as

(3.5) $$\begin{eqnarray}\displaystyle \lim _{R\rightarrow \infty }U_{d}^{(Ca)} & = & \displaystyle \frac{(9\unicode[STIX]{x1D706}^{2}-3\unicode[STIX]{x1D706}+24)M(2+3\unicode[STIX]{x1D706})}{20(-3\unicode[STIX]{x1D706}-2)^{2}(1+\unicode[STIX]{x1D706})}\nonumber\\ \displaystyle & & \displaystyle -\,\underbrace{\frac{((108\unicode[STIX]{x1D706}^{2}-15\unicode[STIX]{x1D706}k-36\unicode[STIX]{x1D706}+20k-72)\unicode[STIX]{x1D6FD}-108\unicode[STIX]{x1D706}^{2}+36\unicode[STIX]{x1D706}+72)\unicode[STIX]{x1D6FD}kM}{20(2+3\unicode[STIX]{x1D706})((3\unicode[STIX]{x1D706}-k+2)\unicode[STIX]{x1D6FD}-3\unicode[STIX]{x1D706}-2)^{2}(1+\unicode[STIX]{x1D706})}}_{\text{correction}\,\,\text{due}\,\,\text{to}\,\,\text{surfactant}}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

3.3 Perfectly dielectric media

Another important case of ( $R,S$ ) combination often employed in EHD studies is when a perfectly dielectric drop is settling in another perfectly dielectric medium. This limit can be mathematically deduced by taking the $R\rightarrow S$ limit. Similar to the perfectly conducting drop case, here also the $U_{d}^{(Re_{E})}$ vanishes due to zero surface charge in the leading order as well as in $O(Re_{E})$ . However, the scenario gets changed when the drop deforms and the expression for the $O(Ca)$ drop velocity for such situations can be obtained as:

(3.6) $$\begin{eqnarray}\displaystyle \lim _{R\rightarrow S}U_{d}^{(Ca)} & = & \displaystyle \frac{3M(3\unicode[STIX]{x1D706}^{2}-\unicode[STIX]{x1D706}+8)(S-1)^{2}}{(20+20\unicode[STIX]{x1D706})(2+3\unicode[STIX]{x1D706})(2+S)^{2}}\nonumber\\ \displaystyle & & \displaystyle -\,\underbrace{\frac{(S-1)^{2}(108\unicode[STIX]{x1D706}^{2}\unicode[STIX]{x1D6FD}-15k\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FD}-36\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FD}+20k\unicode[STIX]{x1D6FD}-108\unicode[STIX]{x1D706}^{2}-72\unicode[STIX]{x1D6FD}+36\unicode[STIX]{x1D706}+72)\unicode[STIX]{x1D6FD}kM}{20(2+3\unicode[STIX]{x1D706})(2+S)^{2}(3\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FD}-k\unicode[STIX]{x1D6FD}+2\unicode[STIX]{x1D6FD}-3\unicode[STIX]{x1D706}-2)^{2}(1+\unicode[STIX]{x1D706})}}_{\text{correction}\,\,\text{due}\,\,\text{to}\,\,\text{surfactant}}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

In figure 11 the effect of variation of the permittivity ratio on the drop velocity has been highlighted for various $\unicode[STIX]{x1D6FD}$ and  $k$ . It shows that the retardation effect of surfactant on the $O(Ca)$ correction to drop velocity increases if either the parameter $S$ is increased or decreased from the point $S=1$ . The vanishing value of this perturbation velocity at $S=1$ , which is evident from (3.6), is also portrayed in both the figures 11(a) and 11(b). Since in this specific case, electrical properties of both the fluids become the same ( $R=S=1$ ), the electric Maxwell stress jump at the interface vanishes. As a consequence charge accumulation on the surface does not take place, resulting in zero shape deformation. Thus the settling velocity reduces to the classical Hadamard–Rybczynski velocity only with a corresponding correction due to surfactant ( $U_{d}^{(0)}$ ).

Figure 11.

$O(Ca)$ velocity correction versus permittivity ratio for (a) various $\unicode[STIX]{x1D6FD}$ with $k=1$ and (b) various $k$ with $\unicode[STIX]{x1D6FD}=0.5$ . The other parameters are $M=5$ , $Ca=0.2$ and $\unicode[STIX]{x1D706}=0.4359$ .