4.1. Differential equations

The pertinent variables in the suspending liquid are the ionic concentration $c^{\pm }$ , the electric potential ${\it\varphi}$ and the velocity field $\boldsymbol{u}$ . These are governed by the differential equations of the standard electrokinetic model.

1. (a) Nernst–Planck ion conservation,

   (4.2) $$\begin{eqnarray}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{j}^{\,\pm }+{\it\alpha}^{\pm }\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}c^{\pm }=0,\end{eqnarray}$$
   wherein the molecular fluxes of cations and anions, respectively normalized by $D_{\ast }^{\pm }\mathscr{C}_{\ast }/a_{\ast }$ , are due to diffusion and electro-migration,
   (4.3) $$\begin{eqnarray}\boldsymbol{j}^{\,\pm }=-\boldsymbol{{\rm\nabla}}c^{\pm }\mp c^{\pm }\boldsymbol{{\rm\nabla}}{\it\varphi}.\end{eqnarray}$$
   The ion drag coefficients
   (4.4) $$\begin{eqnarray}{\it\alpha}^{\pm }=\frac{{\it\epsilon}_{\ast }{\it\varphi}_{\ast }^{2}}{{\it\mu}_{\ast }D_{\ast }^{\pm }}\end{eqnarray}$$
   appearing in (4.2) are independent of both the drop size $a_{\ast }$ and reference concentration $\mathscr{C}_{\ast }$ . Moreover, because of the Stokes–Einstein relations, they are actually independent of the liquid viscosity ${\it\mu}^{\ast }$ ; rather, they are simply proportional to the size of the ions. For typical ions in aqueous solutions, ${\it\alpha}^{\pm }\lesssim 0.5$ (Saville Reference Saville1977).

2. (b) Poisson’s equation,

   (4.5) $$\begin{eqnarray}-2{\it\delta}^{2}{\rm\nabla}^{2}{\it\varphi}=c^{+}-c^{-},\end{eqnarray}$$
   wherein
   (4.6) $$\begin{eqnarray}{\it\delta}=\frac{1}{{\it\kappa}_{\ast }a_{\ast }}\end{eqnarray}$$
   is the dimensionless Debye thickness, in which the Debye width $1/{\it\kappa}_{\ast }$ is defined by
   (4.7) $$\begin{eqnarray}{\it\kappa}_{\ast }^{2}=\frac{2ze_{\ast }\mathscr{C}_{\ast }}{{\it\epsilon}_{\ast }{\it\varphi}_{\ast }}.\end{eqnarray}$$

3. (c) Stokes equations, subject to Coulomb body forces,

   (4.8a,b ) $$\begin{eqnarray}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}=0,\quad \boldsymbol{{\rm\nabla}}p={\rm\nabla}^{2}\boldsymbol{u}+{\rm\nabla}^{2}{\it\varphi}\boldsymbol{{\rm\nabla}}{\it\varphi},\end{eqnarray}$$
   where $p$ is the pressure field.

Within the drop we denote the ionic concentrations by $\bar{c}^{\pm }$ , the electric potential by $\bar{{\it\varphi}}$ , and the fluid velocity by $\bar{\boldsymbol{u}}$ . These fields are governed by the following differential equations.

1. (a) Nernst–Planck ion conservation,

   (4.9) $$\begin{eqnarray}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\bar{\boldsymbol{j}}^{\,\pm }+{\it\alpha}^{\pm }\bar{\boldsymbol{u}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\bar{c}^{\pm }=0,\end{eqnarray}$$
   where the molecular fluxes are again normalized by $D_{\ast }^{\pm }\mathscr{C}_{\ast }/a_{\ast }$ :
   (4.10) $$\begin{eqnarray}\bar{\boldsymbol{j}}^{\,\pm }=M(\mp \bar{c}^{\pm }\boldsymbol{{\rm\nabla}}\bar{{\it\varphi}}-\boldsymbol{{\rm\nabla}}\bar{c}^{\pm }).\end{eqnarray}$$

2. (b) Poisson’s equation,

   (4.11) $$\begin{eqnarray}-2{\it\delta}^{2}S^{-1}{\rm\nabla}^{2}\bar{{\it\varphi}}=\bar{c}^{+}-\bar{c}^{-}.\end{eqnarray}$$

3. (c) Stokes equations, subject to Coulomb body forces,

   (4.12a,b ) $$\begin{eqnarray}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\bar{\boldsymbol{u}}=0,\quad \boldsymbol{{\rm\nabla}}\bar{p}=M^{-1}{\rm\nabla}^{2}\bar{\boldsymbol{u}}+S^{-1}{\rm\nabla}^{2}\bar{{\it\varphi}}\boldsymbol{{\rm\nabla}}\bar{{\it\varphi}},\end{eqnarray}$$
   wherein $\bar{p}$ is the drop-phase pressure field.

4.2. Alternative formulation

An alternative description is obtained through the use of the mean (‘salt’) concentration (normalized by $\mathscr{C}_{\ast }$ ) and volumetric charge density (normalized by $2ze_{\ast }\mathscr{C}_{\ast }$ )

(4.13a,b ) $$\begin{eqnarray}c={\textstyle \frac{1}{2}}(c^{+}+c^{-}),\quad q={\textstyle \frac{1}{2}}(c^{+}-c^{-}),\end{eqnarray}$$

instead of the ionic concentrations $c^{\pm }$ . The ‘salt flux’ and ‘current density,’ defined by

(4.14a,b ) $$\begin{eqnarray}\boldsymbol{j}=\frac{\boldsymbol{j}^{+}+\boldsymbol{j}^{-}}{2},\quad \boldsymbol{i}=\frac{\boldsymbol{j}^{+}-\boldsymbol{j}^{-}}{2},\end{eqnarray}$$

then adopt the forms (see (4.3))

(4.15a,b ) $$\begin{eqnarray}\boldsymbol{j}=-\boldsymbol{{\rm\nabla}}c-q\boldsymbol{{\rm\nabla}}{\it\varphi},\quad \boldsymbol{i}=-\boldsymbol{{\rm\nabla}}q-c\boldsymbol{{\rm\nabla}}{\it\varphi}.\end{eqnarray}$$

(Because $\boldsymbol{j}^{\,\pm }$ are normalized by different quantities, the quantities $\boldsymbol{j}$ and $\boldsymbol{i}$ do not possess the meaning of true flux quantities. Nonetheless, their use is convenient.) Mutual addition and subtraction of (4.2) yields the respective salt- and charge-conservation equations,

(4.16) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{j}+\frac{{\it\alpha}^{+}+{\it\alpha}^{-}}{2}\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}c+\frac{{\it\alpha}^{+}-{\it\alpha}^{-}}{2}\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}q=0, & \displaystyle\end{eqnarray}$$

(4.17) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{i}+\frac{{\it\alpha}^{+}-{\it\alpha}^{-}}{2}\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}c+\frac{{\it\alpha}^{+}+{\it\alpha}^{-}}{2}\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}q=0. & \displaystyle\end{eqnarray}$$

(4.18) $$\begin{eqnarray}q=-{\it\delta}^{2}{\rm\nabla}^{2}{\it\varphi}.\end{eqnarray}$$

Within the drop, the corresponding quantities

(4.19a,b ) $$\begin{eqnarray}\bar{\boldsymbol{j}}=\frac{\bar{\boldsymbol{j}}^{+}+\bar{\boldsymbol{j}}^{-}}{2},\quad \bar{\boldsymbol{i}}=\frac{\bar{\boldsymbol{j}}^{+}-\bar{\boldsymbol{j}}^{-}}{2}\end{eqnarray}$$

adopt the respective forms (see (4.10))

(4.20a,b ) $$\begin{eqnarray}\bar{\boldsymbol{j}}=M(-\boldsymbol{{\rm\nabla}}\bar{c}-\bar{q}\boldsymbol{{\rm\nabla}}\bar{{\it\varphi}}),\quad \bar{\boldsymbol{i}}=M(-\boldsymbol{{\rm\nabla}}\bar{q}-\bar{c}\boldsymbol{{\rm\nabla}}\bar{{\it\varphi}}).\end{eqnarray}$$

Mutual addition and subtraction of (4.9) yields here the analogs of (4.16) and (4.17),

(4.21) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\bar{\boldsymbol{j}}+\frac{{\it\alpha}^{+}+{\it\alpha}^{-}}{2}\bar{\boldsymbol{u}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\bar{c}+\frac{{\it\alpha}^{+}-{\it\alpha}^{-}}{2}\bar{\boldsymbol{u}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\bar{q}=0, & \displaystyle\end{eqnarray}$$

(4.22) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\bar{\boldsymbol{i}}+\frac{{\it\alpha}^{+}-{\it\alpha}^{-}}{2}\bar{\boldsymbol{u}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\bar{c}+\frac{{\it\alpha}^{+}+{\it\alpha}^{-}}{2}\bar{\boldsymbol{u}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\bar{q}=0. & \displaystyle\end{eqnarray}$$

(4.23) $$\begin{eqnarray}\bar{q}=-{\it\delta}^{2}S^{-1}{\rm\nabla}^{2}\bar{{\it\varphi}}.\end{eqnarray}$$

4.3. Interfacial boundary conditions

In writing the interfacial boundary conditions we denote the surface concentrations of cations and anions, normalized by $2\mathscr{C}_{\ast }/{\it\kappa}_{\ast }$ , by ${\it\Gamma}^{\pm }$ . With that definition, we find from (4.7) that

(4.24) $$\begin{eqnarray}{\it\Gamma}^{+}-{\it\Gamma}^{-}=\text{interfacial charge density (normalized by }{\it\epsilon}_{\ast }{\it\kappa}_{\ast }{\it\varphi}_{\ast }\text{)}.\end{eqnarray}$$

The boundary conditions are written using spherical polar coordinates $(r,{\it\theta},{\it\varpi})$ , $r=0$ coinciding with the drop centre and ${\it\theta}=0$ pointing in the applied-field direction. Because of axial symmetry, all scalar fields are independent of the azimuthal angle  ${\it\varpi}$ . Moreover, the velocity fields admit the axially symmetry forms

(4.25a,b ) $$\begin{eqnarray}\boldsymbol{u}=u\hat{\boldsymbol{e}}_{r}+v\hat{\boldsymbol{e}}_{{\it\theta}},\quad \bar{\boldsymbol{u}}=\bar{u}\hat{\boldsymbol{e}}_{r}+\bar{v}\hat{\boldsymbol{e}}_{{\it\theta}},\end{eqnarray}$$

with velocity components that are independent of ${\it\varpi}$ . Similar forms apply to the ionic fluxes, the salt fluxes, and the current densities.

The conditions consist of the following.

1. (a) Ion conservation,

   (4.26) $$\begin{eqnarray}\hat{\boldsymbol{e}}_{r}\boldsymbol{\cdot }(\boldsymbol{j}^{\,\pm }-\bar{\boldsymbol{j}}^{\,\pm })+2{\it\delta}{\it\alpha}^{\pm }\boldsymbol{{\rm\nabla}}_{s}\boldsymbol{\cdot }({\it\Gamma}^{\pm }\boldsymbol{u})=0,\end{eqnarray}$$
   where the first term represents molecular flux from the bulk, and the second term represents surface convection (in which $\boldsymbol{{\rm\nabla}}_{s}$ is the surface-gradient operator).

2. (b) Electric-potential continuity,

   (4.27) $$\begin{eqnarray}{\it\varphi}=\bar{{\it\varphi}}.\end{eqnarray}$$

3. (c) The boundary form of Gauss law,

   (4.28) $$\begin{eqnarray}{\it\Gamma}^{+}-{\it\Gamma}^{-}=-{\it\delta}\left(\frac{\partial {\it\varphi}}{\partial r}-S^{-1}\frac{\partial \bar{{\it\varphi}}}{\partial r}\right).\end{eqnarray}$$

4. (d) Impermeability,

   (4.29) $$\begin{eqnarray}u=\bar{u}=0.\end{eqnarray}$$

5. (e) Velocity continuity,

   (4.30) $$\begin{eqnarray}v=\bar{v}.\end{eqnarray}$$

6. (f) Shear-stress balance,

   (4.31) $$\begin{eqnarray}\left(\frac{\partial v}{\partial r}-v\right)-M^{-1}\left(\frac{\partial \bar{v}}{\partial r}-\bar{v}\right)-{\it\delta}^{-1}\left({\it\Gamma}^{+}-{\it\Gamma}^{-}\right)\frac{\partial {\it\varphi}}{\partial {\it\theta}}-{\it\delta}^{-1}\frac{\partial }{\partial {\it\theta}}({\it\Gamma}^{+}+{\it\Gamma}^{-})=0,\end{eqnarray}$$
   where the last two terms respectively represent the electric shear stress and the Marangoni effect (see Baygents & Saville Reference Baygents and Saville1991).

In addition to conditions (4.26)–(4.31), the surface concentrations ${\it\Gamma}^{\pm }$ are subject to kinetic conditions governing ionic adsorption and desorption. Following Baygents & Saville (Reference Baygents, Saville and Wang1989) we assume fast kinetics with a linear concentration dependence,

(4.32) $$\begin{eqnarray}{\it\Gamma}^{\pm }=K^{\pm }c^{\pm }=\bar{K}^{\pm }\bar{c}^{\pm },\end{eqnarray}$$

wherein $(K^{\pm },\bar{K}^{\pm })$ are adsorption coefficients, normalized by $2/{\it\kappa}_{\ast }$ . As will become evident, $O(1)$ values of $(K^{\pm },\bar{K}^{\pm })$ correspond to $O(1)$ zeta potentials.

4.4. Additional conditions

The boundary conditions at $r=1$ are supplemented by appropriate far-field conditions. Thus, at large distances away from the drop the concentrations approach a unity value (in agreement with definition of $\mathscr{C}_{\ast }$ as the equilibrium concentration)

(4.33a,b ) $$\begin{eqnarray}c\rightarrow 1,\quad q\rightarrow 0\end{eqnarray}$$

and the electric field becomes uniform

(4.34) $$\begin{eqnarray}{\it\varphi}\sim -{\it\beta}r\cos {\it\theta},\end{eqnarray}$$

wherein

(4.35) $$\begin{eqnarray}{\it\beta}=\frac{a_{\ast }\mathscr{E}_{\ast }}{{\it\varphi}_{\ast }}\end{eqnarray}$$

is the dimensionless magnitude of the applied field.

In contrast to standard electrohydrodynamic descriptions, the presence here of equilibrium charge necessitates the allowing of possible drop motion, say with velocity $u^{\infty }$ , relative to the otherwise quiescent liquid. In the comoving frame in which the preceding description is provided, this motion implies the far-field condition

(4.36) $$\begin{eqnarray}\boldsymbol{u}\rightarrow -u^{\infty }\hat{\boldsymbol{\imath }}\end{eqnarray}$$

where $\hat{\boldsymbol{\imath }}$ is a unit vector in the applied-field direction. The velocity $u^{\infty }$ is not a priori prescribed. Rather, it is determined from the condition that the drop is force free,

(4.37) $$\begin{eqnarray}\oint \text{d}A\,\hat{\boldsymbol{n}}\boldsymbol{\cdot }\left\{-p\unicode[STIX]{x1D644}+\boldsymbol{{\rm\nabla}}\boldsymbol{u}+(\boldsymbol{{\rm\nabla}}\boldsymbol{u})^{\dagger }+\boldsymbol{{\rm\nabla}}{\it\varphi}\boldsymbol{{\rm\nabla}}{\it\varphi}-{\textstyle \frac{1}{2}}\boldsymbol{{\rm\nabla}}{\it\varphi}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}{\it\varphi}\unicode[STIX]{x1D644}\right\}=\mathbf{0},\end{eqnarray}$$

where both Newtonian and Maxwell stresses are accounted for. Here, $\unicode[STIX]{x1D644}$ is the identity dyadic and $\dagger$ denotes tensor transposition. Note that condition (4.37) is trivially satisfied in the fore–aft symmetric Taylor–Melcher description.

6.1. Expansions

Consider now the asymptotic expansions in the exterior Debye layer (with similar expansions holding in the interior layer). We postulate the following expansion of the ionic concentrations

(6.5) $$\begin{eqnarray}C^{\pm }=C_{0,0}^{\pm }+{\it\delta}{\it\beta}C_{1,-1}^{\pm }+\cdots \,,\end{eqnarray}$$

with similar expansions for $C$ and $Q$ . We also postulate the electric-potential expansion

(6.6) $$\begin{eqnarray}{\it\Phi}={\it\beta}{\it\Phi}_{0,-1}+{\it\Phi}_{0,0}+{\it\delta}{\it\beta}{\it\Phi}_{1,-1}+\cdots \,,\end{eqnarray}$$

and the following expansion of radial ionic fluxes

(6.7) $$\begin{eqnarray}\hat{\boldsymbol{e}}_{r}\boldsymbol{\cdot }\boldsymbol{J}^{\pm }={\it\beta}J_{0,-1}^{\pm }+{\it\delta}{\it\beta}^{2}J_{1,-2}^{\pm }+{\it\delta}{\it\beta}J_{1,-1}^{\pm }+\cdots \,,\end{eqnarray}$$

which imply, in turn, for the radial salt flux and current density,

(6.8) $$\begin{eqnarray}\displaystyle & \hat{\boldsymbol{e}}_{r}\boldsymbol{\cdot }\boldsymbol{J}={\it\beta}J_{0,-1}+{\it\delta}{\it\beta}^{2}J_{1,-2}+{\it\delta}{\it\beta}J_{1,-1}+\cdots \,, & \displaystyle\end{eqnarray}$$

(6.9) $$\begin{eqnarray}\displaystyle & \hat{\boldsymbol{e}}_{r}\boldsymbol{\cdot }\boldsymbol{I}={\it\beta}I_{0,-1}+{\it\delta}{\it\beta}^{2}I_{1,-2}+{\it\delta}{\it\beta}I_{1,-1}+\cdots \,. & \displaystyle\end{eqnarray}$$

(6.10) $$\begin{eqnarray}V={\it\beta}^{2}V_{0,-2}+{\it\beta}V_{0,-1}+{\it\delta}{\it\beta}^{2}V_{1,-2}+\cdots \,.\end{eqnarray}$$

The continuity equation (4.8a ) in conjunction with the impermeability condition (4.29) then implies an $O({\it\delta}{\it\beta}^{2})$ radial velocity

(6.11) $$\begin{eqnarray}U\sim O({\it\delta}{\it\beta}^{2}).\end{eqnarray}$$

Note that the prefactors in the various expansions of velocity and flux-like variables always involve ${\it\beta}$ ; this is plausible since in the absence of an applied field the system is in equilibrium with no flow or net fluxes. It is also worth noting that the asymptotic limit (5.1) and (5.2) does not determine the magnitude of the prefactor ${\it\delta}{\it\beta}^{2}$ relative to unity; this however does not pose any restriction on the resulting analysis.

In addition to the expansions within the fluid domains, we also need to present asymptotic expansions of the interfacial ionic concentrations. In view of the kinetic conditions (4.32), we anticipate similar expansions as those of the Debye-layer concentrations, see (6.5):

(6.12) $$\begin{eqnarray}{\it\Gamma}^{\pm }={\it\Gamma}_{0,0}^{\pm }+{\it\delta}{\it\beta}{\it\Gamma}_{1,-1}^{\pm }+\cdots \,.\end{eqnarray}$$

Note that $\mathscr{Q}$ and $\bar{\mathscr{Q}}$ , and hence also ${\it\Lambda}$ , possess similar expansions.

6.2. Preliminary results

From the $O({\it\beta})$ balance of Poisson’s equation we find $\partial ^{2}{\it\Phi}_{0,-1}/\partial Z^{2}=0$ , implying that ${\it\Phi}_{0,-1}$ must grow linearly with $Z$ : ${\it\Phi}_{0,-1}=A({\it\theta})Z+B({\it\theta})$ . Asymptotic matching with the $O({\it\beta})$ bulk potential reveals that $A=0$ ; as ${\it\Phi}_{0,-1}$ is independent of $Z$ , it is simply an extrapolation of the bulk potential ${\it\varphi}_{-1}$ . Similarly, $\bar{{\it\Phi}}_{0,-1}$ is an extrapolation of the bulk potential $\bar{{\it\varphi}}_{-1}$ within the drop. We write these results in the form

(6.13a,b ) $$\begin{eqnarray}{\it\Phi}_{0,-1}\equiv {\it\varphi}_{-1},\quad \bar{{\it\Phi}}_{0,-1}\equiv \bar{{\it\varphi}}_{-1};\end{eqnarray}$$

hereafter, bulk variables appearing in Debye-scale equations are understood to be evaluated at the macroscale boundary $r=1$ . The microscale continuity condition (4.27) at $Z=0$ then furnishes the effective continuity condition

(6.14) $$\begin{eqnarray}{\it\varphi}_{-1}=\bar{{\it\varphi}}_{-1}.\end{eqnarray}$$

With transverse non-uniformities in the electric potential appearing only at $O(1)$ , the radial Debye-layer electric fields retain their $O({\it\delta}^{-1})$ equilibrium scaling, just as in standard electrokinetic phenomena under moderate applied fields (Yariv Reference Yariv2009).

Consider now the momentum balance in the radial direction, where the $O({\it\delta}^{-1}{\it\beta}^{2})$ viscous stresses are dominated by the $O({\it\delta}^{-3})$ Coulomb body forces. This implies that the $Z$ -derivatives of $P$ and $\bar{P}$ are $O({\it\delta}^{-2})$ : the $O({\it\delta}^{-2})$ pressure field is accordingly transversely non-uniform. Moreover, since the pressure variations in the bulk are at most $O({\it\beta}^{2})$ , asymptotically smaller than ${\it\delta}^{-2}$ , large- $Z$ matching excludes the possibility of a transversely-uniform Debye-layer pressure term which is $\gg {\it\delta}^{-2}$ .

The momentum balances in the tangential direction, at leading $O({\it\delta}^{-2}{\it\beta}^{2})$ , is accordingly unaffected by the Debye-layer pressure; they merely read

(6.15a,b ) $$\begin{eqnarray}\frac{\partial ^{2}V_{0,-2}}{\partial Z^{2}}=0,\quad \frac{\partial ^{2}\bar{V}_{0,-2}}{\partial Z^{2}}=0.\end{eqnarray}$$

Asymptotic matching with the leading $O({\it\beta}^{2})$ bulk flow then reveals that $V_{0,-2}$ and $\bar{V}_{0,-2}$ are transversely uniform; just as the leading-order potentials, they constitute extrapolations of the corresponding bulk variables (see (4.25)):

(6.16a,b ) $$\begin{eqnarray}V_{0,-2}\equiv v_{-2},\quad \bar{V}_{0,-2}\equiv \bar{v}_{-2}.\end{eqnarray}$$

The microscale continuity condition (4.30) at $Z=0$ then results in the effective continuity condition

(6.17) $$\begin{eqnarray}v_{-2}=\bar{v}_{-2}\quad \text{at}~r=1,\end{eqnarray}$$

governing the bulk fields. This condition is supplemented by the effective impermeability condition

(6.18) $$\begin{eqnarray}u_{-2}=\bar{u}_{-2}=0\quad \text{at}~r=1,\end{eqnarray}$$

which simply follows from the Debye scaling (6.11) of the radial velocities.

With the transversely uniform leading-order velocities (6.16), the continuity equation implies that the leading $O({\it\delta}{\it\beta}^{2})$ radial component must vary linearly with $Z$ . It then follows from the radial momentum balance that the pressure associated with the leading-order flow is only $O({\it\delta}{\it\beta}^{2})$ , asymptotically smaller than the $O({\it\delta}^{-2})$ pressure associated with the combination of $O(1)$ volume-charge density and $O({\it\delta}^{-1})$ transverse electric field in the Debye layer, which we consider next.

6.3. Leading-order Debye cloud

The presumed absence of $O({\it\delta}^{-1})$ radial ionic fluxes implies (see (6.7))

(6.19a,b ) $$\begin{eqnarray}\frac{\partial C_{0,0}^{\pm }}{\partial Z}\pm C_{0,0}^{\pm }\frac{\partial {\it\Phi}_{0,0}}{\partial Z}=0,\quad \frac{\partial \bar{C}_{0,0}^{\pm }}{\partial Z}\pm \bar{C}_{0,0}^{\pm }\frac{\partial \bar{{\it\Phi}}_{0,0}}{\partial Z}=0.\end{eqnarray}$$

Integration in conjunction with large- $Z$ asymptotic matching furnishes the Boltzmann distributions

(6.20a,b ) $$\begin{eqnarray}C_{0,0}^{\pm }=\text{e}^{\mp {\it\Psi}},\quad \bar{C}_{0,0}^{\pm }=\bar{c}_{0}\text{e}^{\mp \bar{{\it\Psi}}},\end{eqnarray}$$

wherein $\bar{c}_{0}$ is the uniform salt concentration within drop (yet to be determined) and

(6.21a,b ) $$\begin{eqnarray}{\it\Psi}={\it\Phi}_{0,0}-{\it\varphi}_{0},\quad \bar{{\it\Psi}}=\bar{{\it\Phi}}_{0,0}-\bar{{\it\varphi}}_{0}\end{eqnarray}$$

are the excess potentials within the suspending liquid and the drop, respectively. In view of the matching requirement, these variables must vanish as $Z\rightarrow \pm \infty$ , respectively. The charge densities associated with (6.20) are

(6.22a,b ) $$\begin{eqnarray}Q_{0,0}=-\sinh \,{\it\Psi},\quad \bar{Q}_{0,0}=-\bar{c}_{0}\sinh \bar{{\it\Psi}}.\end{eqnarray}$$

Substitution into the $O(1)$ balances of Poisson’s equations

(6.23a,b ) $$\begin{eqnarray}\frac{\partial ^{2}{\it\Psi}}{\partial Z^{2}}=-Q_{0,0},\quad \frac{\partial ^{2}\bar{{\it\Psi}}}{\partial Z^{2}}=-S\bar{Q}_{0,0},\end{eqnarray}$$

followed by integration (using the decay conditions at $\pm \infty$ ) yields the first-order equations

(6.24a,b ) $$\begin{eqnarray}\frac{\partial {\it\Psi}}{\partial Z}=-2\sinh \frac{{\it\Psi}}{2},\quad \frac{\partial \bar{{\it\Psi}}}{\partial Z}=2\sqrt{S\bar{c}_{0}}\sinh \frac{\bar{{\it\Psi}}}{2}.\end{eqnarray}$$

We define the zeta potentials as the leading-order voltage drops between the interface and the respective bulk values,

(6.25a,b ) $$\begin{eqnarray}{\it\zeta}={\it\Psi}(Z=0),\quad \bar{{\it\zeta}}=\bar{{\it\Psi}}(Z=0).\end{eqnarray}$$

Using (6.20), the ionic concentrations about $Z=0$ are

(6.26a,b ) $$\begin{eqnarray}C_{0}^{\pm }(Z=0^{+})=\text{e}^{\mp {\it\zeta}},\quad \bar{C}_{0}^{\pm }(Z=0^{+})=\bar{c}_{0}\text{e}^{\mp \bar{{\it\zeta}}}.\end{eqnarray}$$

The kinetic conditions (4.32) thus read at $O(1)$

(6.27) $$\begin{eqnarray}{\it\Gamma}_{0,0}^{\pm }=K^{\pm }\text{e}^{\mp {\it\zeta}}=\bar{K}^{\pm }\bar{c}_{0}\text{e}^{\mp \bar{{\it\zeta}}},\end{eqnarray}$$

providing four equations for the five unknowns ${\it\Gamma}_{0,0}^{\pm }$ , ${\it\zeta}$ , $\bar{{\it\zeta}}$ and $\bar{c}_{0}$ . An additional equation is provided by the $O(1)$ balance of Gauss law’s (4.28)

(6.28) $$\begin{eqnarray}{\it\Gamma}_{0,0}^{+}-{\it\Gamma}_{0,0}^{-}=-\frac{\partial {\it\Psi}}{\partial Z}+S^{-1}\frac{\partial \bar{{\it\Psi}}}{\partial Z}\quad \text{at}~Z=0,\end{eqnarray}$$

which, upon substitution of (6.24), becomes

(6.29) $$\begin{eqnarray}{\it\Gamma}_{0,0}^{+}-{\it\Gamma}_{0,0}^{-}=2\left(\sinh \frac{{\it\zeta}}{2}+\sqrt{\bar{c}_{0}/S}\sinh \frac{\bar{{\it\zeta}}}{2}\right).\end{eqnarray}$$

Since (6.27) and (6.29) are algebraic equations, ${\it\Gamma}_{0,0}^{\pm }$ and the zeta potentials ${\it\zeta}$ and $\bar{{\it\zeta}}$ are independent of ${\it\theta}$ (and then so are ${\it\Psi}$ and $\bar{{\it\Psi}}$ ). These variables, together with $\bar{c}_{0}$ , depend only upon the adsorption coefficients $(K^{\pm },\bar{K}^{\pm })$ and the permittivity ratio $S$ . Specifically, (6.27) readily yield

(6.30) $$\begin{eqnarray}\bar{c}_{0}=\sqrt{\frac{K^{+}K^{-}}{\bar{K}^{+}\bar{K}^{-}}}.\end{eqnarray}$$

Finally, transverse integration of (6.22) using (6.24) yields

(6.31a,b ) $$\begin{eqnarray}\mathscr{Q}_{0,0}=-2\sinh \frac{{\it\zeta}}{2},\quad \bar{\mathscr{Q}}_{0,0}=-2\sqrt{\bar{c}_{0}/S}\sinh \frac{{\it\zeta}}{2}.\end{eqnarray}$$

Substitution of these results and (6.29) into (6.4) yields zero apparent charge at $O(1)$

(6.32) $$\begin{eqnarray}{\it\Lambda}_{0,0}=0,\end{eqnarray}$$

representing the expected screening of the interfacial charge by the surrounding Debye layers.

Of course, the effective conditions (6.14) and (6.17) and (6.18) are insufficient to determine the leading-order electric field and fluid flow. To obtain the requisite additional boundary conditions, we must go beyond leading-order analysis in the Debye layers.

7.1. Effective current continuity

Within the Debye layer, non-zero transverse ionic fluxes appear at $O({\it\beta})$ . The $O({\it\delta}^{-1}{\it\beta})$ balances of the Nernst–Planck equations (4.2) read

(7.1a,b ) $$\begin{eqnarray}\frac{\partial J_{0,-1}^{\pm }}{\partial Z}=0,\quad \frac{\partial \bar{J}_{0,-1}^{\pm }}{\partial Z}=0.\end{eqnarray}$$

Thus, the $O({\it\beta})$ fluxes are transversely uniform (but may depend upon ${\it\theta}$ ). Consider now the leading $O({\it\beta})$ balances of the surface conservation equations (4.26). Since the advection term is only $O({\it\delta}{\it\beta}^{2})$ , it is subdominant to the $O({\it\beta})$ diffusive fluxes (see (5.2)). We accordingly find:

(7.2) $$\begin{eqnarray}J_{0,-1}^{\pm }=\bar{J}_{0,-1}^{\pm }\quad \text{at}~Z=0.\end{eqnarray}$$

In view of the transversal uniformity of $J_{0,-1}^{\pm }$ and $\bar{J}_{0,-1}^{\pm }$ , the left-hand side may be evaluated at any $Z~({>}0)$ and the right-hand side may be evaluated at any $Z~({<}0)$ . Mutual subtraction thus implies

(7.3) $$\begin{eqnarray}I_{0,-1}=\bar{I}_{0,-1}\end{eqnarray}$$

where, again, the left-hand side may be evaluated at any $Z({>}0)$ and the right-hand side may be evaluated at any $Z({<}0)$ . This current-density constancy, in conjunction with the requisite matching with the outer currents, yields an effective current-continuity condition

(7.4) $$\begin{eqnarray}\hat{\boldsymbol{e}}_{r}\boldsymbol{\cdot }\boldsymbol{i}_{-1}=\hat{\boldsymbol{e}}_{r}\boldsymbol{\cdot }\bar{\boldsymbol{i}}_{-1}.\end{eqnarray}$$

Specifically, substitution of (5.10) and (5.12) yields the effective condition

(7.5) $$\begin{eqnarray}\frac{\partial {\it\varphi}_{-1}}{\partial r}=M\bar{c}_{0}\frac{\partial \bar{{\it\varphi}}_{-1}}{\partial r}.\end{eqnarray}$$

With the effective conditions (6.14) and (7.5), the outer potentials ${\it\varphi}_{-1}$ and $\bar{{\it\varphi}}_{-1}$ are completely determined. Note that charge convection does not appear in the effective surface-charge balance (7.5).

7.2. Effective Gauss-type condition

Consider now Poisson’s equation (4.18) at $O({\it\delta}{\it\beta})$ :

(7.6a,b ) $$\begin{eqnarray}-\frac{\partial ^{2}{\it\Phi}_{1,-1}}{\partial Z^{2}}=Q_{1,-1},\quad -S^{-1}\frac{\partial ^{2}\bar{{\it\Phi}}_{1,-1}}{\partial Z^{2}}=\bar{Q}_{1,-1}.\end{eqnarray}$$

We integrate these equations with respect to $Z$ , the first from 0 to $\infty$ and the second from $-\infty$ to 0. Use of the matching requirements

(7.7a,b ) $$\begin{eqnarray}\frac{\partial {\it\Phi}_{1,-1}}{\partial Z}(Z\rightarrow \infty )\rightarrow \frac{\partial {\it\varphi}_{-1}}{\partial r},\quad \frac{\partial \bar{{\it\Phi}}_{1,-1}}{\partial Z}(Z\rightarrow -\infty )\rightarrow \frac{\partial \bar{{\it\varphi}}_{-1}}{\partial r},\end{eqnarray}$$

then yields

(7.8) $$\begin{eqnarray}\displaystyle & \displaystyle -\frac{\partial {\it\varphi}_{-1}}{\partial r}+\frac{\partial {\it\Phi}_{1,-1}}{\partial Z}(Z=0)=\mathscr{Q}_{1,-1}, & \displaystyle\end{eqnarray}$$

(7.9) $$\begin{eqnarray}\displaystyle & \displaystyle -S^{-1}\frac{\partial \bar{{\it\Phi}}_{1,-1}}{\partial Z}(Z=0)+S^{-1}\frac{\partial \bar{{\it\varphi}}_{-1}}{\partial r}=\bar{\mathscr{Q}}_{1,-1}. & \displaystyle\end{eqnarray}$$

$Z=0$

$O({\it\delta}{\it\beta})$

(7.10) $$\begin{eqnarray}{\it\Gamma}_{1,-1}^{+}-{\it\Gamma}_{1,-1}^{-}=-\frac{\partial {\it\Phi}_{1,-1}}{\partial Z}+S^{-1}\frac{\partial \bar{{\it\Phi}}_{1,-1}}{\partial Z}\quad \text{at}~Z=0.\end{eqnarray}$$

Use of (6.4) thus provides the macroscale apparent surface charge at $O({\it\delta}{\it\beta})$

(7.11) $$\begin{eqnarray}{\it\Lambda}_{1,-1}=\frac{\partial {\it\varphi}_{-1}}{\partial r}-S^{-1}\frac{\partial \bar{{\it\varphi}}_{-1}}{\partial r}.\end{eqnarray}$$

In view of (6.32), this is the leading-order apparent net charge.

7.3. Flow

Since $V_{0,-2}$ is independent of $Z$ , the first non-trivial balance of the tangential momentum balance is associated with the leading $O({\it\beta})$ correction to the Debye-scale velocity. Upon making use of (6.13), this balance yields at $O({\it\delta}^{-2}{\it\beta})$ :

(7.12a,b ) $$\begin{eqnarray}\frac{\partial ^{2}V_{0,-1}}{\partial Z^{2}}=-\frac{\text{d}^{2}{\it\Psi}}{\text{d}Z^{2}}\frac{\partial {\it\varphi}_{-1}}{\partial {\it\theta}},\quad M^{-1}\frac{\partial ^{2}\bar{V}_{0,-1}}{\partial Z^{2}}=-S^{-1}\frac{\text{d}^{2}\bar{{\it\Psi}}}{\text{d}Z^{2}}\frac{\partial {\it\varphi}_{-1}}{\partial {\it\theta}}.\end{eqnarray}$$

(Note that at this order there are no Debye-scale tangential pressure gradients.) A single integration yields

(7.13a,b ) $$\begin{eqnarray}\frac{\partial V_{0,-1}}{\partial Z}=-\frac{\text{d}{\it\Psi}}{\text{d}Z}\frac{\partial {\it\varphi}_{-1}}{\partial {\it\theta}},\quad M^{-1}\frac{\partial \bar{V}_{0,-1}}{\partial Z}=-S^{-1}\frac{\text{d}\bar{{\it\Psi}}}{\text{d}Z}\frac{\partial \bar{{\it\varphi}}_{-1}}{\partial {\it\theta}},\end{eqnarray}$$

where the integration constants (functions of ${\it\theta}$ ) must vanish by asymptotic matching with the $O({\it\beta}^{2})$ shear rate in the bulk. Consider now the leading $O({\it\delta}^{-1}{\it\beta})$ balance of the shear-stress jump condition (4.31),

(7.14) $$\begin{eqnarray}\frac{\partial V_{0,-1}}{\partial Z}-M^{-1}\frac{\partial \bar{V}_{0,-1}}{\partial Z}=({\it\Gamma}_{0,0}^{+}-{\it\Gamma}_{0,0}^{-})\frac{\partial {\it\varphi}_{-1}}{\partial {\it\theta}}\quad \text{at}~Z=0.\end{eqnarray}$$

Upon substitution of (7.13) and Gauss’s condition (6.28), it is evident that this condition is trivially satisfied.

We accordingly address the leading $O({\it\delta}^{-1}{\it\beta}^{2})$ correction of the tangential momentum balances, associated with transverse variations of the $O({\it\delta}{\it\beta}^{2})$ tangential velocities. In view of (6.13) and the $O({\it\delta}{\it\beta})$ balance of Poisson’s equation, and given the absence of $O({\it\delta}^{-1}{\it\beta}^{2})$ tangential pressure gradients, these balances read

(7.15a,b ) $$\begin{eqnarray}\frac{\partial ^{2}V_{1,-2}}{\partial Z^{2}}=Q_{1,-1}\frac{\partial {\it\varphi}_{-1}}{\partial {\it\theta}},\quad M^{-1}\frac{\partial ^{2}\bar{V}_{1,-2}}{\partial Z^{2}}=\bar{Q}_{1,-1}\frac{\partial \bar{{\it\varphi}}_{-1}}{\partial {\it\theta}}.\end{eqnarray}$$

Respective integration of these equations from 0 to $\infty$ and from $-\infty$ to 0, in conjunction with the matching conditions

(7.16a,b ) $$\begin{eqnarray}\frac{\partial V_{1,-2}}{\partial Z}(Z\rightarrow \infty )=\frac{\partial v_{-2}}{\partial r},\quad \frac{\partial \bar{V}_{1,-2}}{\partial Z}(Z\rightarrow -\infty )=\frac{\partial \bar{v}_{-2}}{\partial r}\end{eqnarray}$$

yields

(7.17a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial v_{-2}}{\partial r}-\frac{\partial V_{1,-2}}{\partial Z}(Z=0)=\mathscr{Q}_{1,-1}\frac{\partial {\it\varphi}_{-1}}{\partial {\it\theta}}, & \displaystyle\end{eqnarray}$$

(7.17b ) $$\begin{eqnarray}\displaystyle & \displaystyle M^{-1}\left[\frac{\partial \bar{V}_{1,-2}}{\partial Z}(Z=0)-\frac{\partial \bar{v}_{-2}}{\partial r}\right]=\bar{\mathscr{Q}}_{1,-1}\frac{\partial \bar{{\it\varphi}}_{-1}}{\partial {\it\theta}}. & \displaystyle\end{eqnarray}$$

We now consider the $O({\it\beta}^{2})$ balance of the shear-stress jump condition (4.31):

(7.18) $$\begin{eqnarray}\frac{\partial V_{-2,1}}{\partial Z}-v_{-2}-M^{-1}\left(\frac{\partial \bar{V}_{-2,1}}{\partial Z}-\bar{v}_{-2}\right)=({\it\Gamma}_{1,-1}^{+}-{\it\Gamma}_{1,-1}^{-})\frac{\partial {\it\varphi}_{-1}}{\partial {\it\theta}}\quad \text{at}~Z=0.\end{eqnarray}$$

Substituting (7.17) while making use of (6.14) and the definition (6.4) furnishes the effective shear-stress jump condition

(7.19) $$\begin{eqnarray}\frac{\partial v_{-2}}{\partial r}-v_{-2}-M^{-1}\left(\frac{\partial \bar{v}_{-2}}{\partial r}-\bar{v}_{-2}\right)={\it\Lambda}_{1,-1}\frac{\partial {\it\varphi}_{-1}}{\partial {\it\theta}},\end{eqnarray}$$

which entail the apparent charge ${\it\Lambda}_{1,-1}$ , itself provided by the effective Gauss condition (7.11). Since condition (7.19) together with (6.17) and (6.18) is sufficient to determine the bulk Stokes flows, we have successfully obtained a closed macroscale model.

7.4. Conductivity ratio

The effective condition (7.5) is expressed in terms of $M\bar{c}_{0}$ . This product is actually equal to the conductivity ratio $R$ used by Melcher & Taylor (Reference Melcher and Taylor1969). To see that, we note that the dimensional current densities in the suspending and drop phases are respectively provided by the expressions

(7.20a,b ) $$\begin{eqnarray}ze_{\ast }\frac{D_{\ast }^{+}\mathscr{C}_{\ast }}{a_{\ast }}\boldsymbol{j}^{+}-ze_{\ast }\frac{D_{\ast }^{-}\mathscr{C}_{\ast }}{a_{\ast }}\boldsymbol{j}^{-},\quad ze_{\ast }\frac{D_{\ast }^{+}\mathscr{C}_{\ast }}{a_{\ast }}\bar{\boldsymbol{j}}^{+}-ze_{\ast }\frac{D_{\ast }^{-}\mathscr{C}_{\ast }}{a_{\ast }}\bar{\boldsymbol{j}}^{-}.\end{eqnarray}$$

(As already mentioned, neither of the dimensionless quantities $\boldsymbol{i}$ nor $\bar{\boldsymbol{i}}$ represents a true current density because of the different non-dimensionalization of cationic and anionic fluxes.) These densities may be expressed in terms of $\boldsymbol{j}$ and $\boldsymbol{i}$ through (4.14). As $\boldsymbol{i}$ is $O({\it\beta})$ while $\boldsymbol{j}$ is asymptotically smaller, we find that at leading $O({\it\beta})$ the ionic fluxes $\boldsymbol{j}^{\,\pm }$ are respectively given by $\pm \boldsymbol{i}$ (and, similarly, $\bar{\boldsymbol{j}}^{\,\pm }$ are respectively given by $\pm \bar{\boldsymbol{i}}$ ). Moreover, at that leading order $\boldsymbol{i}=-\boldsymbol{{\rm\nabla}}{\it\varphi}$ and $\bar{\boldsymbol{i}}=-M\bar{c}_{0}\boldsymbol{{\rm\nabla}}\bar{{\it\varphi}}$ , see (5.10) and (5.12). Thus, the dimensional current densities (7.20) in the suspending and drop phases are respectively proportional to the dimensional electric fields, $-({\it\varphi}_{\ast }/a_{\ast })\boldsymbol{{\rm\nabla}}{\it\varphi}$ and $-({\it\varphi}_{\ast }/a_{\ast })\boldsymbol{{\rm\nabla}}\bar{{\it\varphi}}$ , the proportionality coefficients being

(7.21a,b ) $$\begin{eqnarray}ze_{\ast }\frac{\mathscr{C}_{\ast }}{{\it\varphi}_{\ast }}(D_{\ast }^{+}+D_{\ast }^{-}),\quad M\bar{c}_{0}ze_{\ast }\frac{\mathscr{C}_{\ast }}{{\it\varphi}_{\ast }}(D_{\ast }^{+}+D_{\ast }^{-}).\end{eqnarray}$$

Accordingly, the respective expressions (7.21) constitute the conductivities of the suspending fluid and the drop. The ratio of the drop conductivity to that of the suspending electrolyte is

(7.22) $$\begin{eqnarray}R=M\bar{c}_{0}.\end{eqnarray}$$

7.5. Electrophoretic velocity

We now consider the leading-order electrophoretic velocity. We first observe that the structure governing the harmonic $O({\it\beta})$ potential ${\it\varphi}_{-1}$ does not allow for a monopole term. Indeed, the problem governing the $O({\it\beta})$ potentials ${\it\varphi}_{-1}$ and $\bar{{\it\varphi}}_{-1}$ is uncoupled to the flow, and is animated by the inhomogeneous far-field condition (5.18a ); so, aside from an additive constant, both potentials must be proportional to $\cos {\it\theta}$ . In the absence of a monopole term, and with a far-field approach to a uniform field, the electric field associated with ${\it\varphi}_{-1}$ cannot result in a net electrostatic force at $O({\it\beta}^{2})$ , see Rivette & Baygents (Reference Rivette and Baygents1996). The force acting on the drop at that asymptotic order in solely contributed by Newtonian stresses associated with leading-order $O({\it\beta}^{2})$ flow (represented by the first two terms in the integrand of (5.19)). Since that flow (as well as the comparable flow within the drop) is governed by the homogeneous Stokes equations (5.16) with a constraint of a zero hydrodynamic force, symmetry arguments imply that $u^{\infty }$ must vanish at $O({\it\beta}^{2})$ :

(7.23) $$\begin{eqnarray}u_{-2}^{\infty }=0.\end{eqnarray}$$

As discussed in § 10.5, drop electrophoresis does appear at a higher asymptotic order.

7.6. Effective normal-stress jump

In the preceding derivation, the drop shape was taken to be spherical. This is tantamount to assuming a large surface-tension coefficient (small capillary number). The preceding results thus correspond to the leading-order description in a formal expansion in the small capillary number. The leading-order deviation from sphericity can be readily determined once the normal-stress jump associated with the leading-order description is calculated (Leal Reference Leal2007).

This approach was indeed used extensively in electrohydrodynamic analyses (Torza, Cox & Mason Reference Torza, Cox and Mason1971), where the normal-stress jump was evaluated using the Taylor–Melcher model. This raises a fundamental question, since the drop deformation is actually affected by the microscale stresses at the genuine interface: is it legitimate to employ the macroscale stresses toward calculation of that deformation?

In appendix B we evaluate the microscale stress jump across the interface and show that it coincides with the one ‘naïvely’ evaluated using the macroscale model. The calculation scheme used in applications of the Taylor–Melcher model is thus substantiated.

9.1. Direct inner–outer matching?

As already mentioned, the microscale model employed herein is essentially that used by Baygents & Saville (Reference Baygents, Saville and Wang1989). The only simplification here is the focus upon binary electrolyte solutions, as opposed to the more general case of $N$ ionic species considered by Baygents & Saville (Reference Baygents, Saville and Wang1989). The limit process (5.1) and (5.2) used to derive the macroscale model is also adopted from Baygents & Saville (Reference Baygents, Saville and Wang1989). In contrast to our successful matching procedure, however, Baygents & Saville (Reference Baygents, Saville and Wang1989) found it impossible to directly match the inner (Debye layer) and outer (bulk) regions. In what follows we discuss the difference between their analysis and ours.

In the course of their matching procedure, Baygents & Saville (Reference Baygents, Saville and Wang1989) obtained $N$ equations relating the dimensionless dipole strength of the drop to the $2N$ ionic mobilities (inside and outside the drop) and $2N$ ionic ‘bulk’ concentrations. They then explain that these equations over-specify the dipole strength ‘since the ionic mobilities and bulk concentrations are fixed quantities’: see equation (4.2), etc., in Baygents & Saville (Reference Baygents, Saville and Wang1989). Given this over-specification, Baygents & Saville (Reference Baygents, Saville and Wang1989) concluded that the inner and outer regions cannot be matched directly and resorted to a separate analysis of an intermediate boundary layer, of thickness $1/{\it\beta}$ , where ‘all three transport modes (convection, diffusion and electromigration) are important’. That layer resembles the diffusive boundary layer appearing in forced-convection heat- and mass-transfer problems (Acrivos & Goddard Reference Acrivos and Goddard1965). The details of that intermediate-layer analysis and its matching with the inner and outer regions are not provided by Baygents & Saville (Reference Baygents, Saville and Wang1989); they nonetheless assert that these procedures eventually results in the familiar quadrupolar flow of Taylor (Reference Taylor1966).

It appears that Baygents & Saville (Reference Baygents, Saville and Wang1989) have overlooked the Einstein–Smoluchowski relation, which implies that the ratio of the ionic conductivities in the respective phases is independent of the specific species $k$ ( $1\leqslant k\leqslant N$ ) considered. This is clearly illustrated in the binary case ( $N=2$ ) addressed in the present paper, where it is readily verified that the two equations governing the dipole strength simply coincide, giving its value (in the present notation) as $(M\bar{c}_{0}-1)/(M\bar{c}_{0}+2)$ . Moreover, since $M\bar{c}_{0}$ coincides with the conductivity ratio $R$ (see (7.22)), this dipole strength turns out identical to that which would be obtained from the electrohydrodynamic model (8.2)–(8.11): see indeed e.g. equation (25) in Melcher & Taylor (Reference Melcher and Taylor1969). It thus seems that the general approach of Baygents & Saville (Reference Baygents, Saville and Wang1989), allowing for arbitrary number $N$ of ionic species, tends to obscure the above-mentioned simplifying points. Indeed, for $N>2$ the ‘bulk’ concentrations are not necessarily identical, significantly complicating the interpretation of equation (4.2) in Baygents & Saville (Reference Baygents, Saville and Wang1989). It is unclear whether the simplifying arguments we have employed may be extended to the general problem of $N>2$ .

9.2. Diffusive boundary layer

Does the present analysis also give rise to the emergence of a diffusive boundary layer? The $O({\it\beta}^{2})$ ionic convection across the $O({\it\delta})$ -wide Debye layer results in an $O({\it\delta}{\it\beta}^{2})$ ‘surface current’ of salt. As this current is generally nonuniform, ionic conservation implies that a comparable flux must be supplied from and into the electro-neutral bulk. It may appear as though this salt flux would give rise to $O({\it\delta}{\it\beta}^{2})$ salt gradients in the bulk; these, however, are incompatible with (5.13). A diffusive boundary is thus formed, where diffusion is balanced by advection; its thickness is accordingly $1/{\it\beta}$ . (By (5.2), this narrow layer is much thicker than the Debye layer.) To accommodate the salt fluxes emerging from the Debye layer, the ratio of the excess-salt concentration in that layer to its $O(1/{\it\beta})$ thickness must be $O({\it\delta}{\it\beta}^{2})$ : that excess concentration thus turns out $O({\it\delta}{\it\beta})$ , and is accordingly asymptotically small.

We have encountered similar incompatibilities in various electrokinetic problems (Yariv, Schnitzer & Frankel Reference Yariv, Schnitzer and Frankel2011; Schnitzer & Yariv Reference Schnitzer and Yariv2012b ; Schnitzer et al. Reference Schnitzer, Frankel and Yariv2013), where they are also resolved by the introduction of an intermediate diffusion layer. In these problems, the presence of the diffusive layer had no effect on the leading-order inner and outer solutions, hence its resolution was not required for a successful leading-order asymptotic matching between them. The situation in the present problem is similar. The structure of the diffusive boundary layer is thus tacitly ignored in the preceding analysis.

10.1. The Baygents–Saville limit

In deriving the macroscale model for the electrokinetic transport we have employed the Baygents–Saville limit (5.1) and (5.2). Following this derivation, the necessities for imposing the restrictions specifying this limit process become evident. The assumption ${\it\beta}\gg 1$ results in uniform salt concentrations in both the suspending liquid and the drop phase. It also results in the domination of the fore–aft symmetric $O({\it\beta}^{2})$ electrohydrodynamic flow pattern over the $O({\it\beta})$ electro-osmotic flow animating the drop phoretic motion. The restriction (5.2) is also necessary for the self-consistency of our asymptotic expansions: when ${\it\beta}$ scales as $1/{\it\delta}$ , tangential ionic advection with the Debye layer become $O({\it\delta}^{-2})$ large, thus necessitating a transversely varying $O({\it\delta}^{-1})$ normal fluxes. The Debye layer is not at quasi-equilibrium, and the entire scheme collapses.

The limitations imposed by the Baygents–Saville limit are naturally met in leaky dielectric electrohydrodynamic phenomena. As drops are typically much larger than colloidal particles, equation (5.1) is easily satisfied for virtually all practical values of the salt concentration $\mathscr{C}_{\ast }$ : see definitions (4.6) and (4.35). Incidentally, these definitions also imply that the product ${\it\delta}{\it\beta}$ is independent of the drop size $a_{\ast }$ : satisfaction of (5.2) thus essentially limits the magnitude of the applied field $\mathscr{E}_{\ast }$ , or, more precisely, of $\mathscr{E}_{\ast }^{2}/\mathscr{C}_{\ast }$ .

Note that, at least until recently (Schnitzer et al. Reference Schnitzer, Frankel and Yariv2013; Schnitzer & Yariv Reference Schnitzer and Yariv2013; Schnitzer et al. Reference Schnitzer, Frankel and Yariv2014), virtually all electrokinetic analyses of drops under electric fields had actually addressed the limit of weak applied fields, see e.g. Ohshima et al. (Reference Ohshima, Healy and White1984) and Baygents & Saville (Reference Baygents and Saville1991).

10.2. Absence of surface-charge convection

The linearity of the electrostatic problem, which enables analytic solution of the macroscale model, follows from the absence of surface convection of charge. In our macroscale model, this absence automatically follows from the Baygents–Saville limit (1.1). Indeed, for the effective $O({\it\delta}{\it\beta}^{2})$ surface current associated with the leading $O({\it\beta}^{2})$ flow to enter the ion conservation equation (4.26) at the $O({\it\beta})$ leading order, ${\it\beta}$ must be $O({\it\delta}^{-1})$ . Such large values of the applied field do not satisfy the restriction (5.2).

In fact, the surface current associated with the leading $O({\it\beta}^{2})$ flow is even smaller than $O({\it\delta}{\it\beta}^{2})$ . Indeed, as the $O({\it\beta}^{2})$ tangential velocity is transversely uniform (see (6.16)), the $O({\it\delta}{\it\beta}^{2})$ surface current is simply provided by its product with the apparent $O(1)$ Debye-layer charge; the latter, however, vanishes by screening (see (6.32)). It may therefore appear that the leading-order surface current is due to the leading $O({\it\delta}{\it\beta})$ apparent charge, and is accordingly $O({\it\delta}^{2}{\it\beta}^{3})$ . Note however that another source for a surface current is the $O({\it\beta})$ electro-osmotic flow, which we did not bother to calculate. As this flow is transversely non-uniform, it results through advection of the $O(1)$ ionic densities in an $O({\it\delta}{\it\beta})$ surface current. The Baygents–Saville restriction (5.2) does not determine which of these two effects dominates. The corrections implied by these currents to the charge-conservation condition (7.5), where the Ohmic currents are $O({\it\beta})$ , are of relative orders $({\it\delta}{\it\beta})^{2}$ and ${\it\delta}$ , respectively.

In the electrohydrodynamic literature, the neglect of charge convection is conventionally associated with the assumption of small electric Reynolds number (see Melcher & Taylor Reference Melcher and Taylor1969). Consider indeed the typical dimensional magnitude of the respective charge conduction and convection terms in a surface-charge balance (e.g. equation (Il) in Melcher & Taylor Reference Melcher and Taylor1969). The former is of order ${\it\sigma}_{\ast }\mathscr{E}_{\ast },{\it\sigma}_{\ast }$ being a typical conductivity, while the latter, represented by a surface-divergence term, is of order ${\it\Lambda}_{\ast }v_{\ast }/a_{\ast }$ , where ${\it\Lambda}_{\ast }$ is the apparent surface-charge density; in the leaky dielectric description, that density is of order ${\it\epsilon}_{\ast }\mathscr{E}_{\ast }$ . It follows that the ratio of charge convection to conduction is of order ${\it\epsilon}_{\ast }v_{\ast }/{\it\sigma}_{\ast }a_{\ast }$ . This is the electric Reynolds number $\mathit{Re}_{e}$ .

Following our derivation of the electrohydrodynamic Taylor–Melcher model as a coarse-grained description, this number is easily estimated. Consider the simple case of equal diffusivities $D_{\ast }^{\pm }=D_{\ast }$ , where ${\it\alpha}^{\pm }$ are identical, $={\it\alpha}$ say (see (4.4)). The bulk conductivity ${\it\sigma}_{\ast }$ is then given by $2ze_{\ast }\mathscr{C}_{\ast }D_{\ast }/{\it\varphi}_{\ast }$ , see (7.21). Using the leaky dielectric velocity scale ${\it\epsilon}_{\ast }a_{\ast }\mathscr{E}_{\ast }^{2}/{\it\mu}_{\ast }$ we readily find using definitions (4.1), (4.4), (4.6) and (4.35) that

(10.1) $$\begin{eqnarray}\mathit{Re}_{e}={\it\alpha}{\it\delta}^{2}{\it\beta}^{2}.\end{eqnarray}$$

Since ${\it\alpha}$ is typically $O(1)$ (see (4.4) etc.), the electric Reynolds number is asymptotically small in the Baygents–Saville limit (1.1) underlying our electrokinetic description: see figure 2.

Figure 2.

The Taylor–Melcher leaky dielectric essentially constitutes a coarse-grained electrokinetic model in the Baygents–Saville limit of strong fields ${\it\beta}\gg 1$ and thin Debye layers, ${\it\delta}\ll 1$ , with $1\ll {\it\beta}\ll {\it\delta}^{-1}$ . This limit corresponds to small electric Reynolds numbers, $\mathit{Re}_{e}\ll 1$ , which is consistent with the absence of surface-charge advection in the derived macroscale model.

In the classical paper of Melcher & Taylor (Reference Melcher and Taylor1969), charge convection is presented in the electrohydrodynamic model (see their Table I). It is then neglected throughout when performing steady analyses of specific configurations: in particular, in the analysis of the flow about a leaky dielectric drop. This neglect is introduced as an additional assumption. As Melcher & Taylor (Reference Melcher and Taylor1969) had no underlying electrokinetic model to provide the magnitude of the bulk conductivity, they could not have obtained the estimate (10.1).

Note that the parts of the analysis of Melcher & Taylor (Reference Melcher and Taylor1969) which deal with instabilities and fluid motion driven by alternating currents allow for finite electric Reynolds number. Moreover, charge convection was incorporated in several electrohydrodynamic models (see Feng & Scott Reference Feng and Scott1996; Xu & Homsy Reference Xu and Homsy2006; Salipante & Vlahovska Reference Salipante and Vlahovska2010) in an attempt to improve upon Taylor’s initial analysis. In view of estimate (10.1), these studies fall outside the Baygents and Saville regime, and hence their electrokinetic interpretation remains unclear.

10.3. The conductivity ratio: a material property?

The macroscale equations depend upon the permittivity ratio $S$ , the viscosity ratio $M$ and the conductivity ratio $R$ , just as in the model outlined by Melcher & Taylor (Reference Melcher and Taylor1969). Melcher & Taylor (Reference Melcher and Taylor1969) consider these parameters as given material ratios. In our scheme there is however a conceptual difference between the permittivity and viscosity ratios $S$ and $M$ and the conductivity ratio $R$ : while the former two indeed represent (ratios between) true material properties (and actually appear in the microscope formulation), the latter is a derived macroscale quantity, which does not appear in the original formulation. In view of expression (8.12), moreover, $R$ depends upon the adsorption coefficients $(K^{\pm },\bar{K}^{\pm })$ (normalized by $2/{\it\kappa}_{\ast }$ ), and accordingly cannot be considered a material property. Putting it differently, while the conductivity of the suspending liquid can be conceptually measured in an experiment, independently of the drop, the same does not hold for the drop conductivity.

This difference with the Taylor–Melcher model should be viewed with caution, however, given our simplistic modelling of charge transfer across the interface and our simplifying assumption of a strong binary electrolyte. A more realistic description of ionic transport would incorporate bulk reactions (see e.g. Saville Reference Saville1997) and allow only several of the ionic species to pass through the interface. Within the framework of such a general model, one can envisage two scenarios where the drop-phase conductivity is effectively a property of the drop, unaffected by the passage of currents. The first is realized in the limit of fast reactions, while the second occurs when the ionic composition in the two liquids consists of a high concentration of ‘inert’ salt ions (for which the adsorption coefficients vanish), which essentially determines the bulk conductivity, and a small concentration of ions which undergo interfacial sorption, thereby enabling the passage of electric current through the interface.

10.4. Highly conducting drops

A familiar feature of the flow field predicted by the leaky-dielectric model is that it vanishes in the limit $R\gg 1$ . This implies that, for sufficiently large $R$ , the (formally subdominant) electrokinetic flow associated with the presence of equilibrium charge becomes comparable with the (formally dominant) leaky-dielectric flow. The field strengths at which this transition occurs, and hence at which the leaky-dielectric cannot be expected to apply, can be easily estimated from our analysis. On the one hand, inspection of the macroscale equations of § 8 reveals that the Taylor–Melcher flow vanishes like $1/R$ as $R\rightarrow \infty$ ; this corresponds to an $O({\it\beta}^{2}/R)$ flow in terms of the electrokinetic scalings of § 4. On the other hand, our analysis reveals that the electrokinetic velocity is $O({\it\beta})$ , and does not vanish with $R$ . Thus, the derived macroscale model collapses when $R\sim O({\it\beta})$ .

10.5. Net drop charge and electrophoresis

We now discuss the net drop charge: namely the total charge held by the interface in the absence of external field. In the Taylor–Melcher description, the dipolar charge distribution clearly implies that such a net charge is absent. Moreover, no diffuse layer accompanies, either explicitly or implicitly, the local interfacial charge. Indeed, in the introduction to his review paper, Saville (Reference Saville1997) wrote that ‘With …leaky dielectrics, diffuse layers associated with equilibrium charge are usually absent’. This claim is refuted here: our electrokinetic analysis naturally allows the presence of equilibrium charge at the interface, which is screened by diffuse-charge layers. The amount of equilibrium charge is simply set by the interfacial kinetics, and would vanish only for specifically chosen combinations of the respective coefficients. In the physicochemical model (4.32), zero drop charge is realized for $K^{+}=K^{-}=\bar{K}^{+}=\bar{K}^{-}$ and $S=1$ : see (6.27) and (6.29). In the general case, then, the charge on the genuine interface is non-zero. In fact, the presence of non-zero interfacial charge is actually necessary for the validity of our asymptotic scheme, which hinges upon the induced charge being a small perturbation.

Our analysis reveals that, remarkably, the electrophoretic motion engendered by the net drop charge constitutes an asymptotic correction to the Taylor–Melcher flow, itself animated by a perturbation to that equilibrium charge. The appearance of electrophoresis at a higher order might be consistent with existing experiments, where the drop indeed tends to move under the electric field. (Such motion was considered as an inevitable inconvenience that interferes with the convenient observation and measurement of electrohydrodynamic drop deformation, see Taylor (Reference Taylor1966) and Vizika & Saville (Reference Vizika and Saville1992).)

Our unified approach provides the means for understating this inevitable motion, which apparently goes hand in hand with the Taylor–Melcher flow. The calculation of drop electrophoresis in our scheme requires continuing the asymptotic analysis towards deriving a higher-order macroscale model. Such an analysis reveals that drop electrophoresis is animated by both shear and a two-sided electro-osmotic slip mechanism. While the latter is already implied in our analysis (see (7.13)), the calculation of the shear contribution entails a complicated calculation, including the resolution of the two diffusive boundary layers. In the limit of a highly viscous drop, one can show that the shear contribution becomes trivial. The drop thus migrates according to a Smoluchowski-type formula, with the zeta potential appearing therein being that of the external Debye layer. Interestingly then, the equilibrium charge in that case cannot be straightforwardly inferred from electrophoretic measurements.

The compatibility of non-zero net charge with the absence of electrophoresis at leading order is clearly a manifestation of the delicate limit process (1.1), representing the strong fields characterizing Taylor–Melcher flows. This compatibility casts doubts regarding the applicability of the scheme developed by Zholkovskij, Masliyah & Czarnecki (Reference Zholkovskij, Masliyah and Czarnecki2002). In that paper, which proposes a different approach to model the transition from electrokinetics to electrohydrodynamics, the Taylor–Melcher description is developed for an arbitrary Debye length. In sharp contrast to Baygents & Saville (Reference Baygents, Saville and Wang1989), this is accomplished via a weak-field approximation. Putting aside the validity of such an approximation in describing Taylor–Melcher flows, it appears that the main shortcoming in the analysis of Zholkovskij et al. (Reference Zholkovskij, Masliyah and Czarnecki2002) is the necessity of assuming zero net charge; as the authors explain, this corresponds to artificially specifying appropriate ratios of the kinetic parameters. Similarly, the recent stability analysis of Gambhire & Thaokar (Reference Gambhire and Thaokar2014) also makes use of an effective weak-field linearization. In that paper the authors avoid the arbitrariness in the kinetic approach of Zholkovskij et al. (Reference Zholkovskij, Masliyah and Czarnecki2002) by focusing upon inert solute which does not adsorb to the genuine interface. While this gives rise to a consistent physical model, the absence of ionic transport through the interface renders it insufficient when attempting to describe the electrokinetics underlying the leaky-dielectric model.