2.1 Set-up

Figure 6 shows the configuration considered here: a Newtonian viscous fluid in the interior domain $\unicode[STIX]{x1D6FA}_{in}$ is separated from another Newtonian viscous fluid in the exterior domain $\unicode[STIX]{x1D6FA}_{ex}$ by an interface denoted by $\unicode[STIX]{x1D6E4}$ . We let the whole region be $\unicode[STIX]{x1D6FA}=\unicode[STIX]{x1D6FA}_{in}\cup \unicode[STIX]{x1D6FA}_{ex}\cup \unicode[STIX]{x1D6E4}$ . The region $\unicode[STIX]{x1D6FA}$ may either be a bounded domain (so that there is a boundary $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ ) or the whole of $\mathbb{R}^{3}$ . Unlike the analysis in Schnitzer & Yariv (Reference Schnitzer and Yariv2015), here the fluid interface $\unicode[STIX]{x1D6E4}$ is not restricted to a spherical surface, and is allowed to be time dependent (and hence a time-dependent $\unicode[STIX]{x1D6FA}_{in}$ ). We shall sometimes use the notation $\unicode[STIX]{x1D6E4}_{t}$ to make this time dependence explicit. In the following, for convenience of the reader, we may reiterate the definition of some variables that were introduced in the § 1.2.1.

Figure 6.

Set-up for the modified Saville model. The interior/exterior medium is denoted by $\unicode[STIX]{x1D6FA}_{in}/\unicode[STIX]{x1D6FA}_{ex}$ . The interface is denoted by $\unicode[STIX]{x1D6E4}$ . The unknown functions in $\unicode[STIX]{x1D6FA}_{in,ex}$ are the cation/anion/salt concentrations $c,a,s$ , velocity and pressure fields $\boldsymbol{u},p$ and the electrostatic potential $\unicode[STIX]{x1D719}$ .

We consider a salt (S) which may dissociate into cations ( $\text{C}^{+}$ ) and anions ( $\text{A}^{-}$ ) in a solvent, as in reaction (1.1). As discussed in § 1, we consider the weak electrolyte case, in which most of the salt does not dissociate into its constituent ions. This may happen, for example, if NaCl is dissolved in a non-polar solvent. When the above dissociation reaction is at equilibrium in the non-polar solvent, there should be considerably more S than $\text{C}^{+}$ or $\text{A}^{-}$ .

In the bulk ( $\unicode[STIX]{x1D6FA}_{in}\cup \unicode[STIX]{x1D6FA}_{ex}$ ) the concentrations of $\text{C}^{+}$ , $\text{A}^{-}$ and S are denoted by $c$ , $a$ and $s$ respectively. The equations satisfied by these variables in $\unicode[STIX]{x1D6FA}_{in}\cup \unicode[STIX]{x1D6FA}_{ex}$ are:

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}c}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{u}c)=\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left(D_{C}\left(\unicode[STIX]{x1D735}c+c\frac{F}{RT}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}\right)\right)+k_{+}s-k_{-}ca, & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}a}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{u}a)=\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left(D_{A}\left(\unicode[STIX]{x1D735}a-a\frac{F}{RT}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}\right)\right)+k_{+}s-k_{-}ca, & \displaystyle\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}s}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{u}s)=\unicode[STIX]{x1D735}\boldsymbol{\cdot }(D_{S}\unicode[STIX]{x1D735}s)-k_{+}s+k_{-}ca. & \displaystyle\end{eqnarray}$$

Here, $\boldsymbol{u}$ is the solvent velocity, $D_{C,A,S}$ are the diffusion coefficients, $F$ is the Faraday constant, $RT$ is the ideal gas constant times absolute temperature, $\unicode[STIX]{x1D719}$ is the electrostatic potential and $k_{\pm }$ are the rates of forward and backward reaction in (1.1). The diffusion coefficients $D_{C,A,S}$ and the reaction rate constants $k_{\pm }$ may differ in $\unicode[STIX]{x1D6FA}_{in}$ and $\unicode[STIX]{x1D6FA}_{ex}$ , but we assume they are constant within each region. We will sometimes use the notation $D_{C,in}$ or $D_{C,ex}$ to refer to the diffusion constant in regions $\unicode[STIX]{x1D6FA}_{in}$ and $\unicode[STIX]{x1D6FA}_{ex}$ respectively. Analogous expressions will be used for $D_{A,S}$ and for $k_{\pm }$ . We need an equation for the fluid velocity $\boldsymbol{u}$ as well as the electrostatic potential $\unicode[STIX]{x1D719}$ . For the electrostatic potential, we have:

(2.4) $$\begin{eqnarray}\displaystyle -\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D716}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719})=F(c-a), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D716}$ is the dielectric constant. For the fluid velocity, we assume Stokes flow:

(2.5) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}\unicode[STIX]{x0394}\boldsymbol{u}-\unicode[STIX]{x1D735}p=F(c-a)\unicode[STIX]{x1D735}\unicode[STIX]{x1D719},\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D707}$ is the viscosity and $p$ is the pressure. The dielectric constant $\unicode[STIX]{x1D716}$ and the viscosity $\unicode[STIX]{x1D707}$ may take different but spatially constant values in $\unicode[STIX]{x1D6FA}_{in}$ and $\unicode[STIX]{x1D6FA}_{ex}$ , and we use the notation $\unicode[STIX]{x1D716}_{k},\unicode[STIX]{x1D707}_{k},k=\text{in},\text{ex}$ to denote the constants in the two regions. We have chosen to ignore the inertial term as is customary in most treatments of the electrohydrodynamics of leaky dielectrics; Saville (Reference Saville1997) estimates the Reynolds number to be of the order of $10^{-4}$ even under high imposed electric fields. Note that the force balance equation in (2.5) can also be written as:

(2.6) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D72E}_{S}(\boldsymbol{u},p)+\unicode[STIX]{x1D72E}_{M}(\unicode[STIX]{x1D719}))=0, & \displaystyle\end{eqnarray}$$

(2.7a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D72E}_{S}(\boldsymbol{u},p)=2\unicode[STIX]{x1D707}\unicode[STIX]{x1D735}_{S}\boldsymbol{u}-pI,\quad 2\unicode[STIX]{x1D735}_{S}\boldsymbol{u}=(\unicode[STIX]{x1D735}\boldsymbol{u}+(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}), & \displaystyle\end{eqnarray}$$

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D72E}_{M}(\unicode[STIX]{x1D719})=\unicode[STIX]{x1D716}(\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}\otimes \unicode[STIX]{x1D735}\unicode[STIX]{x1D719}-{\textstyle \frac{1}{2}}|\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}|^{2}I). & \displaystyle\end{eqnarray}$$

The tensor $\unicode[STIX]{x1D72E}_{S}$ is the Stokes stress tensor and the tensor $\unicode[STIX]{x1D72E}_{M}$ is known as the Maxwell stress tensor.

The above equations, posed in the bulk $\unicode[STIX]{x1D6FA}_{in,ex}$ , are the same as those proposed in Saville (Reference Saville1997). However, in Saville (Reference Saville1997), the specification for interfacial conditions for the ionic concentrations is incomplete. The author proceeds using heuristic arguments to simplify and ignore certain terms to arrive at the Taylor–Melcher model. In contrast, here, we shall completely specify the interface conditions at $\unicode[STIX]{x1D6E4}$ , and will use the resulting system of equations as the starting point for all subsequent discussion. Given this, we shall refer to our model as the modified Saville model. At $\unicode[STIX]{x1D6E4}$ , the boundary conditions for the electrostatic potential are

(2.9a,b ) $$\begin{eqnarray}\displaystyle [\unicode[STIX]{x1D719}]=0,\quad \left[\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}\boldsymbol{n}}\right]=0, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{n}$ is the normal on $\unicode[STIX]{x1D6E4}$ (pointing from region $\unicode[STIX]{x1D6FA}_{in}$ to $\unicode[STIX]{x1D6FA}_{ex}$ ) and $[w]$ is the jump in the value of $w$ across the interface $\unicode[STIX]{x1D6E4}$ :

(2.10) $$\begin{eqnarray}\displaystyle [w]\equiv w|_{\unicode[STIX]{x1D6E4}_{in}}-w|_{\unicode[STIX]{x1D6E4}_{ex}}, & & \displaystyle\end{eqnarray}$$

with $w_{\unicode[STIX]{x1D6E4}_{in,ex}}$ denoting the value of $w$ evaluated at the $\unicode[STIX]{x1D6FA}_{in,ex}$ face of $\unicode[STIX]{x1D6E4}$ . For the fluid equations, we have the following interface conditions.

(2.11) $$\begin{eqnarray}\displaystyle [\boldsymbol{u}]=0,\quad [(\unicode[STIX]{x1D72E}_{S}(\boldsymbol{u},p)+\unicode[STIX]{x1D72E}_{M}(\unicode[STIX]{x1D719}))\boldsymbol{n}]=-\unicode[STIX]{x1D6FE}_{\ast }\unicode[STIX]{x1D705}\boldsymbol{n}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}_{\ast }$ is the surface tension coefficient and $\unicode[STIX]{x1D705}$ is the sum of the principal curvatures of the surface $\unicode[STIX]{x1D6E4}$ . The above boundary condition on the stress, with the help of (2.9), can also be recast as:

(2.12) $$\begin{eqnarray}\displaystyle [\unicode[STIX]{x1D72E}_{S}(\boldsymbol{u},p)\boldsymbol{n}]=-\!\left[\frac{\unicode[STIX]{x1D716}}{2}\left(\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}\boldsymbol{n}}\right)^{2}-|\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D719}|^{2}\right)\right]\boldsymbol{n}-\unicode[STIX]{x1D6FE}_{\ast }\unicode[STIX]{x1D705}\boldsymbol{n}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6E4}}$ denotes the surface gradient operator on the interface $\unicode[STIX]{x1D6E4}$ . Finally, we impose the kinematic (no-slip) condition; the interface moves with the local fluid velocity.

On both sides of the interface $\unicode[STIX]{x1D6E4}$ , we set

(2.13) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \left[-D_{C}\left(\unicode[STIX]{x1D735}c+c\frac{F}{RT}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}\right)\boldsymbol{\cdot }\boldsymbol{n}\right]=0,\quad l_{C}c|_{\unicode[STIX]{x1D6E4}_{ex}}=c|_{\unicode[STIX]{x1D6E4}_{in}},\\ \displaystyle \left[-D_{A}\left(\unicode[STIX]{x1D735}a-a\frac{F}{RT}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}\right)\boldsymbol{\cdot }\boldsymbol{n}\right]=0,\quad l_{A}a|_{\unicode[STIX]{x1D6E4}_{ex}}=a|_{\unicode[STIX]{x1D6E4}_{in}},\\ \displaystyle [-D_{S}\unicode[STIX]{x1D735}s\boldsymbol{\cdot }\boldsymbol{n}]=0,\quad l_{S}s|_{\unicode[STIX]{x1D6E4}_{ex}}=s|_{\unicode[STIX]{x1D6E4}_{in}},\end{array}\right\} & & \displaystyle\end{eqnarray}$$

where $l_{C,A,S}$ are positive constants known as partition coefficients. These boundary conditions are the same as those imposed in Zholkovskij et al. (Reference Zholkovskij, Masliyah and Czarnecki2002), and were missing in the electrodiffusion model proposed in Saville (Reference Saville1997). We refer the reader back to the discussion surrounding equation (1.5) for a thermodynamic interpretation of the partition coefficients $l_{C,A,S}$ . The partition coefficients $l_{C,A,S}$ must satisfy the following relations:

(2.14) $$\begin{eqnarray}\displaystyle \frac{l_{C}l_{A}}{l_{S}}=\frac{K_{eq,in}}{K_{eq,ex}},\quad \text{with }K_{eq,i}=\frac{k_{+,i}}{k_{-,i}},i=in,ex, & & \displaystyle\end{eqnarray}$$

where $k_{\pm ,i},i=in,ex$ are the rate constants in $\unicode[STIX]{x1D6FA}_{i},i=in,ex$ respectively. The above is needed for thermodynamic consistency; the requirement that the cation, the anion and the salt each have a defined energy level leads to the above restriction, and guarantees that the system of equations, as a whole, satisfies a free energy identity. The details of this are discussed in appendix A.

If $\unicode[STIX]{x1D6FA}$ is bounded, we must impose boundary conditions on the outer boundary $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ . Here, the boundary conditions are

(2.15a-e ) $$\begin{eqnarray}\displaystyle c=c_{\ast },\quad a=a_{\ast },\quad s=s_{\ast },\quad \boldsymbol{u}=0,\quad \unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{b}. & & \displaystyle\end{eqnarray}$$

Assuming electroneutral boundary conditions, the concentrations $c_{\ast },a_{\ast }$ and $s_{\ast }$ satisfy the following equations:

(2.16) $$\begin{eqnarray}\displaystyle k_{+,ex}s_{\ast }-k_{-,ex}c_{\ast }a_{\ast }=0,\quad c_{\ast }=a_{\ast }, & & \displaystyle\end{eqnarray}$$

where $k_{\pm ,ex}$ denotes the reaction rate constants in $\unicode[STIX]{x1D6FA}_{ex}$ . We may interpret $\unicode[STIX]{x1D719}_{b}$ as the externally imposed voltage, and may allow $\unicode[STIX]{x1D719}_{b}$ to depend on time.

If $\unicode[STIX]{x1D6FA}=\mathbb{R}^{3}$ , we may set:

(2.17a-c ) $$\begin{eqnarray}\displaystyle \lim _{|\boldsymbol{x}|\rightarrow \infty }c=c_{\ast },\quad \lim _{|\boldsymbol{x}|\rightarrow \infty }a=a_{\ast },\quad \lim _{|\boldsymbol{x}|\rightarrow \infty }s=s_{\ast }, & & \displaystyle\end{eqnarray}$$

where $c_{\ast },a_{\ast }$ and $s_{\ast }$ are to satisfy (2.16). For the voltage and velocity field, we may set

(2.18a,b ) $$\begin{eqnarray}\displaystyle \lim _{|\boldsymbol{x}|\rightarrow \infty }(\unicode[STIX]{x1D719}-\unicode[STIX]{x1D719}_{b})=0,\quad \lim _{|\boldsymbol{x}|\rightarrow \infty }\boldsymbol{u}=0, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D719}_{b}(\boldsymbol{x},t)$ is a given function that prescribes the behaviour of the voltage at infinity.

2.2 Non-dimensionalization

We now non-dimensionalize the above equations. Let the quantities $\widehat{\cdot }$ denote the dimensionless quantities. We set:

(2.19) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle c=c_{\ast }\widehat{c},\quad a=c_{\ast }\widehat{a},\quad s=s_{\ast }\widehat{s},\quad \unicode[STIX]{x1D719}=\frac{RT}{F}\widehat{\unicode[STIX]{x1D719}},\quad \unicode[STIX]{x1D719}_{b}=\frac{RT}{F}\widehat{\unicode[STIX]{x1D719}_{b}},\\ \displaystyle D_{C,A,S}=D_{\ast }\widehat{D}_{C,A,S},\quad \unicode[STIX]{x1D716}=\unicode[STIX]{x1D716}_{\ast }\widehat{\unicode[STIX]{x1D716}},\quad \unicode[STIX]{x1D707}=\unicode[STIX]{x1D707}_{\ast }\widehat{\unicode[STIX]{x1D707}},\quad \boldsymbol{x}=L\widehat{\boldsymbol{x}},\quad t=t_{0}\widehat{t}=\frac{L}{u_{\ast }}\widehat{t},\quad \unicode[STIX]{x1D705}=\frac{\widehat{\unicode[STIX]{x1D705}}}{L},\\ \displaystyle \boldsymbol{u}=u_{\ast }\widehat{\boldsymbol{u}}=\frac{\unicode[STIX]{x1D716}_{\ast }(RT/F)^{2}}{\unicode[STIX]{x1D707}_{\ast }L}\widehat{\boldsymbol{u}},\quad p=\frac{\unicode[STIX]{x1D716}_{\ast }(RT/F)^{2}}{L^{2}}\widehat{p},\end{array}\right\} & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $D_{\ast }$ , $\unicode[STIX]{x1D716}_{\ast }$ , $\unicode[STIX]{x1D707}_{\ast }$ and $L$ are the characteristic diffusion coefficient, dielectric constant, viscosity and length respectively. We have taken the characteristic voltage to be the thermal voltage $RT/F$ rather than the characteristic magnitude of the externally imposed voltage. The latter scaling will be discussed in § 4.3. Using the above non-dimensionalization, we have, in $\unicode[STIX]{x1D6FA}_{in}\cup \unicode[STIX]{x1D6FA}_{ex}$ :

(2.20) $$\begin{eqnarray}\displaystyle & \displaystyle Pe\left(\frac{\unicode[STIX]{x2202}c}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{u}c)\right)=\unicode[STIX]{x1D735}\boldsymbol{\cdot }(D_{C}(\unicode[STIX]{x1D735}c+c\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}))+\frac{k_{ex}k}{\unicode[STIX]{x1D6FC}}\left(s-\frac{ca}{K}\right), & \displaystyle\end{eqnarray}$$

(2.21) $$\begin{eqnarray}\displaystyle & \displaystyle Pe\left(\frac{\unicode[STIX]{x2202}a}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{u}a)\right)=\unicode[STIX]{x1D735}\boldsymbol{\cdot }(D_{A}(\unicode[STIX]{x1D735}a-a\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}))+\frac{k_{ex}k}{\unicode[STIX]{x1D6FC}}\left(s-\frac{ca}{K}\right), & \displaystyle\end{eqnarray}$$

(2.22) $$\begin{eqnarray}\displaystyle & \displaystyle Pe\left(\frac{\unicode[STIX]{x2202}s}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{u}s)\right)=\unicode[STIX]{x1D735}\boldsymbol{\cdot }(D_{S}\unicode[STIX]{x1D735}s)-k_{ex}k\left(s-\frac{ca}{K}\right), & \displaystyle\end{eqnarray}$$

(2.23) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D716}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719})=(c-a), & \displaystyle\end{eqnarray}$$

(2.24) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FF}^{2}(\unicode[STIX]{x1D707}\unicode[STIX]{x0394}\boldsymbol{u}-\unicode[STIX]{x1D735}p)=(c-a)\unicode[STIX]{x1D735}\unicode[STIX]{x1D719},\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0, & \displaystyle\end{eqnarray}$$

where we have dropped $\widehat{\cdot }$ for notational simplicity. $Pe$ is the Péclet number

(2.25a-e ) $$\begin{eqnarray}\displaystyle Pe=\frac{u_{\ast }L}{D_{\ast }},\!\quad k_{ex}=\frac{k_{+,ex}L^{2}}{D_{\ast }},\!\quad \unicode[STIX]{x1D6FC}=\frac{c_{\ast }}{s_{\ast }}=\frac{k_{+,ex}}{k_{-,ex}c_{\ast }},\!\quad \unicode[STIX]{x1D6FF}=\frac{r_{D}}{L},\!\quad r_{D}=\sqrt{\frac{\unicode[STIX]{x1D716}_{\ast }RT/F}{Fc_{\ast }}},\qquad \quad ~\ & & \displaystyle\end{eqnarray}$$

and

(2.26) $$\begin{eqnarray}\displaystyle k=\left\{\begin{array}{@{}ll@{}}k_{+,in}/k_{+,ex}\quad & \text{in }\unicode[STIX]{x1D6FA}_{in},\\ 1\quad & \text{in }\unicode[STIX]{x1D6FA}_{ex},\end{array}\right.,\quad K=\left\{\begin{array}{@{}ll@{}}\displaystyle K_{in}\equiv K_{eq,in}/K_{eq,ex}=\frac{l_{C}l_{A}}{l_{S}}\quad & \text{in }\unicode[STIX]{x1D6FA}_{in},\\ 1\quad & \text{in }\unicode[STIX]{x1D6FA}_{ex}.\end{array}\right. & & \displaystyle\end{eqnarray}$$

Note that the last equality comes from (2.16), and the expression for $K_{i}$ is derived from the thermodynamic restrictions (2.14). The dimensionless boundary conditions at the interface are

(2.27a,b ) $$\begin{eqnarray}\displaystyle [-D_{C}(\unicode[STIX]{x1D735}c+c\unicode[STIX]{x1D735}\unicode[STIX]{x1D719})\boldsymbol{\cdot }\boldsymbol{n}]=0,\quad l_{C}c|_{\unicode[STIX]{x1D6E4}_{ex}}=c|_{\unicode[STIX]{x1D6E4}_{in}}, & & \displaystyle\end{eqnarray}$$

(2.28a,b ) $$\begin{eqnarray}\displaystyle [-D_{A}(\unicode[STIX]{x1D735}a-a\unicode[STIX]{x1D735}\unicode[STIX]{x1D719})\boldsymbol{\cdot }\boldsymbol{n}]=0,\quad l_{A}a|_{\unicode[STIX]{x1D6E4}_{ex}}=a|_{\unicode[STIX]{x1D6E4}_{in}}, & & \displaystyle\end{eqnarray}$$

(2.29a,b ) $$\begin{eqnarray}\displaystyle [-D_{S}\unicode[STIX]{x1D735}s\boldsymbol{\cdot }\boldsymbol{n}]=0,\quad l_{S}s|_{\unicode[STIX]{x1D6E4}_{ex}}=s|_{\unicode[STIX]{x1D6E4}_{in}}, & & \displaystyle\end{eqnarray}$$

(2.30) $$\begin{eqnarray}\displaystyle & \displaystyle [\unicode[STIX]{x1D716}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}\boldsymbol{\cdot }\boldsymbol{n}]=0,\quad [\unicode[STIX]{x1D719}]=0, & \displaystyle\end{eqnarray}$$

(2.31) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FF}^{2}[\unicode[STIX]{x1D6F4}(\boldsymbol{u},p)\boldsymbol{n}]=-\unicode[STIX]{x1D6FF}^{2}\left[\frac{\unicode[STIX]{x1D716}}{2}\left(\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}\boldsymbol{n}}\right)^{2}-|\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D719}|^{2}\right)\right]\boldsymbol{n}-\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D705}\boldsymbol{n}, & \displaystyle\end{eqnarray}$$

(2.32) $$\begin{eqnarray}\displaystyle & \displaystyle [\boldsymbol{u}]=0, & \displaystyle\end{eqnarray}$$

with the dimensionless surface tension coefficient $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{\ast }/c_{\ast }RTL$ . The dimensionless boundary conditions at $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ are:

(2.33a-c ) $$\begin{eqnarray}\displaystyle c=a=s=1,\quad \unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{b},\quad \boldsymbol{u}=0. & & \displaystyle\end{eqnarray}$$

When $\unicode[STIX]{x1D6FA}=\mathbb{R}^{3}$ , the above must be replaced by appropriate limits as $|\boldsymbol{x}|\rightarrow \infty$ as in (2.17) and (2.18).

4.1 Derivation of the Taylor–Melcher model

As discussed above, the cases $l_{C}=l_{A}$ and $l_{C}\neq l_{A}$ are fundamentally different. In this section, we consider the case $l_{C}=l_{A}$ . Under this assumption, we consider the limit $\unicode[STIX]{x1D6FF}\rightarrow 0$ in the charge diffusion model to derive the TM model. In the calculations to follow, we scale $Pe$ and $\unicode[STIX]{x1D6FE}$ with respect to $\unicode[STIX]{x1D6FF}$ as follows:

(4.1a,b ) $$\begin{eqnarray}\displaystyle Pe=\frac{\unicode[STIX]{x1D712}}{\unicode[STIX]{x1D6FF}^{2}},\quad \unicode[STIX]{x1D6FE}=\widehat{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6FF}^{2}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D712}$ and $\widehat{\unicode[STIX]{x1D6FE}}$ are constant. As we shall see, this scaling yields the full TM model with surface charge convection for arbitrary interfacial geometry. In fact, there are three time scales in the TM model, the Maxwell–Wagner charge relaxation time scale, the electrohydrodynamic time scale and the capillary time scale (Salipante & Vlahovska Reference Salipante and Vlahovska2010). All terms in the TM model will be important only when these three time scales are of the same order, and the above scaling ensures this. Different distinguished limits can be obtained depending on how we scale the two dimensionless numbers $Pe$ and $\unicode[STIX]{x1D6FE}$ . The scaling of $Pe$ with respect to $\unicode[STIX]{x1D6FF}$ determines whether surface charge convection will be important. This is summarized in table 1. $Pe=O(\unicode[STIX]{x1D6FF}^{-1})$ leads to the TM model with surface charge convection as an $O(\unicode[STIX]{x1D6FF})$ correction. For smaller $Pe$ , surface charge convection is negligible. We shall not present these calculations here since they are quite similar to (and simpler than) the calculations presented here. In appendix A, we shall also see that (4.1) is the natural one as dictated by the free energy identity.

Table 1.

The presence of surface charge convection in the limiting TM model depending on the scaling of $Pe$ or $Pe_{E}$ with respect to $\unicode[STIX]{x1D6FF}$ or $\unicode[STIX]{x1D6FF}_{E}$ . The dimensionless constants $Pe_{E}$ and $\unicode[STIX]{x1D6FF}_{E}$ pertain to asymptotics under strong electric fields discussed in § 4.3, and are defined in (4.79) and (4.82).

We expand $q$ in powers of $\unicode[STIX]{x1D6FF}$ as:

(4.2) $$\begin{eqnarray}\displaystyle q=q_{0}+\unicode[STIX]{x1D6FF}q_{1}+\unicode[STIX]{x1D6FF}^{2}q_{2}+\cdots \,, & & \displaystyle\end{eqnarray}$$

and likewise for other variables $\unicode[STIX]{x1D719}$ and $\boldsymbol{u}$ .

Let us first consider equations in the bulk or outer layer. We see from (3.17) that:

(4.3) $$\begin{eqnarray}\displaystyle q_{0}=q_{1}=0. & & \displaystyle\end{eqnarray}$$

This is also compatible with (3.10). The leading non-trivial equations we obtain from (3.17), (3.10) and (3.18) are:

(4.4) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D712}\left(\frac{\unicode[STIX]{x2202}q_{2}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{u}_{0}q_{2})\right)=\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70E}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{0}), & \displaystyle\end{eqnarray}$$

(4.5) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D716}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{0})=q_{2}, & \displaystyle\end{eqnarray}$$

(4.6) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D707}\unicode[STIX]{x0394}\boldsymbol{u}_{0}-\unicode[STIX]{x1D735}p_{0}=q_{2}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{0},\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{0}=0, & \displaystyle\end{eqnarray}$$

where the conductivity $\unicode[STIX]{x1D70E}$ is given by:

(4.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}=\unicode[STIX]{x1D6F4}(q_{0}=0)=(D_{C}+D_{A})\sqrt{S}=\left\{\begin{array}{@{}ll@{}}(D_{C,in}+D_{A,in})\sqrt{l_{C}l_{A}}\quad & \text{in }\unicode[STIX]{x1D6FA}_{in},\\ D_{C,ex}+D_{A,ex}\quad & \text{in }\unicode[STIX]{x1D6FA}_{ex},\end{array}\right. & & \displaystyle\end{eqnarray}$$

where we used the relation (3.9), (3.21) and (2.14) in the last equality. We shall sometimes use the notation $\unicode[STIX]{x1D70E}_{k},k=\text{in},\text{ex}$ to denote the value of $\unicode[STIX]{x1D70E}$ in the two regions. In this section, $l_{C}=l_{A}$ by assumption, and therefore, $\sqrt{S}=l_{C}$ in $\unicode[STIX]{x1D6FA}_{in}$ .

In order to obtain interface conditions for the above equations in the outer layer, we perform a boundary layer analysis near $\unicode[STIX]{x1D6E4}$ . Small $\unicode[STIX]{x1D6FF}$ in (3.17) implies a boundary layer of thickness $\unicode[STIX]{x1D6FF}$ near $\unicode[STIX]{x1D6E4}$ (see figure 7). To study this boundary layer, we introduce an curvilinear coordinate system $(\unicode[STIX]{x1D709},\boldsymbol{\unicode[STIX]{x1D702}})=(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702}^{1},\unicode[STIX]{x1D702}^{2})$ fitted to the interface $\unicode[STIX]{x1D6E4}$ (figure 8). Let $\boldsymbol{\unicode[STIX]{x1D702}}$ be a local coordinate system on the surface $\unicode[STIX]{x1D6E4}$ so that $\boldsymbol{x}=\boldsymbol{X}(\boldsymbol{\unicode[STIX]{x1D702}},t)$ is the Cartesian coordinate position of the evolving interface. We let points of fixed $\boldsymbol{\unicode[STIX]{x1D702}}$ move with the normal velocity of the interface:

(4.8a,b ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{X}}{\unicode[STIX]{x2202}t}=u_{\bot }(\boldsymbol{\unicode[STIX]{x1D702}},t)\boldsymbol{n}(\boldsymbol{\unicode[STIX]{x1D702}},t),\quad u_{\bot }(\boldsymbol{\unicode[STIX]{x1D702}},t)=\boldsymbol{u}(\boldsymbol{X}(\boldsymbol{\unicode[STIX]{x1D702}},t),t)\boldsymbol{\cdot }\boldsymbol{n}(\boldsymbol{\unicode[STIX]{x1D702}},t), & & \displaystyle\end{eqnarray}$$

where, $\boldsymbol{n}$ is the outward pointing unit normal (pointing from $\unicode[STIX]{x1D6FA}_{in}$ to $\unicode[STIX]{x1D6FA}_{ex}$ ). The map ${\mathcal{T}}$ :

(4.9) $$\begin{eqnarray}\displaystyle {\mathcal{T}}(\unicode[STIX]{x1D709},\boldsymbol{\unicode[STIX]{x1D702}},t)\mapsto \boldsymbol{X}(\boldsymbol{\unicode[STIX]{x1D702}},t)+\unicode[STIX]{x1D709}\boldsymbol{n}(\boldsymbol{\unicode[STIX]{x1D702}},t) & & \displaystyle\end{eqnarray}$$

defines the desired local curvilinear coordinate system. The coordinate $\unicode[STIX]{x1D709}$ is thus the signed distance function from the interface, where $\unicode[STIX]{x1D709}>0$ is on the $\unicode[STIX]{x1D6FA}_{ex}$ side and $\unicode[STIX]{x1D709}<0$ on the $\unicode[STIX]{x1D6FA}_{in}$ side. We use an arbitrary point $\boldsymbol{x}=\boldsymbol{x}_{\ast }\in \unicode[STIX]{x1D6E4}$ at time $t=t_{\ast }$ as the origin in the $(\unicode[STIX]{x1D709},\boldsymbol{\unicode[STIX]{x1D702}})$ coordinate system in the following boundary layer analysis.

Figure 8.

The curvilinear coordinate system. $\unicode[STIX]{x1D709}$ is perpendicular to the surface $\unicode[STIX]{x1D6E4}$ , where $\unicode[STIX]{x1D709}<0$ is on the $\unicode[STIX]{x1D6FA}_{in}$ side. The $\unicode[STIX]{x1D709}=0$ surface corresponds to $\unicode[STIX]{x1D6E4}$ and $\boldsymbol{\unicode[STIX]{x1D702}}=(\unicode[STIX]{x1D702}^{1},\unicode[STIX]{x1D702}^{2})$ is the coordinate system fitted to $\unicode[STIX]{x1D6E4}$ .

In an abuse of notation, $q,\unicode[STIX]{x1D719}$ and other scalar functions will be seen interchangeably as functions of $(\unicode[STIX]{x1D709},\boldsymbol{\unicode[STIX]{x1D702}},t)$ as well as of $(\boldsymbol{x},t)$ . For the velocity field $\boldsymbol{u}$ , we introduce the velocity functions $(u,v^{1},v^{2})$ adapted to the curvilinear coordinate system:

(4.10) $$\begin{eqnarray}\displaystyle \boldsymbol{u}(\boldsymbol{X},t)=u(\unicode[STIX]{x1D709},\boldsymbol{\unicode[STIX]{x1D702}},t)\frac{\unicode[STIX]{x2202}{\mathcal{T}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}+v^{i}(\unicode[STIX]{x1D709},\boldsymbol{\unicode[STIX]{x1D702}},t)\frac{\unicode[STIX]{x2202}{\mathcal{T}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}, & & \displaystyle\end{eqnarray}$$

where ${\mathcal{T}}$ is the local coordinate map (4.9). In the above and henceforth, we shall use the summation convention for repeated indices. The function $u$ is the fluid velocity normal to level sets of the signed distance function of $\unicode[STIX]{x1D6E4}$ and $v^{1},v^{2}$ are the fluid velocities tangent to the same level sets.

First we introduce the stretched boundary layer coordinate system $\unicode[STIX]{x1D709}^{\prime }\equiv \unicode[STIX]{x1D709}/\unicode[STIX]{x1D6FF}$ close to the interface, with the inner layer variables denoted by $\widetilde{\cdot }$ :

(4.11) $$\begin{eqnarray}\displaystyle \widetilde{q}(\unicode[STIX]{x1D709}^{\prime },\boldsymbol{\unicode[STIX]{x1D702}},t)=q(\unicode[STIX]{x1D709}/\unicode[STIX]{x1D6FF},\boldsymbol{\unicode[STIX]{x1D702}},t), & & \displaystyle\end{eqnarray}$$

and likewise for $\unicode[STIX]{x1D719},u,v_{1}$ and $v_{2}$ . We then expand each of these variables in powers of $\unicode[STIX]{x1D6FF}$ as follows:

(4.12) $$\begin{eqnarray}\displaystyle \widetilde{q}=\widetilde{q}_{0}+\unicode[STIX]{x1D6FF}\widetilde{q}_{1}+\unicode[STIX]{x1D6FF}^{2}\widetilde{q}_{2}\cdots \,. & & \displaystyle\end{eqnarray}$$

Expressions resulting from the Stokes equation (3.18) in inner layer coordinates are discussed in appendix B.

In the calculations to follow, the symbol $[\cdot ]$ , applied to an inner layer variable, is the jump in the value across $\unicode[STIX]{x1D709}^{\prime }=0$ ( $[\cdot ]=\cdot |_{\unicode[STIX]{x1D709}^{\prime }=0-}-\cdot |_{\unicode[STIX]{x1D709}^{\prime }=0+}$ ). When applied to an outer layer variable, the symbol $[\cdot ]$ denotes the jump in the value across $\unicode[STIX]{x1D709}=0$ : $[\cdot ]=\cdot |_{\unicode[STIX]{x1D709}=0-}-\cdot |_{\unicode[STIX]{x1D709}=0+}$ .

From equations (3.17) and (3.18) (see (B 20)) we have:

(4.13a-c ) $$\begin{eqnarray}\displaystyle -\frac{\unicode[STIX]{x2202}^{2}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}=\widetilde{q}_{0},\quad \widetilde{q}_{0}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=0\rightarrow \widetilde{q}_{0}=0,\quad \frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=0. & & \displaystyle\end{eqnarray}$$

The interface conditions (2.30) yield

(4.14) $$\begin{eqnarray}\displaystyle [\widetilde{\unicode[STIX]{x1D719}}_{0}]=0. & & \displaystyle\end{eqnarray}$$

Given the matching conditions

(4.15) $$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\widetilde{\unicode[STIX]{x1D719}}_{0}=\lim _{\unicode[STIX]{x1D709}\rightarrow 0\pm }\unicode[STIX]{x1D719}_{0}\equiv \unicode[STIX]{x1D719}_{\pm }, & & \displaystyle\end{eqnarray}$$

we see that

(4.16) $$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D719}}_{0}\equiv \lim _{\unicode[STIX]{x1D709}\rightarrow 0\pm }\unicode[STIX]{x1D719}_{0}\quad \text{for }-\!\infty <\unicode[STIX]{x1D709}^{\prime }<\infty . & & \displaystyle\end{eqnarray}$$

In particular, this implies that

(4.17) $$\begin{eqnarray}\displaystyle [\unicode[STIX]{x1D719}_{0}]=0. & & \displaystyle\end{eqnarray}$$

This shows that the voltage, to leading order, must be continuous across the boundary layer. Expressions (4.13) and (4.16) are compatible with the following additional equations and boundary conditions that we obtain as leading-order equations from (2.30) and (3.23):

(4.18a,b ) $$\begin{eqnarray}\displaystyle \left[\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right]=0,\quad l_{C}\widetilde{q}_{0}|_{\unicode[STIX]{x1D709}^{\prime }=0+}=\widetilde{q}_{0}|_{\unicode[STIX]{x1D709}^{\prime }=0-}. & & \displaystyle\end{eqnarray}$$

Note in particular that $\widetilde{q}_{0}\equiv 0$ is compatible with (3.15) only because $l_{C}=l_{A}$ (in which case (3.15) reduces to (3.23)).

We now turn to equations at the next order. From (3.18) we obtain

(4.19a,b ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\widetilde{u}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=0,\quad \unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}^{2}\widetilde{v}_{0}^{i}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}=\widetilde{q}_{0}g^{ij}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}=0,\quad i=1,2, & & \displaystyle\end{eqnarray}$$

where we used (4.13) in the second expression. This shows that

(4.20) $$\begin{eqnarray}\displaystyle \widetilde{u}_{0}=\left\{\begin{array}{@{}ll@{}}U_{-}\quad & \text{for }\unicode[STIX]{x1D709}^{\prime }<0,\\ U_{+}\quad & \text{for }\unicode[STIX]{x1D709}^{\prime }>0,\end{array}\right.\quad \widetilde{v}_{0}^{i}=\left\{\begin{array}{@{}ll@{}}V_{-}^{i}\unicode[STIX]{x1D709}^{\prime }+W_{-}^{i}\quad & \text{for }\unicode[STIX]{x1D709}^{\prime }<0,\\ V_{+}^{i}\unicode[STIX]{x1D709}^{\prime }+W_{+}^{i}\quad & \text{for }\unicode[STIX]{x1D709}^{\prime }>0,\end{array}\right. & & \displaystyle\end{eqnarray}$$

where $U_{\pm },V_{\pm }^{i},W_{\pm }^{i}$ do not depend on $\unicode[STIX]{x1D709}^{\prime }$ . Condition (2.32) at the interface $\unicode[STIX]{x1D709}^{\prime }=0$ gives rise to:

(4.21) $$\begin{eqnarray}\displaystyle [\widetilde{u}_{0}]=[\widetilde{v}_{0}^{i}]=0. & & \displaystyle\end{eqnarray}$$

The matching conditions at $\unicode[STIX]{x1D709}^{\prime }=\pm \infty$ are given by:

(4.22a,b ) $$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\widetilde{u}_{0}=\lim _{\unicode[STIX]{x1D709}\rightarrow 0\pm }u_{0},\quad \lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\widetilde{v}_{0}^{i}=\lim _{\unicode[STIX]{x1D709}\rightarrow 0\pm }v_{0}^{i}. & & \displaystyle\end{eqnarray}$$

Implicit in the above matching conditions is that all of the above limits exist. In particular, the limits of $\widetilde{v}_{0}^{i}$ at $\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty$ must exist, and therefore, $V_{\pm }^{i}=0$ in (4.20). Using (4.21), we thus conclude that:

(4.23a,b ) $$\begin{eqnarray}\displaystyle \widetilde{u}_{0}\equiv \lim _{\unicode[STIX]{x1D709}\rightarrow 0\pm }u_{0}\quad \text{and}\quad \widetilde{v}_{0}^{i}\equiv \lim _{\unicode[STIX]{x1D709}\rightarrow 0\pm }v_{0}^{i},i=1,2,\quad \text{for }-\!\infty <\unicode[STIX]{x1D709}^{\prime }<\infty . & & \displaystyle\end{eqnarray}$$

In particular, this implies that:

(4.24) $$\begin{eqnarray}\displaystyle [u_{0}]=[v_{0}^{i}]=0,\quad i=1,2. & & \displaystyle\end{eqnarray}$$

This shows that the velocity field in the outer layer is continuous across the boundary layer to leading order.

We need further interface conditions to solve the outer layer equations (4.4)–(4.6). We first consider the incompressibility condition in (3.18) and (2.32), from which we obtain (see (B 8)):

(4.25a,b ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\widetilde{u}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+\unicode[STIX]{x1D705}\widetilde{u}_{0}+\frac{1}{\sqrt{|g|}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}(\sqrt{|g|}\widetilde{v}_{0}^{i})=0,\quad [\widetilde{u}_{1}]=0, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D705}$ is the sum of the two principal curvatures of the surface $\unicode[STIX]{x1D6E4}$ and $|g|$ is the determinant of the metric tensor $g_{ij}$ associated with the interface:

(4.26a,b ) $$\begin{eqnarray}\displaystyle g_{ij}=\frac{\unicode[STIX]{x2202}\boldsymbol{X}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}\cdot \frac{\unicode[STIX]{x2202}\boldsymbol{X}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}},\quad |g|=\det (g_{ij}). & & \displaystyle\end{eqnarray}$$

By (4.22), $\widetilde{u}_{0}$ and $\widetilde{v}_{0}^{i}$ do not depend on $\unicode[STIX]{x1D709}^{\prime }$ and are equal to the outer layer values $u_{0}$ and $v_{0}^{i}$ respectively. Thus:

(4.27) $$\begin{eqnarray}\displaystyle \widetilde{u}_{1}=-\!\left(\unicode[STIX]{x1D705}u_{0}+\frac{1}{\sqrt{|g|}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}(\sqrt{|g|}v_{0}^{i})\right)\unicode[STIX]{x1D709}^{\prime }+\widetilde{u}_{\bot ,1}, & & \displaystyle\end{eqnarray}$$

where $\widetilde{u}_{\bot ,1}$ is the value of $\widetilde{u}_{1}$ at $\unicode[STIX]{x1D709}^{\prime }=0$ .

We first focus on the equations for $q$ and $\unicode[STIX]{x1D719}$ . From (3.10) and (3.17), we obtain the following equations:

(4.28) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D712}\left(\frac{\unicode[STIX]{x2202}\widetilde{q}_{1}}{\unicode[STIX]{x2202}t}+(\widetilde{u}_{1}-\widetilde{u}_{\bot ,1})\frac{\unicode[STIX]{x2202}\widetilde{q}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+\widetilde{v}_{0}^{i}\frac{\unicode[STIX]{x2202}\widetilde{q}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}\right)=-\frac{\unicode[STIX]{x2202}\widetilde{J}_{q,1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}, & \displaystyle\end{eqnarray}$$

(4.29) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{J}_{q,1}=-\frac{1}{2}(D_{C}+D_{A})\frac{\unicode[STIX]{x2202}\widetilde{q}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}-\unicode[STIX]{x1D70E}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}, & \displaystyle\end{eqnarray}$$

(4.30) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}^{2}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}=\widetilde{q}_{1}, & \displaystyle\end{eqnarray}$$

with interface conditions:

(4.31a,b ) $$\begin{eqnarray}\displaystyle [\widetilde{\unicode[STIX]{x1D719}}_{1}]=\left[\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right]=[\widetilde{J}_{q,1}]=0,\quad l_{C}\widetilde{q}_{1}|_{\unicode[STIX]{x1D709}^{\prime }=0+}=\widetilde{q}_{1}|_{\unicode[STIX]{x1D709}^{\prime }=0-}. & & \displaystyle\end{eqnarray}$$

The matching conditions for $\widetilde{q}_{1}$ are, by (4.3):

(4.32) $$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\widetilde{q}_{1}=0. & & \displaystyle\end{eqnarray}$$

The above, together with matching conditions for $\widetilde{\unicode[STIX]{x1D719}}_{1}$ , to be discussed shortly, should be sufficient to match the outer and inner layer solutions. However, the impossibility of obtaining closed-form solutions for the inner layer equations makes the matching procedure difficult without additional assumptions. We assume that $\widetilde{q}_{1}$ is integrable:

(4.33) $$\begin{eqnarray}\displaystyle \int _{-\infty }^{\infty }|\widetilde{q}_{1}|\,\text{d}\unicode[STIX]{x1D709}^{\prime }<\infty . & & \displaystyle\end{eqnarray}$$

Furthermore, we assume that:

(4.34a,b ) $$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\frac{\unicode[STIX]{x2202}\widetilde{q}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=0,\quad \lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\unicode[STIX]{x1D709}^{\prime }\widetilde{q}_{1}=0. & & \displaystyle\end{eqnarray}$$

The above two conditions in (4.34) are likely to be consequences of (4.33), (4.32) together with the fact that $\widetilde{q}_{1}$ satisfies (4.28)–(4.30), and hence, is redundant.

To obtain matching conditions for $\widetilde{\unicode[STIX]{x1D719}}_{1}$ , we invoke Kaplun’s matching procedure by introducing an intermediate coordinate system $\unicode[STIX]{x1D709}_{m}$ which scales as $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D708}}\unicode[STIX]{x1D709}_{m},0<\unicode[STIX]{x1D708}<1$ . The result of this analysis is that:

(4.35) $$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=\lim _{\unicode[STIX]{x1D709}\rightarrow 0\pm }\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}. & & \displaystyle\end{eqnarray}$$

The derivation of these conditions relies on (4.30) and (4.33), which ensures that the limits $\lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }$ exist. An immediate consequence of the above is that we may integrate (4.30) from $\unicode[STIX]{x1D709}^{\prime }=-\infty$ to $\infty$ to find that

(4.36a,b ) $$\begin{eqnarray}\displaystyle \left[\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right]=q_{\unicode[STIX]{x1D6E4}},\quad q_{\unicode[STIX]{x1D6E4}}\equiv \int _{-\infty }^{\infty }\widetilde{q}_{1}\,\text{d}\unicode[STIX]{x1D709}^{\prime }. & & \displaystyle\end{eqnarray}$$

Next, integrate equation (4.28) from $\unicode[STIX]{x1D709}=0$ to $\infty$ . Let us first consider the left-hand side. We have:

(4.37) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{0}^{\infty }\left(\frac{\unicode[STIX]{x2202}\widetilde{q}_{1}}{\unicode[STIX]{x2202}t}+(\widetilde{u}_{1}-\widetilde{u}_{\bot ,1})\frac{\unicode[STIX]{x2202}\widetilde{q}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+\widetilde{v}_{0}^{i}\frac{\unicode[STIX]{x2202}\widetilde{q}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}\right)\,\text{d}\unicode[STIX]{x1D709}^{\prime }\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{\unicode[STIX]{x2202}q_{\unicode[STIX]{x1D6E4}}^{+}}{\unicode[STIX]{x2202}t}+\left(\unicode[STIX]{x1D705}u_{0}+\frac{1}{\sqrt{|g|}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}(\sqrt{|g|}v_{0}^{i})\right)q_{\unicode[STIX]{x1D6E4}}^{+}+v_{0}^{i}\frac{\unicode[STIX]{x2202}q_{\unicode[STIX]{x1D6E4}}^{+}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}},\quad q_{\unicode[STIX]{x1D6E4}}^{+}\equiv \int _{0}^{\infty }\widetilde{q}_{1}\,\text{d}\unicode[STIX]{x1D709}^{\prime }.\qquad\end{eqnarray}$$

Note that $q_{\unicode[STIX]{x1D6E4}}^{+}$ is well defined thanks to (4.33). The second term in the integrand was integrated by parts, where we used (4.27) and (4.34). In the last term in the integrand, we replaced $\widetilde{v}_{0}^{i}$ with the outer layer value using (4.22). Let us now turn to the right-hand side of (4.28). We integrate to obtain:

(4.38) $$\begin{eqnarray}\displaystyle \int _{0}^{\infty }-\frac{\unicode[STIX]{x2202}\widetilde{J}_{q,1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\,\text{d}\unicode[STIX]{x1D709}^{\prime }=\left.\unicode[STIX]{x1D70E}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right|_{\unicode[STIX]{x1D709}=0+}+\widetilde{J}_{q,1}|_{\unicode[STIX]{x1D709}^{\prime }=0+}, & & \displaystyle\end{eqnarray}$$

where we used (4.34) and (4.35). Combining the above two equations, we have:

(4.39) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}\left(\frac{\unicode[STIX]{x2202}q_{\unicode[STIX]{x1D6E4}}^{+}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D705}u_{0}q_{\unicode[STIX]{x1D6E4}}^{+}+\frac{1}{\sqrt{|g|}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}(\sqrt{|g|}v_{0}^{i}q_{\unicode[STIX]{x1D6E4}}^{+})\right)=\left.\unicode[STIX]{x1D70E}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right|_{\unicode[STIX]{x1D709}=0+}+\widetilde{J}_{q,1}|_{\unicode[STIX]{x1D709}^{\prime }=0+}. & & \displaystyle\end{eqnarray}$$

We may perform a similar calculation on the $\unicode[STIX]{x1D709}^{\prime }<0$ side and use (4.31) to conclude that:

(4.40) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}\left(\frac{\unicode[STIX]{x2202}q_{\unicode[STIX]{x1D6E4}}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D705}u_{0}q_{\unicode[STIX]{x1D6E4}}+\frac{1}{\sqrt{|g|}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}(\sqrt{|g|}v_{0}^{i}q_{\unicode[STIX]{x1D6E4}})\right)=-\!\left[\unicode[STIX]{x1D70E}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right], & & \displaystyle\end{eqnarray}$$

where $q_{\unicode[STIX]{x1D6E4}}$ was defined in (4.36).

To obtain the stress boundary conditions, let us first focus on the velocity field component $u$ . Equation (3.18) yields (B 22), which together with (4.13) and (4.23) gives:

(4.41) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}^{2}\widetilde{u}_{1}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}-\frac{\unicode[STIX]{x2202}\widetilde{p}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}-\widetilde{q}_{1}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=0. & & \displaystyle\end{eqnarray}$$

The interface conditions (2.32) and (2.31) at $\unicode[STIX]{x1D709}^{\prime }=0$ yield (see (B 15)):

(4.42) $$\begin{eqnarray}\displaystyle \left[2\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}\widetilde{u}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}-\widetilde{p}_{0}\right]=-\!\left[\frac{\unicode[STIX]{x1D716}}{2}\left(\left(\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right)^{2}-g^{ij}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}\right)\right]-\widehat{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D705}, & & \displaystyle\end{eqnarray}$$

where $g^{ij}$ are the components of the inverse of the metric tensor $g_{ij}$ . The matching conditions at $\unicode[STIX]{x1D709}^{\prime }=\pm \infty$ for $p_{0}$ are

(4.43) $$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\widetilde{p}_{0}=\lim _{\unicode[STIX]{x1D709}\rightarrow 0\pm }p_{0}. & & \displaystyle\end{eqnarray}$$

Note that the incompressibility condition in the outer layer, at $\unicode[STIX]{x1D709}=0\pm$ , can be expressed as:

(4.44) $$\begin{eqnarray}\displaystyle \left.\left(\frac{\unicode[STIX]{x2202}u_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}+\unicode[STIX]{x1D705}u_{0}+\frac{1}{\sqrt{|g|}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}(\sqrt{|g|}v_{0}^{i})\right)\right|_{\unicode[STIX]{x1D709}=0\pm }=0. & & \displaystyle\end{eqnarray}$$

Comparing this expression with (4.27) yields:

(4.45) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\widetilde{u}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=\left.\frac{\unicode[STIX]{x2202}u_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right|_{\unicode[STIX]{x1D709}=0\pm }. & & \displaystyle\end{eqnarray}$$

We see from (4.27) that $\widetilde{u}_{1}$ is a linear function of $\unicode[STIX]{x1D709}^{\prime }$ . This observation may be used together with (4.41) to obtain:

(4.46) $$\begin{eqnarray}\displaystyle 2\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}^{2}\widetilde{u}_{1}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}-\frac{\unicode[STIX]{x2202}\widetilde{p}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=\widetilde{q}_{1}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}. & & \displaystyle\end{eqnarray}$$

Let us integrate the above from $\unicode[STIX]{x1D709}^{\prime }=0$ to $\unicode[STIX]{x1D709}^{\prime }=\infty$ . We have:

(4.47) $$\begin{eqnarray}\displaystyle \left.\left(2\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}u_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}-p_{0}\right)\right|_{\unicode[STIX]{x1D709}=0+}-\left.\left(2\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}\widetilde{u}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}-\widetilde{p}_{0}\right)\right|_{\unicode[STIX]{x1D709}^{\prime }=0+}=\int _{0}^{\infty }\widetilde{q}_{1}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\,\text{d}\unicode[STIX]{x1D709}^{\prime }, & & \displaystyle\end{eqnarray}$$

where we used (4.45) to obtain the first term in the above. Note that the right-hand side of the above is indeed integrable thanks to our assumption (4.33). Indeed, using (4.30), we have:

(4.48) $$\begin{eqnarray}\displaystyle \int _{0}^{\infty }\widetilde{q}_{1}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\,\text{d}\unicode[STIX]{x1D709}^{\prime } & = & \displaystyle -\int _{0}^{\infty }\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}^{2}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\,\text{d}\unicode[STIX]{x1D709}^{\prime }\nonumber\\ \displaystyle & = & \displaystyle \left.\frac{\unicode[STIX]{x1D716}}{2}\left(\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right)^{2}\right|_{\unicode[STIX]{x1D709}^{\prime }=0+}-\lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \infty }\frac{\unicode[STIX]{x1D716}}{2}\left(\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right)^{2}\nonumber\\ \displaystyle & = & \displaystyle \left.\frac{\unicode[STIX]{x1D716}}{2}\left(\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right)^{2}\right|_{\unicode[STIX]{x1D709}^{\prime }=0+}-\left.\frac{\unicode[STIX]{x1D716}}{2}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right)^{2}\right|_{\unicode[STIX]{x1D709}=0+}.\end{eqnarray}$$

In the last equality, we used (4.35), which in turn relied on condition (4.33). Integrating (4.46) with from $\unicode[STIX]{x1D709}^{\prime }=0$ to $-\infty$ and following the same procedure, we obtain a similar relation on the $\unicode[STIX]{x1D709}^{\prime }<0$ side. If we combine the calculations at the $\unicode[STIX]{x1D709}^{\prime }>0$ and $\unicode[STIX]{x1D709}^{\prime }<0$ sides, we find that:

(4.49) $$\begin{eqnarray}\displaystyle \left[2\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}u_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}-p_{0}+\frac{\unicode[STIX]{x1D716}}{2}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right)^{2}\right]=\left[2\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}\widetilde{u}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}-\widetilde{p}_{0}+\frac{\unicode[STIX]{x1D716}}{2}\left(\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right)^{2}\right]. & & \displaystyle\end{eqnarray}$$

We may now combine this with (4.42) and (4.16) to find that:

(4.50) $$\begin{eqnarray}\displaystyle \left[2\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}u_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}-p_{0}+\frac{\unicode[STIX]{x1D716}}{2}\left(\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right)^{2}-g^{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}\right)\right]=-\widehat{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D705}. & & \displaystyle\end{eqnarray}$$

This is the normal stress balance condition for the outer variables (see (B 15)).

Next we turn to the equations for $v^{i}$ . Equation (3.18) yields (B 26), which together with (4.23) and (4.13) gives:

(4.51) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}^{2}\widetilde{v}_{1}^{i}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}=q_{1}g^{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}},\quad i=1,2. & & \displaystyle\end{eqnarray}$$

We have used (4.16) to replace the inner layer variable $\widetilde{\unicode[STIX]{x1D719}}_{0}$ with the outer layer variable $\unicode[STIX]{x1D719}_{0}$ . The interface conditions (2.31) at $\unicode[STIX]{x1D709}^{\prime }=0$ gives (see (B 15)):

(4.52) $$\begin{eqnarray}\displaystyle [\widetilde{v}_{1}^{i}]=0,\quad \left[\unicode[STIX]{x1D707}\left(\frac{\unicode[STIX]{x2202}\widetilde{v}_{1}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+g^{ij}\frac{\unicode[STIX]{x2202}\widetilde{u}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}\right)\right]=0. & & \displaystyle\end{eqnarray}$$

Using Kaplun’s matching procedure as we did for $\widetilde{\unicode[STIX]{x1D719}}_{1}$ , we obtain, with the help of (4.33),

(4.53) $$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\frac{\unicode[STIX]{x2202}\widetilde{v}_{1}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=\lim _{\unicode[STIX]{x1D709}\rightarrow 0\pm }\frac{\unicode[STIX]{x2202}v_{0}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}. & & \displaystyle\end{eqnarray}$$

We may now integrate (4.51) over $0<\unicode[STIX]{x1D709}^{\prime }<\infty$ to obtain

(4.54) $$\begin{eqnarray}\displaystyle \left.\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}v_{0}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right|_{\unicode[STIX]{x1D709}=0+}-\left.\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}\widetilde{v}_{1}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right|_{\unicode[STIX]{x1D709}^{\prime }=0+}=\left(\int _{0}^{\infty }\widetilde{q}_{1}\,\text{d}\unicode[STIX]{x1D709}^{\prime }\right)g^{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}, & & \displaystyle\end{eqnarray}$$

where we used (4.53). We may likewise integrate (4.51) over $-\infty <\unicode[STIX]{x1D709}^{\prime }<0$ and combine this with the above to find that

(4.55) $$\begin{eqnarray}\displaystyle \left[\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}v_{0}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right]-\left[\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}\widetilde{v}_{1}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right]=-q_{\unicode[STIX]{x1D6E4}}g^{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}, & & \displaystyle\end{eqnarray}$$

where $q_{\unicode[STIX]{x1D6E4}}$ was defined in (4.36). The above, together with (4.53) and (4.23) yields

(4.56) $$\begin{eqnarray}\displaystyle \left[\unicode[STIX]{x1D707}\left(\frac{\unicode[STIX]{x2202}v_{0}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}+g^{ij}\frac{\unicode[STIX]{x2202}u_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}\right)\right]=-q_{\unicode[STIX]{x1D6E4}}g^{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}. & & \displaystyle\end{eqnarray}$$

This is the tangential stress balance condition for the outer variables (see (B 15)). This concludes our discussion of the interfacial boundary conditions. At the outer boundary $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ (or at $|\boldsymbol{x}|=\infty$ ), there is no boundary layer, and we simply obtain the conditions:

(4.57a-c ) $$\begin{eqnarray}\displaystyle \boldsymbol{u}_{0}=0,\quad q_{2}=0,\quad \unicode[STIX]{x1D719}_{0}=\unicode[STIX]{x1D719}_{b}\quad \text{at }\boldsymbol{x}\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}\text{ or }|\boldsymbol{x}|=\infty . & & \displaystyle\end{eqnarray}$$

Summary: Let us now collect our results. Dropping the subscript $0$ and renaming $q_{2}$ as $q_{\unicode[STIX]{x1D6FA}}$ in (4.4)–(4.6), we have:

(4.58) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}q_{\unicode[STIX]{x1D6FA}}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{u}q_{\unicode[STIX]{x1D6FA}})=\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\widehat{\unicode[STIX]{x1D70E}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}),\quad \widehat{\unicode[STIX]{x1D70E}}=\frac{\unicode[STIX]{x1D70E}}{\unicode[STIX]{x1D712}}, & \displaystyle\end{eqnarray}$$

(4.59) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D716}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719})=q_{\unicode[STIX]{x1D6FA}}, & \displaystyle\end{eqnarray}$$

(4.60) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D707}\unicode[STIX]{x0394}\boldsymbol{u}-\unicode[STIX]{x1D735}p=\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D72E}_{S}(\boldsymbol{u},p)=q_{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719},\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0. & \displaystyle\end{eqnarray}$$

The boundary conditions for (4.59) are given by (4.17) and (4.36):

(4.61) $$\begin{eqnarray}\displaystyle [\unicode[STIX]{x1D719}]=0,\quad \left[\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}\boldsymbol{n}}\right]=q_{\unicode[STIX]{x1D6E4}}, & & \displaystyle\end{eqnarray}$$

where $q_{\unicode[STIX]{x1D6E4}}$ satisfies equation (4.40), which in the original coordinates, can be written as:

(4.62) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}^{\bot }q_{\unicode[STIX]{x1D6E4}}+\unicode[STIX]{x1D705}u_{\bot }q_{\unicode[STIX]{x1D6E4}}+\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6E4}}\boldsymbol{\cdot }(\boldsymbol{u}_{\Vert }q_{\unicode[STIX]{x1D6E4}})=-\!\left[\widehat{\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}\boldsymbol{n}}\right],\quad u_{\bot }=\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{n},\boldsymbol{u}_{\Vert }=\boldsymbol{u}-u_{\bot }\boldsymbol{n}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6E4}}\boldsymbol{\cdot }$ is the surface divergence operator and $\unicode[STIX]{x2202}_{t}^{\bot }q_{\unicode[STIX]{x1D6E4}}$ is the time derivative of $q_{\unicode[STIX]{x1D6E4}}$ taken along trajectories that travel with the normal velocity of the interface. The boundary conditions for (4.60) are (4.24), (4.50) and (4.56), which we may rewrite as:

(4.63) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle [\boldsymbol{u}]=0,\\ \displaystyle [\unicode[STIX]{x1D72E}_{S}(\boldsymbol{u},p)\boldsymbol{n}]=-\!\left[\frac{\unicode[STIX]{x1D716}}{2}\left(\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}\boldsymbol{n}}\right)^{2}-|\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D719}|^{2}\right)\right]\boldsymbol{n}-q_{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D719}-\widehat{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D705}\boldsymbol{n}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

With the help of (4.61), the last stress boundary condition can also be written as follows:

(4.64) $$\begin{eqnarray}\displaystyle [(\unicode[STIX]{x1D72E}_{S}(\boldsymbol{u},p)+\unicode[STIX]{x1D72E}_{M}(\unicode[STIX]{x1D719}))\boldsymbol{n}]=-\widehat{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D705}\boldsymbol{n}. & & \displaystyle\end{eqnarray}$$

The outer boundary conditions are given by (4.57) (with our modified notation). The final observation to make is that (4.58) can be rewritten as follows using (4.59) and the incompressibility condition in (4.60):

(4.65) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}q_{\unicode[STIX]{x1D6FA}}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}q_{\unicode[STIX]{x1D6FA}}=-\frac{1}{\unicode[STIX]{x1D70F}}q_{\unicode[STIX]{x1D6FA}},\quad \unicode[STIX]{x1D70F}=\frac{\unicode[STIX]{x1D716}}{\widehat{\unicode[STIX]{x1D70E}}}. & & \displaystyle\end{eqnarray}$$

This makes clear that equation (4.58) requires no boundary condition and that $q_{\unicode[STIX]{x1D6FA}}$ decays exponentially along fluid particle trajectories. After an initial transient, therefore, bulk charge is absent. In other words, the subset of phase space characterized by $q_{\unicode[STIX]{x1D6FA}}\equiv 0$ is invariant and exponentially attracting. We may thus set $q_{\unicode[STIX]{x1D6FA}}\equiv 0$ in (4.58)–(4.60) to find:

(4.66) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x0394}\unicode[STIX]{x1D719}=0, & \displaystyle\end{eqnarray}$$

(4.67) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D707}\unicode[STIX]{x0394}\boldsymbol{u}-\unicode[STIX]{x1D735}p=0,\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0. & \displaystyle\end{eqnarray}$$

We have recovered the TM model with surface charge convection for arbitrary interface geometry. The unknown variables are listed in the figure 9.

Figure 9.

Set-up for the TM model. The unknown functions are now the velocity and pressure fields $\boldsymbol{u},p$ and electrostatic potential $\unicode[STIX]{x1D719}$ in $\unicode[STIX]{x1D6FA}_{in,ex}$ and the surface charge density $q_{\unicode[STIX]{x1D6E4}}$ on $\unicode[STIX]{x1D6E4}$ . The bulk charge $q_{\unicode[STIX]{x1D6FA}}$ , shown in parentheses above, can be set identically to $0$ so that we recover the classical TM model with surface charge advection (see argument below (4.65)).

4.2 Structure of the boundary layer

The above derivation gives us additional information about the Taylor–Melcher regime beyond what the TM model provides. Indeed, we now have the equations for the charge distribution within the inner space charge layer. Suppose we have solved the dynamic TM model with suitable initial condition for $q_{\unicode[STIX]{x1D6E4}}$ and initial interface geometry. We may then find the charge distribution $\widetilde{q}_{1}$ inside the interfacial charge layer by solving the following equation obtained from (4.27) to (4.30):

(4.68) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\widetilde{q}_{1}}{\unicode[STIX]{x2202}t}-\left(\unicode[STIX]{x1D705}u+\frac{1}{\sqrt{|g|}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}(\sqrt{|g|}v^{i})\right)\unicode[STIX]{x1D709}^{\prime }\frac{\unicode[STIX]{x2202}\widetilde{q}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+v^{i}\frac{\unicode[STIX]{x2202}\widetilde{q}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}=D_{q}\frac{\unicode[STIX]{x2202}^{2}\widetilde{q}_{1}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}-\frac{1}{\unicode[STIX]{x1D70F}}\widetilde{q}_{1},\quad D_{q}=\frac{D_{C}+D_{A}}{2\unicode[STIX]{x1D712}}, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}$ is as in (4.65). This must be solved under the following conditions obtained from (4.31), (4.32) and (4.36):

(4.69a-c ) $$\begin{eqnarray}\displaystyle l_{C}\widetilde{q}_{1}|_{\unicode[STIX]{x1D709}^{\prime }=0+}=\widetilde{q}_{1}|_{\unicode[STIX]{x1D709}^{\prime }=0-},\quad \lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\widetilde{q}_{1}=0,\quad \int _{-\infty }^{\infty }\widetilde{q}_{1}\,\text{d}\unicode[STIX]{x1D709}^{\prime }=q_{\unicode[STIX]{x1D6E4}}. & & \displaystyle\end{eqnarray}$$

The functions $u,v^{i}$ and $q_{\unicode[STIX]{x1D6E4}}$ are solutions to the TM model and are thus known functions. If we take as our initial condition $q_{\unicode[STIX]{x1D6E4}}=0$ , it would be reasonable to take $\widetilde{q}_{1}\equiv 0$ as the initial condition for (4.68). Once $\widetilde{q}_{1}$ is known, we may also obtain $\widetilde{\unicode[STIX]{x1D719}}_{1}$ and $\widetilde{p}_{0}$ using (4.30), (4.31), (4.35), (4.27), (4.43) and (4.46).

To gain insight into the structure of the interfacial charge distribution, let us consider a steady state of the TM model. The interface is stationary ( $u=0$ ) and the interfacial charge does not change in time ( $\unicode[STIX]{x2202}\widetilde{q}_{1}/\unicode[STIX]{x2202}t=0$ ). Equation (4.68) reduces to:

(4.70) $$\begin{eqnarray}\displaystyle -\left(\frac{1}{\sqrt{|g|}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}(\sqrt{|g|}v^{i})\right)\unicode[STIX]{x1D709}^{\prime }\frac{\unicode[STIX]{x2202}\widetilde{q}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+v^{i}\frac{\unicode[STIX]{x2202}\widetilde{q}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}=D_{q}\frac{\unicode[STIX]{x2202}^{2}\widetilde{q}_{1}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}-\frac{1}{\unicode[STIX]{x1D70F}}\widetilde{q}_{1}. & & \displaystyle\end{eqnarray}$$

To simplify further, let us consider this equation at stagnation points, where $v_{1}=v_{2}=0$ .

(4.71a,b ) $$\begin{eqnarray}\displaystyle D_{q}\frac{\unicode[STIX]{x2202}^{2}\widetilde{q}_{1}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}-\unicode[STIX]{x1D706}\unicode[STIX]{x1D709}^{\prime }\frac{\unicode[STIX]{x2202}\widetilde{q}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}-\frac{1}{\unicode[STIX]{x1D70F}}\widetilde{q}_{1}=0,\quad \unicode[STIX]{x1D706}=-\frac{1}{\sqrt{|g|}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}(\sqrt{|g|}v^{i})=-\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6E4}}\boldsymbol{\cdot }\boldsymbol{u}_{\Vert }.\qquad & & \displaystyle\end{eqnarray}$$

It can be shown that such stagnation points always exist when the interface $\unicode[STIX]{x1D6E4}$ is homeomorphic to a sphere for topological reasons. We have now only to solve the above ordinary differential equation in $\unicode[STIX]{x1D709}^{\prime }$ under conditions (4.69). First, define

(4.72a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}_{max}=\max (\unicode[STIX]{x1D70F}_{in},\unicode[STIX]{x1D70F}_{ex}),\quad \unicode[STIX]{x1D70F}_{k}=\frac{\unicode[STIX]{x1D716}_{k}}{\widehat{\unicode[STIX]{x1D70E}}_{k}}=\frac{\unicode[STIX]{x1D712}\unicode[STIX]{x1D716}_{k}}{\unicode[STIX]{x1D70E}_{k}},\quad k=in,ex. & & \displaystyle\end{eqnarray}$$

We let $D_{q,k},k=in,ex$ denote the value of $D_{q}$ in the interior ( $\unicode[STIX]{x1D709}^{\prime }<0$ ) and exterior ( $\unicode[STIX]{x1D709}^{\prime }>0$ ) fluid respectively. Note that $\unicode[STIX]{x1D706}$ is the same on both sides of the interface since the fluid velocity is continuous across the interface. We have the following result, which is proved in appendix C.

Proposition 1. Consider equation (4.71) under condition (4.69) and assume $q_{\unicode[STIX]{x1D6E4}}\neq 0$ . There is a unique solution if and only if $\unicode[STIX]{x1D706}\unicode[STIX]{x1D70F}_{max}<1$ . If $\unicode[STIX]{x1D706}\unicode[STIX]{x1D70F}_{max}\geqslant 1$ , there is no solution. When there is a solution, $\widetilde{q}_{1}$ is either positive or negative everywhere (depending of the sign of $q_{\unicode[STIX]{x1D6E4}}$ ) and has the following behaviour. If $\unicode[STIX]{x1D706}>0$

(4.73) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \widetilde{q}_{1}(\unicode[STIX]{x1D709}^{\prime })=q_{\unicode[STIX]{x1D6E4}}C_{in}^{+}|\unicode[STIX]{x1D709}^{\prime }|^{-(\unicode[STIX]{x1D706}\unicode[STIX]{x1D70F}_{in})^{-1}}(1+O({\unicode[STIX]{x1D709}^{\prime }}^{-2}))\quad \text{as }\unicode[STIX]{x1D709}^{\prime }\rightarrow -\infty ,\\ \displaystyle \widetilde{q}_{1}(\unicode[STIX]{x1D709}^{\prime })=q_{\unicode[STIX]{x1D6E4}}C_{ex}^{+}{\unicode[STIX]{x1D709}^{\prime }}^{-(\unicode[STIX]{x1D706}\unicode[STIX]{x1D70F}_{ex})^{-1}}(1+O({\unicode[STIX]{x1D709}^{\prime }}^{-2}))\quad \text{as }\unicode[STIX]{x1D709}^{\prime }\rightarrow \infty .\end{array}\right\} & & \displaystyle\end{eqnarray}$$

If $\unicode[STIX]{x1D706}=0$ ,

(4.74) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \widetilde{q}_{1}(\unicode[STIX]{x1D709}^{\prime })=\frac{q_{\unicode[STIX]{x1D6E4}}l_{C}}{\sqrt{D_{q,in}\unicode[STIX]{x1D70F}_{in}}l_{C}+\sqrt{D_{q,ex}\unicode[STIX]{x1D70F}_{ex}}}\exp \left(-\frac{|\unicode[STIX]{x1D709}^{\prime }|}{\sqrt{D_{q,in}\unicode[STIX]{x1D70F}_{in}}}\right)\quad \text{for }\unicode[STIX]{x1D709}^{\prime }<0,\\ \displaystyle \widetilde{q}_{1}(\unicode[STIX]{x1D709}^{\prime })=\frac{q_{\unicode[STIX]{x1D6E4}}}{\sqrt{D_{q,in}\unicode[STIX]{x1D70F}_{in}}l_{C}+\sqrt{D_{q,ex}\unicode[STIX]{x1D70F}_{ex}}}\exp \left(-\frac{\unicode[STIX]{x1D709}^{\prime }}{\sqrt{D_{q,ex}\unicode[STIX]{x1D70F}_{ex}}}\right)\quad \text{for }\unicode[STIX]{x1D709}^{\prime }>0.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

If $\unicode[STIX]{x1D706}<0$ ,

(4.75) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \widetilde{q}_{1}(\unicode[STIX]{x1D709}^{\prime })=q_{\unicode[STIX]{x1D6E4}}C_{in}^{-}|\unicode[STIX]{x1D709}^{\prime }|^{-(|\unicode[STIX]{x1D706}|\unicode[STIX]{x1D70F}_{in})^{-1}-1}\exp \left(-\frac{|\unicode[STIX]{x1D706}|}{2D_{q,in}}{\unicode[STIX]{x1D709}^{\prime }}^{2}\right)(1+O({\unicode[STIX]{x1D709}^{\prime }}^{-2}))\quad \text{as }\unicode[STIX]{x1D709}^{\prime }\rightarrow -\infty ,\\ \displaystyle \widetilde{q}_{1}(\unicode[STIX]{x1D709}^{\prime })=q_{\unicode[STIX]{x1D6E4}}C_{ex}^{-}{\unicode[STIX]{x1D709}^{\prime }}^{-(|\unicode[STIX]{x1D706}|\unicode[STIX]{x1D70F}_{ex})^{-1}-1}\exp \left(-\frac{|\unicode[STIX]{x1D706}|}{2D_{q,ex}}{\unicode[STIX]{x1D709}^{\prime }}^{2}\right)(1+O({\unicode[STIX]{x1D709}^{\prime }}^{-2}))\quad \text{as }\unicode[STIX]{x1D709}^{\prime }\rightarrow \infty .\end{array}\right\}\qquad & & \displaystyle\end{eqnarray}$$

In the above, $C_{k}^{\pm },k=\text{in},\text{ex}$ are positive constants that depend only on $\unicode[STIX]{x1D706}\unicode[STIX]{x1D70F}_{k},\unicode[STIX]{x1D706}/D_{q,k}$ and $l_{C}$ . The above also satisfy the second condition in (4.34).

When

(4.76) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}=-\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6E4}}\boldsymbol{\cdot }\boldsymbol{u}_{\Vert }<\frac{1}{\unicode[STIX]{x1D70F}_{max}} & & \displaystyle\end{eqnarray}$$

at a stagnation point, the above Proposition states that there is a unique inner layer charge distribution consistent with the TM model. However, the charge distribution does not exhibit the familiar exponential decay with distance from the interface. If $\unicode[STIX]{x1D706}>0$ the decay is only algebraic. Convective charge accumulation leads to a broader space charge layer. If $\unicode[STIX]{x1D706}<0$ at a stagnation point, charge distribution decay is faster than exponential.

At a stagnation point, if $q_{\unicode[STIX]{x1D6E4}}\neq 0$ and

(4.77) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}=-\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6E4}}\boldsymbol{\cdot }\boldsymbol{u}_{\Vert }\geqslant \frac{1}{\unicode[STIX]{x1D70F}_{max}}, & & \displaystyle\end{eqnarray}$$

an inner layer charge distribution consistent with the steady state of the TM model does not exist. Indeed, when $\unicode[STIX]{x1D706}\unicode[STIX]{x1D70F}_{max}\geqslant 1$ , the asymptotic behaviour for $\widetilde{q}_{1}$ in (4.73) makes it impossible for $\widetilde{q}_{1}$ to satisfy (4.33) since:

(4.78) $$\begin{eqnarray}\displaystyle \int _{1}^{\infty }|\unicode[STIX]{x1D709}^{\prime }|^{-(\unicode[STIX]{x1D706}\unicode[STIX]{x1D70F}_{max})^{-1}}\,\text{d}\unicode[STIX]{x1D709}^{\prime }=\infty . & & \displaystyle\end{eqnarray}$$

The picture that emerges is that convective charge accumulation overwhelms bulk charge dissipation when $\unicode[STIX]{x1D706}\unicode[STIX]{x1D70F}_{max}\geqslant 1$ , and the boundary layer is destroyed (the reader is referred back to § 1.2.3 and figure 4 for a heuristic discussion). Our analysis here is confined to stagnation points, but similar statements may hold for points at which $\boldsymbol{u}_{\Vert }\neq 0$ .

Figure 10.

A flow field plot indicating regions of positive and negative surface divergence $\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6E4}}\boldsymbol{\cdot }\boldsymbol{u}_{\Vert }$ . Plotted is the flow field around a sphere of the quadrupole vortex of Taylor (horizontal axis is the axis of rotation and is parallel to the imposed electric field direction) (Taylor Reference Taylor1966). On the sphere, $\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6E4}}\boldsymbol{\cdot }\boldsymbol{u}_{\Vert }<0$ in the region in thick red and $\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6E4}}\boldsymbol{\cdot }\boldsymbol{u}_{\Vert }>0$ in the region in thin blue. The charge distribution in the region $\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6E4}}\boldsymbol{\cdot }\boldsymbol{u}_{\Vert }<0$ is expected to decay algebraically with distance from the interface.

Consider gradually increasing the imposed electric field on a leaky dielectric droplet. Suppose the increase in field strength is slow enough that the system is at steady state at each instant. As the imposed electric field is increased, surface convection will be stronger and the space charge layer where $\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6E4}}\boldsymbol{\cdot }\boldsymbol{u}_{\Vert }<0$ will broaden (see figure 10). At a certain threshold field strength, condition (4.76) at a stagnation point (or a similar condition at non-stagnation points) may be violated. At this threshold, the TM model will cease to be valid. The boundary layer will be destroyed possibly resulting in a non-zero charge distribution extending into the bulk. Recall that the derivation of the stress balance boundary condition also required condition (4.33). Stress balance across the interface may become impossible as the threshold field strength is reached.

4.3 Large imposed voltage

Given that the imposed voltage is large in most experimental set-ups, it is of interest to ask whether our analysis may be extended to this case. Here, we discuss the necessary scaling, state the result and only sketch our derivation, since the details are almost exactly the same as our foregoing analysis.

Let $E_{\ast }$ be the representative imposed electric field strength. Modify the scaling of $\boldsymbol{u},p,\unicode[STIX]{x1D719}$ in (2.19) by replacing the thermal voltage $RT/F$ with the imposed representative voltage $E_{\ast }L$ . Define the following constants:

(4.79) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{E}=\frac{\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x1D714}}=\frac{r_{D}/\unicode[STIX]{x1D714}}{L}=\frac{r_{E}}{L},\quad \unicode[STIX]{x1D714}=\sqrt{\frac{RT/F}{E_{\ast }L}},\quad r_{E}=\sqrt{\frac{\unicode[STIX]{x1D716}_{\ast }E_{\ast }L}{Fc_{\ast }}}. & & \displaystyle\end{eqnarray}$$

The length $r_{E}$ is defined by replacing the thermal voltage $RT/F$ in the definition of the Debye length with the externally imposed voltage $E_{\ast }L$ . In the notation of Saville (Reference Saville1977), Baygents & Saville (Reference Baygents and Saville1990), Schnitzer & Yariv (Reference Schnitzer and Yariv2015), $\unicode[STIX]{x1D6FD}\equiv \unicode[STIX]{x1D714}^{-2}$ is the dimensionless imposed field strength. With $\unicode[STIX]{x1D714}=1$ , we are back to the original scaling. When $\unicode[STIX]{x1D714}\ll 1$ , we can perform a similar analysis to that presented above provided

(4.80) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{E}\ll \unicode[STIX]{x1D714}\ll 1. & & \displaystyle\end{eqnarray}$$

This is equivalent to the parametric ordering

(4.81) $$\begin{eqnarray}\displaystyle 1\ll \unicode[STIX]{x1D714}^{-2}=\unicode[STIX]{x1D6FD}\ll \unicode[STIX]{x1D6FF}^{-1} & & \displaystyle\end{eqnarray}$$

assumed in Schnitzer & Yariv (Reference Schnitzer and Yariv2015). This leads to an inner layer of thickness $r_{E}$ and an inner–inner layer of thickness $r_{D}$ (figure 11). The presence of two layers is similar to Baygents & Saville (Reference Baygents and Saville1990) except that there the thickness of the wider layer is $L/\unicode[STIX]{x1D6FD}$ . Outside of these boundary layers, the TM model is valid. The magnitude of surface charge convection is determined as in table 1, depending on the magnitude of $Pe_{E}=Pe\unicode[STIX]{x1D714}^{-2}$ with respect to $\unicode[STIX]{x1D6FF}_{E}$ .

We now briefly discuss the modifications needed to the foregoing analysis. The dimensionless equations of the modified Saville model in § 2.2 change as follows. Equation (2.20) becomes:

(4.82) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle Pe_{E}\left(\frac{\unicode[STIX]{x2202}c}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{u}c)\right)=\unicode[STIX]{x1D735}\boldsymbol{\cdot }(D_{C}(\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D735}c+c\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}))+\frac{k_{E}}{\unicode[STIX]{x1D6FC}}\left(s-\frac{ca}{K}\right),\\ \displaystyle Pe_{E}=Pe\unicode[STIX]{x1D714}^{-2},\quad k_{E}=k\unicode[STIX]{x1D714}^{2}\end{array}\right\} & & \displaystyle\end{eqnarray}$$

and the interface conditions for the solutes change accordingly. In equations (2.23), (2.24) and its attendant interface conditions, $\unicode[STIX]{x1D6FF}_{E}$ replaces every instance of $\unicode[STIX]{x1D6FF}$ .

We first take the limit $\unicode[STIX]{x1D6FC}\rightarrow 0$ in the modified Saville model to obtain the charge diffusion model as in § 3. Equation (3.10) for $q$ in the new scaling takes the following form:

(4.83) $$\begin{eqnarray}\displaystyle Pe_{E}\left(\frac{\unicode[STIX]{x2202}q}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{u}q)\right)=-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{J}_{q},\quad \boldsymbol{J}_{q}=-\unicode[STIX]{x1D6F4}(q)\left(\frac{\unicode[STIX]{x1D714}^{2}}{\sqrt{4S+q^{2}}}\unicode[STIX]{x1D735}q+\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}\right), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6F4}(q)$ is the same as in (3.12). Equations (3.17), (3.18) and their boundary conditions remain the same except that $\unicode[STIX]{x1D6FF}$ should be replaced by $\unicode[STIX]{x1D6FF}_{E}$ .

We next take the limit $\unicode[STIX]{x1D6FF}_{E}\rightarrow 0$ scaling

(4.84a,b ) $$\begin{eqnarray}\displaystyle Pe_{E}\equiv Pe\unicode[STIX]{x1D714}^{-2}=\unicode[STIX]{x1D712}_{E}\unicode[STIX]{x1D6FF}_{E}^{-2},\quad \unicode[STIX]{x1D6FE}=\widehat{\unicode[STIX]{x1D6FE}}_{E}\unicode[STIX]{x1D6FF}_{E}^{2} & & \displaystyle\end{eqnarray}$$

as in (4.1). The boundary layer coordinate is scaled so that $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6FF}_{E}\unicode[STIX]{x1D709}^{\prime }$ . The resulting TM model is exactly the same as in (4.58)–(4.63) except that the constant $\widehat{\unicode[STIX]{x1D70E}}$ should be replaced by $\widehat{\unicode[STIX]{x1D70E}}_{E}=\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D712}_{E}$ and $\widehat{\unicode[STIX]{x1D6FE}}$ by $\widehat{\unicode[STIX]{x1D6FE}}_{E}$ . The equation in the interfacial layer (4.68) takes the form:

(4.85) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \frac{\unicode[STIX]{x2202}\widetilde{q}_{1}}{\unicode[STIX]{x2202}t}-\left(\unicode[STIX]{x1D705}u+\frac{1}{\sqrt{|g|}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}(\sqrt{|g|}v^{i})\right)\unicode[STIX]{x1D709}^{\prime }\frac{\unicode[STIX]{x2202}\widetilde{q}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+v^{i}\frac{\unicode[STIX]{x2202}\widetilde{q}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}=\unicode[STIX]{x1D714}^{2}D_{q,E}\frac{\unicode[STIX]{x2202}^{2}\widetilde{q}_{1}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}-\frac{1}{\unicode[STIX]{x1D70F}_{E}}\widetilde{q}_{1},\\ \displaystyle D_{q,E}=\frac{D_{C}+D_{A}}{2\unicode[STIX]{x1D712}_{E}},\quad \unicode[STIX]{x1D70F}_{E}=\frac{\unicode[STIX]{x1D716}}{\widehat{\unicode[STIX]{x1D70E}}_{E}}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

The boundary conditions at $\unicode[STIX]{x1D709}^{\prime }=\pm \infty$ and $\unicode[STIX]{x1D709}^{\prime }=0\pm$ remain the same.

Let us finally let $\unicode[STIX]{x1D714}\rightarrow 0$ . The constant $\unicode[STIX]{x1D714}$ only appears inside the boundary layer equations, so it only affects the behaviour in the charge layer of width $\unicode[STIX]{x1D6FF}_{E}$ (or $r_{E}$ in physical dimensions). If we let $\unicode[STIX]{x1D714}\rightarrow 0$ in (4.85), we lose the second-order spatial derivative, making it impossible to satisfy the boundary conditions. This indicates the presence of a boundary layer of width $\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}_{E}=\unicode[STIX]{x1D6FF}$ (or $r_{D}$ in physical dimensions). Confining our analysis to stagnation points as in § 4.2, we obtain, in place of (4.71)

(4.86) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}^{2}D_{q,E}\frac{\unicode[STIX]{x2202}^{2}\widetilde{q}_{1}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}-\unicode[STIX]{x1D706}\unicode[STIX]{x1D709}^{\prime }\frac{\unicode[STIX]{x2202}\widetilde{q}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}-\frac{1}{\unicode[STIX]{x1D70F}_{E}}\widetilde{q}_{1}=0,\quad \unicode[STIX]{x1D706}=-\frac{1}{\sqrt{|g|}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}(\sqrt{|g|}v^{i})=-\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6E4}}\boldsymbol{\cdot }\boldsymbol{u}_{\Vert }. & & \displaystyle\end{eqnarray}$$

This equation is in fact just a rescaled version of (4.86) and can be solved in exactly the same way. Taking the limit as $\unicode[STIX]{x1D714}\rightarrow 0$ in the resulting expression, one sees that the charge $\widetilde{q}_{1}$ will be identically equal to $0$ in the inner layer (and not the inner–inner layer) when $\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6E4}}\boldsymbol{\cdot }\boldsymbol{u}_{\Vert }>0$ .

Figure 11.

The boundary layer structure when the externally imposed voltage is large. There are two layers, the inner layer of thickness $r_{E}$ ( $\unicode[STIX]{x1D6FF}_{E}$ in dimensionless units) and an inner–inner layer of thickness $r_{D}$ ( $\unicode[STIX]{x1D6FF}$ in dimensionless units).

5.1 Leading-order equations for the initial layer with $l_{C}\neq l_{A}$

We now consider the case $l_{C}\neq l_{A}$ . In this case, there is a double layer at the interface $\unicode[STIX]{x1D6E4}$ as was suggested at the end of § 3. We consider the limit $\unicode[STIX]{x1D6FF}\rightarrow 0$ as in § 4. Here, the dimensionless parameters are scaled so that

(5.1a,b ) $$\begin{eqnarray}\displaystyle Pe=O(\unicode[STIX]{x1D6FF}^{0}),\quad \unicode[STIX]{x1D6FE}=\widehat{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6FF}. & & \displaystyle\end{eqnarray}$$

These scalings are different from the scalings (4.1) used in § 4. As will shall see, the scaling of the surface tension strength $\unicode[STIX]{x1D6FE}$ is chosen so we can obtain a well-posed limiting problem. The scaling of $Pe$ is chosen for analytical feasibility; the choice in (4.1) leads to a difficult analytical problem. We shall return to this point in § 6.1. We expand the variables $q,\unicode[STIX]{x1D719},\boldsymbol{u}$ and $p$ in powers of $\unicode[STIX]{x1D6FF}$ as follows:

(5.2) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle q=q_{0}+\unicode[STIX]{x1D6FF}q_{1}+\unicode[STIX]{x1D6FF}^{2}q_{2}+\cdots \,,\\ \displaystyle \unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{0}+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}_{1}+\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D719}_{2}+\cdots \,,\\ \displaystyle \boldsymbol{u}=\frac{1}{\unicode[STIX]{x1D6FF}}\boldsymbol{u}_{-1}+\boldsymbol{u}_{0}+\unicode[STIX]{x1D6FF}\boldsymbol{u}_{1}+\cdots \,,\\ \displaystyle p=\frac{1}{\unicode[STIX]{x1D6FF}^{2}}p_{-2}+\frac{1}{\unicode[STIX]{x1D6FF}}p_{-1}+p_{0}+\cdots \,.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

As we shall see, the stronger surface tension (see scaling of $\unicode[STIX]{x1D6FE}$ in (5.1) compared with (4.1)) and the presence of the Galvani potential necessitates velocity and pressure fields of order $1/\unicode[STIX]{x1D6FF}$ (and $1/\unicode[STIX]{x1D6FF}^{2}$ ). We shall see, in fact, that $\boldsymbol{u}_{-1}$ represents the initial fluid velocity transient.

We first consider the equations in the outer layer. Substituting the expansions in (5.2) into (3.17), we find that (4.3) holds exactly as before. Plugging this into (3.10), we find that

(5.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70E}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{0})=0\quad \text{and hence }\unicode[STIX]{x0394}\unicode[STIX]{x1D719}_{0}=0. & & \displaystyle\end{eqnarray}$$

The above expression corresponds to (4.4) of the § 4.1, except for the absence of the advection term thanks to our assumption that $Pe=O(1)$ . In the implication above, we used the fact that $D_{C,A}$ and hence $\unicode[STIX]{x1D70E}$ is spatially constant (within $\unicode[STIX]{x1D6FA}_{in}$ and $\unicode[STIX]{x1D6FA}_{ex}$ respectively). Equation (4.3) and (5.3), applied to (3.17) gives us:

(5.4) $$\begin{eqnarray}\displaystyle q_{2}=0. & & \displaystyle\end{eqnarray}$$

We remark that the derivation of (5.3) and (5.4) is different from the way we derived (4.66) and $q_{\unicode[STIX]{x1D6FA}}=q_{2}=0$ in § 4.1, where these were derived as a consequence of (4.4) and (4.5) (see (4.65)).

Since $q_{0}=q_{1}=0$ by (4.3), we conclude that

(5.5) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}p_{-2}=0, & \displaystyle\end{eqnarray}$$

(5.6) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D707}\unicode[STIX]{x0394}\boldsymbol{u}_{-1}-\unicode[STIX]{x1D735}p_{-1}=0,\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{-1}=0. & \displaystyle\end{eqnarray}$$

From the first equation, we see that $p_{-2}$ is constant within $\unicode[STIX]{x1D6FA}_{in}$ and $\unicode[STIX]{x1D6FA}_{ex}$ respectively. We set

(5.7) $$\begin{eqnarray}\displaystyle p_{-2}=\left\{\begin{array}{@{}ll@{}}p_{-}\quad & \text{in }\unicode[STIX]{x1D6FA}_{in},\\ p_{+}\quad & \text{in }\unicode[STIX]{x1D6FA}_{ex}.\end{array}\right. & & \displaystyle\end{eqnarray}$$

We now turn to equations in the inner layer. We introduce a curvilinear boundary layer coordinate system as in the previous section. From (3.10), we have:

(5.8) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\widetilde{J}_{q,0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=0, & \displaystyle\end{eqnarray}$$

(5.9) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{J}_{q,0}=-\unicode[STIX]{x1D6F4}(\widetilde{q}_{0})\left(\frac{1}{\sqrt{4S+\widetilde{q}_{0}^{\,2}}}\frac{\unicode[STIX]{x2202}\widetilde{q}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right). & \displaystyle\end{eqnarray}$$

From (3.17), we have:

(5.10) $$\begin{eqnarray}\displaystyle -\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}^{2}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}=\widetilde{q}_{0}. & & \displaystyle\end{eqnarray}$$

The above two equations are satisfied for $\unicode[STIX]{x1D709}^{\prime }>0$ and $\unicode[STIX]{x1D709}^{\prime }<0$ . The interface conditions at $\unicode[STIX]{x1D709}^{\prime }=0$ are given by (2.30), (3.14) and (3.15):

(5.11) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle [\widetilde{\unicode[STIX]{x1D719}}_{0}]=\left[\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right]=[\widetilde{J}_{q,0}]=0,\\ \displaystyle l_{C}(\widetilde{q}_{0}+\sqrt{4+\widetilde{q}_{0}^{2}})|_{\unicode[STIX]{x1D709}^{\prime }=0+}=(\widetilde{q}_{0}+\sqrt{4S_{in}+\widetilde{q}_{0}^{2}})|_{\unicode[STIX]{x1D709}^{\prime }=0-}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Here, $\cdot |_{\unicode[STIX]{x1D709}^{\prime }=0\pm }$ denotes the limiting value of quantity of interest from the positive and negative sides of $\unicode[STIX]{x1D709}^{\prime }=0$ respectively and $[\cdot ]=\cdot |_{\unicode[STIX]{x1D709}^{\prime }=0-}-\cdot |_{\unicode[STIX]{x1D709}^{\prime }=0+}$ . The matching conditions must be that, at $\unicode[STIX]{x1D709}\rightarrow \pm \infty$ , the boundary layer values matches the limiting values in the outer layer.

(5.12a,b ) $$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\widetilde{\unicode[STIX]{x1D719}}_{0}=\lim _{\unicode[STIX]{x1D709}\rightarrow 0\pm }\unicode[STIX]{x1D719}_{0}\equiv \unicode[STIX]{x1D719}_{\pm },\quad \lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\widetilde{q}_{0}=\lim _{\unicode[STIX]{x1D709}\rightarrow 0\pm }q_{0}=0. & & \displaystyle\end{eqnarray}$$

The last equality follows from (4.3). We now solve the above system of equations. From (5.8), we see that $\widetilde{J}_{q,0}$ is a constant. From (5.9), we see that

(5.13) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}(\ln (\widetilde{q}_{0}+\sqrt{4S+\widetilde{q}_{0}^{2}})+\widetilde{\unicode[STIX]{x1D719}}_{0})=-\frac{\widetilde{J}_{q,0}}{\unicode[STIX]{x1D6F4}(\widetilde{q}_{0})}. & & \displaystyle\end{eqnarray}$$

Given (5.12), we conclude that

(5.14) $$\begin{eqnarray}\displaystyle \left|\int _{0}^{\pm \infty }\frac{\widetilde{J}_{q,0}}{\unicode[STIX]{x1D6F4}(\widetilde{q}_{0})}\,\text{d}\unicode[STIX]{x1D709}^{\prime }\right|<\infty . & & \displaystyle\end{eqnarray}$$

Since $\widetilde{q}_{0}$ is assumed to go to $0$ as $\unicode[STIX]{x1D709}^{\prime }\rightarrow \infty$ , $\unicode[STIX]{x1D6F4}(\widetilde{q}_{0})$ remains bounded (and positive) for all $\unicode[STIX]{x1D709}^{\prime }$ , and thus the above integral is unbounded unless $\widetilde{J}_{q,0}=0$ . Thus, $\widetilde{J}_{q,0}=0$ and

(5.15) $$\begin{eqnarray}\displaystyle \ln (\widetilde{q}_{0}+\sqrt{4S+\widetilde{q}_{0}^{2}})+\widetilde{\unicode[STIX]{x1D719}}_{0}=\ln (2\sqrt{S})+\unicode[STIX]{x1D719}_{\pm }, & & \displaystyle\end{eqnarray}$$

where the $+$   $(-)$ sign is valid for $\unicode[STIX]{x1D709}^{\prime }>0$ $(\unicode[STIX]{x1D709}^{\prime }<0)$ . Solving the above for $\widetilde{q}_{0}$ we obtain

(5.16) $$\begin{eqnarray}\displaystyle \widetilde{q}_{0}=-2\sqrt{S}\sinh (\widetilde{\unicode[STIX]{x1D719}}_{0}-\unicode[STIX]{x1D719}_{\pm }). & & \displaystyle\end{eqnarray}$$

Substituting the above into (5.10) we obtain the following equation for $\widetilde{\unicode[STIX]{x1D719}}_{0}$

(5.17) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}^{2}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}=2\sqrt{S}\sinh (\widetilde{\unicode[STIX]{x1D719}}_{0}-\unicode[STIX]{x1D719}_{\pm }). & & \displaystyle\end{eqnarray}$$

This equation can be solved analytically to yield:

(5.18) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \widetilde{\unicode[STIX]{x1D719}}_{0}=\unicode[STIX]{x1D719}_{\pm }+2\ln \left(\frac{1+A_{\pm }\exp (\mp \unicode[STIX]{x1D706}_{\pm }\unicode[STIX]{x1D709}^{\prime })}{1-A_{\pm }\exp (\mp \unicode[STIX]{x1D706}_{\pm }\unicode[STIX]{x1D709}^{\prime })}\right),\\ \displaystyle \unicode[STIX]{x1D706}_{+}=\sqrt{2}\unicode[STIX]{x1D716}_{ex}^{-1/2},\quad \unicode[STIX]{x1D706}_{-}=\sqrt{2}\unicode[STIX]{x1D716}_{in}^{-1/2}S_{in}^{1/4},\end{array}\right\} & & \displaystyle\end{eqnarray}$$

where again the $+$   $(-)$ sign corresponds to the expression for $\unicode[STIX]{x1D709}^{\prime }>0$ $(\unicode[STIX]{x1D709}^{\prime }<0)$ . The ratio between the Debye layer thicknesses on the two sides of the interface is thus given by:

(5.19) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D706}_{+}}{\unicode[STIX]{x1D706}_{-}}=\sqrt{\frac{\unicode[STIX]{x1D716}_{in}}{\unicode[STIX]{x1D716}_{ex}\sqrt{S}}}. & & \displaystyle\end{eqnarray}$$

Using condition (5.12) to determine the constants $\unicode[STIX]{x1D719}_{\pm }$ and $A_{\pm }$ , we find

(5.20) $$\begin{eqnarray}\displaystyle [\unicode[STIX]{x1D719}_{0}]=\unicode[STIX]{x1D719}_{-}-\unicode[STIX]{x1D719}_{+}=\ln \left(\frac{l_{C}}{\sqrt{S_{in}}}\right)=\frac{1}{2}\ln \left(\frac{l_{C}}{l_{A}}\right)\equiv \unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}}, & & \displaystyle\end{eqnarray}$$

and

(5.21) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle A_{+}=\frac{\sqrt{\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D70C}+\unicode[STIX]{x1D701})}-\sqrt{1+\unicode[STIX]{x1D70C}\unicode[STIX]{x1D701}}}{\sqrt{\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D70C}+\unicode[STIX]{x1D701})}+\sqrt{1+\unicode[STIX]{x1D70C}\unicode[STIX]{x1D701}}},\quad A_{-}=\frac{\sqrt{\unicode[STIX]{x1D70C}+\unicode[STIX]{x1D701}}-\sqrt{\unicode[STIX]{x1D70C}(1+\unicode[STIX]{x1D70C}\unicode[STIX]{x1D701})}}{\sqrt{\unicode[STIX]{x1D70C}+\unicode[STIX]{x1D701}}+\sqrt{\unicode[STIX]{x1D70C}(1+\unicode[STIX]{x1D70C}\unicode[STIX]{x1D701})}},\\ \displaystyle \unicode[STIX]{x1D70C}=\sqrt{\frac{l_{C}}{\sqrt{S_{in}}}},\quad \unicode[STIX]{x1D701}=\sqrt{\frac{\unicode[STIX]{x1D716}_{ex}}{\unicode[STIX]{x1D716}_{in}\sqrt{S_{in}}}}=\frac{}{.}\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Note that in (5.20) we recovered (3.26). We further note that there is a jump in voltage across the interface $\unicode[STIX]{x1D6E4}$ if $l_{C}\neq l_{A}$ . Define the interior and exterior voltage differences $\unicode[STIX]{x1D719}_{in,ex}^{\unicode[STIX]{x1D6E5}}$ :

(5.22) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D719}_{in}^{\unicode[STIX]{x1D6E5}}=\widetilde{\unicode[STIX]{x1D719}}_{0}(-\infty )-\widetilde{\unicode[STIX]{x1D719}}_{0}(0)=-2\ln \left(\frac{1+A_{-}}{1-A_{-}}\right),\\ \displaystyle \unicode[STIX]{x1D719}_{ex}^{\unicode[STIX]{x1D6E5}}=\widetilde{\unicode[STIX]{x1D719}}_{0}(0)-\widetilde{\unicode[STIX]{x1D719}}_{0}(\infty )=2\ln \left(\frac{1+A_{+}}{1-A_{+}}\right).\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Clearly $\unicode[STIX]{x1D719}_{in}^{\unicode[STIX]{x1D6E5}}+\unicode[STIX]{x1D719}_{ex}^{\unicode[STIX]{x1D6E5}}=\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}}$ . It is not difficult to see from (5.21) that $\unicode[STIX]{x1D719}_{in,ex}^{\unicode[STIX]{x1D6E5}}$ have the same sign. Noting that $\ln \unicode[STIX]{x1D70C}=\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}}/2$ , we may expand $A_{\pm }$ and hence $\unicode[STIX]{x1D719}_{in,ex}^{\unicode[STIX]{x1D6E5}}$ in terms of $\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}}$ to find:

(5.23a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{in}^{\unicode[STIX]{x1D6E5}}=\frac{\unicode[STIX]{x1D701}}{1+\unicode[STIX]{x1D701}}\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}}+O((\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}})^{2}),\quad \unicode[STIX]{x1D719}_{ex}^{\unicode[STIX]{x1D6E5}}=\frac{1}{1+\unicode[STIX]{x1D701}}\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}}+O((\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}})^{2}). & & \displaystyle\end{eqnarray}$$

The parameter $\unicode[STIX]{x1D701}$ is thus the ratio between $\unicode[STIX]{x1D719}_{in}^{\unicode[STIX]{x1D6E5}}$ and $\unicode[STIX]{x1D719}_{ex}^{\unicode[STIX]{x1D6E5}}$ for small $\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}}$ . We may also calculate the amount of charge that accumulates on the interface:

(5.24) $$\begin{eqnarray}\displaystyle \int _{-\infty }^{0}\widetilde{q}_{0}\,\text{d}\unicode[STIX]{x1D709}^{\prime }=-\int _{0}^{\infty }\widetilde{q}_{0}\,\text{d}\unicode[STIX]{x1D709}^{\prime }=((\unicode[STIX]{x1D716}_{in}\unicode[STIX]{x1D706}_{-})^{-1}+(\unicode[STIX]{x1D716}_{ex}\unicode[STIX]{x1D706}_{+})^{-1})^{-1}\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}}+O((\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}})^{2}). & & \displaystyle\end{eqnarray}$$

For small $\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}}$ , therefore, the interface can be viewed as two capacitor in series, with capacitance $\unicode[STIX]{x1D716}_{in}\unicode[STIX]{x1D706}_{-}$ and $\unicode[STIX]{x1D716}_{ex}\unicode[STIX]{x1D706}_{+}$ . We also note that

(5.25) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}=\left.\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}\right|_{\unicode[STIX]{x1D709}=0+}=\left.\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}\right|_{\unicode[STIX]{x1D709}=0-}. & & \displaystyle\end{eqnarray}$$

The latter inequality is true even if $\unicode[STIX]{x1D719}_{0}$ has a jump, thanks to (5.20) which says that the magnitude of the jump $\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}}$ is independent of $\unicode[STIX]{x1D702}^{i}$ .

We now focus on the fluid equations. Since the expansions for the velocity and pressure start at $O(\unicode[STIX]{x1D6FF}^{-1})$ and $O(\unicode[STIX]{x1D6FF}^{-2})$ , the calculations in appendix B do not apply, but the calculations there are easily modified for the case at hand. For $\widetilde{u}_{-1}$ we have:

(5.26a,b ) $$\begin{eqnarray}\displaystyle -\frac{\unicode[STIX]{x2202}\widetilde{p}_{-2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=\widetilde{q}_{0}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }},\quad \frac{\unicode[STIX]{x2202}\widetilde{u}_{-1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=0, & & \displaystyle\end{eqnarray}$$

with interface conditions:

(5.27) $$\begin{eqnarray}\displaystyle [\widetilde{u}_{-1}]=0,\quad \left[2\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}\widetilde{u}_{-1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}-\widetilde{p}_{-2}\right]=0. & & \displaystyle\end{eqnarray}$$

The matching conditions are:

(5.28a,b ) $$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\widetilde{u}_{-1}=\lim _{\unicode[STIX]{x1D709}\rightarrow 0\pm }u_{-1},\quad \lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\widetilde{p}_{-2}=\lim _{\unicode[STIX]{x1D709}\rightarrow 0\pm }p_{-2}\equiv p_{\pm }. & & \displaystyle\end{eqnarray}$$

We thus see that

(5.29) $$\begin{eqnarray}\displaystyle \widetilde{u}_{-1}=\lim _{\unicode[STIX]{x1D709}\rightarrow 0\pm }u_{-1}, & & \displaystyle\end{eqnarray}$$

and therefore,

(5.30) $$\begin{eqnarray}\displaystyle [u_{-1}]=0. & & \displaystyle\end{eqnarray}$$

For $\widetilde{p}_{-2}$ , we see that

(5.31) $$\begin{eqnarray}\displaystyle \widetilde{p}_{-2}=\left\{\begin{array}{@{}ll@{}}p_{+}+(\unicode[STIX]{x1D716}_{ex}/2)(\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime })^{2}\quad & \text{for }\unicode[STIX]{x1D709}^{\prime }>0,\\ p_{-}+(\unicode[STIX]{x1D716}_{in}/2)(\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime })^{2}\quad & \text{for }\unicode[STIX]{x1D709}^{\prime }<0.\end{array}\right. & & \displaystyle\end{eqnarray}$$

Since $p_{\pm }$ does not depend $\unicode[STIX]{x1D702}^{i}$ (see (5.7)), $\widetilde{p}_{-2}$ does not depend on $\boldsymbol{\unicode[STIX]{x1D702}}$ . Given (5.27) and (5.29), $p_{-2}$ is continuous at $\unicode[STIX]{x1D709}^{\prime }=0$ and therefore,

(5.32) $$\begin{eqnarray}\displaystyle \left.\left(p_{+}-\frac{\unicode[STIX]{x1D716}_{ex}}{2}\left(\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right)^{2}\right)\right|_{\unicode[STIX]{x1D709}^{\prime }=0+}=\left.\left(p_{-}-\frac{\unicode[STIX]{x1D716}_{in}}{2}\left(\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right)^{2}\right)\right|_{\unicode[STIX]{x1D709}^{\prime }=0-}. & & \displaystyle\end{eqnarray}$$

After some algebra, we find that:

(5.33) $$\begin{eqnarray}\displaystyle [p_{-2}]=p_{-}-p_{+}=\frac{\unicode[STIX]{x1D716}_{in}}{2}\left(\frac{4\unicode[STIX]{x1D706}_{-}A_{-}}{1-A_{-}^{2}}\right)^{2}-\frac{\unicode[STIX]{x1D716}_{ex}}{2}\left(\frac{4\unicode[STIX]{x1D706}_{+}A_{+}}{1-A_{+}^{2}}\right)^{2}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D706}_{\pm },A_{\pm }$ are defined in (5.18) and (5.21). When $l_{C}=l_{A}$ , $A_{\pm }=0$ and thus there is no pressure difference to leading order. Otherwise, there will in general be a pressure difference.

We may also consider the equations for $\widetilde{v}_{-1}^{i}$ . This analysis, the details of which we omit, yields:

(5.34) $$\begin{eqnarray}\displaystyle \widetilde{v}_{-1}^{i}=\lim _{\unicode[STIX]{x1D709}\rightarrow 0\pm }v_{-1}^{i},\quad i=1,2. & & \displaystyle\end{eqnarray}$$

In particular, we have:

(5.35) $$\begin{eqnarray}\displaystyle [v_{-1}^{i}]=0,\quad i=1,2. & & \displaystyle\end{eqnarray}$$

We now turn to equations at the next order to obtain the boundary conditions for (5.3). From (3.10) and (3.17) we have:

(5.36) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\widetilde{J}_{q,1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=0, & \displaystyle\end{eqnarray}$$

(5.37) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{J}_{q,1}=-\unicode[STIX]{x1D6F4}(\widetilde{q}_{0})\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\left(\frac{\widetilde{q}_{1}}{\sqrt{4S+\widetilde{q}_{0}^{2}}}+\widetilde{\unicode[STIX]{x1D719}}_{1}\right), & \displaystyle\end{eqnarray}$$

(5.38) $$\begin{eqnarray}\displaystyle & \displaystyle -\widetilde{q}_{1}=\unicode[STIX]{x1D716}\left(\frac{\unicode[STIX]{x2202}^{2}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}+\unicode[STIX]{x1D705}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right), & \displaystyle\end{eqnarray}$$

with interface conditions from (2.30), (3.14) and (3.15) given by:

(5.39) $$\begin{eqnarray}\displaystyle [\widetilde{\unicode[STIX]{x1D719}}_{1}]=\left[\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right]=[\widetilde{J}_{q,1}]=\left[\frac{\widetilde{q}_{1}}{\sqrt{4S+\widetilde{q}_{0}^{2}}}\right]=0. & & \displaystyle\end{eqnarray}$$

We see from (5.36) and the above that $\widetilde{J}_{q,1}$ is constant throughout $-\infty <\unicode[STIX]{x1D709}^{\prime }<\infty$ . Let this constant be equal to $J_{0}$ . From (5.37) and (5.38), we obtain the equation:

(5.40) $$\begin{eqnarray}\displaystyle -\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}\left(\frac{\widetilde{q}_{1}}{\sqrt{4S+\widetilde{q}_{0}^{2}}}\right)+\widetilde{q}_{1}=-\unicode[STIX]{x1D716}\unicode[STIX]{x1D705}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+\unicode[STIX]{x1D716}J_{0}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\left(\frac{1}{\unicode[STIX]{x1D6F4}(\widetilde{q}_{0})}\right), & & \displaystyle\end{eqnarray}$$

with interface conditions:

(5.41) $$\begin{eqnarray}\displaystyle \left[\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\left(\frac{\widetilde{q}_{1}}{\sqrt{4S+\widetilde{q}_{0}^{2}}}\right)\right]=\left[\frac{\widetilde{q}_{1}}{\sqrt{4S+\widetilde{q}_{0}^{2}}}\right]=0. & & \displaystyle\end{eqnarray}$$

It is not difficult to see that this equation for $\widetilde{q}_{1}$ has a unique bounded solution that decays exponentially to $0$ as $\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty$ . It will be useful later to have a somewhat more explicit form for $\widetilde{q}_{1}$ and $\widetilde{\unicode[STIX]{x1D719}}_{1}$ . Define the functions $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D705}}$ and $\unicode[STIX]{x1D713}_{J}$ as being solutions to the following equations:

(5.42) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D705}}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}+\sqrt{4S+\widetilde{q}_{0}^{2}}\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D705}}=-\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}, & \displaystyle\end{eqnarray}$$

(5.43) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D713}_{J}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}+\sqrt{4S+\widetilde{q}_{0}^{2}}\unicode[STIX]{x1D713}_{J}=\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\left(\frac{1}{\unicode[STIX]{x1D6F4}(\widetilde{q}_{0})}\right), & \displaystyle\end{eqnarray}$$

with interface conditions:

(5.44) $$\begin{eqnarray}\displaystyle [\unicode[STIX]{x1D713}_{\cdot }]=\left[\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}_{\cdot }}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right]=0, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D713}_{\cdot }$ is a place holder for either $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D705}}$ and $\unicode[STIX]{x1D713}_{J}$ . We also require that both $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D705}}$ and $\unicode[STIX]{x1D713}_{J}$ decay to $0$ at $\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty$ . Note here that $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D705}}$ and $\unicode[STIX]{x1D713}_{J}$ depend solely on $\widetilde{q}_{0}$ and $\widetilde{\unicode[STIX]{x1D719}}_{0}$ and hence, like $\widetilde{q}_{0}$ and $\widetilde{\unicode[STIX]{x1D719}}_{0}$ , are universal functions that only depend on the parameters of the system. Then, we have:

(5.45) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{q}_{1}=\sqrt{4S+\widetilde{q}_{0}^{2}}(\unicode[STIX]{x1D705}\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D705}}+J_{0}\unicode[STIX]{x1D713}_{J}), & \displaystyle\end{eqnarray}$$

(5.46) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=-\unicode[STIX]{x1D705}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D705}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}-J_{0}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}_{J}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}-\frac{J_{0}}{\unicode[STIX]{x1D6F4}(\widetilde{q}_{0})}. & \displaystyle\end{eqnarray}$$

We can now find $\widetilde{\unicode[STIX]{x1D719}}_{1}$ the above together with the first of the interface conditions in (5.39). To relate $J_{0}$ to the outer layer variables, we use Kaplun’s matching procedure to find that:

(5.47) $$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D709}\rightarrow \pm 0}\unicode[STIX]{x1D70E}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}=\lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\unicode[STIX]{x1D70E}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=-J_{0}. & & \displaystyle\end{eqnarray}$$

In particular, we have:

(5.48) $$\begin{eqnarray}\displaystyle \left[\unicode[STIX]{x1D70E}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right]=0. & & \displaystyle\end{eqnarray}$$

Let us now turn to the equations for the velocities. For $u$ , we have:

(5.49) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}^{2}\widetilde{u}_{0}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}-\frac{\unicode[STIX]{x2202}\widetilde{p}_{-1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=\widetilde{q_{1}}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+\widetilde{q}_{0}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}, & \displaystyle\end{eqnarray}$$

(5.50) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\widetilde{u}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+\unicode[STIX]{x1D705}\widetilde{u}_{-1}+\frac{1}{\sqrt{|g|}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}(\sqrt{|g|}\widetilde{v}_{-1}^{i})=0, & \displaystyle\end{eqnarray}$$

with interface conditions:

(5.51) $$\begin{eqnarray}\displaystyle [u_{0}]=0,\quad \left[2\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}\widetilde{u}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}-\widetilde{p}_{-1}+\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right]=-\widehat{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D705}. & & \displaystyle\end{eqnarray}$$

Taking the derivative of (5.50) with respect to $\unicode[STIX]{x1D709}^{\prime }$ and using (5.29) and (5.34), we see that

(5.52) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}^{2}\widetilde{u}_{0}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}=0. & & \displaystyle\end{eqnarray}$$

Using the above and (5.38) as well as (5.10), we have:

(5.53) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\left(2\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}\widetilde{u}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}-\widetilde{p}_{-1}+\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right)=-\unicode[STIX]{x1D705}\unicode[STIX]{x1D716}\left(\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right)^{2}. & & \displaystyle\end{eqnarray}$$

We obtain the following from the usual matching procedure. We first have

(5.54) $$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\frac{\unicode[STIX]{x2202}\widetilde{u}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=\lim _{\unicode[STIX]{x1D709}\rightarrow 0\pm }\frac{\unicode[STIX]{x2202}u_{-1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}, & & \displaystyle\end{eqnarray}$$

together with

(5.55) $$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\frac{\unicode[STIX]{x2202}\widetilde{u}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=\lim _{\unicode[STIX]{x1D709}\rightarrow 0\pm }\frac{\unicode[STIX]{x2202}u_{-1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}},\quad [u_{0}]=0. & & \displaystyle\end{eqnarray}$$

We also have:

(5.56) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \left[2\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}u_{-1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}-p_{-1}\right]=-\widehat{\unicode[STIX]{x1D6FE}}_{eff}\unicode[STIX]{x1D705},\quad \widehat{\unicode[STIX]{x1D6FE}}_{eff}=\widehat{\unicode[STIX]{x1D6FE}}-\left\langle \unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }},\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right\rangle ,\\ \displaystyle \left\langle \unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }},\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right\rangle =\frac{8\unicode[STIX]{x1D706}_{+}\unicode[STIX]{x1D716}_{ex}A_{+}^{2}}{1-A_{+}^{2}}+\frac{8\unicode[STIX]{x1D706}_{-}\unicode[STIX]{x1D716}_{in}A_{-}^{2}}{1-A_{-}^{2}},\end{array}\right\} & & \displaystyle\end{eqnarray}$$

where $\langle \cdot ,\cdot \rangle$ is the standard inner product on $L^{2}(\mathbb{R})$ , the space of square integrable functions on the real line and $\unicode[STIX]{x1D706}_{\pm },A_{\pm }$ are defined in (5.18) and (5.21). For $v^{i}$ we have:

(5.57) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}^{2}\widetilde{v}_{0}^{i}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}-g^{ij}\frac{\unicode[STIX]{x2202}\widetilde{p}_{-2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}=\widetilde{q}_{0}g^{ij}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}. & & \displaystyle\end{eqnarray}$$

From (5.31), $\widetilde{p}_{-2}$ does not depend on $\boldsymbol{\unicode[STIX]{x1D702}}$ . Thus, we have:

(5.58) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}^{2}\widetilde{v}_{0}^{i}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}=\widetilde{q}_{0}g^{ij}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}. & & \displaystyle\end{eqnarray}$$

This is supplemented with the interface condition:

(5.59) $$\begin{eqnarray}\displaystyle [v_{0}^{i}]=0,\quad \left[\unicode[STIX]{x1D707}\left(\frac{\unicode[STIX]{x2202}\widetilde{v}_{0}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+g^{ij}\frac{\unicode[STIX]{x2202}\widetilde{u}_{-1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}\right)\right]=0. & & \displaystyle\end{eqnarray}$$

Note that (5.57) together with (5.10) yields:

(5.60) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}\left(\unicode[STIX]{x1D707}\widetilde{v}_{0}^{i}+\unicode[STIX]{x1D716}\widetilde{\unicode[STIX]{x1D719}}_{0}g^{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}\right)=0, & & \displaystyle\end{eqnarray}$$

where we used (5.25). Thus,

(5.61) $$\begin{eqnarray}\displaystyle \widetilde{v}_{0}^{i}=-\frac{\unicode[STIX]{x1D716}}{\unicode[STIX]{x1D707}}\widetilde{\unicode[STIX]{x1D719}}_{0}g^{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}+C_{\pm }\unicode[STIX]{x1D709}^{\prime }+B_{\pm }, & & \displaystyle\end{eqnarray}$$

where the $+$ $(-)$ sign is chosen when $\unicode[STIX]{x1D709}^{\prime }>0$ ( $\unicode[STIX]{x1D709}^{\prime }<0$ ), where $B_{\pm },C_{\pm }$ are constants to be determined. The interface conditions (5.59) yield:

(5.62a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{ex}C_{+}=\unicode[STIX]{x1D707}_{in}C_{-},\quad B_{+}-\frac{\unicode[STIX]{x1D716}_{ex}}{\unicode[STIX]{x1D707}_{ex}}\widetilde{\unicode[STIX]{x1D719}}_{0}(0)g^{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}=B_{-}-\frac{\unicode[STIX]{x1D716}_{in}}{\unicode[STIX]{x1D707}_{in}}\widetilde{\unicode[STIX]{x1D719}}_{0}(0)g^{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}. & & \displaystyle\end{eqnarray}$$

The usual matching procedure yields:

(5.63a,b ) $$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D709}\rightarrow 0\pm }\frac{\unicode[STIX]{x2202}v_{-1}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}=\lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\frac{\unicode[STIX]{x2202}\widetilde{v}_{0}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=C_{\pm },\quad \lim _{\unicode[STIX]{x1D709}\rightarrow 0\pm }v_{0}^{i}=\lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\left(-\frac{\unicode[STIX]{x1D716}}{\unicode[STIX]{x1D707}}\widetilde{\unicode[STIX]{x1D719}}_{0}g^{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}+B_{\pm }\right).\qquad & & \displaystyle\end{eqnarray}$$

In particular, we have:

(5.64) $$\begin{eqnarray}\displaystyle \left[\unicode[STIX]{x1D707}\left(\frac{\unicode[STIX]{x2202}v_{-1}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}+g^{ij}\frac{\unicode[STIX]{x2202}u_{-1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}\right)\right]=0, & & \displaystyle\end{eqnarray}$$

and

(5.65) $$\begin{eqnarray}\displaystyle [v_{0}^{i}]=-\!\left(\frac{\unicode[STIX]{x1D716}_{in}}{\unicode[STIX]{x1D707}_{in}}\unicode[STIX]{x1D719}_{in}^{\unicode[STIX]{x1D6E5}}+\frac{\unicode[STIX]{x1D716}_{ex}}{\unicode[STIX]{x1D707}_{ex}}\unicode[STIX]{x1D719}_{ex}^{\unicode[STIX]{x1D6E5}}\right)g^{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D719}_{in,ex}^{\unicode[STIX]{x1D6E5}}$ were defined in (5.22). As remarked below (5.22), $\unicode[STIX]{x1D719}_{in,ex}^{\unicode[STIX]{x1D6E5}}$ have the same sign, and thus, there is a jump in the velocity $v_{0}^{i}$ across the interface $\unicode[STIX]{x1D6E4}$ so long as $l_{C}\neq l_{A}$ and $\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}/\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}\neq 0$ . This velocity slip is essentially a liquid–liquid version of the Smoluchowski slip formula for liquid–solid interfaces. This slip arises due to the fact that there are charges of opposite sign on the two sides of the interface $\unicode[STIX]{x1D6E4}$ (see figure 5).

We may now collect our results. The governing equations for the leading-order electrostatic potential and fluid velocity are:

(5.66) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70E}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{0})=0, & & \displaystyle\end{eqnarray}$$

(5.67a,b ) $$\begin{eqnarray}\displaystyle [\unicode[STIX]{x1D719}_{0}]=\ln \left(\frac{l_{C}}{\sqrt{S_{in}}}\right)=\frac{1}{2}\ln \left(\frac{l_{C}}{l_{A}}\right),\quad \left[\unicode[STIX]{x1D70E}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right]=0, & & \displaystyle\end{eqnarray}$$

(5.68) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D707}\unicode[STIX]{x0394}\boldsymbol{u}_{-1}-\unicode[STIX]{x1D735}p_{-1}=0,\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{-1}=0, & \displaystyle\end{eqnarray}$$

(5.69) $$\begin{eqnarray}\displaystyle & \displaystyle [\boldsymbol{u}_{-1}]=0,\quad [\unicode[STIX]{x1D6F4}(\boldsymbol{u}_{-1},p_{-1})\boldsymbol{n}]=-\widehat{\unicode[STIX]{x1D6FE}}_{eff}\unicode[STIX]{x1D705}\boldsymbol{n}. & \displaystyle\end{eqnarray}$$

The effective tension $\widehat{\unicode[STIX]{x1D6FE}}_{eff}$ is defined as

(5.70a,b ) $$\begin{eqnarray}\displaystyle \widehat{\unicode[STIX]{x1D6FE}}_{eff}=\widehat{\unicode[STIX]{x1D6FE}}-\left\langle \unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }},\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right\rangle ,\quad \left\langle \unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }},\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right\rangle =\frac{8\unicode[STIX]{x1D706}_{+}\unicode[STIX]{x1D716}_{ex}A_{+}^{2}}{1-A_{+}^{2}}+\frac{8\unicode[STIX]{x1D706}_{-}\unicode[STIX]{x1D716}_{in}A_{-}^{2}}{1-A_{-}^{2}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D706}_{\pm },A_{\pm }$ are defined in (5.18) and (5.21). The equations for voltage $\unicode[STIX]{x1D719}_{0}$ and the fluid velocity $\boldsymbol{u}_{-1}$ are thus completely decoupled, except that the effective surface tension constant is modified to $\widehat{\unicode[STIX]{x1D6FE}}_{eff}$ . We point out that the equation for $\boldsymbol{u}_{-1}$ is well posed only if $\widehat{\unicode[STIX]{x1D6FE}}_{eff}>0$ , suggesting that the above asymptotic procedure may not be valid if $\widehat{\unicode[STIX]{x1D6FE}}_{eff}<0$ .

Assume that $\widehat{\unicode[STIX]{x1D6FE}}_{eff}>0$ . To leading order, the fluid will undergo Stokesian motion without any influence from the electric field. An initially deformed droplet will approach a sphere (if initially homeomorphic to a sphere). The dynamics described by $\boldsymbol{u}_{-1}$ thus represents the initial layer, after which the droplet is approximately spherical and $\boldsymbol{u}_{0}$ becomes the leading-order term of the velocity field.

5.2 Perturbation from sphere

We now consider the dynamics of the droplet after the initial layer and the droplet is approximately spherical. This amounts to obtaining equations for $\boldsymbol{u}_{0}$ . We expand the curvature $\unicode[STIX]{x1D705}$ in terms of $\unicode[STIX]{x1D6FF}$ :

(5.71) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}=\unicode[STIX]{x1D705}_{0}+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D705}_{1}+\cdots \,. & & \displaystyle\end{eqnarray}$$

We note that, strictly speaking, this expansion had to be applied to all preceding calculations. It can be checked that this will not have made any difference in our calculations thus far (since this would only amount to changing all instances of $\unicode[STIX]{x1D705}$ to $\unicode[STIX]{x1D705}_{0}$ in the foregoing calculations). Assuming that an equilibrium steady state is reached ( $\widehat{\unicode[STIX]{x1D6FE}}_{eff}>0$ )

(5.72) $$\begin{eqnarray}\displaystyle u_{-1}=0,\quad p_{-1}=\left\{\begin{array}{@{}ll@{}}0\quad & \text{in }\unicode[STIX]{x1D6FA}_{in}\\ \widehat{\unicode[STIX]{x1D6FE}}_{eff}\unicode[STIX]{x1D705}_{0}\quad & \text{in }\unicode[STIX]{x1D6FA}_{ex}\end{array}\right.,\quad [p_{-1}]=\widehat{\unicode[STIX]{x1D6FE}}_{eff}\unicode[STIX]{x1D705}_{0}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D705}_{0}$ is the constant curvature of the spherical droplet and we have normalized the pressure in $\unicode[STIX]{x1D6FA}_{ex}$ to be $0$ .

Given (5.4) (and (4.3)), the equation for $\boldsymbol{u}_{0}$ in the outer layer is given by:

(5.73) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}\unicode[STIX]{x0394}\boldsymbol{u}_{0}-\unicode[STIX]{x1D735}p_{0}=0,\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{0}=0. & & \displaystyle\end{eqnarray}$$

We already have the interfacial conditions (5.55) and (5.65). We have only to obtain the stress jump conditions across the interface. As before, the variables with $\widetilde{\cdot }$ are for the inner layer.

From (5.72) and using (5.29), (5.34), we have:

(5.74) $$\begin{eqnarray}\displaystyle \widetilde{u}_{-1}=\widetilde{v}_{-1}^{i}=0. & & \displaystyle\end{eqnarray}$$

We caution that $\widetilde{p}_{-1}$ is not necessarily constant within the inner layer. We also have:

(5.75) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\widetilde{u}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=0,\quad \lim _{\unicode[STIX]{x1D709}^{\prime }\rightarrow \pm \infty }\frac{\unicode[STIX]{x2202}\widetilde{v}_{0}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=0, & & \displaystyle\end{eqnarray}$$

where we used (5.54) in the first equality and (5.63) in the second. Let us now consider the equations for $u$ . From (3.18) using (5.74) and (5.75) we have (see (B 8) and (B 22)):

(5.76) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}^{2}\widetilde{u}_{1}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}-\frac{\unicode[STIX]{x2202}\widetilde{p}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=\widetilde{q}_{0}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+\widetilde{q}_{1}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+\widetilde{q}_{2}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}, & \displaystyle\end{eqnarray}$$

(5.77) $$\begin{eqnarray}\displaystyle & \displaystyle 0=\frac{\unicode[STIX]{x2202}\widetilde{u}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+\unicode[STIX]{x1D705}_{0}\widetilde{u}_{0}+\frac{1}{\sqrt{|g|}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}(\sqrt{|g|}\widetilde{v}_{0}^{i}). & \displaystyle\end{eqnarray}$$

From (2.31) and (5.25), we have (see (B 15))

(5.78) $$\begin{eqnarray}\displaystyle \left[2\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}\widetilde{u}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}-p_{0}+\frac{\unicode[STIX]{x1D716}}{2}\left(\left(\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right)^{2}+2\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}-g^{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}\right)\right]=-\widehat{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D705}_{1}. & & \displaystyle\end{eqnarray}$$

Take the derivative of (5.77) with respect to $\unicode[STIX]{x1D709}^{\prime }$ , multiply by $\unicode[STIX]{x1D707}$ and add to (5.76). Using (5.74), we obtain:

(5.79) $$\begin{eqnarray}\displaystyle 2\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}^{2}\widetilde{u}_{1}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}-\frac{\unicode[STIX]{x2202}\widetilde{p}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=-\frac{\unicode[STIX]{x1D707}}{\sqrt{|g|}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}\left(\sqrt{|g|}\frac{\unicode[STIX]{x2202}\widetilde{v}_{0}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right)+\widetilde{q}_{0}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+\widetilde{q}_{1}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+\widetilde{q}_{2}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}. & & \displaystyle\end{eqnarray}$$

Note that $\widetilde{q}_{2}$ satisfies:

(5.80) $$\begin{eqnarray}\displaystyle -\widetilde{q}_{2}=\unicode[STIX]{x1D716}\left(\frac{\unicode[STIX]{x2202}^{2}\widetilde{\unicode[STIX]{x1D719}}_{2}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}+\unicode[STIX]{x1D705}_{0}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+\unicode[STIX]{x1D705}_{1}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+\unicode[STIX]{x1D6E5}_{\boldsymbol{\unicode[STIX]{x1D702}}}\widetilde{\unicode[STIX]{x1D719}}_{0}\right), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6E5}_{\boldsymbol{\unicode[STIX]{x1D702}}}$ is the Laplace–Beltrami operator on the interface:

(5.81) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}_{\boldsymbol{\unicode[STIX]{x1D702}}}\unicode[STIX]{x1D719}_{0}=\frac{1}{\sqrt{|g|}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}\left(\sqrt{|g|}g^{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}\right). & & \displaystyle\end{eqnarray}$$

Using (5.60), (5.63) and (5.75), we have:

(5.82) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}\widetilde{v}_{0}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}g^{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}=0. & & \displaystyle\end{eqnarray}$$

From this, we see that

(5.83) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D707}}{\sqrt{|g|}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}\left(\sqrt{|g|}\frac{\unicode[STIX]{x2202}\widetilde{v}_{0}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right)+\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\unicode[STIX]{x1D6E5}_{\boldsymbol{\unicode[STIX]{x1D702}}}\unicode[STIX]{x1D719}_{0}=0. & & \displaystyle\end{eqnarray}$$

Using the above relations, (5.10) and (5.38), equation (5.79) can be rewritten as:

(5.84) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\left(2\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}\widetilde{u}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}-p_{0}+\frac{\unicode[STIX]{x1D716}}{2}\left(\left(\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right)^{2}+2\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right)\right)\nonumber\\ \displaystyle & & \displaystyle \quad =-2\unicode[STIX]{x1D705}_{0}\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}-\unicode[STIX]{x1D705}_{1}\unicode[STIX]{x1D716}\left(\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right)^{2}.\end{eqnarray}$$

We may now apply the usual matching procedure to obtain:

(5.85) $$\begin{eqnarray}\displaystyle & & \displaystyle \left[2\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}u_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}-p_{0}+\frac{\unicode[STIX]{x1D716}}{2}\left(\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right)^{2}-g^{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}\right)\right]\nonumber\\ \displaystyle & & \displaystyle \quad =-\widehat{\unicode[STIX]{x1D6FE}}_{eff}\unicode[STIX]{x1D705}_{1}+2\unicode[STIX]{x1D705}_{0}\left(\frac{\unicode[STIX]{x1D716}_{in}}{\unicode[STIX]{x1D70E}_{in}}\unicode[STIX]{x1D719}_{in}^{\unicode[STIX]{x1D6E5}}+\frac{\unicode[STIX]{x1D716}_{ex}}{\unicode[STIX]{x1D70E}_{ex}}\unicode[STIX]{x1D719}_{ex}^{\unicode[STIX]{x1D6E5}}-I_{J}\right)J_{0}-2\unicode[STIX]{x1D705}_{0}^{2}I_{\unicode[STIX]{x1D705}},\end{eqnarray}$$

where $\widehat{\unicode[STIX]{x1D6FE}}_{eff}$ was defined in (5.56), $\unicode[STIX]{x1D719}_{in,ex}^{\unicode[STIX]{x1D6E5}}$ were defined in (5.22) and

(5.86) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle I_{\unicode[STIX]{x1D705}}=\left\langle \unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }},\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D705}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right\rangle ,\quad I_{J}=\left\langle \unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }},\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}_{J}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+R_{\unicode[STIX]{x1D6E5}}\right\rangle ,\\ \displaystyle R_{\unicode[STIX]{x1D6E5}}=\left\{\begin{array}{@{}ll@{}}\unicode[STIX]{x1D6F4}(\widetilde{q}_{0})^{-1}-\unicode[STIX]{x1D70E}_{in}^{-1}\quad & \text{for }\unicode[STIX]{x1D709}^{\prime }<0,\\ \unicode[STIX]{x1D6F4}(\widetilde{q}_{0})^{-1}-\unicode[STIX]{x1D70E}_{ex}^{-1}\quad & \text{for }\unicode[STIX]{x1D709}^{\prime }>0.\end{array}\right.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

In the above derivation, we used (5.46), (5.42) and (5.43). Note that $I_{\unicode[STIX]{x1D705}}$ and $I_{J}$ depend only on the parameters of the system of equations.

We now turn to the equations for $v^{i}$ . We have (see (B 27)):

(5.87) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}\left(\frac{\unicode[STIX]{x2202}^{2}\widetilde{v}_{1}^{i}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}+2\unicode[STIX]{x1D705}_{0}\frac{\unicode[STIX]{x2202}\widetilde{v}_{0}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right)-g^{ij}\frac{\unicode[STIX]{x2202}p_{-1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}=\widetilde{q}_{0}g^{ij}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}+\widetilde{q}_{1}g^{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}, & & \displaystyle\end{eqnarray}$$

where we used (5.25) in the last term. The interface condition is given by:

(5.88) $$\begin{eqnarray}\displaystyle \left[\unicode[STIX]{x1D707}\left(\frac{\unicode[STIX]{x2202}\widetilde{v}_{1}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}+g^{ij}\frac{\unicode[STIX]{x2202}\widetilde{u}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}\right)\right]=0. & & \displaystyle\end{eqnarray}$$

From (5.52) and (5.53), we have:

(5.89) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\left(\frac{\unicode[STIX]{x2202}\widetilde{p}_{-1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}-\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}^{2}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right)=0. & & \displaystyle\end{eqnarray}$$

Recalling that $p_{-1}$ is spatially constant in the outer layer by assumption, we see that

(5.90) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\widetilde{p}_{-1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}-\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}^{2}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{i}}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}=0. & & \displaystyle\end{eqnarray}$$

Plugging this back into (5.87) and using (5.10), we obtain:

(5.91) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}\left(\frac{\unicode[STIX]{x2202}^{2}\widetilde{v}_{1}^{i}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}+2\unicode[STIX]{x1D705}_{0}\frac{\unicode[STIX]{x2202}\widetilde{v}_{0}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\right)=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\left(\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}g^{ij}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}\right)+\widetilde{q}_{1}g^{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}. & & \displaystyle\end{eqnarray}$$

Further using (5.82) and (5.38),

(5.92) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}^{2}\widetilde{v}_{1}^{i}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\left(\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}g^{ij}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}\right)+\left(\unicode[STIX]{x1D716}\unicode[STIX]{x1D705}_{0}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}-\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}^{2}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}\right)g^{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}. & & \displaystyle\end{eqnarray}$$

Let us integrate both sides from $\unicode[STIX]{x1D709}^{\prime }=-\infty$ to $\infty$ . The first term on the right yields:

(5.93) $$\begin{eqnarray}\displaystyle \int _{-\infty }^{\infty }\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}\left(\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}g^{ij}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}\right)\,\text{d}\unicode[STIX]{x1D709}^{\prime }=\left[\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}g^{ij}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}\right]=0, & & \displaystyle\end{eqnarray}$$

where we used the continuity of $\unicode[STIX]{x1D716}\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }$ and $\widetilde{\unicode[STIX]{x1D719}}_{1}$ across $\unicode[STIX]{x1D709}^{\prime }=0$ (see (5.11) and (5.39)) in the second equality. Let us turn to the second term on the right-hand side of (5.92). Noting that $g^{ij}\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}/\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}$ does not depend on $\unicode[STIX]{x1D709}^{\prime }$ , it is sufficient to compute the following.

(5.94) $$\begin{eqnarray}\displaystyle \int _{-\infty }^{\infty }\left(\unicode[STIX]{x1D716}\unicode[STIX]{x1D705}_{0}\frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D719}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{\prime }}-\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}^{2}\widetilde{\unicode[STIX]{x1D719}}_{1}}{\unicode[STIX]{x2202}{\unicode[STIX]{x1D709}^{\prime }}^{2}}\right)\,\text{d}\unicode[STIX]{x1D709}^{\prime }=-\unicode[STIX]{x1D705}_{0}(\unicode[STIX]{x1D716}_{in}\unicode[STIX]{x1D719}_{in}^{\unicode[STIX]{x1D6E5}}+\unicode[STIX]{x1D716}_{ex}\unicode[STIX]{x1D719}_{ex}^{\unicode[STIX]{x1D6E5}})-\left(\frac{\unicode[STIX]{x1D716}_{in}}{\unicode[STIX]{x1D70E}_{in}}-\frac{\unicode[STIX]{x1D716}_{ex}}{\unicode[STIX]{x1D70E}_{ex}}\right)J_{0}. & & \displaystyle\end{eqnarray}$$

In the above, we used the definitions of $\unicode[STIX]{x1D719}_{in,ex}^{\unicode[STIX]{x1D6E5}}$ in (5.22), as well as (5.47) and (5.11) for the second integrand. The matching procedure for (5.92) can now be completed using the above together with (5.88), from which we find that

(5.95) $$\begin{eqnarray}\displaystyle \left[\unicode[STIX]{x1D707}\left(\frac{\unicode[STIX]{x2202}v_{0}^{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}+g^{ij}\frac{\unicode[STIX]{x2202}u_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}\right)\right]=\left(\unicode[STIX]{x1D705}_{0}(\unicode[STIX]{x1D716}_{in}\unicode[STIX]{x1D719}_{in}^{\unicode[STIX]{x1D6E5}}+\unicode[STIX]{x1D716}_{ex}\unicode[STIX]{x1D719}_{ex}^{\unicode[STIX]{x1D6E5}})+\left(\frac{\unicode[STIX]{x1D716}_{in}}{\unicode[STIX]{x1D70E}_{in}}-\frac{\unicode[STIX]{x1D716}_{ex}}{\unicode[STIX]{x1D70E}_{ex}}\right)J_{0}\right)g^{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{j}}. & & \displaystyle\end{eqnarray}$$

We have thus obtained the stress boundary conditions (5.85) and (5.95) for $u_{0}$ and $v_{0}^{i}$ respectively. Together with (5.55) and (5.65), these conditions constitute the interface conditions for the Stokes equation (5.73). It is important to note that, when the voltage jump $[\unicode[STIX]{x1D719}_{0}]$ is equal to $0$ (or $l_{C}=l_{A}$ ), the above interface conditions reduce to those of the TM model without convection. In this sense, the above calculation generalizes the TM model (without convection, applied to a sphere) in the presence of an interfacial double layer charge density.

5.3 Spherical drop under uniform electric field

We now apply the foregoing calculations to the case of a spherical drop under a uniform electric field. The calculation here parallels that of Schnitzer & Yariv (Reference Schnitzer and Yariv2015). Assuming the viscous drop maintains a spherical shape with a dimensionless radius of $1$ , we use spherical polar coordinates for our calculation. The voltage is given by:

(5.96) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}=\left\{\begin{array}{@{}ll@{}}\displaystyle -Er\cos \unicode[STIX]{x1D703}\frac{3\unicode[STIX]{x1D70E}_{ex}}{2\unicode[STIX]{x1D70E}_{ex}+\unicode[STIX]{x1D70E}_{in}}+\frac{1}{2}\ln (l_{C}/l_{A})\quad & \text{for }r<1,\\ \displaystyle -E\cos \unicode[STIX]{x1D703}\left(r+\frac{\unicode[STIX]{x1D70E}_{ex}-\unicode[STIX]{x1D70E}_{in}}{2\unicode[STIX]{x1D70E}_{ex}+\unicode[STIX]{x1D70E}_{in}}r^{-2}\right)\quad & \text{for }r\geqslant 1,\end{array}\right. & & \displaystyle\end{eqnarray}$$

where $r$ is the radial coordinate and $\unicode[STIX]{x1D703}$ is the polar angle. The electric field is pointing in the $\unicode[STIX]{x1D703}=0$ direction and is of magnitude $E$ . In the above and in what follows, we have omitted the subscript $0$ indicating the order in the asymptotic expansion. We shall only be interested in calculating the order $0$ quantities. Let $u_{r}$ and $u_{\unicode[STIX]{x1D703}}$ be the velocities in the $r$ and $\unicode[STIX]{x1D703}$ directions. The azimuthal component of the flow will be $0$ by symmetry. We suppose the velocity field is $0$ in the far field. Using (5.55) and (5.65), at the interface $r=1$ , we have the conditions:

(5.97) $$\begin{eqnarray}\displaystyle & \displaystyle [u_{r}]=0, & \displaystyle\end{eqnarray}$$

(5.98) $$\begin{eqnarray}\displaystyle & \displaystyle [u_{\unicode[STIX]{x1D703}}]=-\!\left(\frac{\unicode[STIX]{x1D716}_{in}}{\unicode[STIX]{x1D707}_{in}}\unicode[STIX]{x1D719}_{in}^{\unicode[STIX]{x1D6E5}}+\frac{\unicode[STIX]{x1D716}_{ex}}{\unicode[STIX]{x1D707}_{ex}}\unicode[STIX]{x1D719}_{ex}^{\unicode[STIX]{x1D6E5}}\right)\frac{3\unicode[STIX]{x1D70E}_{ex}E}{2\unicode[STIX]{x1D70E}_{ex}+\unicode[STIX]{x1D70E}_{in}}\sin \unicode[STIX]{x1D703}. & \displaystyle\end{eqnarray}$$

We now turn to the stress jump conditions. Let $\unicode[STIX]{x1D6F4}_{rr}$ and $\unicode[STIX]{x1D6F4}_{r\unicode[STIX]{x1D703}}$ be the $rr$ and $r\unicode[STIX]{x1D703}$ components of the fluid stress, which can be expressed using $u_{r}$ and $u_{\unicode[STIX]{x1D703}}$ as:

(5.99a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F4}_{rr}=2\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}u_{r}}{\unicode[STIX]{x2202}r}-p,\quad \unicode[STIX]{x1D6F4}_{r\unicode[STIX]{x1D703}}=\unicode[STIX]{x1D707}\left(r\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(\frac{u_{\unicode[STIX]{x1D703}}}{r}\right)+\frac{1}{r}\frac{\unicode[STIX]{x2202}u_{r}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\right). & & \displaystyle\end{eqnarray}$$

The stress jump conditions (5.85) and (5.95) yield:

(5.100) $$\begin{eqnarray}\displaystyle [\unicode[STIX]{x1D6F4}_{rr}] & = & \displaystyle \frac{1}{2}\left(\frac{3\unicode[STIX]{x1D70E}_{ex}E}{2\unicode[STIX]{x1D70E}_{ex}+\unicode[STIX]{x1D70E}_{in}}\right)^{2}\left(\unicode[STIX]{x1D716}_{in}-\unicode[STIX]{x1D716}_{ex}+\left(\unicode[STIX]{x1D716}_{ex}\left(\left(\frac{\unicode[STIX]{x1D70E}_{in}}{\unicode[STIX]{x1D70E}_{ex}}\right)^{2}+1\right)-2\unicode[STIX]{x1D716}_{in}\right)\cos ^{2}\unicode[STIX]{x1D703}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\frac{12\unicode[STIX]{x1D70E}_{ex}\unicode[STIX]{x1D70E}_{in}E}{2\unicode[STIX]{x1D70E}_{ex}+\unicode[STIX]{x1D70E}_{in}}\left(\frac{\unicode[STIX]{x1D716}_{in}}{\unicode[STIX]{x1D70E}_{in}}\unicode[STIX]{x1D719}_{in}^{\unicode[STIX]{x1D6E5}}+\frac{\unicode[STIX]{x1D716}_{ex}}{\unicode[STIX]{x1D70E}_{ex}}\unicode[STIX]{x1D719}_{ex}^{\unicode[STIX]{x1D6E5}}-I_{J}\right)\cos \unicode[STIX]{x1D703}-8I_{\unicode[STIX]{x1D705}},\end{eqnarray}$$

(5.101) $$\begin{eqnarray}\displaystyle [\unicode[STIX]{x1D6F4}_{r\unicode[STIX]{x1D703}}]=\left(\frac{3\unicode[STIX]{x1D70E}_{ex}E}{2\unicode[STIX]{x1D70E}_{ex}+\unicode[STIX]{x1D70E}_{in}}\right)^{2}\left(\unicode[STIX]{x1D716}_{in}-\unicode[STIX]{x1D716}_{ex}\left(\frac{\unicode[STIX]{x1D70E}_{in}}{\unicode[STIX]{x1D70E}_{ex}}\right)\right)\sin \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D703}+\frac{6\unicode[STIX]{x1D70E}_{ex}E}{2\unicode[STIX]{x1D70E}_{ex}+\unicode[STIX]{x1D70E}_{in}}(\unicode[STIX]{x1D716}_{in}\unicode[STIX]{x1D719}_{in}^{\unicode[STIX]{x1D6E5}}+\unicode[STIX]{x1D716}_{ex}\unicode[STIX]{x1D719}_{ex}^{\unicode[STIX]{x1D6E5}})\sin \unicode[STIX]{x1D703}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

We have used the fact that $\unicode[STIX]{x1D705}_{0}=2$ for a unit sphere. Since we assume the drop shape is spherical, $\unicode[STIX]{x1D705}_{1}=0$ . This constraint should lead to restrictions on the parameter values. We shall later restore the surface tension term $\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D705}_{1}$ .

We now introduce the Stokes streamfunction $\unicode[STIX]{x1D713}(r,\unicode[STIX]{x1D703})$ to solve the above interface problem, which is related to $u_{r}$ and $u_{\unicode[STIX]{x1D703}}$ via

(5.102a,b ) $$\begin{eqnarray}\displaystyle u_{r}=\frac{1}{r^{2}\sin \unicode[STIX]{x1D703}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}},\quad u_{\unicode[STIX]{x1D703}}=-\frac{1}{r\sin \unicode[STIX]{x1D703}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}r}. & & \displaystyle\end{eqnarray}$$

We use the well-known separation of variables solution (see, for example, Leal (Reference Leal2007)). From the above interface conditions, the Stokes streamfunction $\unicode[STIX]{x1D713}$ should be written as:

(5.103) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}(r,\unicode[STIX]{x1D703})=\left\{\begin{array}{@{}ll@{}}\displaystyle \mathop{\sum }_{n=1}^{2}(A_{n}r^{n+3}+B_{n}r^{n+1})Q_{n}(\cos \unicode[STIX]{x1D703})\quad & \text{if }r<1,\\ \displaystyle \mathop{\sum }_{n=1}^{2}(C_{n}r^{2-n}+D_{n}r^{-n})Q_{n}(\cos \unicode[STIX]{x1D703})\quad & \text{if }r\geqslant 1,\end{array}\right. & & \displaystyle\end{eqnarray}$$

where

(5.104a,b ) $$\begin{eqnarray}\displaystyle Q_{1}(z)=1-z^{2},\quad Q_{2}(z)=z-z^{3}. & & \displaystyle\end{eqnarray}$$

The corresponding pressure is computed as

(5.105) $$\begin{eqnarray}\displaystyle p(r,\unicode[STIX]{x1D703})=\left\{\begin{array}{@{}ll@{}}\unicode[STIX]{x1D707}_{in}(\frac{7}{2}A_{2}r^{2}+20A_{1}\cos \unicode[STIX]{x1D703}+\frac{21}{2}A_{2}r^{2}\cos 2\unicode[STIX]{x1D703})\quad & \text{if }r<1,\\ \unicode[STIX]{x1D707}_{ex}(C_{2}r^{-3}+2C_{1}r^{-2}\cos \unicode[STIX]{x1D703}+3C_{2}r^{-3}\cos 2\unicode[STIX]{x1D703})\quad & \text{if }r\geqslant 1.\end{array}\right. & & \displaystyle\end{eqnarray}$$

Let us determine the coefficients in the above expressions. Plugging in the above expression into (5.97), we obtain:

(5.106) $$\begin{eqnarray}\displaystyle & & \displaystyle 2(A_{1}+B_{1})\cos \unicode[STIX]{x1D703}-(A_{2}+B_{2})(1-3\cos ^{2}\unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \quad =2(C_{1}+D_{1})\cos \unicode[STIX]{x1D703}-(C_{2}+D_{2})(1-3\cos ^{2}\unicode[STIX]{x1D703})=u_{r}|_{r=1}.\end{eqnarray}$$

Since we assume that the drop maintains its spherical shape, $u_{r}$ at $r=1$ must be proportional to $\cos \unicode[STIX]{x1D703}$ . Therefore, we have:

(5.107a,b ) $$\begin{eqnarray}\displaystyle A_{1}+B_{1}=C_{1}+D_{1},\quad A_{2}+B_{2}=C_{2}+D_{2}=0. & & \displaystyle\end{eqnarray}$$

From (5.98), we obtain:

(5.108) $$\begin{eqnarray}\displaystyle & & \displaystyle (-4A_{1}-2B_{1}+C_{1}-D_{1})\sin \unicode[STIX]{x1D703}+(-5A_{2}-3B_{2}-2D_{2})\sin \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D703}\nonumber\\ \displaystyle & & \displaystyle \quad =-\left(\frac{\unicode[STIX]{x1D716}_{in}}{\unicode[STIX]{x1D707}_{in}}\unicode[STIX]{x1D719}_{in}^{\unicode[STIX]{x1D6E5}}+\frac{\unicode[STIX]{x1D716}_{ex}}{\unicode[STIX]{x1D707}_{ex}}\unicode[STIX]{x1D719}_{ex}^{\unicode[STIX]{x1D6E5}}\right)\frac{3E\unicode[STIX]{x1D70E}_{ex}}{2\unicode[STIX]{x1D70E}_{in}+\unicode[STIX]{x1D70E}_{ex}}\sin \unicode[STIX]{x1D703}.\end{eqnarray}$$

Thus,

(5.109) $$\begin{eqnarray}\displaystyle & \displaystyle -4A_{1}-2B_{1}+C_{1}-D_{1}=-\!\left(\frac{\unicode[STIX]{x1D716}_{in}}{\unicode[STIX]{x1D707}_{in}}\unicode[STIX]{x1D719}_{in}^{\unicode[STIX]{x1D6E5}}+\frac{\unicode[STIX]{x1D716}_{ex}}{\unicode[STIX]{x1D707}_{ex}}\unicode[STIX]{x1D719}_{ex}^{\unicode[STIX]{x1D6E5}}\right)\frac{3E\unicode[STIX]{x1D70E}_{ex}}{2\unicode[STIX]{x1D70E}_{ex}+\unicode[STIX]{x1D70E}_{in}}, & \displaystyle\end{eqnarray}$$

(5.110) $$\begin{eqnarray}\displaystyle & \displaystyle -5A_{2}-3B_{2}-2D_{2}=0. & \displaystyle\end{eqnarray}$$

Likewise, from (5.100) and (5.101), we obtain the equations:

(5.111) $$\begin{eqnarray}\displaystyle & \displaystyle -12\unicode[STIX]{x1D707}_{in}A_{1}+\unicode[STIX]{x1D707}_{ex}(6C_{1}+12D_{1})=\frac{12\unicode[STIX]{x1D70E}_{ex}\unicode[STIX]{x1D70E}_{in}E}{2\unicode[STIX]{x1D70E}_{ex}+\unicode[STIX]{x1D70E}_{in}}\left(\frac{\unicode[STIX]{x1D716}_{in}}{\unicode[STIX]{x1D70E}_{in}}\unicode[STIX]{x1D719}_{in}^{\unicode[STIX]{x1D6E5}}+\frac{\unicode[STIX]{x1D716}_{ex}}{\unicode[STIX]{x1D70E}_{ex}}\unicode[STIX]{x1D719}_{ex}^{\unicode[STIX]{x1D6E5}}-I_{J}\right), & \displaystyle\end{eqnarray}$$

(5.112) $$\begin{eqnarray}\displaystyle & \displaystyle -3(\unicode[STIX]{x1D707}_{in}(A_{2}-2B_{2})-\unicode[STIX]{x1D707}_{ex}(6C_{2}+8D_{2}))=\frac{1}{2}\left(\frac{3\unicode[STIX]{x1D70E}_{ex}E}{2\unicode[STIX]{x1D70E}_{ex}+\unicode[STIX]{x1D70E}_{in}}\right)^{2}\left(\unicode[STIX]{x1D716}_{ex}\left(\left(\frac{\unicode[STIX]{x1D70E}_{in}}{\unicode[STIX]{x1D70E}_{ex}}\right)^{2}+1\right)-2\unicode[STIX]{x1D716}_{in}\right), & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(5.113) $$\begin{eqnarray}\displaystyle & \displaystyle -6\unicode[STIX]{x1D707}_{in}A_{1}+6\unicode[STIX]{x1D707}_{ex}D_{1}=\frac{6\unicode[STIX]{x1D70E}_{ex}E}{2\unicode[STIX]{x1D70E}_{ex}+\unicode[STIX]{x1D70E}_{in}}(\unicode[STIX]{x1D716}_{in}\unicode[STIX]{x1D719}_{in}^{\unicode[STIX]{x1D6E5}}+\unicode[STIX]{x1D716}_{ex}\unicode[STIX]{x1D719}_{ex}^{\unicode[STIX]{x1D6E5}}) & \displaystyle\end{eqnarray}$$

(5.114) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D707}_{in}(16A_{2}+6B_{2})+\unicode[STIX]{x1D707}_{ex}(6C_{2}+16D_{2})=\left(\frac{3\unicode[STIX]{x1D70E}_{ex}E}{2\unicode[STIX]{x1D70E}_{ex}+\unicode[STIX]{x1D70E}_{in}}\right)^{2}\!\left(\unicode[STIX]{x1D716}_{in}-\unicode[STIX]{x1D716}_{ex}\!\left(\frac{\unicode[STIX]{x1D70E}_{in}}{\unicode[STIX]{x1D70E}_{ex}}\right)\!\right)\!.\quad & \displaystyle\end{eqnarray}$$

We may solve the above for the eight constants. The five equations for $A_{2},B_{2},C_{2}$ and $D_{2}$ , are exactly the same as those discussed in Taylor (Reference Taylor1966), and this results in a condition on the parameters for solvability. This leads to the discriminating function for prolate/oblate deformation. We note that $l_{A}\neq l_{C}$ only affects $A_{1},B_{1},C_{1}$ and $D_{1}$ . Thus the voltage jump due to $l_{A}\neq l_{C}$ does not affect the discriminating function at leading order. However, we expect the migration speed of the drop $V_{mgr}$ to depend on such voltage jump because, from (5.106), $V_{mgr}=2(A_{1}+B_{1})$ . Solving for the constants $A_{1},B_{1},C_{1}$ and $D_{1}$ , we obtain:

(5.115) $$\begin{eqnarray}\displaystyle V_{mgr} & = & \displaystyle \frac{2(5\unicode[STIX]{x1D716}_{in}\unicode[STIX]{x1D70E}_{ex}\unicode[STIX]{x1D707}_{ex}\unicode[STIX]{x1D719}_{in}^{\unicode[STIX]{x1D6E5}}+\unicode[STIX]{x1D716}_{ex}(4\unicode[STIX]{x1D70E}_{in}(\unicode[STIX]{x1D707}_{in}+\unicode[STIX]{x1D707}_{ex})-\unicode[STIX]{x1D70E}_{ex}(\unicode[STIX]{x1D707}_{in}+2\unicode[STIX]{x1D707}_{ex}))\unicode[STIX]{x1D719}_{ex}^{\unicode[STIX]{x1D6E5}})}{\unicode[STIX]{x1D707}_{ex}(3\unicode[STIX]{x1D707}_{in}+2\unicode[STIX]{x1D707}_{ex})(2\unicode[STIX]{x1D70E}_{ex}+\unicode[STIX]{x1D70E}_{in})}E\nonumber\\ \displaystyle & & \displaystyle -\,\frac{8(\unicode[STIX]{x1D707}_{in}+\unicode[STIX]{x1D707}_{ex})\unicode[STIX]{x1D70E}_{in}\unicode[STIX]{x1D70E}_{ex}I_{J}}{\unicode[STIX]{x1D707}_{ex}(3\unicode[STIX]{x1D707}_{in}+2\unicode[STIX]{x1D707}_{ex})(2\unicode[STIX]{x1D70E}_{ex}+\unicode[STIX]{x1D70E}_{in})}E.\end{eqnarray}$$

Using (5.23) and noting that $I_{J}=O((\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}})^{2})$ we have:

(5.116) $$\begin{eqnarray}\displaystyle V_{mgr}=\frac{2(5\unicode[STIX]{x1D716}_{in}\unicode[STIX]{x1D70E}_{ex}\unicode[STIX]{x1D707}_{ex}\unicode[STIX]{x1D701}+4\unicode[STIX]{x1D716}_{ex}\unicode[STIX]{x1D70E}_{in}(\unicode[STIX]{x1D707}_{in}+\unicode[STIX]{x1D707}_{ex})-\unicode[STIX]{x1D716}_{ex}\unicode[STIX]{x1D70E}_{ex}(\unicode[STIX]{x1D707}_{in}+2\unicode[STIX]{x1D707}_{ex}))}{\unicode[STIX]{x1D707}_{ex}(3\unicode[STIX]{x1D707}_{in}+2\unicode[STIX]{x1D707}_{ex})(2\unicode[STIX]{x1D70E}_{ex}+\unicode[STIX]{x1D70E}_{in})(1+\unicode[STIX]{x1D701})}\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}}E+O((\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}})^{2}). & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

If $\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}}$ is small, the droplet migrates in the direction of the sign of $\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}}E$ when

(5.117) $$\begin{eqnarray}\displaystyle Q_{mgr}=\frac{5\unicode[STIX]{x1D716}_{rt}\unicode[STIX]{x1D701}+4\unicode[STIX]{x1D70E}_{rt}(1+\unicode[STIX]{x1D707}_{rt})}{2+\unicode[STIX]{x1D707}_{rt}}>1,\quad \unicode[STIX]{x1D716}_{rt}=\frac{\unicode[STIX]{x1D716}_{in}}{\unicode[STIX]{x1D716}_{ex}},\quad \unicode[STIX]{x1D70E}_{rt}=\frac{\unicode[STIX]{x1D70E}_{in}}{\unicode[STIX]{x1D70E}_{ex}},\quad \unicode[STIX]{x1D707}_{rt}=\frac{\unicode[STIX]{x1D707}_{in}}{\unicode[STIX]{x1D707}_{ex}}. & & \displaystyle\end{eqnarray}$$

The direction of droplet migration, therefore, is not necessarily equal to $\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}}E$ as might have been expected. This is analogous to the conclusions of Baygents & Saville (Reference Baygents and Saville1991), in which the electrophoresis of conducting drops was studied using an electrodiffusion model.

We now numerically evaluate the flow field. The only non-trivial detail is the evaluation of the term $I_{J}$ defined in (5.86), which in turn requires the numerical solution of $\unicode[STIX]{x1D713}_{J}$ and $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D705}}$ which satisfies (5.43) and (5.42) respectively. The technical difficulty here is that these differential equations are posed on the real line, and a straight forward discretization does not work. We take a smooth map of the real line onto a finite interval and discretize the transformed equations to obtain a numerical solution. The details of this computation are omitted.

Flow fields around a migrating drop are shown in figure 12. The flow fields are similar in its appearance to the quadrupole vortex flow field photograph reported in Taylor (Reference Taylor1966). In his paper, Taylor reports that the droplet drifts, and the flow field shows an asymmetry not explained by the TM model.

Figure 12.

Flow field around a migrating droplet, in a frame moving with the droplet. Rotation axis is horizontal. $D_{A/C,in,ex}=1$ , $\unicode[STIX]{x1D707}_{in,ex}=1$ , $l_{C}=1.01$ , $l_{A}=1/l_{C}$ . In (a), $\unicode[STIX]{x1D716}_{in}=1$ , $\unicode[STIX]{x1D716}_{ex}=2$ and in (b), $\unicode[STIX]{x1D716}_{in}=2$ , $\unicode[STIX]{x1D716}_{ex}=1$ so that the flow field is consistent with a prolate/oblate deformation in figure (a)/(b). Electric field is in the direction of the arrow. $\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}}>0$ , $Q_{mgr}>1$ in both figures so that the droplet is moving in the direction of the arrow.

6.1 Scaling and asymptotics

First, we note that any asymptotic calculation is only as good as the expansion ansatz ((4.2) for § 4 and (5.2) for § 5). It is therefore important future work to validate of our conclusions by computational or analytic means. Computational verification will necessarily require numerical methods for the modified Saville or charge diffusion models. Numerical methods in Tomar et al. (Reference Tomar, Gerlach, Biswas, Alleborn, Sharma, Durst, Welch and Delgado2007), López-Herrera, Popinet & Herrada (Reference López-Herrera, Popinet and Herrada2011), Berry, Davidson & Harvie (Reference Berry, Davidson and Harvie2013), Hu et al. (Reference Hu, Lai and Young2015) may be of particular interest in this regard.

We have chosen particular scalings for the dimensionless variables to obtain limiting models. In § 3, we took the limit $\unicode[STIX]{x1D6FC}\rightarrow 0$ before taking the limit $\unicode[STIX]{x1D6FF}\rightarrow 0$ , as dictated by (1.7). The scaling $\unicode[STIX]{x1D6FC}\sim \unicode[STIX]{x1D6FF}$ , for example, may be appropriate in certain situations (see § 6.2). In § 4.1, equation (4.1) was chosen to obtain the TM model. As we saw in table 1 other scalings for $Pe$ (when $\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6E5}}=0$ ) lead to the variants of the TM model. Other distinguished scalings may be worth exploring. In § 5, scaling (5.1) was chosen for analytical feasibility, and other scalings may likewise be worthy of further study. Our analysis there requires that the surface tension coefficient $\unicode[STIX]{x1D6FE}$ be greater than twice the electrostatic energy stored in the electric double layer (see (5.56) and the discussion at the end of § 5.1). When this condition is violated, it does not seem to be possible to take the limit $\unicode[STIX]{x1D6FF}\rightarrow 0$ . This may indicate the presence of an interfacial electrochemical/electrohydrodynamic instability.

Scaling of dimensionless variables should be guided by their magnitudes in experimental set-ups, an important subject to which we now turn.

6.2 Dimensionless parameters and surface charge convection

We discuss the size of dimensionless parameters that featured in our analysis. Let the representative constants be:

(6.1) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}T=298~\text{K},\quad \unicode[STIX]{x1D707}_{\ast }=1~\text{Pa}~\text{s},\quad L=1~\text{mm},\quad \unicode[STIX]{x1D716}_{\ast }=4\unicode[STIX]{x1D716}_{0},\\ \unicode[STIX]{x1D70E}_{\ast }=10^{-9}~\text{S}~\text{m}^{-1},\quad \unicode[STIX]{x1D6FE}_{\ast }=1~\text{mN}~\text{m}^{-1},\end{array}\right\} & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D716}_{0}$ is the permittivity of vacuum and $\unicode[STIX]{x1D70E}_{\ast }$ is the representative conductivity. All values follow Saville (Reference Saville1997) except $\unicode[STIX]{x1D6FE}_{\ast }$ which corresponds to the order of magnitude in Salipante & Vlahovska (Reference Salipante and Vlahovska2010). Following Saville (Reference Saville1997) we set the representative ionic radius $r_{ion}$ to be:

(6.2) $$\begin{eqnarray}\displaystyle r_{ion}=0.25~\text{nm}. & & \displaystyle\end{eqnarray}$$

Assuming the Stokes–Einstein relation, we may set:

(6.3a,b ) $$\begin{eqnarray}\displaystyle D_{\ast }=\frac{k_{B}T}{6\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}_{\ast }r_{ion}},\quad \unicode[STIX]{x1D70E}_{\ast }=D_{\ast }\frac{F^{2}}{RT}c_{\ast }, & & \displaystyle\end{eqnarray}$$

where $k_{B}$ is the Boltzmann constant. The expression for $\unicode[STIX]{x1D70E}_{\ast }$ allows for the estimation of $c_{\ast }$ . The above yields the following values for the dimensionless parameters:

(6.4a-c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}=1.8\times 10^{-4},\quad Pe=2.7\times 10^{-2},\quad \unicode[STIX]{x1D6FE}=1.3. & & \displaystyle\end{eqnarray}$$

In addition, Saville (Reference Saville1997) uses the above to estimate the parameter $\unicode[STIX]{x1D6FC}$ as:

(6.5) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}\approx 10^{-4}. & & \displaystyle\end{eqnarray}$$

We note that many of the above parameters may easily vary by a factor of $10$ – $100$ ( $\unicode[STIX]{x1D70E}_{\ast }$ for example) in either direction, and thus, should be taken as a rough estimate. Since the identity of the conducting ions and their reactions in leaky dielectrics are unknown, the estimate for $r_{ion}$ and $\unicode[STIX]{x1D6FC}$ above are necessarily uncertain (Saville Reference Saville1997).

We now discuss the implications of the above values for the applicability of our analysis. Our most important assumption is the ordering $\unicode[STIX]{x1D6FC}\ll \unicode[STIX]{x1D6FF}\ll 1$ as in (1.7). The above indicates that $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FF}$ are comparable. Note, however, that $\unicode[STIX]{x1D6FF}$ can be made larger if we assume that $\unicode[STIX]{x1D70E}_{\ast }$ is smaller ( $\unicode[STIX]{x1D70E}_{\ast }=10^{-11}$ – $10^{-12}~\text{S}~\text{m}^{-1}$ in Salipante & Vlahovska (Reference Salipante and Vlahovska2010)) or $L$ is smaller. In addition, if the ions present in the leaky dielectrics are generated by the dissociation of the solvent itself as discussed in § 3 (e.g. $\text{H}^{+}$ and $\text{OH}^{-}$ in pure water), the ratio $\unicode[STIX]{x1D6FC}$ can be significantly smaller (approximately $10^{-9}$ for pure water). The assumption $\unicode[STIX]{x1D6FC}\ll \unicode[STIX]{x1D6FF}\ll 1$ is thus likely to be reasonable for many systems. It would still be of interest to consider the limit $\unicode[STIX]{x1D6FC}\approx \unicode[STIX]{x1D6FF}\ll 1$ , possibly a more appropriate scaling in certain systems.

In (6.4), $Pe$ is at most $O(1)$ with respect to $\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D6FE}$ is much larger than $O(\unicode[STIX]{x1D6FF})$ . According to table 1, this indicates that the TM model without surface charge convection will be valid to leading order. The large value of $\unicode[STIX]{x1D6FE}$ with respect to $\unicode[STIX]{x1D6FF}^{2}$ indicates that there will be little deformation from the sphere; a formal analysis of this situation will be analogous to the calculation in § 5.

Recall from § 4.3 that our analysis can be extended to the case of large imposed voltage provided (4.80) is satisfied. Given the estimate of $\unicode[STIX]{x1D6FF}\approx 10^{-4}$ in (6.4), $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D714}^{-2}=10^{3}$ may be the largest reasonable value for the dimensionless electric field magnitude for our derivation to hold. In this case, the dimensionless parameters of (4.79) and (4.82) are:

(6.6a,b ) $$\begin{eqnarray}\displaystyle Pe_{E}=Pe\unicode[STIX]{x1D6FD}=2.7\times 10,\quad \unicode[STIX]{x1D6FF}_{E}=\unicode[STIX]{x1D6FF}\sqrt{\unicode[STIX]{x1D6FD}}=5.6\times 10^{-3}. & & \displaystyle\end{eqnarray}$$

Given that these are rough estimates, it is possible that $Pe_{E}=O(\unicode[STIX]{x1D6FF}_{E}^{-1})$ in certain systems. In this case, table 1 indicates that we may obtain surface charge convection as an $O(\unicode[STIX]{x1D6FF}_{E})$ correction. The surface tension coefficient $\unicode[STIX]{x1D6FE}$ is still much larger than $\unicode[STIX]{x1D6FF}_{E}$ , so that deformation from the sphere may be small.

The above considerations lead to the question of whether there are any leaky dielectric systems for which surface charge convection can be obtained as a leading-order term. According to (4.84), we would need:

(6.7) $$\begin{eqnarray}\displaystyle Pe_{E}=Pe\unicode[STIX]{x1D6FD}\approx \unicode[STIX]{x1D6FF}_{E}^{-2}=\unicode[STIX]{x1D6FF}^{-2}\unicode[STIX]{x1D6FD}^{-1}, & & \displaystyle\end{eqnarray}$$

which implies:

(6.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}\approx \sqrt{Pe^{-1}}\unicode[STIX]{x1D6FF}^{-1}. & & \displaystyle\end{eqnarray}$$

This same scaling can be obtained just from the TM model. Balancing of the convective and conduction terms in the full TM model (see (4.61) and (4.62)) amounts to equating the electrohydrodynamic time with the Maxwell–Wagner relaxation time (Salipante & Vlahovska Reference Salipante and Vlahovska2010):

(6.9) $$\begin{eqnarray}\displaystyle t_{EHD}=\frac{\unicode[STIX]{x1D707}_{\ast }}{\unicode[STIX]{x1D716}_{\ast }E_{\ast }^{2}}\approx \frac{\unicode[STIX]{x1D716}_{\ast }}{\unicode[STIX]{x1D70E}_{\ast }}=t_{MW}. & & \displaystyle\end{eqnarray}$$

This leads to the scaling of $\unicode[STIX]{x1D6FD}$ in (6.8). The ratio $Re_{E}=t_{MW}/t_{EHD}$ is sometimes referred to as the electric Reynolds number. We may thus say that the scaling (6.8) corresponds to the regime where $Re_{E}=O(1)$ .

Since we need (4.81) for the validity of the asymptotic analysis, equation (6.8) implies that

(6.10) $$\begin{eqnarray}\displaystyle Pe\gg 1 & & \displaystyle\end{eqnarray}$$

must be satisfied. Using (6.3), we may compute $Pe$ as:

(6.11) $$\begin{eqnarray}\displaystyle Pe=\frac{r_{ion}}{r_{B}},\quad r_{B}=\frac{q_{el}^{2}}{6\unicode[STIX]{x03C0}\unicode[STIX]{x1D716}_{\ast }k_{B}T}, & & \displaystyle\end{eqnarray}$$

where $q_{el}$ is the elementary charge. The length $r_{B}$ is the Bjerrum length (up to a factor of $2/3$ ), which is of the order of 0.5–10 nm depending on the dielectric constant. On the other hand, ion size is expected to be in the subnanometre range. Thus, barring the use of exotic materials, $Pe$ is typically never larger than $1$ .

Surface charge convection will thus be negligible at thermal voltages, but will become appreciable as the imposed voltage becomes larger. At very large voltages for which the electric Reynolds number is $O(1)$ , however, our derivation of the TM model may break down. Our theory is thus consistent with reports that surface charge convection is important for large imposed voltages (Xu & Homsy Reference Xu and Homsy2006; Salipante & Vlahovska Reference Salipante and Vlahovska2010; Lanauze et al. Reference Lanauze, Walker and Khair2015; Vlahovska Reference Vlahovska2016; Das & Saintillan Reference Das and Saintillan2017b ; Sengupta et al. Reference Sengupta, Walker and Khair2017), but experimentally imposed voltages are often higher than the range over which our asymptotic calculations are guaranteed to be valid. It is, however, quite possible that validity of our asymptotics does not deteriorate too much even at such high voltages.

It is clear that the TM model itself (and thus, any asymptotic derivation of the TM model) fails when and after geometric singularities arise. Recall that the charge diffusion model of § 3 depended only on the smallness of $\unicode[STIX]{x1D6FC}$ , and is thus potentially valid at very large field strengths. Direct analysis and simulation of the charge diffusion model may help resolve the physics of charge convection and singularity formation under strong fields.

6.3 Surface electrochemistry, Galvani potential and electromigration

It is well known that many liquid–liquid interfaces have a spontaneous voltage jump known as the Galvani potential (GP) (Girault & Schiffrin Reference Girault and Schiffrin1989; Reymond et al. Reference Reymond, Fermın, Lee and Girault2000). In its presence, as shown in § 5, a suspended leaky dielectric droplet will migrate under an imposed electric field. Our analysis yields an explicit formula for droplet migration velocity. We have, unfortunately, not been able to obtain an estimate for the electromigration velocity primarily because of the absence of independent measurements of the GP. We may, in turn, be able to use the measured electromigration velocity to obtain an estimate of the GP based on our formula. We also point out that the electromigration velocity formulae for droplets described in Booth (Reference Booth1951), Baygents & Saville (Reference Baygents and Saville1991), Pascall & Squires (Reference Pascall and Squires2011) are different from those obtained here. Although this is not surprising given the different modelling assumptions, it would be interesting to clarify the interrelation among these calculations.

The properties of the EDL and the resulting GP, which underpins our analysis of droplet migration, should depend on the details of the surface electrochemistry at the liquid–liquid interface. Our interfacial boundary conditions for ionic concentrations are the simplest possible. Unlike Saville (Reference Saville1997), Schnitzer & Yariv (Reference Schnitzer and Yariv2015), the modified Saville model does not incorporate surface ionic concentrations. We have assumed that the anions and cations can move across the liquid–liquid interface. It is possible that electric current flows across the interface via Faradaic reactions rather than by simple drift of ions. A more sophisticated model for surface electrochemistry may thus be needed for a better understanding of droplet electromigration and related phenomena (Reymond et al. Reference Reymond, Fermın, Lee and Girault2000).