2.1 Thermodynamic analysis

The generation power $W$ and conversion efficiency $\unicode[STIX]{x1D709}$ can be written as

(2.3a,b ) $$\begin{eqnarray}W=I\unicode[STIX]{x0394}\unicode[STIX]{x1D719}\quad \text{and}\quad \unicode[STIX]{x1D709}=\frac{I\unicode[STIX]{x0394}\unicode[STIX]{x1D719}}{Q(-\unicode[STIX]{x0394}p)}.\end{eqnarray}$$

The generation power is maximized by $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}=-\unicode[STIX]{x0394}pL_{12}/2L_{22}$ , which is exactly half of the streaming potential. The maximum generation power and the corresponding conversion efficiency are given by (Xuan & Li Reference Xuan and Li2006)

(2.4a,b ) $$\begin{eqnarray}W_{max\,W}=\frac{I_{s}\unicode[STIX]{x0394}\unicode[STIX]{x1D719}_{s}}{4}=\frac{\unicode[STIX]{x1D6FC}}{4}L_{11}(\unicode[STIX]{x0394}p)^{2}\quad \text{and}\quad \unicode[STIX]{x1D709}_{max\,W}=\frac{I_{s}\unicode[STIX]{x0394}\unicode[STIX]{x1D719}_{s}}{4Q(-\unicode[STIX]{x0394}p)}=\frac{\unicode[STIX]{x1D6FC}}{2(2-\unicode[STIX]{x1D6FC})},\end{eqnarray}$$

where the subscript $max\,W$ indicates the condition of maximum generation power, $\unicode[STIX]{x1D6FC}=L_{12}L_{21}/L_{11}L_{22}$ is the figure of merit for electrokinetic energy conversion, and $0\leqslant \unicode[STIX]{x1D6FC}<1$ due to the positive definiteness of  $\unicode[STIX]{x1D647}$ . On the other hand, when the voltage satisfies $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}=-\unicode[STIX]{x0394}p(1-\sqrt{1-\unicode[STIX]{x1D6FC}})L_{11}/L_{12}$ , the conversion efficiency is maximized. At this condition, the generation power and efficiency are given by (Xuan & Li Reference Xuan and Li2006)

(2.5a,b ) $$\begin{eqnarray}W_{max\,\unicode[STIX]{x1D709}}=L_{11}\frac{\sqrt{1-\unicode[STIX]{x1D6FC}}(1-\sqrt{1-\unicode[STIX]{x1D6FC}})^{2}}{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x0394}p)^{2}\quad \text{and}\quad \unicode[STIX]{x1D709}_{max\,\unicode[STIX]{x1D709}}=\frac{(1-\sqrt{1-\unicode[STIX]{x1D6FC}})^{2}}{\unicode[STIX]{x1D6FC}},\end{eqnarray}$$

where the subscript $max$ indicates the condition of maximum generation efficiency.

2.2 Electric circuit analysis

Consider such a fluidic device connected to an electrical resistor (with resistance  $R_{L}$ ) in a circuit (see figure 1 b). The efficiency is defined as the electrical power, $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}^{2}/R_{L}$ , consumed by the load divided by the input mechanical pumping power, $Q(-\unicode[STIX]{x0394}p)$ . It is convenient to define a rescaled load resistance, $\unicode[STIX]{x1D6FA}=R_{L}/R_{ch}$ , relative to the channel resistance, $R_{ch}=1/L_{22}$ . The maximum electrical power can be achieved at $\unicode[STIX]{x1D6FA}=1$ . At this condition, the maximum power and the corresponding conversion efficiency are the same as in (2.4). Furthermore, the optimal efficiency and the corresponding generation power are found to be

(2.6a,b ) $$\begin{eqnarray}W_{max\,\unicode[STIX]{x1D709}}=L_{11}\frac{\sqrt{1-\unicode[STIX]{x1D6FC}}(1-\sqrt{1-\unicode[STIX]{x1D6FC}})^{2}}{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x0394}p)^{2}\quad \text{and}\quad \unicode[STIX]{x1D709}_{max\,\unicode[STIX]{x1D709}}=\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D6FC}+2(\sqrt{1-\unicode[STIX]{x1D6FC}}+1-\unicode[STIX]{x1D6FC})},\end{eqnarray}$$

for $\unicode[STIX]{x1D6FA}=1/\sqrt{1-\unicode[STIX]{x1D6FC}}$ (van der Heyden et al. Reference van der Heyden, Bonthuis, Stein, Meyer and Dekker2006). It is easy to verify that the efficiencies in (2.5) and (2.6) are the same, although they are in different forms.

Figure 1.

(a) Schematic illustration of electrokinetic energy conversion in a nanochannel filled with two-layer fluids. A positive external pressure gradient $-\unicode[STIX]{x0394}p/l=-(p_{out}-p_{in})/l$ is applied across the nanochannel in the $x$ -direction, which drives the flow of mobile ions and thus induces a streaming current $I_{s}$ and a streaming electric field of strength $E_{s}=-\unicode[STIX]{x0394}\unicode[STIX]{x1D719}/l=-(\unicode[STIX]{x1D6F7}_{out}-\unicode[STIX]{x1D6F7}_{in})/l$ . This electric field, in turn, generates a current (conduction current  $I_{c}$ ) to flow back against the direction of the pressure-driven flow. (b) Equivalent electric circuit of the considered system connected to an external load resistor with adjustable resistance  $R_{L}$ .

Interestingly, we find that the results (i.e. the electrical power and efficiency under the conditions of maximum output power and/or maximum efficiency) are the same whether based on thermodynamic or electric circuit analyses. Table 1 summarizes the electrical powers and conversion efficiencies obtained in different ways.

Table 1.

Summary of electrical power and conversion efficiency under the maximum power generation and/or maximum conversion efficiency conditions, based on thermodynamic and electric circuit analyses, respectively.

a Indicates the condition of maximum power generation.

b Indicates the condition of maximum conversion efficiency, and the approximation is for small  $\unicode[STIX]{x1D6FC}$ .

Comparing the two generation powers $W_{max\,W}$ and $W_{max\,\unicode[STIX]{x1D709}}$ in (2.4) and (2.5), we find that they are identical up to $O(\unicode[STIX]{x1D6FC}^{2})$ for small $\unicode[STIX]{x1D6FC}$ by means of

(2.7) $$\begin{eqnarray}\frac{\sqrt{1-\unicode[STIX]{x1D6FC}}(1-\sqrt{1-\unicode[STIX]{x1D6FC}})^{2}}{\unicode[STIX]{x1D6FC}}=\frac{\unicode[STIX]{x1D6FC}}{4}-\frac{\unicode[STIX]{x1D6FC}^{3}}{64}+O(\unicode[STIX]{x1D6FC}^{4}).\end{eqnarray}$$

Similarly, the two conversion efficiencies $\unicode[STIX]{x1D709}_{max\,W}$ and $\unicode[STIX]{x1D709}_{max\,\unicode[STIX]{x1D709}}$ are identical up to $O(\unicode[STIX]{x1D6FC}^{2})$ by means of

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{(1-\sqrt{1-\unicode[STIX]{x1D6FC}})^{2}}{\unicode[STIX]{x1D6FC}}=\frac{\unicode[STIX]{x1D6FC}}{4}+\frac{\unicode[STIX]{x1D6FC}^{2}}{8}+\frac{5\unicode[STIX]{x1D6FC}^{3}}{64}+O(\unicode[STIX]{x1D6FC}^{4}), & \displaystyle\end{eqnarray}$$

(2.9) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x1D6FC}}{2(2-\unicode[STIX]{x1D6FC})}=\frac{\unicode[STIX]{x1D6FC}}{4}+\frac{\unicode[STIX]{x1D6FC}^{2}}{8}+\frac{\unicode[STIX]{x1D6FC}^{3}}{16}+O(\unicode[STIX]{x1D6FC}^{4}). & \displaystyle\end{eqnarray}$$

Note that the upper bound of $\unicode[STIX]{x1D709}_{max\,W}$ is $1/2$ while the upper bound of $\unicode[STIX]{x1D709}_{max\,\unicode[STIX]{x1D709}}$ is 1. However, for typical electrokinetic energy converters, the value of $\unicode[STIX]{x1D6FC}$ is small (e.g.  $\unicode[STIX]{x1D6FC}<0.5$ ), so the above two extreme powers (or efficiencies) are essentially close to each other (Xuan & Li Reference Xuan and Li2006).

A few researchers (e.g. Daiguji et al. Reference Daiguji, Yang, Szeri and Majumdar2004b ; Olthuis et al. Reference Olthuis, Schippers, Eijkel and van den Berg2005; Berli Reference Berli2010; Chanda et al. Reference Chanda, Sinha and Das2014) calculated the input volume flow rate  $Q$ in the definition of the efficiency $\unicode[STIX]{x1D709}$ as the purely pressure-driven volume flow rate, i.e.  $Q=L_{11}(-\unicode[STIX]{x0394}p)$ . In this situation, the maximum efficiency that can be achieved is

(2.10) $$\begin{eqnarray}\unicode[STIX]{x1D709}_{max}=\frac{I_{s}\unicode[STIX]{x0394}\unicode[STIX]{x1D719}_{s}}{4L_{11}(\unicode[STIX]{x0394}p)^{2}}=\frac{\unicode[STIX]{x1D6FC}}{4}.\end{eqnarray}$$

It is easy to see that the efficiency $\unicode[STIX]{x1D709}_{max}$ is the leading term of the above efficiency $\unicode[STIX]{x1D709}_{max\,W}$ or $\unicode[STIX]{x1D709}_{max\,\unicode[STIX]{x1D709}}$ by using (2.8) and (2.9). For small  $\unicode[STIX]{x1D6FC}$ , these extreme efficiencies achieved under three different conditions are nearly identical, i.e.  $\unicode[STIX]{x1D709}_{max}\approx \unicode[STIX]{x1D709}_{max\,W}\approx \unicode[STIX]{x1D709}_{max\,\unicode[STIX]{x1D709}}$ .

3.1 Electrical double-layer electric potential distribution

We assume that the fluids contain binary symmetric electrolyte ions with valence $z_{+}=-z_{-}=z$ . Owing to the interaction between the electrolyte ions and the charged walls as well as the liquid–liquid interface, EDLs are formed. In addition, in a two-fluid system, an unequal partitioning of micro-ions between the two liquid phases is induced by the difference in the dielectric constant (Ohshima et al. Reference Ohshima, Nomura, Kamaya and Ueda1985; Leunissen et al. Reference Leunissen, Zwanikken, Van Roij, Chaikin and Van Blaaderen2007; Gopmandal & Ohshima Reference Gopmandal and Ohshima2017; Saha et al. Reference Saha, Gopmandal and Ohshima2017). Considering the ion partitioning effect, the Boltzmann distribution satisfied by the electrolyte ions can be written as

(3.1) $$\begin{eqnarray}\left.\begin{array}{@{}ll@{}}\displaystyle n_{\pm }=n_{\infty }\exp \left(\mp \frac{ez\unicode[STIX]{x1D713}}{k_{B}T}\right), & 0<y<h_{1},\\ \displaystyle n_{\pm }=b_{\pm }n_{\infty }\exp \left(\mp \frac{ez\unicode[STIX]{x1D713}}{k_{B}T}\right), & -h_{2}<y<0.\end{array}\right\}\end{eqnarray}$$

Here, $\unicode[STIX]{x1D713}$ is the EDL electric potential, $n_{\pm }$ are the concentrations of the electrolyte ions, $b_{\pm }$ are the ion partition coefficients of the electrolyte ions with $n_{\pm }(0-)=b_{\pm }n_{\pm }(0+)$ , $n_{\infty }$ is the bulk ionic concentration at $\unicode[STIX]{x1D713}=0$ , $e$ is the electronic charge, $k_{B}$ is the Boltzmann constant and $T$ is the absolute temperature.

The equilibrium electric potential $\unicode[STIX]{x1D713}$ relative to the bulk electrolyte and the local volumetric net charge density $\unicode[STIX]{x1D70C}_{e}=ez(n_{+}-n_{-})$ can be related through the Poisson equation as

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D713}=-\frac{\unicode[STIX]{x1D70C}_{e}}{\unicode[STIX]{x1D700}_{i}},\quad i=1,2.\end{eqnarray}$$

For the fully developed flow, the distribution of the electric potential can be regarded as a function of $y$ , i.e.  $\unicode[STIX]{x1D713}=\unicode[STIX]{x1D713}(y)$ .

To solve the electric potential distributions within fluids 1 and 2, appropriate boundary conditions must be imposed. In previous studies, three different models have been adopted for the electrostatic properties of the surface: constant zeta potential (Yang et al. Reference Yang, Lu, Kostiuk and Kwok2003; Davidson & Xuan Reference Davidson and Xuan2008a ,Reference Davidson and Xuan b ), constant surface charge density (Daiguji, Yang & Majumdar Reference Daiguji, Yang and Majumdar2004a ; Daiguji et al. Reference Daiguji, Yang, Szeri and Majumdar2004b ; Stein et al. Reference Stein, Kruithof and Dekker2004; Chang & Yang Reference Chang and Yang2010) and local chemical equilibrium (Behrens & Grier Reference Behrens and Grier2001; van der Heyden et al. Reference van der Heyden, Bonthuis, Stein, Meyer and Dekker2007; Wang & Kang Reference Wang and Kang2010). The surface charge essentially varies with the bulk ionic concentration, the pH value and the temperature of the solution, and the double-layer interactions. This implies that the chemical equilibrium model may be the most suitable one to describe the solid–liquid interface, compared with the other two models. But there are many parameters that need to be specified beforehand, such as the surface density $\unicode[STIX]{x1D6E4}$ of chargeable sites, pH value, equilibrium constant $p$ K describing the dissociation of SiOH groups and Stern layer phenomenological capacity $C$ (Behrens & Grier Reference Behrens and Grier2001). In order to avoid excessive complexity, the condition of constant surface charge density is used, which yields reasonable results and is much better than assuming a constant zeta potential (Stein et al. Reference Stein, Kruithof and Dekker2004; van der Heyden, Stein & Dekker Reference van der Heyden, Stein and Dekker2005). Furthermore, the surface charges are related to the electric potential $\unicode[STIX]{x1D713}(y)$ through Gauss’ law:

(3.3a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{1}=\unicode[STIX]{x1D700}_{1}\left.\frac{\text{d}\unicode[STIX]{x1D713}}{\text{d}y}\right|_{y=h_{1}}\quad \text{and}\quad \unicode[STIX]{x1D70E}_{2}=-\unicode[STIX]{x1D700}_{2}\left.\frac{\text{d}\unicode[STIX]{x1D713}}{\text{d}y}\right|_{y=-h_{2}},\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{1}$ and $\unicode[STIX]{x1D70E}_{2}$ are the surface charge densities of the upper plate and lower plate, respectively.

At the liquid–liquid interface $(y=0)$ , we impose the potential difference $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}$ and Gauss’ law for the electrical displacement (Choi et al. Reference Choi, Sharma, Qian, Lim and Joo2011; Jian et al. Reference Jian, Su, Chang, Liu and He2014):

(3.3c,d ) $$\begin{eqnarray}\unicode[STIX]{x1D713}|_{y=0^{-}}-\unicode[STIX]{x1D713}|_{y=0^{+}}=\unicode[STIX]{x0394}\unicode[STIX]{x1D713}\quad \text{and}\quad \unicode[STIX]{x1D700}_{2}\left.\frac{\text{d}\unicode[STIX]{x1D713}}{\text{d}y}\right|_{y=0^{-}}-\unicode[STIX]{x1D700}_{1}\left.\frac{\text{d}\unicode[STIX]{x1D713}}{\text{d}y}\right|_{y=0^{+}}=\unicode[STIX]{x1D70E}_{s}.\end{eqnarray}$$

Here $\unicode[STIX]{x1D70E}_{s}$ denotes the interface charge density, which implies that the normal component of the electric displacement vector is discontinuous; $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}$ and $\unicode[STIX]{x1D70E}_{s}$ are considered to be independent parameters (Choi et al. Reference Choi, Sharma, Qian, Lim and Joo2011). In addition, the total charges in the channel can be expressed as

(3.4) $$\begin{eqnarray}Q_{c}=A\left(\int _{-h_{2}}^{h_{1}}\unicode[STIX]{x1D70C}_{e}\,\text{d}y+\unicode[STIX]{x1D70E}_{1}+\unicode[STIX]{x1D70E}_{2}+\unicode[STIX]{x1D70E}_{s}\right),\end{eqnarray}$$

where $A$ is the area of the plate. Substituting (3.3) into (3.4) yields $Q_{c}=0$ . This verifies the condition of electroneutrality in the channel.

We introduce the following non-dimensional variables:

(3.5) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle [Y,H_{1},H_{2}]=\frac{1}{h}[y,h_{1},h_{2}],\quad [\unicode[STIX]{x1D6F9},\unicode[STIX]{x0394}\unicode[STIX]{x1D6F9}]=\frac{ze}{k_{B}T}[\unicode[STIX]{x1D713},\unicode[STIX]{x0394}\unicode[STIX]{x1D713}],\\ \displaystyle [\overline{\unicode[STIX]{x1D70E}}_{1},\overline{\unicode[STIX]{x1D70E}}_{2},\overline{\unicode[STIX]{x1D70E}}_{s}]=\frac{zeh}{\unicode[STIX]{x1D700}_{1}k_{B}T}[\unicode[STIX]{x1D70E}_{1},\unicode[STIX]{x1D700}_{r}^{-1}\unicode[STIX]{x1D70E}_{2},\unicode[STIX]{x1D70E}_{s}],\end{array}\right\}\end{eqnarray}$$

and $K=h/\unicode[STIX]{x1D706}_{D}$ with the Debye screening length $\unicode[STIX]{x1D706}_{D}=\sqrt{\unicode[STIX]{x1D700}_{1}k_{B}T/2n_{\infty }z^{2}e^{2}}$ , where $\unicode[STIX]{x1D700}_{r}=\unicode[STIX]{x1D700}_{2}/\unicode[STIX]{x1D700}_{1}$ denotes the permittivity ratio. With these scaled variables, (3.1) and (3.2) reduce to

(3.6) $$\begin{eqnarray}\frac{\text{d}^{2}\unicode[STIX]{x1D6F9}}{\text{d}Y^{2}}=\left\{\begin{array}{@{}ll@{}}K^{2}\sinh \unicode[STIX]{x1D6F9}, & 0<Y<H_{1},\\ (2\unicode[STIX]{x1D700}_{r})^{-1}K^{2}[-b_{+}\exp (-\unicode[STIX]{x1D6F9})+b_{-}\exp \unicode[STIX]{x1D6F9}], & -H_{2}<Y<0.\end{array}\right.\end{eqnarray}$$

The corresponding dimensionless boundary conditions become

(3.7) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \overline{\unicode[STIX]{x1D70E}}_{1}=\left.\frac{\text{d}\unicode[STIX]{x1D6F9}}{\text{d}Y}\right|_{Y=H_{1}},\quad \overline{\unicode[STIX]{x1D70E}}_{2}=-\left.\frac{\text{d}\unicode[STIX]{x1D6F9}}{\text{d}Y}\right|_{Y=-H_{2}},\quad \unicode[STIX]{x1D6F9}|_{Y=0^{-}}-\unicode[STIX]{x1D6F9}|_{Y=0^{+}}=\unicode[STIX]{x0394}\unicode[STIX]{x1D6F9},\\ \displaystyle \left.\unicode[STIX]{x1D700}_{r}\frac{\text{d}\unicode[STIX]{x1D6F9}}{\text{d}Y}\right|_{Y=0^{-}}-\left.\frac{\text{d}\unicode[STIX]{x1D6F9}}{\text{d}Y}\right|_{Y=0^{+}}=\overline{\unicode[STIX]{x1D70E}}_{s}.\end{array}\right\}\end{eqnarray}$$

Equation (3.6) with the boundary conditions (3.7) is solved iteratively through a second-order central difference scheme. The iteration procedure starts with a guessed value of the potential. Here we assume that the partition coefficients for cations and anions are identical, i.e.  $b_{+}=b_{-}=b$ . The detailed scheme and algorithm can be found in appendix A. Meanwhile, the convergence and validity of the algorithm and the dependence of the results on the number of grid points are also analysed.

3.2 Flow field coupled with electrokinetic interaction

The flow maintained by constant pressure gradient $-\text{d}p/\text{d}x=-\unicode[STIX]{x0394}p/l$ is assumed to be one-dimensional, steady and fully developed; the body force is the electric field force $\unicode[STIX]{x1D70C}_{e}E_{s}=\unicode[STIX]{x1D70C}_{e}(-\unicode[STIX]{x0394}\unicode[STIX]{x1D719})/l$ . Thus in such a two-fluid system the governing equation for the velocity can be described by

(3.8) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{i}\frac{\text{d}^{2}u}{\text{d}y^{2}}-\frac{\unicode[STIX]{x0394}p}{l}+\unicode[STIX]{x1D70C}_{e}E_{s}=0,\quad i=1,2,\end{eqnarray}$$

where $u(y)$ is the axial velocity along the positive $x$ -direction. At the solid–liquid interface, the no-slip conditions are given by $u|_{y=h_{1}}=u|_{y=-h_{2}}=0$ . The continuity of the velocity as well as the balance of the total stresses including the Maxwell stress and shear stress between the two layers should be applied at the liquid–liquid interface (Jian et al. Reference Jian, Su, Chang, Liu and He2014)

(3.9a,b ) $$\begin{eqnarray}u|_{y=0^{+}}=u|_{y=0^{-}},\quad \left.\left(\unicode[STIX]{x1D707}_{1}\frac{\text{d}u}{\text{d}y}-\unicode[STIX]{x1D700}_{1}E_{s}\frac{\text{d}\unicode[STIX]{x1D713}}{\text{d}y}\right)\right|_{y=0^{+}}=\left.\left(\unicode[STIX]{x1D707}_{2}\frac{\text{d}u}{\text{d}y}-\unicode[STIX]{x1D700}_{2}E_{s}\frac{\text{d}\unicode[STIX]{x1D713}}{\text{d}y}\right)\right|_{y=0^{-}}.\end{eqnarray}$$

Combining the governing equations in different fluids with the related boundary conditions, the analytical solution of the velocity distribution can be derived as

(3.10) $$\begin{eqnarray}u=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{\unicode[STIX]{x0394}p}{2l\unicode[STIX]{x1D707}_{1}}[y^{2}-h_{1}^{2}+A_{2}(y-h_{1})]+\frac{\unicode[STIX]{x0394}\unicode[STIX]{x1D719}}{l\unicode[STIX]{x1D707}_{1}}[A_{1}(y-h_{1})+\unicode[STIX]{x1D700}_{1}(\unicode[STIX]{x1D713}_{1,w}-\unicode[STIX]{x1D713})], & 0\leqslant y\leqslant h_{1},\\ \displaystyle \frac{\unicode[STIX]{x0394}p}{2l\unicode[STIX]{x1D707}_{2}}[y^{2}-h_{2}^{2}+A_{2}(y+h_{2})]+\frac{\unicode[STIX]{x0394}\unicode[STIX]{x1D719}}{l\unicode[STIX]{x1D707}_{2}}[A_{1}(y+h_{2})+\unicode[STIX]{x1D700}_{2}(\unicode[STIX]{x1D713}_{2,w}-\unicode[STIX]{x1D713})], & -h_{2}\leqslant y\leqslant 0,\end{array}\right.\end{eqnarray}$$

where

(3.11a,b ) $$\begin{eqnarray}A_{1}=\frac{\unicode[STIX]{x1D707}_{r}\unicode[STIX]{x1D700}_{1}(\unicode[STIX]{x1D713}_{1,w}-\unicode[STIX]{x1D713}(0+))-\unicode[STIX]{x1D700}_{2}(\unicode[STIX]{x1D713}_{2,w}-\unicode[STIX]{x1D713}(0-))}{\unicode[STIX]{x1D707}_{r}h_{1}+h_{2}},\quad A_{2}=\frac{h_{2}^{2}-\unicode[STIX]{x1D707}_{r}h_{1}^{2}}{\unicode[STIX]{x1D707}_{r}h_{1}+h_{2}},\end{eqnarray}$$

$\unicode[STIX]{x1D707}_{r}=\unicode[STIX]{x1D707}_{2}/\unicode[STIX]{x1D707}_{1}$ denotes the viscosity ratio, and $\unicode[STIX]{x1D713}_{1,w}$ and $\unicode[STIX]{x1D713}_{2,w}$ are the wall zeta potentials of the upper plate and lower plate, respectively.

Integrating the velocity (3.10) over the channel cross-section gives the volumetric flow rate $Q$ and then yields the expressions for the phenomenological coefficients in (2.1) as

(3.12) $$\begin{eqnarray}L_{11}=\frac{w}{2l}\left[\frac{2h_{1}^{3}}{3\unicode[STIX]{x1D707}_{1}}+\frac{2h_{2}^{3}}{3\unicode[STIX]{x1D707}_{2}}+A_{2}\left(\frac{h_{1}^{2}}{2\unicode[STIX]{x1D707}_{1}}-\frac{h_{2}^{2}}{2\unicode[STIX]{x1D707}_{2}}\right)\right]\end{eqnarray}$$

and

(3.13) $$\begin{eqnarray}L_{12}=\frac{w}{l}\left[\frac{\unicode[STIX]{x1D700}_{1}}{\unicode[STIX]{x1D707}_{1}}\int _{0}^{h_{1}}(\unicode[STIX]{x1D713}(y)-\unicode[STIX]{x1D713}_{1,w})\,\text{d}y+\frac{\unicode[STIX]{x1D700}_{2}}{\unicode[STIX]{x1D707}_{2}}\int _{-h_{2}}^{0}(\unicode[STIX]{x1D713}(y)-\unicode[STIX]{x1D713}_{2,w})\,\text{d}y+A_{1}\left(\frac{h_{1}^{2}}{2\unicode[STIX]{x1D707}_{1}}-\frac{h_{2}^{2}}{2\unicode[STIX]{x1D707}_{2}}\right)\right].\end{eqnarray}$$

According to the Nernst–Planck equation, the transport of ions in the channel is described in terms of convection and migration resulting from the pressure difference and electric potential gradient (Wang & Kang Reference Wang and Kang2010), respectively. In addition, the presence of mobile ions along the liquid–liquid interface in such a two-fluid system should be included for the transport of ions. Thus, the current through the channel cross-section is expressed as

(3.14) $$\begin{eqnarray}I=w\int _{-h_{2}}^{h_{1}}\unicode[STIX]{x1D70C}_{e}u\,\text{d}y+w\unicode[STIX]{x1D70E}_{s}u(0)+w\int _{-h_{2}}^{h_{1}}\mathop{\sum }_{i}ez_{i}n_{i}\unicode[STIX]{x1D708}_{i}E_{s}\,\text{d}y+G_{w}(-\unicode[STIX]{x0394}\unicode[STIX]{x1D719}).\end{eqnarray}$$

Here, the first term is the contribution from convection; the second one characterizes the contribution of mobile ions at the interface; the third is the electro-migration of ions; the last term is added to account for the conductance of the Stern layer near the wall that is denoted by $G_{w}$ ; the ion mobility $\unicode[STIX]{x1D708}_{i}=ez_{i}D_{i}/k_{B}T$ (i.e. the Nernst–Einstein equation); and $D_{i}$ is the diffusivity of the ion. The diffusivity and mobility of the ion may vary in fluids 1 and 2. Based on the Stokes–Einstein relationship (Masliyah & Bhattacharjee Reference Masliyah and Bhattacharjee2006), we propose that the diffusivity of the cations and anions in different fluids is inversely proportional to the viscosity of the fluid, i.e.  $D_{+,1}/D_{+,2}=D_{-,1}/D_{-,2}=\unicode[STIX]{x1D707}_{r}$ . Furthermore, we have considered identical values of the diffusivity ratio $\unicode[STIX]{x1D708}_{r}$ of the cations to the anions in both fluids 1 and 2, i.e.  $D_{+,1}/D_{-,1}=D_{+,2}/D_{-,2}=\unicode[STIX]{x1D708}_{r}$ .

Substituting the velocity (3.10) into (3.14) yields the other two phenomenological coefficients $L_{21}$ and $L_{22}$ (see appendix B for the detailed derivation); here $L_{21}$ is found to be the same as $L_{12}$ , which shows the validity of the Onsager reciprocal relation for two-fluid systems; and $L_{22}$ is given by

(3.15) $$\begin{eqnarray}L_{22}=\frac{w\unicode[STIX]{x1D700}_{1}^{2}}{lh\unicode[STIX]{x1D707}_{2}}\left(\frac{k_{B}T}{ze}\right)^{2}\left[\unicode[STIX]{x1D707}_{r}\,f_{1}+\unicode[STIX]{x1D700}_{r}\,f_{2}-\overline{A}_{1}^{2}(H_{2}+\unicode[STIX]{x1D707}_{r}H_{1})-\frac{ez\unicode[STIX]{x1D707}_{2}\unicode[STIX]{x1D708}_{-,1}}{2\unicode[STIX]{x1D700}_{1}k_{B}T}K^{2}f_{3}\right]+G_{w},\end{eqnarray}$$

with

(3.16) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle f_{1}=H_{1}\overline{\unicode[STIX]{x1D70E}}_{1}^{2}+2K^{2}\left[\int _{0}^{H_{1}}\cosh (\unicode[STIX]{x1D6F9})\,\text{d}Y-H_{1}\cosh (\unicode[STIX]{x1D6F9}_{1,w})\right],\\ \displaystyle f_{2}=H_{2}\overline{\unicode[STIX]{x1D70E}}_{2}^{2}+2B^{2}\left[\int _{-H_{2}}^{0}\cosh (\unicode[STIX]{x1D6F9})\,\text{ d }Y-H_{2}\cosh (\unicode[STIX]{x1D6F9}_{2,w},)\right]\\ \displaystyle f_{3}=\int _{0}^{H_{1}}[\unicode[STIX]{x1D708}_{r}\exp (-\unicode[STIX]{x1D6F9})+\exp \unicode[STIX]{x1D6F9}]\,\text{d}Y+b\unicode[STIX]{x1D707}_{r}^{-1}\int _{-H_{2}}^{0}[\unicode[STIX]{x1D708}_{r}\exp (-\unicode[STIX]{x1D6F9})+\exp \unicode[STIX]{x1D6F9}]\,\text{ d }Y,\\ \displaystyle B=\sqrt{\frac{b}{\unicode[STIX]{x1D700}_{r}}}K,\quad \overline{A}_{1}=\frac{ezh}{\unicode[STIX]{x1D700}_{1}k_{B}T}A_{1}.\end{array}\right\}\end{eqnarray}$$

Here, the non-dimensional height $(H_{1},H_{2})$ , electric potential $(\unicode[STIX]{x1D6F9},\unicode[STIX]{x1D6F9}_{1,w},\unicode[STIX]{x1D6F9}_{2,w})$ and surface charge density $(\overline{\unicode[STIX]{x1D70E}}_{1},\overline{\unicode[STIX]{x1D70E}}_{2})$ have been used. The expressions for $L_{11}$ and $L_{12}$ are readily rewritten in terms of these non-dimensional variables. It can be observed that the interface charge density $\unicode[STIX]{x1D70E}_{s}$ is absent in the expressions for $L_{11}$ , $L_{12}$ and  $L_{22}$ , though it appears in expression (3.14) for the current. This indicates that $\unicode[STIX]{x1D70E}_{s}$ only has an indirect effect on electrokinetic energy conversion by affecting the distribution of electric potential in the EDLs (see figure 3). This can be explained by the fact that the motion of mobile interfacial charges is compensated by the motion of the counter-ions in the interface Stern layer.

4.1 Oil phase with large viscosity

In the first scenario, fluid 2 is considered to be an oil phase with large viscosity and low permittivity, such as cyclohexyl bromide (CHB) with $\unicode[STIX]{x1D700}_{2}=7.9\times (8.854\times 10^{-12})~\text{C}^{2}~\text{J}^{-1}~\text{m}^{-1}$ and $\unicode[STIX]{x1D707}_{2}=2.269\times 10^{-3}~\text{kg}~\text{m}^{-1}~\text{s}^{-1}$ (Gopmandal & Ohshima Reference Gopmandal and Ohshima2017), compared with the aqueous phase in fluid 1. That implies $\unicode[STIX]{x1D700}_{r}<1$ and $\unicode[STIX]{x1D707}_{r}>1$ . Furthermore, the value of the ion partition coefficient $b$ is considered to be less than unity, as the electrolyte ions always tend to remain in the medium with high dielectric constant (Ohshima et al. Reference Ohshima, Nomura, Kamaya and Ueda1985).

In figure 3, we display the dimensionless net charge density and electrostatic potential profiles in the two-fluid system. It can be found that the net charge density in the oil phase is much less than that in the aqueous phase due to the ion partitioning effect that reduces the salt concentration in the oil layer. Thus, the EDL effect in the oil layer is not remarkable relative to the aqueous solution. In addition, there is a peak near the liquid–liquid interface, which depends on the interface charge density  $\unicode[STIX]{x1D70E}_{s}$ . The magnitudes of net charge density and potential in the two-fluid system increase significantly with the increase of  $|\unicode[STIX]{x1D70E}_{s}|$ .

Figure 3.

Profiles of the dimensionless (a) net charge density and (b) electrostatic potential in a two-fluid system for different interface charge density, where $h=100$  nm,  $n_{\infty }=1$  mM,  $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}=0$  V, the partition coefficient $b=0.1$ , and the dimensionless thicknesses of the aqueous phase and oil phase are 0.8 and 0.2 respectively.

Figure 4 displays the variations of the dimensionless velocity $U=2\unicode[STIX]{x1D707}_{1}lu/(-\unicode[STIX]{x0394}p)$ with the interface charge density for different interface locations, where $H_{1}=1-H_{2}$ represents the dimensionless thickness of the aqueous layer. Note that $\unicode[STIX]{x1D70E}_{2}=0~\text{mC}~\text{m}^{-2}$ , and thus the Stern conductance near the oil–glass/silica surface can be disregarded; thereby the wall Stern conductance $G_{w}$ of the whole two-layer fluid system is reduced. In this paper, $G_{w}$ is assumed to be 11 pS. Comparing figure 4(a) with figure 4(b), it can be seen that an increase in the relative thickness of the oil layer leads to a decrease in the velocity across the channel because of the large viscosity of the oil (CHB). The role of interfacial charges is also investigated. In the case of a thin oil layer (i.e. large  $H_{1}$ ), the interfacial charge has a retardation effect on the fluid velocity; while for a thick oil layer (i.e. small  $H_{1}$ ), the opposite effect (i.e. enhanced effect) is observed.

Figure 4.

Profiles of the dimensionless velocity $U$ across the channel for interface charge density $\unicode[STIX]{x1D70E}_{s}=0$ (dashed line), $-5$ (dash-dotted line) and $-10$ (solid line) with (a) relatively thick oil layer $(H_{1}=0.1)$ and (b) relatively thin oil layer $(H_{1}=0.9)$ . The other parameters are $h=100$  nm,  $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}=40$  mV,  $n_{\infty }=1$  mM,  $G_{w}=11$  pS, the ratio $\unicode[STIX]{x1D708}_{r}=0.25$ and the ion partition coefficient $b=0.1$ . Insets show local enlarged drawings.

As stated in § 2, the figure of merit $\unicode[STIX]{x1D6FC}$ gauges the performance of electrokinetic energy converters, and the maximum generation efficiency is a monotonically increasing function of  $\unicode[STIX]{x1D6FC}$ . The variations of $\unicode[STIX]{x1D6FC}$ against $\unicode[STIX]{x1D70E}_{s}$ are demonstrated in figure 5(a) for different values of dimensionless thickness  $H_{1}$ . It can be seen that the presence of a relatively thin oil layer (large  $H_{1}$ ) tends to increase the value of $\unicode[STIX]{x1D6FC}$ with the interfacial charge, thereby improving the energy conversion efficiency. While, for a relatively thick oil layer (small  $H_{1}$ ), the opposite trend can be observed with the interfacial charge. This is essentially related to the effect of interfacial charge on the velocity profile, and will be explained from the perspective of energy transfer and dissipation later. Furthermore, for the case of a thin oil layer (e.g.  $H_{2}=1-H_{1}<0.2$ ), the value of $\unicode[STIX]{x1D6FC}$ is more than that in single-phase flow for moderate and large interface charge densities (e.g.  $|\unicode[STIX]{x1D70E}_{s}|>4~\text{mC}~\text{m}^{-2}$ ). This implies that the efficiency of such a two-fluid energy converter may exceed that of a single-phase flow system. The efficiency ratio can be defined by the ratio of maximum efficiency in binary flow to that in single-phase flow (i.e.  $\unicode[STIX]{x1D709}_{max}^{b}/\unicode[STIX]{x1D709}_{max}^{s}$ ), and the results obtained in our scenario are demonstrated in figure 5(b). From these figures, we can see that the maximum efficiency in the two-fluid flow will increase with the increase of $|\unicode[STIX]{x1D70E}_{s}|$ for a thin oil layer. The maximum efficiency can be up to 1.5 times that of the single-phase flow system in the range of  $\unicode[STIX]{x1D70E}_{s}$ .

Figure 5.

(a) Variations of $\unicode[STIX]{x1D6FC}$ against the interface charge density for different values of dimensionless thickness  $H_{1}$ . Here $h=100$  nm,  $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}=-40$  mV,  $n_{\infty }=1$  mM,  $G_{w}=11$  pS, the ratio $\unicode[STIX]{x1D708}_{r}=0.25$ and the partition coefficient $b=0.01$ . Note that the value of $\unicode[STIX]{x1D6FC}$ in single-phase flow is also plotted for comparison. (b) Same as (a) but for the variations of the efficiency ratio. (c) Variations of the efficiency ratio with $\unicode[STIX]{x1D70E}_{s}$ for different values of $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}$ . Here $h=100$  nm,  $n_{\infty }=1$  mM,  $G_{w}=11$  pS,  $\unicode[STIX]{x1D708}_{r}=0.25$ ,  $b=0.01$ and $H_{1}=0.9$ . (d) Same as (b) but the bulk ionic concentration $n_{\infty }=10$  mM instead of 1 mM, and the range of $\unicode[STIX]{x1D70E}_{s}$ is also different due to different ionic concentration.

Figure 5(c) illustrates how the energy conversion efficiency ratio varies with the interface charge density $\unicode[STIX]{x1D70E}_{s}$ for different interface potential difference $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}$ at fixed $H_{1}=0.9$ . We find that a positive $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}$ tends to increase the conversion efficiency and a negative one reduces the efficiency. This can be explained by associating the interface potential difference $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}$ with the distribution of the charges. To be specific, a positive $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}$ will increase the net charge density in the EDL formed at the liquid–liquid interface, thereby increasing the streaming current and the conversion efficiency; while a negative $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}$ has the opposite effect (data not shown). In the high-salt regime, the efficiency of a single-fluid system decreases significantly because of the power dissipation caused by ionic conductance (van der Heyden et al. Reference van der Heyden, Bonthuis, Stein, Meyer and Dekker2007). In contrast, the ion partitioning effect in a two-fluid system reduces the salt concentration within the oil layer, thereby reducing the electrical conductivity. Therefore, the combined effects of interfacial charge and ion partitioning may be used to improve the performance of electrokinetic energy converters at high salt. Figure 5(d) displays the variations of the efficiency ratio at the bulk ionic concentration $n_{\infty }=10$  mM. In our calculations, the efficiency of the two-fluid energy converter can be up to twice that of the single-phase flow system when a thin oil layer is added. Similarly, the presence of interfacial charge in the two-fluid flow also augments the maximum generation power. The maximum powers can be up to 1.5 and 2 times that of the single-phase flow system at $n_{\infty }=1$  mM and 10 mM, respectively. The corresponding figures are not shown here.

Figure 6.

Variations of (a) the electrical Joule heating dissipation (denoted by $P_{E}$ ) and (b) viscous dissipation (denoted by $P_{V}$ ) with the interface charge density $\unicode[STIX]{x1D70E}_{s}$ for different values of dimensionless thickness $H_{1}$ in two-fluid flows. (c,d) Variations of the ratio of (c) the electrical Joule heating dissipation to the power input ( $P_{E}/P_{in}$ ) and (d) viscous dissipation to the power input ( $P_{V}/P_{in}$ ) with  $\unicode[STIX]{x1D70E}_{s}$ . Here $h=100$  nm,  $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}=40$  mV,  $n_{\infty }=1$  mM,  $G_{w}=11$  pS,  $\unicode[STIX]{x0394}p=-4$  bar, the ratio $\unicode[STIX]{x1D708}_{r}=0.25$ and the partition coefficient $b=0.01$ . The symbol represents the corresponding value in single-phase flow of water solution, which is a constant independent of the interfacial charges.

From the viewpoint of energy transfer, the input power, produced by the pressure work, is converted into three parts: output power, viscous dissipation and electrical Joule heating dissipation. In order to further investigate the performance of the two-fluid energy converter, we show the variations of the viscous dissipation (denoted by $P_{V}$ ) and electrical Joule heating dissipation (denoted by $P_{E}$ ) with $\unicode[STIX]{x1D70E}_{s}$ for different values of dimensionless thickness $H_{1}$ in figure 6. The power consumed by viscous and electrical Joule heating can be derived by a combined thermodynamic and fluid mechanics analysis on the system, which is given in appendix C. From figure 6, it can be seen that viscous dissipation consumes most of the power (more than 90 % of the total power input), whether it is in binary or single flows (figure 6 d). This is a major factor limiting the efficiency of electrokinetic energy converters. In contrast to the single-phase flows of water solution, the addition of a thin oil layer reduces viscous dissipation for moderate or high interfacial charge density (figure 6 b,d), due to the retardation effect of the interfacial charge on the fluid velocity (see figure 4 b). While in the case of a relatively thick oil layer, the ratio of the viscous dissipation to the input power increases with the interfacial charge, because of the enhanced effect of the interfacial charge on the fluid velocity (see figure 4 a). Thus, a thin oil layer tends to increase the energy conversion efficiency with the interfacial charge, and a thick oil layer has the opposite trend, which explains the effect of interface location on the energy conversion efficiency in figure 5. It is noteworthy that, although the presence of interfacial charges leads to an increase in electrical Joule heating dissipation, the amount of the increased power consumption is less than that of the reduced viscous dissipation in the case of thin oil layers (e.g.  $H_{1}=0.9$ in figure 6 c,d). Therefore, the total consumption of power is reduced and the efficiency is improved.

4.2 Oil phase with small viscosity

Figure 7.

(a) Profiles of the dimensionless velocity $U$ across the channel for different values of interface charge density $\unicode[STIX]{x1D70E}_{s}$ and the dimensionless thicknesses  $H_{1}$ . (b) Variations of the figure of merit $\unicode[STIX]{x1D6FC}$ with the interface charge density $\unicode[STIX]{x1D70E}_{s}$ against different values of dimensionless thickness  $H_{1}$ . Here $h=100$  nm,  $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}=40$  mV,  $n_{\infty }=1$  mM,  $G_{w}=11$  pS, the ratio $\unicode[STIX]{x1D708}_{r}=0.25$ and the partition coefficient $b=0.01$ . Note that the value of $\unicode[STIX]{x1D6FC}$ in single-phase flow is also plotted in this panel for comparison.

In the second scenario, we consider fluid 2 to be an oil phase with low viscosity and permittivity, such as octane with $\unicode[STIX]{x1D700}_{2}=2.0\times (8.854\times 10^{-12})~\text{C}^{2}~\text{J}^{-1}~\text{m}^{-1}$ and $\unicode[STIX]{x1D707}_{2}=0.566\times 10^{-3}~\text{kg}~\text{m}^{-1}~\text{s}^{-1}$ (Lee et al. Reference Lee, Barbulovic-Nad, Wu, Xuan and Li2006), relative to the aqueous phase in fluid 1. That implies $\unicode[STIX]{x1D707}_{r}<1$ . Figure 7(a) displays the profiles of the dimensionless velocity $U$ for different interface charge densities and interface locations (characterized by  $H_{1}$ ). Because of the relatively small viscosity of the oil (octane), the increase in the thickness of the oil layer (i.e. small  $H_{1}$ ) leads to an increase of the velocity across the channel. That is different from the case of $\unicode[STIX]{x1D707}_{r}>1$ (see figure 4). On the other hand, the interfacial charge has different effects (enhanced or weakened) on the fluid velocity for different interface locations, which is similar to the case of $\unicode[STIX]{x1D707}_{r}>1$ . The variations of the figure of merit $\unicode[STIX]{x1D6FC}$ with $\unicode[STIX]{x1D70E}_{s}$ are demonstrated in figure 7(b) against different values of  $H_{1}$ . Compared with the single-phase flow, the presence of a relatively thin oil layer (e.g.  $H_{2}=1-H_{1}<0.1$ ) can increase the value of $\unicode[STIX]{x1D6FC}$ for large interfacial charges, thus improving the energy conversion efficiency.

Variations of the ratio of maximum efficiency in binary flow to that in single-phase flow are demonstrated in figure 8 at different salt concentrations in the case of $\unicode[STIX]{x1D707}_{r}<1$ . One finds that the maximum efficiency can be up to 1.5 times that of the single-phase flow system for a relatively thin oil layer at the salt concentration $n_{\infty }=1$  mM. While, in the high-salt regime (e.g.  $n_{\infty }=10$  mM), the efficiency of the two-fluid energy converter can be up to 2 times that of the single-phase flow system due to the combined effects of interfacial charge and ion partitioning. Meanwhile, the maximum generation power in the two-fluid flow is also augmented due to the presence of interfacial charges. The maximum power will increase 1.5 and 2 times compared with the single-phase flow system at $n_{\infty }=1$  mM and 10 mM, respectively, when a relatively thin oil layer is added (the corresponding figures are not shown here). These results indicate that whether $\unicode[STIX]{x1D707}_{r}>1$ or $\unicode[STIX]{x1D707}_{r}<1$ , the influence of interfacial charges on electrokinetic energy conversion are similar in the limit of a relatively thin oil layer.

Figure 8.

Variations of the efficiency ratio in two-fluid flow to that in single-phase flow in the case of $\unicode[STIX]{x1D707}_{r}<1$ . The values of the parameters are the same as in figure 5 except for (a)  $n_{\infty }=1$  mM and (b)  $n_{\infty }=10$  mM.

Furthermore, we investigate the performance of a two-fluid energy converter from the viewpoint of energy transfer for the case of $\unicode[STIX]{x1D707}_{r}<1$ . In figure 9, we show the variations of the viscous dissipation $P_{V}$ and electrical Joule heating dissipation $P_{E}$ with the interface charge density for different interface locations. In contrast to the case of $\unicode[STIX]{x1D707}_{r}>1$ , the addition of an oil layer results in an increase of viscous dissipation (figure 9 b) because of the increased velocity. But the ratio of viscous dissipation to total power input is reduced for thin oil layers and large interface charge density (see figures 9 d and 11 b). This is essentially due to the retardation effect of interfacial charges on the fluid velocity in the case of a thin oil layer (see figure 7 a). Note that at the same time the presence of interfacial charges leads to an increase in electrical Joule heating dissipation for thin oil layers and large interface charge density (see figure 9 a,c). However, the increased ratio of the Joule heating dissipation to the power input is less than the reduced one of the viscous dissipation to the power input. Therefore, a conclusion similar to the case of $\unicode[STIX]{x1D707}_{r}>1$ can be obtained, i.e. the addition of a thin oil layer reduces the total consumption of power and improves the efficiency.

Figure 9.

Variations of (a) the electrical Joule heating dissipation $P_{E}$ and (b) viscous dissipation $P_{V}$ with the interface charge density $\unicode[STIX]{x1D70E}_{s}$ for different values of dimensionless thickness $H_{1}$ in the case of $\unicode[STIX]{x1D707}_{r}<1$ . (c,d) Variations of the ratio of (c) the electrical Joule heating dissipation to the power input ( $P_{E}/P_{in}$ ) and (d) viscous dissipation to the power input ( $P_{V}/P_{in}$ ). Here the values of other parameters are the same as in figure 6. The symbol represents the corresponding value in single-phase flow of water solution, which is a constant independent of the interfacial charges.