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Derivation of an effective dispersion model for electro-osmotic flow involving free boundaries in a thin strip

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Abstract

Since dispersion is one of the key parameters in solute transport, its accurate modeling is essential to avoid wrong predictions of flow and transport behavior. In this research, we derive new effective dispersion models which are valid also in evolving geometries. To this end, we consider reactive ion transport under dominate flow conditions (i.e. for high Peclet number) in a thin, potentially evolving strip. Electric charges and the induced electric potential (the zeta potential) give rise to electro-osmotic flow in addition to pressure-driven flow. At the pore-scale a mathematical model in terms of coupled partial differential equations is introduced. If applicable, the free boundary, i.e. the interface between an attached layer of immobile chemical species and the fluid is taken into account via the thickness of the layer. To this model, a formal limiting procedure is applied and the resulting upscaled models are investigated for dispersive effects. In doing so, we emphasize the cross-coupling effects of hydrodynamic dispersion (Taylor–Aris dispersion) and dispersion created by electro-osmotic flow. Moreover, we study the limit of small and large Debye length. Our results improve the understanding of fundamentals of flow and transport processes, since we can now explicitly calculate the dispersion coefficient even in evolving geometries. Further research may certainly address the situation of clogging by means of numerical studies. Finally, improved predictions of breakthrough curves as well as facilitated modeling of mixing and separation processes are possible.

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Acknowledgements

This research was inspired by fruitful discussions with Kundan Kumar, University of Bergen, Norway. The personal exchange was kindly supported by the German Academic Exchange Service in the framework of the German–Norwegian collaborative research support scheme 2016.

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Correspondence to Nadja Ray.

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Appendix: Upscaling reactive flow and transport under an external electric field and double layer potential in the fixed geometry setting

Appendix: Upscaling reactive flow and transport under an external electric field and double layer potential in the fixed geometry setting

The proof of Theorem 1, i.e. for the fixed geometry case is conducted.

1.1 A.1 Poisson–Boltzmann equation and Boltzmann distribution in the double layer

Lemma 5

In the double layer, the (stationary) Boltzmann distribution holds for the concentrations and the corresponding free charge density obeys the following linearized representation. Moreover, we obtain the following expression for the double layer potential:

$$\begin{aligned}&c^{\pm ,\mathrm{{DL}}}_\varepsilon = c^{\pm ,\mathrm{{DL}}}_0 (y) = c_\infty \mathrm{{e}}^{\mp \phi ^\mathrm{{DL}}_0} \quad \mathrm{in}\,\varOmega ,\\&c^{+,\mathrm{{DL}}}_0 - c^{-,\mathrm{{DL}}}_0 = c_\infty \left( \mathrm{{e}}^{- \phi ^\mathrm{{DL}}_0} - \mathrm{{e}}^{ \phi ^\mathrm{{DL}}_0} \right) = -2c_\infty \sinh (\phi _0^\mathrm{{DL}}) \quad \mathrm{in}\,\varOmega ,\\&\phi ^\mathrm{{DL}}_\varepsilon = \phi ^\mathrm{{DL}}_0 (y) = \zeta \frac{\cosh (\kappa y)}{\cosh (\kappa )} \quad \mathrm{in}\,\varOmega . \end{aligned}$$

Proof

Due to the assumptions on the fixed geometry, the model equations, cf. Sect. 2.2.2, may be reformulated as follows: For the double layer potential it holds

$$\begin{aligned}&-{\varepsilon ^2}\partial _{xx} \phi ^\mathrm{{DL}}_\varepsilon - \partial _{yy} \phi ^\mathrm{{DL}}_\varepsilon = c^{+,\mathrm{{DL}}}_\varepsilon - c^{-,\mathrm{{DL}}}_\varepsilon \quad \mathrm{in}\,\varOmega ,\\&\phi ^\mathrm{{DL}}_\varepsilon = \zeta \quad \mathrm{on}\, \varGamma ^\pm ,\\&{\varepsilon }\partial _x\phi ^\mathrm{{DL}}_\varepsilon = 0 \quad \mathrm{on}\,\varGamma ^\mathrm{{in}},\\&-{\varepsilon }\partial _x\phi ^\mathrm{{DL}}_\varepsilon = 0 \quad \mathrm{on}\,\varGamma ^\mathrm{{out}}. \end{aligned}$$

For the transport of the ion concentrations in the double layer, it holds

$$\begin{aligned}&\partial _t c^{\pm ,\mathrm{{DL}}}_\varepsilon - \frac{1}{\varepsilon } D\partial _{yy} c^{\pm ,\mathrm{{DL}}}_\varepsilon \mp \frac{1}{\varepsilon }\partial _y \left( c^{\pm ,\mathrm{{DL}}}_\varepsilon \partial _y\phi ^\mathrm{{DL}}_\varepsilon \right) = 0 \quad \mathrm{in}\,\varOmega \\&D\partial _y c^{\pm ,\mathrm{{DL}}}_\varepsilon \pm c^{\pm ,\mathrm{{DL}}}_\varepsilon \partial _y\phi ^\mathrm{{DL}}_\varepsilon = 0 \quad \mathrm{on}\,\varGamma ^{\pm },\\&c^{\pm ,\mathrm{{DL}}}_\varepsilon = c^{\pm ,\mathrm{{DL}}}_l \quad \mathrm{on}\,\varGamma ^\mathrm{{in}},\\&\frac{1}{\varepsilon }D\partial _y c^{\pm ,\mathrm{{DL}}}_\varepsilon \cdot \nu _r = 0 \quad \mathrm{on}\,\varGamma ^\mathrm{{out}}. \end{aligned}$$

The lowest order transport equation reads

$$\begin{aligned} D\partial _{yy} c^{\pm ,\mathrm{{DL}}}_0 \pm \partial _y \left( c^{\pm ,\mathrm{{DL}}}_0\partial _y\phi ^\mathrm{{DL}}_0\right)&= 0 , \end{aligned}$$

which is clearly satisfied by the Boltzmann distribution

$$\begin{aligned} c^{\pm ,\mathrm{{DL}}}_0 = c_\infty \mathrm{{e}}^{\mp \phi ^\mathrm{{DL}}_0}. \end{aligned}$$
(17)

Turning now to the Poisson equation in lowest order and considering (for simplicity) the linearization of the free charge density, we obtain the Poisson-Boltzmann equation with Debye length \(\kappa ^2:=2c_\infty \).

$$\begin{aligned}&- \partial _{yy} \phi ^\mathrm{{DL}}_0 = c^{+,\mathrm{{DL}}}_0 - c^{-,\mathrm{{DL}}}_0 = -2c_\infty \sinh (\phi ^\mathrm{{DL}}_0) = -\kappa ^2\phi ^\mathrm{{DL}}_0 \quad \mathrm{in}\,\varOmega , \end{aligned}$$
(18a)
$$\begin{aligned}&\phi ^\mathrm{{DL}}_0 = \zeta \quad \mathrm{on}\, \varGamma ^\pm . \end{aligned}$$
(18b)

The definition of the Debye length is slightly different from the usual one, cf. e.g. [2], since further physical constants are set to 1 throughout the paper. Then,

$$\begin{aligned} \phi ^\mathrm{{DL}}_0=\zeta \frac{\cosh (\kappa y)}{\cosh (\kappa )} \end{aligned}$$

solves (18) and also satisfies the boundary conditions on the inflow and outflow boundary \(\partial _x\phi ^\mathrm{{DL}}_0 = 0\). An analytical solution of the non-linearized Poisson-Boltzmann equation is provided in [42].

We presume that the Boltzmann distribution holds in general for some constant reference value \(c_\infty \). Then, the Taylor expansion yields an expression for the higher order concentration terms:

$$\begin{aligned} c^{\pm ,\mathrm{{DL}}}_\varepsilon = c_\infty \mathrm{{e}}^{\mp \phi ^\mathrm{{DL}}_\varepsilon } = c_\infty \left( \mathrm{{e}}^{\mp \phi ^\mathrm{{DL}}_0} \mp \varepsilon \mathrm{{e}}^{\mp \phi ^\mathrm{{DL}}_0}\phi ^\mathrm{{DL}}_1 \pm \cdots \right) =: c^{\pm ,\mathrm{{DL}}}_0 + \varepsilon c^{\pm ,\mathrm{{DL}}}_1 + \cdots . \end{aligned}$$

Considering the Poisson equation in first order, inserting the expansion for \(c^{\pm ,\mathrm{{DL}}}_1\) and the linearization of the free charge density in first order, gives

$$\begin{aligned}&- \partial _{yy} \phi ^\mathrm{{DL}}_1 = c^{+,\mathrm{{DL}}}_1 - c^{-,\mathrm{{DL}}}_1 = -2c_\infty \sinh (\phi ^\mathrm{{DL}}_0)\phi ^\mathrm{{DL}}_1 = -2c_\infty \phi ^\mathrm{{DL}}_0\phi ^\mathrm{{DL}}_1 \quad \mathrm{in}\,\varOmega ,\\&\phi ^\mathrm{{DL}}_1 = 0 \quad \mathrm{on}\, \varGamma ^\pm ,\\&\partial _x\phi ^\mathrm{{DL}}_1 = 0 \quad \mathrm{on}\,\varGamma ^\mathrm{{in}},\\&-\partial _x\phi ^\mathrm{{DL}}_1 = 0 \quad \mathrm{on}\,\varGamma ^\mathrm{{out}}. \end{aligned}$$

which is satisfied by \(\phi ^\mathrm{{DL}}_1 =0\). This immediately verifies \(c^{\pm ,\mathrm{{DL}}}_1=0\).

With this results the next order expansion of the transport equation

$$\begin{aligned} \partial _t c^{\pm ,\mathrm{{DL}}}_0 = 0 \end{aligned}$$

is directly fulfilled since \(c^{\pm ,\mathrm{{DL}}}_0\) is stationary.

Similar investigations may be made for the higher order terms. In summary, this yields \(\phi ^\mathrm{{DL}}_\varepsilon =\phi ^\mathrm{{DL}}_0\) and \(c^{\pm ,\mathrm{{DL}}}_\varepsilon =c^{\pm ,\mathrm{{DL}}}_0\). \(\square \)

1.2 A.2 Model assumptions and model reformulation in the bulk

Assumption 1

We assume pointwise electroneutrality for the bulk concentrations, i.e. \(c^+_\varepsilon - c^-_\varepsilon =0\).

Remark 4

The assumption of electroneutrality as stated in Assumption 1 does not hold for the ion concentrations near the double layer, i.e. in general \(c^{+,\mathrm{{DL}}}_\varepsilon - c^{-,\mathrm{{DL}}}_\varepsilon \ne 0\).

Assumption 1 and the assumption on the (fixed) geometrical setting lead to the following equations for the external electrostatic potential:

$$\begin{aligned}&-\partial _{xx} \phi ^\mathrm{{ex}}_\varepsilon - \frac{1}{\varepsilon ^2}\partial _{yy} \phi ^\mathrm{{ex}}_\varepsilon = 0 \quad \mathrm{in}\,\varOmega ,\\&-\frac{1}{\varepsilon }\partial _y\phi ^\mathrm{{ex}}_\varepsilon = 0 \quad \mathrm{on}\,\varGamma ^\pm ,\\&\phi ^\mathrm{{ex}}_\varepsilon = \phi _l \quad \mathrm{on}\,\varGamma ^\mathrm{{in}},\\&\phi ^\mathrm{{ex}}_\varepsilon = \phi _r \quad \mathrm{on}\,\varGamma ^\mathrm{{out}}. \end{aligned}$$

We directly conclude

Lemma 6

For the electric potential and its gradient, it holds

$$\begin{aligned}&\phi ^\mathrm{{ex}}_\varepsilon = \phi ^\mathrm{{ex}}_0(x) = \frac{\phi _r - \phi _l}{L}x + \phi _l =: -E x + \phi _l \quad \mathrm{in}\,\varOmega ,\\&\nabla \phi ^\mathrm{{ex}}_\varepsilon = (-E,0)^\mathrm{{T}} \quad \mathrm{in}\,\varOmega . \end{aligned}$$

Hence, we obtain the following reformulation of the transport and Stokes equations from Sect. 2.2.1:

Transport equation for mobile ions

$$\begin{aligned}&\partial _t c^\pm _\varepsilon + \partial _x(v^{(1)}_\varepsilon c^\pm _\varepsilon ) + \frac{1}{\varepsilon }\partial _y(v^{(2)}_\varepsilon c^\pm _\varepsilon ) - \varepsilon D\partial _{xx} c^\pm _\varepsilon - \frac{1}{\varepsilon } D\partial _{yy} c^\pm _\varepsilon \pm \varepsilon E\partial _xc^\pm _\varepsilon \mp \varepsilon \partial _y(c^\pm _\varepsilon \partial _y\phi ^\mathrm{{DL}}_\varepsilon ) = 0 \quad \mathrm{in}\,\varOmega ,\\&- ( - v^{(2)}_\varepsilon c^\pm _\varepsilon + D\partial _y c^\pm _\varepsilon \pm \varepsilon ^2 c^\pm _\varepsilon \partial _y\phi ^\mathrm{{DL}}_\varepsilon ) = \pm \varepsilon f(c^\pm _\varepsilon ,c^\mathrm{{im}}_\varepsilon ) \quad \mathrm{on}\,\varGamma ^\pm . \end{aligned}$$

Incompressible Stokes equations for fluid flow and pressure

$$\begin{aligned}&-\varepsilon ^2 \partial _{xx} v^{(1)}_\varepsilon - \partial _{yy} v^{(1)}_\varepsilon + \partial _x p_\varepsilon = E (c^{+,\mathrm{{DL}}}_\varepsilon - c^{-,\mathrm{{DL}}}_\varepsilon ) \quad \mathrm{in}\,\varOmega ,\\&-\varepsilon ^2 \partial _{xx} v^{(2)}_\varepsilon - \partial _{yy} v^{(2)}_\varepsilon + \frac{1}{\varepsilon }\partial _y p_\varepsilon = - \varepsilon (c^{+,\mathrm{{DL}}}_\varepsilon - c^{-,\mathrm{{DL}}}_\varepsilon )\partial _y\phi ^\mathrm{{DL}}_\varepsilon \quad \mathrm{in}\,\varOmega ,\\&\partial _x v^{(1)}_\varepsilon + \frac{1}{\varepsilon }\partial _y v^{(2)}_\varepsilon = 0 \quad \mathrm{in}\,\varOmega ,\\&v^{(1)}_\varepsilon , v^{(2)}_\varepsilon = 0 \quad \mathrm{on}\,\varGamma ^\pm . \end{aligned}$$

Remark 5

The scaling of the double layer potential in the Stokes equations leads to pressures \(p_0\) and \(p_1\) that are independent of y, which is reasonable for very narrow channels, cf. Chap. 8.2.2. in [2].

1.3 A.3 Lowest order

In this section, we consider the lowest order model equations. With the scaling chosen for the double layer potential, we directly conclude

Lemma 7

The zeroth order terms of the ion concentrations and the pressure field are macroscopic quantities. Moreover, the vertical component of the velocity field vanishes in zeroth order, i.e.

$$\begin{aligned}&c^\pm _0 = c^\pm _0(t,x) \quad \mathrm{in}\,\varOmega ,\\&p_0 = p_0(t,x) \quad \mathrm{in}\,\varOmega ,\\&v^{(2)}_0 = v^{(2)}_0(t,x)\equiv 0 \quad \mathrm{in}\,\varOmega . \end{aligned}$$

1.4 A.4 Zeroth order

In this section, we investigate the zeroth order equations and derive

Lemma 8

The first order pressure term is a macroscopic quantity, i.e.

$$\begin{aligned} p_1 =p_1(t,x) \quad \mathrm{in}\,\varOmega . \end{aligned}$$

The zeroth order of the horizontal velocity component and its mean have two distinct components, cf. Fig. 2. First the pressure-driven parabolic profile leading to Darcy’s law and second an electro-osmotic term, i.e.

$$\begin{aligned}&v^{(1)}_0(y) = -\frac{1}{2}\partial _x p_0 (1-y^2) - E\zeta \left( 1-\frac{\cosh (\kappa y)}{\cosh (\kappa )}\right) \quad \mathrm{in}\,\varOmega ,\\&{\bar{v}}^{(1)}_0 := \frac{1}{2}\int _{-1}^1 v^{(1)}_0 \mathrm{{d}}y = -\frac{1}{3}\partial _x p_0 - E\zeta \left( 1-\frac{\tanh (\kappa )}{\kappa }\right) \quad \mathrm{in}\,[0,L],\\&\partial _x {\bar{v}}^{(1)}_0 = 0 \quad \mathrm{in}\,[0,L]. \end{aligned}$$

For the first order bulk ion concentrations, it holds

$$\begin{aligned} c_1^\pm =&-\frac{1}{D} \partial _t c^\mathrm{{im}}_0 \left( \frac{1}{2}y^2 + b_2(x) \right) -\frac{1}{2D}\partial _x p_0 \partial _xc^\pm _0 \left( \frac{1}{6}y^2 - \frac{1}{12}y^4 + b_4(x)\right) \\&+ \frac{E\zeta }{D}\partial _xc^\pm _0 \left( \frac{\cosh (\kappa y)}{\kappa ^2\cosh (\kappa )} - \frac{1}{2}\frac{\tanh (\kappa )}{\kappa }y^2 + b_6(x)\right) . \end{aligned}$$

Proof

Due to the scaling of the double layer term, from Stokes equations, we directly obtain \(p_1=p_1(t,x)\).

We further consider the Stokes equations in zeroth order and obtain

$$\begin{aligned}&-\partial _{yy} v^{(1)}_0 + \partial _x p_0 = E (c^{+,\mathrm{{DL}}}_0 - c^{-,\mathrm{{DL}}}_0) = -\kappa ^2 E\zeta \frac{\cosh (\kappa y)}{\cosh (\kappa )} \quad \mathrm{in}\,\varOmega , \end{aligned}$$
(19a)
$$\begin{aligned}&v^{(1)}_0 = 0 \quad \mathrm{on}\, \varGamma ^\pm . \end{aligned}$$
(19b)

Integrating twice with respect to y and by means of the no-slip boundary condition \(v^{(1)}_0=0\) on \(\varGamma ^\pm \), the statement of Lemma 8 holds for \(v^{(1)}_0\) and its mean \({\bar{v}}^{(1)}_0\).

Taking additionally the mean \(\frac{1}{2}\int _{-1}^1\cdot \mathrm{{d}}y\) of the incompressibility condition \(\partial _x v^{(1)}_0+\partial _y v^{(2)}_1=0\), applying the Gauß theorem and incorporating the no-slip boundary condition \(v^{(2)}_1=0\) on \(\varGamma ^{\pm }\), yields the incompressibility condition as stated in Lemma 8.

Finally, we consider the following zeroth order transport equations with complementing boundary conditions of first order:

$$\begin{aligned}&\partial _t c^\pm _0 + \partial _x(v^{(1)}_0c^\pm _0) + \partial _y(v^{(2)}_1 c^\pm _0 + v^{(2)}_0 c^\pm _1) - D\partial _{yy} c^\pm _1 = 0 \quad \mathrm{in}\,\varOmega , \end{aligned}$$
(20a)
$$\begin{aligned}&-(- v^{(2)}_1 c^\pm _0 - v^{(2)}_0 c^\pm _1 + D\partial _yc_1^{\pm }) = \pm f(c^\pm _0,c^\mathrm{{im}}_0) = \pm \partial _t c^\mathrm{{im}}_0 \quad \mathrm{on}\,\varGamma ^\pm , \end{aligned}$$
(20b)

To obtain the last equality in (20b), we inserted the evolution equation for \(c^\mathrm{{im}}_\varepsilon \) in zeroth order.

Taking the mean \(\tfrac{1}{2}\int _{-1}^1 \cdot \mathrm{{d}}y\) and incorporating the boundary condition leads with \(c^\pm _0=c^\pm _0(t,x)\), cf. Lemma 7, to:

$$\begin{aligned} \partial _t c^\pm _0 + \partial _t c^\mathrm{{im}}_0 + \partial _x\left( \frac{1}{2}\int _{-1}^1 (v^{(1)}_0 c^\pm _0) \mathrm{{d}}y\right) = 0. \end{aligned}$$
(21)

In order to determine the first order concentrations \(c^\pm _1\), we first consider another equivalent formulation of (20). In doing so, we incorporate Lemma 7, the incompressibility condition and the no-slip boundary condition for the velocity, to obtain

$$\begin{aligned}&\partial _t c^\pm _0 + v^{(1)}_0\partial _xc^\pm _0 - D\partial _{yy} c^\pm _1 = 0 \quad \mathrm{in}\,\varOmega , \end{aligned}$$
(22a)
$$\begin{aligned}&- D\partial _yc_1^{\pm } = \pm \partial _t c^\mathrm{{im}}_0 \quad \mathrm{on}\,\varGamma ^\pm . \end{aligned}$$
(22b)

At this point, we already would obtain \(c_1^\pm \) by integrating (22a) with respect to y; however with an unwanted dependence on the time derivative of the zeroth order concentrations. Therefore, taking again the mean \(\tfrac{1}{2}\int _{-1}^1 \cdot \mathrm{{d}}y\) and incorporating the boundary condition leads with \(c^\pm _0=c^\pm _0(t,x)\), cf. Lemma 7, to:

$$\begin{aligned} \partial _t c^\pm _0 + \partial _t c^\mathrm{{im}}_0 + \frac{1}{2}\left( \int _{-1}^1v^{(1)}_0\mathrm{{d}}y \right) \partial _xc^\pm _0 = \partial _t c^\pm _0 + \partial _t c^\mathrm{{im}}_0 + {\bar{v}}^{(1)}_0 \partial _xc^\pm _0= 0. \end{aligned}$$
(23)

We now subtract (22a)−(23) to obtain an equation for \(c^\pm _1\) and supplementing boundary conditions

$$\begin{aligned}&-\partial _t c^\mathrm{{im}}_0 + \left( v^{(1)}_0 - {\bar{v}}^{(1)}_0\right) \partial _xc^\pm _0 - D\partial _{yy} c^\pm _1 = 0 \quad \mathrm{in}\,\varOmega ,\\&-D\partial _y c^\pm _1 = \pm \partial _t c^\mathrm{{im}}_0 \quad \mathrm{on}\,\varGamma ^\pm . \end{aligned}$$

Now it is possible to determine \(c_1^{\pm }\) by integration with respect to y

$$\begin{aligned} c^\pm _1 =&-\frac{1}{D} \left( \int \int \partial _t c^\mathrm{{im}}_0 \mathrm{{d}}y \, \mathrm{{d}}y \right) + \frac{1}{D}\partial _xc^\pm _0 \left( \int \int \left( v^{(1)}_0 - {\bar{v}}^{(1)}_0 \right) \mathrm{{d}}y \, \mathrm{{d}}y \right) \\ =&-\frac{1}{D} \partial _t c^\mathrm{{im}}_0 \left( \frac{1}{2}y^2 + b_1(x)y + b_2(x) \right) -\frac{1}{2D}\partial _xc^\pm _0\partial _x p_0 \left( \frac{1}{6}y^2 - \frac{1}{12}y^4 + b_3(x)y + b_4(x)\right) \\&+ \frac{E\zeta }{D}\partial _xc^\pm _0 \left( \frac{\cosh (\kappa y)}{\kappa ^2\cosh (\kappa )} - \frac{1}{2} \frac{\tanh (\kappa )}{\kappa }y^2 + b_5(x) y + b_6(x)\right) . \end{aligned}$$

The boundary condition \(-D\partial _yc^\pm _1 = \pm \partial _tc^\mathrm{{im}}_0\) on \(\varGamma ^\pm \) implies \(\partial _tc^\mathrm{{im}}_0 b_1 + \tfrac{1}{2}\partial _xc^\pm _0\partial _x p_0 b_3 - E\zeta \partial _xc^\pm _0 b_5=0\) and therefore the statement from Lemma 8 holds. \(\square \)

1.5 First order

Finally, we consider the first order equations to derive

Lemma 9

The horizontal component of the velocity up to order one and its mean have two distinct components, cf. Fig. 2. First the pressure-driven parabolic profile leading to Darcy’s law and second an electro-osmotic term, i.e.

$$\begin{aligned} v^{(1)}_e(y)&= -\frac{1}{2}\partial _x p_e (1-y^2) - E\zeta \left( 1-\frac{\cosh (\kappa y)}{\cosh (\kappa )}\right) =: v^{(1)}_{e,P}(y) + v^{(1)}_{e,EO}(y) \quad \mathrm{in}\, \varOmega ,\\ {\bar{v}}^{(1)}_e&:= \frac{1}{2}\int _{-1}^1 v^{(1)}_e \mathrm{{d}}y = -\frac{1}{3}\partial _x p_e - E\zeta \left( 1-\frac{\tanh (\kappa )}{\kappa }\right) =: {\bar{v}}^{(1)}_{e,P} + {\bar{v}}^{(1)}_{e,EO} \quad \mathrm{in}\, [0,L],\\ \partial _x{\bar{v}}^{(1)}_e&= 0 \quad \mathrm{in}\, [0,L]. \end{aligned}$$

The transport equation for the bulk ion concentrations up to first order include pressure-driven dispersive effects as well as dispersion effects stemming from electro-osmosis.

$$\begin{aligned}&\partial _t ({\bar{c}}^\pm _e + c^\mathrm{{im}}_e) + \partial _x({\bar{v}}^{(1)}_e {\bar{c}}^\pm _e)\\&\quad - \varepsilon \partial _{x} \left( D \left( 1 + \frac{1}{D^2}\left[ \frac{2}{105}\left( {\bar{v}}^{(1)}_{e,P}\right) ^2-\frac{2F^{(1)}(\kappa )}{15\kappa }{\bar{v}}^{(1)}_{e,P}{\bar{v}}^{(1)}_{e,EO}+\frac{F^{(0)}(\kappa )}{12\kappa ^2}\left( {\bar{v}}^{(1)}_{e,EO}\right) ^2\right] \right) \partial _{x} {\bar{c}}^\pm _e\right) \\&\quad + \varepsilon \partial _{x}\left[ \frac{1}{15D}{\bar{v}}^{(1)}_{e,P}-\frac{F^{(2)}(\kappa )}{3D\kappa }{\bar{v}}^{(1)}_{e,EO}\right] \partial _tc^\mathrm{{im}}_e \pm \varepsilon \partial _x (E{\bar{c}}^\pm _e) \ = \ 0. \end{aligned}$$

The evolution equation for the immobile species up to order one reads

$$\begin{aligned}&\Big (1 + \varepsilon \frac{1}{3} \frac{\partial _1 f({\bar{c}}^\pm _e, c^\mathrm{{im}}_e)}{D}\Big ) \partial _t c^\mathrm{{im}}_e \\&\quad = f({\bar{c}}^\pm _e,c^\mathrm{{im}}_e) + \varepsilon \,\left( \frac{1}{15}\frac{{\bar{v}}^{(1)}_{e,P}}{D} +\frac{E\zeta }{D} \left( \frac{1}{\kappa ^2}-\frac{1}{\kappa ^4}-\frac{1}{3}\frac{\tanh (\kappa )}{\kappa }\right) \right) \partial _1 f({\bar{c}}^\pm _e, c^\mathrm{{im}}_e)\partial _x {\bar{c}}^\pm _e \quad \mathrm{in}\,[0,L]. \end{aligned}$$

Proof

For the Stokes equations in first order with supplementing no-slip boundary condition, we have (due to \(c^{\pm ,\mathrm{{DL}}}_1=0\), cf. Sect. A.1)

$$\begin{aligned}&-\partial _{yy} v^{(1)}_1 + \partial _x p_1 = E (c^{+,\mathrm{{DL}}}_1 - c^{-,\mathrm{{DL}}}_1) = 0 \quad \mathrm{in}\,\varOmega , \end{aligned}$$
(24a)
$$\begin{aligned}&v^{(1)}_1 = 0 \quad \mathrm{on}\, \varGamma ^\pm . \end{aligned}$$
(24b)

In order to get an equation for the velocity up to order one, we define \(v^{(1)}_e = v^{(1)}_0 + \varepsilon v^{(1)}_1\), \(p_e = p_0 + \varepsilon p_1\) and add (19) + \(\varepsilon \) (24). Integrating the resulting equation twice with respect to y, we obtain

$$\begin{aligned} v^{(1)}_e = \int \int \left( \partial _x p_e - E (c^{+,\mathrm{{DL}}}_0 - c^{-,\mathrm{{DL}}}_0)\right) \mathrm{{d}}y \, \mathrm{{d}}y = \int \int \left( \partial _x p_e +2 Ec_\infty \sinh (\varPhi ^\mathrm{{DL}}_0)\right) \mathrm{{d}}y \, \mathrm{{d}}y. \end{aligned}$$

Taking into account the no-slip boundary condition \(v^{(1)}_e = 0\) on \(\varGamma ^\pm \) for the velocity and bearing in mind that \(p_e=p_e(t,x)\) (cf. Lemmas 7 and 8), the statement of Lemma 9 follows for \(v^{(1)}_e\) and its mean \( {\bar{v}}^{(1)}_e\).

The incompressibility condition \(\partial _x {\bar{v}}_1^{(1)}=0\) for the averaged quantity \({\bar{v}}^{(1)}_1\) is obtained along the same lines as the incompressibility condition for the averaged quantity \({\bar{v}}^{(1)}_0\) in the proof of Lemma 8. Consequently, together with the statement of Lemma 8, it holds \(\partial _x {\bar{v}}_\mathrm{{e}}^{(1)}=0\).

Finally, we consider the transport equations in first order and supplementing boundary conditions in second order

$$\begin{aligned}&\partial _t c^\pm _1 + \partial _x(v^{(1)}_0c^\pm _1 + v^{(1)}_1c^\pm _0) + \partial _y(v^{(2)}_0c^\pm _2 + v^{(2)}_1c^\pm _1 + v^{(2)}_2c^\pm _0)\\&\quad - D\partial _{xx} c^\pm _0 - D\partial _{yy} c^\pm _2 \pm \partial _x(E c^\pm _0) \mp \partial _y(c^\pm _0\partial _y\phi ^\mathrm{{DL}}_0) =0 \quad \mathrm{in}\,\varOmega ,\\&\quad -\left( - v^{(2)}_0c^\pm _2 - v^{(2)}_1c^\pm _1 - v^{(2)}_2c^\pm _0 + D\partial _{y} c_2 \pm \partial _y(c^\pm _0\partial _y\phi ^\mathrm{{DL}}_0) \right) = \pm \partial _t c^\mathrm{{im}}_1 \quad \mathrm{on}\, \varGamma ^\pm . \end{aligned}$$

We multiply the transport equation with \(\varepsilon \) and take the mean \(\tfrac{1}{2}\int _{-1}^1 \cdot \mathrm{{d}}y \). We furthermore apply Gauß theorem and incorporate the boundary condition. Finally, we define \({\bar{c}}^\pm _1 := \tfrac{1}{2}\int _{-1}^1 c_1 \mathrm{{d}}y\) to obtain

$$\begin{aligned} \varepsilon \partial _t {\bar{c}}^\pm _1 + \varepsilon \partial _t c^\mathrm{{im}}_1 + \varepsilon \partial _x\left( \frac{1}{2}\int _{-1}^1 (v^{(1)}_0c^\pm _1 + v^{(1)}_1 c^\pm _0) \mathrm{{d}}y\right) - \varepsilon D\partial _{xx} c^\pm _0 \pm \varepsilon \partial _x (Ec^\pm _0) =0. \end{aligned}$$
(25)

In order to obtain an equation up to first order for the concentrations, we define \({\bar{c}}^\pm _e =c^\pm _0 + \varepsilon {\bar{c}}^\pm _1\) and \(c^\mathrm{{im}}_e =c^\mathrm{{im}}_0 + \varepsilon c^\mathrm{{im}}_1\) which fulfill by adding (21) and (25) and further terms in order \(O(\varepsilon ^2)\)

$$\begin{aligned}&\partial _t {\bar{c}}^\pm _e + \partial _t c^\mathrm{{im}}_e + \partial _x\left( \frac{1}{2}\int _{-1}^1 (v^{(1)}_0 c^\pm _0) \mathrm{{d}}y\right) + \varepsilon \partial _x\left( \frac{1}{2}\int _{-1}^1 (v^{(1)}_0c^\pm _1 + v^{(1)}_1 c^\pm _0) \mathrm{{d}}y\right) \nonumber \\&\quad - \varepsilon D\partial _{xx} {\bar{c}}^\pm _e \pm \varepsilon \partial _x (E{\bar{c}}^\pm _e) =0. \end{aligned}$$
(26)

The most sophisticated part is now to calculate the remaining velocity terms. This is done by means of combing and adding suitable terms of order \(O(\varepsilon ^2)\), cf. [13]:

$$\begin{aligned}&\partial _x\left( \frac{1}{2}\int _{-1}^1 (v^{(1)}_0 c^\pm _0) \mathrm{{d}}y\right) + \varepsilon \partial _x\left( \frac{1}{2}\int _{-1}^1 (v^{(1)}_0c^\pm _1 + v^{(1)}_1 c^\pm _0) \mathrm{{d}}y\right) \\&\quad = \partial _x\left( \frac{1}{2}\int _{-1}^1 (v^{(1)}_e c^\pm _0) \mathrm{{d}}y\right) + \varepsilon \partial _x\left( \frac{1}{2}\int _{-1}^1 (v^{(1)}_e c^\pm _1) \mathrm{{d}}y\right) . \end{aligned}$$

Moreover, adding the following zero \(\partial _x ({\bar{v}}^{(1)}_e {\bar{c}}^\pm _e ) - \partial _x ({\bar{v}}^{(1)}_e {\bar{c}}^\pm _e )\), we obtain

$$\begin{aligned}&\partial _x ({\bar{v}}^{(1)}_e {\bar{c}}^\pm _e ) - \partial _x ({\bar{v}}^{(1)}_e {\bar{c}}^\pm _e ) + \partial _x\left( \frac{1}{2}\int _{-1}^1 (v^{(1)}_e c^\pm _0) \mathrm{{d}}y\right) + \varepsilon \partial _x\left( \frac{1}{2}\int _{-1}^1 (v^{(1)}_e c^\pm _1) \mathrm{{d}}y\right) \\&\quad = \partial _x ({\bar{v}}^{(1)}_e {\bar{c}}^\pm _e) - \varepsilon \partial _x ({\bar{v}}^{(1)}_e {\bar{c}}^\pm _1 ) + \varepsilon \partial _x\left( \frac{1}{2}\int _{-1}^1 (v^{(1)}_e c^\pm _1) \mathrm{{d}}y\right) . \end{aligned}$$

Finally, the last two terms need to be calculated in order \(O(\varepsilon ^2)\), i.e. it sufficies to calculate:

$$\begin{aligned}&- \varepsilon \partial _x ({\bar{v}}^{(1)}_0 {\bar{c}}^\pm _1 ) + \varepsilon \partial _x\left( \frac{1}{2}\int _{-1}^1 (v^{(1)}_0 c^\pm _1) \mathrm{{d}}y\right) =\varepsilon \partial _x\left( \frac{1}{2}\int _{-1}^1 (v^{(1)}_0 - {\bar{v}}^{(1)}_0) c^\pm _1 \,\mathrm{{d}}y\right) \\&\quad = \frac{\varepsilon }{D} \partial _x\bigg (\left[ \frac{1}{15}{\bar{v}}^{(1)}_{e,P}-\frac{F^{(2)}(\kappa )}{3\kappa }{\bar{v}}^{(1)}_{e,EO}\right] \partial _tc^\mathrm{{im}}_0\\&\qquad -\left[ \frac{2}{105}\left( {\bar{v}}^{(1)}_{e,P}\right) ^2-\frac{2F^{(1)}(\kappa )}{15\kappa }{\bar{v}}^{(1)}_{e,P}{\bar{v}}^{(1)}_{e,EO}+\frac{F^{(0)}(\kappa )}{12\kappa ^2}\left( {\bar{v}}^{(1)}_{e,EO}\right) ^2\right] \partial _xc^{\pm }_0 \bigg ) \end{aligned}$$

with the \(\kappa \)-depending functions

$$\begin{aligned}&F^{(0)}(\kappa ) = \frac{4(6 +\kappa ^2)\sinh ^2(\kappa ) - 9\kappa \sinh (2\kappa ) - 6\kappa ^2}{(\kappa \cosh (\kappa ) - \sinh (\kappa ))^2}\,,\\&F^{(1)}(\kappa ) = \frac{(- \kappa ^4 + 15\kappa ^2 + 45)\sinh (\kappa ) -45\kappa \cosh (\kappa )}{\kappa ^3(\kappa \cosh (\kappa ) - \sinh (\kappa ))} \\&\text {and} \\&F^{(2)}(\kappa ) = \frac{(- \kappa ^2 - 3)\sinh (\kappa ) + 3\kappa \cosh (\kappa )}{\kappa (\kappa \cosh (\kappa ) - \sinh (\kappa ))}. \end{aligned}$$

It is remarkable to note that the result is indeed independent on the integration constants \(b_2(x), b_4(x)\) and \(b_6(x)\).

Integrating this result into (26), we obtain up to order \(O(\varepsilon ^2)\) the statement of Lemma 9 for the transport of the bulk ion concentrations.

For the immobile species the statement of Lemma 9 follows due to Taylor expansion and the adding of terms of order \(O(\varepsilon ^2)\) from

$$\begin{aligned} \partial _t c^\mathrm{{im}}_e&= f({\bar{c}}^\pm _e,c^\mathrm{{im}}_e) + \varepsilon (c^\pm _1\Big |_{y=\pm 1} - {\bar{c}}^{\pm }_1)\partial _1 f({\bar{c}}^\pm _e, c^\mathrm{{im}}_e). \end{aligned}$$
(27)

\(\square \)

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Ray, N., Schulz, R. Derivation of an effective dispersion model for electro-osmotic flow involving free boundaries in a thin strip. J Eng Math 119, 167–197 (2019). https://doi.org/10.1007/s10665-019-10024-8

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