# Start-Up Electroosmotic Flow of Multi-Layer Immiscible Maxwell Fluids in a Slit Microchannel

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## Abstract

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## 1. Introduction

## 2. Mathematical Modeling

#### 2.1. Physical Model Description

#### 2.2. Governing and Constitutive Equations

#### 2.3. Simplified Mathematical Model

- Fully-developed flow [35].
- Impermeable interfaces between the fluids and ideally polarizable to electric charges [5,54,55,56]. The electric double layers at the liquid-liquid interfaces are composed of two diffuse charge layers separated by a central compact layer, characterized by a potential difference drop due to the orientation of the solvent molecules; also, the continuity of electrical displacements on both sides of the central inner layer, in absence of ions in the inner layer is considered [5,55,56,57,58].
- Flat interfaces between the fluids [44,59,60,61]. This is assumed when considering that there is: (i) creeping flow for low Reynolds numbers, being $R{e}_{n}(={\rho}_{n}H{u}_{HS}/{\eta}_{0,n})\ll 1$, resulting in parallel flows with laminar fluid interfaces [43], and (ii) uniform zeta potentials along the microchannel. Here, the characteristic velocity of flow is the well-known Helmholtz-Smoluchowski velocity defined by ${u}_{HS}=-{\u03f5}_{\mathrm{ref}}{\zeta}_{w}{E}_{x}/{\eta}_{\mathrm{ref}}$, where the subscript “$\mathrm{ref}$” indicates physical properties referred to electrolytes in aqueous solutions at 298.15 K (25 ${}^{\circ}$C) [5,62].
- The gravitational forces are neglected [60].
- Long microchannel neglecting any end effects; hence, the electric potential ${\Phi}_{n}$, is the algebraic sum of the potential due to the electric double layer, ${\psi}_{n}$, and the potential due to the imposed electric field, $\varphi $, as [5]:$${\Phi}_{n}(x,y)={\psi}_{n}\left(y\right)+\varphi \left(x\right),$$
- The local distribution of the free charges, that is, ions, is governed by the electrical potential into the electric double layer, ${\psi}_{n}$, through the Boltzmann distribution as [5]$${\rho}_{e,n}=-2{z}_{n}e{n}_{0,n}sinh\left(\frac{{z}_{n}e{\psi}_{n}}{{k}_{B}{T}_{n}}\right),$$
- There is no imposed pressure gradient on microchannel.
- The electric double layers do not overlap.

#### 2.4. Dimensionless Mathematical Model

## 3. Solution Methodology

#### 3.1. Electric Potential Distribution

#### 3.2. Velocity Distribution

## 4. Results and Discussion

#### 4.1. Solution Validation

#### 4.2. Velocity Profiles

## 5. Tracking of the Velocity

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Dimensionless velocity profiles of an electroosmotic flow obtained by Yang et al. [10] with $n=1$, against the results of the present investigation with three fluid layers, $n=3$ (${\overline{y}}_{1}=$2/3 and ${\overline{y}}_{2}=$4/3). The other parameters are ${\overline{\rho}}_{n}={\overline{\eta}}_{n}={\overline{\u03f5}}_{n}=$1, and $\Delta {\overline{\psi}}_{n}=$0.

**Figure 3.**Dimensionless velocity profiles of an electroosmotic flow obtained by Escandón et al. [16] with $n=1$, against the results of the present investigation with four fluid layers, $n=4$ (${\overline{y}}_{1}=$1/2, ${\overline{y}}_{2}=$1.0, and ${\overline{y}}_{3}=$3/2). The other parameters are ${\overline{\rho}}_{n}={\overline{\eta}}_{n}={\overline{\u03f5}}_{n}=$1, and $\Delta {\overline{\psi}}_{n}=$0, for: (

**a**) ${\overline{\lambda}}_{n}=0.12$ and (

**b**) ${\overline{\lambda}}_{n}=2.5$.

**Figure 4.**Dimensionless electric potential and velocity profiles of an electroosmotic flow with $n=3$, (${\overline{y}}_{1}=2/3$, ${\overline{y}}_{2}=4/3$), ${\overline{\kappa}}_{n}=20$, ${\overline{\rho}}_{n}={\overline{\eta}}_{n}={\overline{\u03f5}}_{n}={\overline{\lambda}}_{n}=$1, for: (

**a**) $\Delta {\overline{\psi}}_{n}=0.5$, (

**b**) $\Delta {\overline{\psi}}_{n}=0$, and (

**c**) $\Delta {\overline{\psi}}_{n}=-0.5$.

**Figure 5.**Dimensionless velocity profiles of an electroosmotic flow with $n=3$, (${\overline{y}}_{1}=2/3$, ${\overline{y}}_{2}=4/3$), ${\overline{\kappa}}_{n}=20$, ${\overline{\rho}}_{n}={\overline{\eta}}_{n}={\overline{\u03f5}}_{n}=$1, and $\Delta {\overline{\psi}}_{n}=0.25$ for: (

**a**) ${\overline{\lambda}}_{n}=0.1$, (

**b**) ${\overline{\lambda}}_{n}=2$, and (

**c**) ${\overline{\lambda}}_{n}=10$.

**Figure 6.**Dimensionless velocity profiles of an electroosmotic flow with $n=3$, (${\overline{y}}_{1}=2/3$, ${\overline{y}}_{2}=4/3$), ${\overline{\kappa}}_{n}=20$, ${\overline{\rho}}_{n}={\overline{\u03f5}}_{n}={\overline{\lambda}}_{n}=$1, and $\Delta {\overline{\psi}}_{n}=0.25$ for: (

**a**) ${\overline{\eta}}_{n}=0.7$, (

**b**) ${\overline{\eta}}_{n}=2$, and (

**c**) ${\overline{\eta}}_{n}=6$.

**Figure 7.**Dimensionless electric potential and velocity profiles of an electroosmotic flow with $n=4$, (${\overline{y}}_{1}=2/3$, ${\overline{y}}_{2}=1$, ${\overline{y}}_{3}=3/2$ ), ${\overline{\kappa}}_{1}=10$, ${\overline{\kappa}}_{2}=20$, ${\overline{\kappa}}_{3}=30$, ${\overline{\kappa}}_{4}=40$, ${\overline{\rho}}_{n}={\overline{\u03f5}}_{n}=$1, ${\overline{\lambda}}_{1}=$0.1, ${\overline{\lambda}}_{2}=$2, ${\overline{\lambda}}_{3}=$1, ${\overline{\lambda}}_{4}=$10, $\Delta {\overline{\psi}}_{1}=0.25$, $\Delta {\overline{\psi}}_{2}=-0.35$, $\Delta {\overline{\psi}}_{3}=-0.25$, and ${\overline{\eta}}_{n}=2$.

**Figure 8.**Tracking of the velocity in the multi-layer flow as a function of the dimensionless time evaluated in the center of the microchannel at $\overline{y}=1$ and with ${\overline{\u03f5}}_{n}=$1 for: (

**a**) effect of $\Delta {\overline{\psi}}_{n}$ (from Figure 4), (

**b**) effect of ${\overline{\lambda}}_{n}$ (from Figure 5), (

**c**) effect of ${\overline{\eta}}_{n}$ (from Figure 6), (

**d**) effect of ${\overline{\rho}}_{n}$ (

**e**) effect of number of layers n, and (

**f**) effect of ${\overline{\kappa}}_{n}$.

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**MDPI and ACS Style**

Escandón, J.; Torres, D.; Hernández, C.; Vargas, R.
Start-Up Electroosmotic Flow of Multi-Layer Immiscible Maxwell Fluids in a Slit Microchannel. *Micromachines* **2020**, *11*, 757.
https://doi.org/10.3390/mi11080757

**AMA Style**

Escandón J, Torres D, Hernández C, Vargas R.
Start-Up Electroosmotic Flow of Multi-Layer Immiscible Maxwell Fluids in a Slit Microchannel. *Micromachines*. 2020; 11(8):757.
https://doi.org/10.3390/mi11080757

**Chicago/Turabian Style**

Escandón, Juan, David Torres, Clara Hernández, and René Vargas.
2020. "Start-Up Electroosmotic Flow of Multi-Layer Immiscible Maxwell Fluids in a Slit Microchannel" *Micromachines* 11, no. 8: 757.
https://doi.org/10.3390/mi11080757