# Combined Pressure-Driven and Electroosmotic Slip Flow through Elliptic Cylindrical Microchannels: The Effect of the Eccentricity of the Channel Cross-Section

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## Abstract

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## 1. Introduction

## 2. Preliminaries on the Elliptic Coordinate System

#### 2.1. Elliptic Coordinates

#### 2.2. Mathieu and Modified Mathieu Functions

## 3. Mathematical Modeling

## 4. Boundary Conditions

## 5. Solutions of the Boundary Value Problem

## 6. Effect of the Eccentricity on Slip Flow

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations and Nomenclature

## Abbreviations

## Nomenclature

A | area of channel cross-section |

a | semi-major axis of an ellipse |

${a}_{q}$ | separation constant of 2-dimensional wave equation in elliptic coordinates |

c | focal length of an ellipse |

${\mathrm{Ce}}_{2m}$ | periodic modified Methieu function corresponding to ${\mathrm{ce}}_{2m}$ |

${\mathrm{ce}}_{2m}$ | periodic Methieu function of integral order |

E | external electric field in z-direction |

${E}^{\ast}$ | dimensionless external electric field in z-direction |

$\overrightarrow{E}$ | vector of external electric field |

e | eccentricity of an ellipse |

${\overrightarrow{f}}_{\mathrm{EOF}}$ | vector of electroosmotic body force |

${h}_{\xi},{h}_{\eta}$ | scalar factors/basic vectors for the elliptic coordinates |

${k}_{B}$ | Boltzmann constant |

l | Navier slip length |

${n}_{0}$ | concentration of ions at bulk |

P | perimeter of the elliptic cross-section |

p | pressure |

${p}_{z}$ | pressure gradient in z-direction |

${p}_{z}^{\ast}$ | dimensionless pressure gradient in z-direction |

${p}^{+}$ | elementary proton charge |

Q | volumetric flow rate per unit area of the channel cross-section |

${Q}^{\ast}$ | dimensionless volumetric flow rate per unit area |

${Q}_{i}^{\ast}$ | dimensionless flow rate in the elliptic channel with the eccentricity equal to i |

${Q}_{0}^{\ast}$ | dimensionless flow rate in the circular channel |

T | fluid absolute temperature |

u | fluid velocity in z-direction |

${u}^{\ast}$ | dimensionless fluid velocity in z-direction |

$\overrightarrow{v}$ | vector of fluid velocity |

$(x,y)$ | Cartesian coordinates |

${z}_{v}$ | valence of ion |

$\delta $ | relative error of ${Q}^{\ast}$ |

$\epsilon $ | fluid permittivity |

$\zeta $ | zeta potential |

$\kappa $ | reciprocal of EDL thickness |

$\mu $ | fluid viscosity |

$(\xi ,\eta )$ | elliptic coordinate system |

${\xi}_{0}$ | boundary interface of the channel |

$\rho $ | fluid density |

${\rho}_{e}$ | ionic charge density of fluid |

$\psi $ | EDL potential |

${\psi}^{\ast}$ | dimensionless EDL potential |

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**Figure 1.**Electroosmotic flow and the electrical double layer (red highlighted area) in an elliptic cross-sectional channel: (

**a**) cross-section view and (

**b**) view along the channel length.

**Figure 4.**Variation of the dimensionless flow rates ${Q}^{\ast}$ with four different values of the slip length $l=0$ (red dotted line), 10 (blue solid line), 100 (green dashed line), and 1000 (black dot-dashed line) $\mathsf{\mu}\mathrm{m}$ on various eccentricities when (

**a**) the area of the channel cross-sections is fixed and (

**b**) the perimeter of the channel cross-sections is fixed.

**Figure 5.**Elliptic cross-sections with three different eccentricities $e=0$ (red dotted line), $0.6$ (blue solid line), and $0.9$ (black dot-dashed line) when (

**a**) the area of the channel cross-sections is fixed and (

**b**) the perimeter of the channel cross-sections is fixed.

**Figure 6.**Velocity profile along the y-axis in the elliptic channel with three different eccentricities $e=0$ (red dotted line), $0.6$ (blue solid line), and $0.9$ (black dot-dashed line) when the area of the channel cross-sections is fixed: (

**a**) $l=0$ $\mathsf{\mu}\mathrm{m}$ and (

**b**) $l=10$ $\mathsf{\mu}\mathrm{m}$.

**Figure 7.**Area of the elliptic cross-section when the perimeter is fixed to $4.127\times {10}^{2}$ $\mathsf{\mu}\mathrm{m}$.

**Figure 8.**Variation of the relative errors of the flow rate with four different values of the slip length $l=0$ (red dotted line), 10 (blue solid line), 100 (green dashed line), and 1000 (black dot-dashed line) $\mathsf{\mu}\mathrm{m}$ on various eccentricities when (

**a**) the area of the channel cross-sections is fixed and (

**b**) the perimeter of the channel cross-sections is fixed.

**Figure 9.**Variation of the relative errors of the flow rate on various eccentricities when the ratios ${p}_{z}^{\ast}/{E}^{\ast}$ are $1/5$ times (red dotted line), $1/2.5$ times (blue solid line), 1 times (green dashed line), $2.5$ times (purple long dashed line), and 5 times (black dot-dashed line) of the ratio used in the investigation in Figure 8 with four different values of the slip length: (

**a**) $l=0$ $\mathsf{\mu}\mathrm{m}$; (

**b**) $l=10$ $\mathsf{\mu}\mathrm{m}$; (

**c**) $l=100$ $\mathsf{\mu}\mathrm{m}$; and (

**d**) $l=1000$ $\mathsf{\mu}\mathrm{m}$.

Name | Symbol | Value | SI Unit |
---|---|---|---|

Fluid viscosity | $\mu $ | $\phantom{-}9.00\times {10}^{-4}$ | Pa s |

Fluid permittivity | $\epsilon $ | $\phantom{-}6.95\times {10}^{-10}$ | F m^{−1} |

Pressure gradient in z-axis | ${p}_{z}$ | $-2.00$ | Pa m^{−1} |

Reciprocal of EDL thickness | $\kappa $ | $\phantom{-}8.00\times {10}^{4}$ | m^{−1} |

Zeta potential | $\zeta $ | $-2.49\times {10}^{-4}$ | $\mathrm{V}$ |

External electric field | E | $\phantom{-}5.00\times {10}^{2}$ | V m^{−1} |

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**MDPI and ACS Style**

Chuchard, P.; Numpanviwat, N.
Combined Pressure-Driven and Electroosmotic Slip Flow through Elliptic Cylindrical Microchannels: The Effect of the Eccentricity of the Channel Cross-Section. *Symmetry* **2022**, *14*, 999.
https://doi.org/10.3390/sym14050999

**AMA Style**

Chuchard P, Numpanviwat N.
Combined Pressure-Driven and Electroosmotic Slip Flow through Elliptic Cylindrical Microchannels: The Effect of the Eccentricity of the Channel Cross-Section. *Symmetry*. 2022; 14(5):999.
https://doi.org/10.3390/sym14050999

**Chicago/Turabian Style**

Chuchard, Pearanat, and Nattakarn Numpanviwat.
2022. "Combined Pressure-Driven and Electroosmotic Slip Flow through Elliptic Cylindrical Microchannels: The Effect of the Eccentricity of the Channel Cross-Section" *Symmetry* 14, no. 5: 999.
https://doi.org/10.3390/sym14050999