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

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 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 |

## References

- Xiang, J.; Cai, Z.; Zhang, Y.; Wang, W. A micro-cam actuated linear peristaltic pump for microfluidic applications. Sens. Actuators A Phys.
**2016**, 251, 20–25. [Google Scholar] [CrossRef] - Bridle, H.; Wang, W.; Gavriilidou, D.; Amalou, F.; Hand, D.P.; Shu, W. Static mode microfluidic cantilevers for detection of waterborne pathogens. Sens. Actuators A Phys.
**2016**, 247, 144–149. [Google Scholar] [CrossRef][Green Version] - Jiemsakul, T.; Manakasettharn, S.; Kanharattanachai, S.; Wanna, Y.; Wangsuya, S.; Pratontep, S. Microfluidic magnetic switching valves based on aggregates of magnetic nanoparticles: Effects of aggregate length and nanoparticle sizes. J. Magn. Magn. Mater.
**2017**, 422, 434–439. [Google Scholar] [CrossRef] - Montessori, A.; Lauricella, M.; La Rocca, M.; Succi, S.; Stolovicki, E.; Ziblat, R.; Weitz, D. Regularized lattice Boltzmann multicomponent models for low capillary and Reynolds microfluidics flows. Comput. Fluids
**2018**, 167, 33–39. [Google Scholar] [CrossRef][Green Version] - Ding, R.; Lisak, G. Sponge-based microfluidic sampling for potentiometric ion sensing. Anal. Chim. Acta
**2019**, 1091, 103–111. [Google Scholar] [CrossRef] [PubMed] - Streck, S.; Clulow, A.J.; Nielsen, H.M.; Rades, T.; Boyd, B.J.; McDowell, A. The distribution of cell-penetrating peptides on polymeric nanoparticles prepared using microfluidics and elucidated with small angle X-ray scattering. J. Colloid Interface Sci.
**2019**, 555, 438–448. [Google Scholar] [CrossRef] - Yan, S.; Chu, F.; Zhang, H.; Yuan, Y.; Huang, Y.; Liu, A.; Wang, S.; Li, W.; Li, S.; Wen, W. Rapid, one-step preparation of SERS substrate in microfluidic channel for detection of molecules and heavy metal ions. Spectrochim. Acta Part A Mol. Biomol. Spectrosc.
**2019**, 220, 117113. [Google Scholar] [CrossRef] - Dejam, M. Hydrodynamic dispersion due to a variety of flow velocity profiles in a porous-walled microfluidic channel. Int. J. Heat Mass Transf.
**2019**, 136, 87–98. [Google Scholar] [CrossRef] - Tian, C.; Tu, Q.; Liu, W.; Wang, J. Recent advances in microfluidic technologies for organ-on-a-chip. TrAC Trends Anal. Chem.
**2019**, 117, 146–156. [Google Scholar] [CrossRef] - Yang, Q.; Ju, D.; Liu, Y.; Lv, X.; Xiao, Z.; Gao, B.; Song, F.; Xu, F. Design of organ-on-a-chip to improve cell capture efficiency. Int. J. Mech. Sci.
**2021**, 209, 106705. [Google Scholar] [CrossRef] - Zhao, K.; Peng, Z.; Jiang, H.; Lv, X.; Li, X.; Deng, Y. Shape-coded hydrogel microparticles integrated with hybridization chain reaction and a microfluidic chip for sensitive detection of multi-target miRNAs. Sens. Actuators B Chem.
**2022**, 361, 131741. [Google Scholar] [CrossRef] - Walczak, R.; Kawa, B.; Adamski, K. Inkjet 3D printed microfluidic device for growing seed root and stalk mechanical characterization. Sens. Actuators A Phys.
**2019**, 297, 111557. [Google Scholar] [CrossRef] - Wei, L.; Fang, G.; Kuang, Z.; Cheng, L.; Wu, H.; Guo, D.; Liu, A. 3D-printed low-cost fabrication and facile integration of flexible epidermal microfluidics platform. Sens. Actuators B Chem.
**2022**, 353, 131085. [Google Scholar] [CrossRef] - Maeki, M.; Uno, S.; Niwa, A.; Okada, Y.; Tokeshi, M. Microfluidic technologies and devices for lipid nanoparticle-based RNA delivery. J. Control. Release
**2022**, 344, 80–96. [Google Scholar] [CrossRef] [PubMed] - Chhabra, R.; Shankar, V. Coulson and Richardson’s Chemical Engineering, 7th ed.; Butterworth-Heinemann: Oxford, UK, 2018. [Google Scholar]
- Hunter, R.J. Zeta Potential in Colloid Science; Academic Press: Cambridge, MA, USA, 1981. [Google Scholar]
- Bhamidimarri, S.P.; Prajapati, J.D.; van den Berg, B.; Winterhalter, M.; Kleinekathöfer, U. Role of Electroosmosis in the Permeation of Neutral Molecules: CymA and Cyclodextrin as an Example. Biophys. J.
**2016**, 110, 600–611. [Google Scholar] [CrossRef] [PubMed][Green Version] - Yoshida, K.; Sato, T.; Eom, S.I.; wan Kim, J.; Yokota, S. A study on an AC electroosmotic micropump using a square pole—Slit electrode array. Sens. Actuators A Phys.
**2017**, 265, 152–160. [Google Scholar] [CrossRef] - Kariminezhad, E.; Elektorowicz, M. Comparison of constant, pulsed, incremental and decremental direct current applications on solid-liquid phase separation in oil sediments. J. Hazard. Mater.
**2018**, 358, 475–483. [Google Scholar] [CrossRef] - Mateos, H.; Valentini, A.; Robles, E.; Brooker, A.; Cioffi, N.; Palazzo, G. Measurement of the zeta-potential of solid surfaces through Laser Doppler Electrophoresis of colloid tracer in a dip-cell: Survey of the effect of ionic strength, pH, tracer chemical nature and size. Colloids Surf. A Physicochem. Eng. Asp.
**2019**, 576, 82–90. [Google Scholar] [CrossRef] - Moschopoulos, P.; Dimakopoulos, Y.; Tsamopoulos, J. Electro-osmotic flow of electrolyte solutions of PEO in microfluidic channels. J. Colloid Interface Sci.
**2020**, 563, 381–393. [Google Scholar] [CrossRef] - Awan, A.U.; Ali, M.; Abro, K.A. Electroosmotic slip flow of Oldroyd-B fluid between two plates with non-singular kernel. J. Comput. Appl. Math.
**2020**, 376, 112885. [Google Scholar] [CrossRef] - Mondal, B.; Mehta, S.K.; Pati, S.; Patowari, P.K. Numerical analysis of electroosmotic mixing in a heterogeneous charged micromixer with obstacles. Chem. Eng. Process. Process Intensif.
**2021**, 168, 108585. [Google Scholar] [CrossRef] - Siddiqui, A.A.; Lakhtakia, A. Debye-Hückel solution for steady electro-osmotic flow of micropolar fluid in cylindrical microcapillary. Appl. Math. Mech.
**2013**, 34, 1305–1326. [Google Scholar] [CrossRef][Green Version] - Tseng, S.; Tai, Y.H.; Hsu, J.P. Ionic current in a pH-regulated nanochannel filled with multiple ionic species. Microfluid. Nanofluid.
**2014**, 17, 933–941. [Google Scholar] [CrossRef] - Goswami, P.; Chakraborty, S. Semi-analytical solutions for electroosmotic flows with interfacial slip in microchannels of complex cross-sectional shapes. Microfluid. Nanofluid.
**2011**, 11, 255–267. [Google Scholar] [CrossRef] - Chuchard, P.; Orankitjaroen, S.; Wiwatanapataphee, B. Study of pulsatile pressure-driven electroosmotic flows through an elliptic cylindrical microchannel with the Navier slip condition. Adv. Differ. Equ.
**2017**, 2017, 160. [Google Scholar] [CrossRef][Green Version] - Arulanandam, S.; Li, D. Liquid transport in rectangular microchannels by electroosmotic pumping. Colloids Surf. A Physicochem. Eng. Asp.
**2000**, 161, 89–102. [Google Scholar] [CrossRef] - Reshadi, M.; Saidi, M.H.; Firoozabadi, B.; Saidi, M.S. Electrokinetic and aspect ratio effects on secondary flow of viscoelastic fluids in rectangular microchannels. Microfluid. Nanofluid.
**2016**, 20, 117. [Google Scholar] [CrossRef] - McLachlan, N.W. Theory and Application of Mathieu Functions; Clarendon Press: Oxford, UK, 1951. [Google Scholar]
- Liu, B.T.; Tseng, S.; Hsu, J.P. Effect of eccentricity on the electroosmotic flow in an elliptic channel. J. Colloid Interface Sci.
**2015**, 460, 81–86. [Google Scholar] [CrossRef] - Numpanviwat, N.; Chuchard, P. Transient Pressure-Driven Electroosmotic Flow through Elliptic Cross-Sectional Microchannels with Various Eccentricities. Computation
**2021**, 9, 27. [Google Scholar] [CrossRef] - Zhu, Y.; Granick, S. Rate-dependent slip of Newtonian liquid at smooth surfaces. Phys. Rev. Lett.
**2001**, 87, 096105. [Google Scholar] [CrossRef][Green Version] - Baudry, J.; Charlaix, E.; Tonck, A.; Mazuyer, D. Experimental Evidence for a Large Slip Effect at a Nonwetting Fluid–Solid Interface. Langmuir
**2001**, 17, 5232–5236. [Google Scholar] [CrossRef] - Bonaccurso, E.; Kappl, M.; Butt, H.J. Hydrodynamic Force Measurements: Boundary Slip of Water on Hydrophilic Surfaces and Electrokinetic Effects. Phys. Rev. Lett.
**2002**, 88, 76103. [Google Scholar] [CrossRef] [PubMed] - Zhang, J.; Kwok, D.Y. Apparent slip over a solid-liquid interface with a no-slip boundary condition. Phys. Rev. E
**2004**, 70, 56701. [Google Scholar] [CrossRef] - Chakraborty, J.; Ray, S.; Chakraborty, S. Role of streaming potential on pulsating mass flow rate control in combined electroosmotic and pressure-driven microfluidic devices. Electrophoresis
**2012**, 33, 419–425. [Google Scholar] [CrossRef] [PubMed] - Li, D. Electrokinetics in Microfluidics; Interface Science and Technology; Elsevier: Amsterdam, The Netherlands, 2004; Volume 2. [Google Scholar]

**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} |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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