Combined Pressure-Driven and Electroosmotic Slip Flow through Elliptic Cylindrical Microchannels: The Effect of the Eccentricity of the Channel Cross-Section
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 |
separation constant of 2-dimensional wave equation in elliptic coordinates | |
c | focal length of an ellipse |
periodic modified Methieu function corresponding to | |
periodic Methieu function of integral order | |
E | external electric field in z-direction |
dimensionless external electric field in z-direction |
vector of external electric field | |
e | eccentricity of an ellipse |
vector of electroosmotic body force | |
scalar factors/basic vectors for the elliptic coordinates | |
Boltzmann constant | |
l | Navier slip length |
concentration of ions at bulk | |
P | perimeter of the elliptic cross-section |
p | pressure |
pressure gradient in z-direction | |
dimensionless pressure gradient in z-direction | |
elementary proton charge | |
Q | volumetric flow rate per unit area of the channel cross-section |
dimensionless volumetric flow rate per unit area | |
dimensionless flow rate in the elliptic channel with the eccentricity equal to i | |
dimensionless flow rate in the circular channel | |
T | fluid absolute temperature |
u | fluid velocity in z-direction |
dimensionless fluid velocity in z-direction | |
vector of fluid velocity | |
Cartesian coordinates | |
valence of ion | |
relative error of | |
fluid permittivity | |
zeta potential | |
reciprocal of EDL thickness | |
fluid viscosity | |
elliptic coordinate system | |
boundary interface of the channel | |
fluid density | |
ionic charge density of fluid | |
EDL potential | |
dimensionless EDL potential |
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Name | Symbol | Value | SI Unit |
---|---|---|---|
Fluid viscosity | Pa s | ||
Fluid permittivity | F m−1 | ||
Pressure gradient in z-axis | Pa m−1 | ||
Reciprocal of EDL thickness | m−1 | ||
Zeta potential | |||
External electric field | E | V m−1 |
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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
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 StyleChuchard, 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