Microfluidics and Nanofluidics

, Volume 11, Issue 3, pp 255–267 | Cite as

Semi-analytical solutions for electroosmotic flows with interfacial slip in microchannels of complex cross-sectional shapes

Research Paper

Abstract

In this article, we investigate the implications of electroosmosis with interfacial slip on electrohydrodynamic transport in microchannels having complex (yet symmetric) cross-sectional shapes, by employing a generic semi-analytical approach. We also devise an approximate technique of flow rate prediction under these conditions, using a combined consideration of electroosmotic slip (under thin electrical double layer limits) and Navier slip conditions (originating out of confinement-induced hydrophobic interactions) at the fluid–solid interface. We further assess the effectiveness of the approximate solutions in perspective of the exact solutions, as a parametric function of the relative thickness of the electrical double layer with respect to the channel hydraulic diameter. We illustrate the underlying consequences through examples of elliptic, polygonal, point star-shaped, and annular microchannel cross sections.

List of symbols

A

Area of the channel cross section (m2)

b

Slip length (m)

Dh

Hydraulic diameter of the channel (m)

E

Electric field (V m−1)

e

Charge of a proton (C)

e

Eccentricity of ellipse

g

Channel boundary

Im

Modified 1st kind Bessel functions of order m

K

Non-dimensional EDL thickness

Km

Modified 2nd kind Bessel functions of order m

kB

Boltzmann constant (J K−1)

n

Normal distance from the surface (m)

n0

Bulk ionic concentration (m−3)

n+

Concentration of cations (m−3)

n

Concentration of anions (m−3)

Q

Volume flow rate in actual situation

Qapp

Volume flow rate in approximate situation

T

Absolute temperature (K)

UHS

Helmholtz–Smoluchowski velocity (m s−1)

u

Velocity in actual situation (m s−1)

uapp

Velocity in approximate situation (m s−1)

z

Valance

Greek symbols

ρ

Fluid density (kg m−3)

μ

Dynamic viscosity (kg m−1 s−1)

ρe

Ionic charge density (C m−3)

ɛ

Permittivity of the medium (C V−1 m−1)

ψ

EDL potential (V)

ζ

Zeta potential (V)

κ

Reciprocal of EDL thickness (m−1)

λ

Eigen value

β

Angle of symmetry (rad.)

References

  1. Bonaccurso E, Butt HJ, Craig VSJ (2003) Surface roughness and hydrodynamic boundary slip of a Newtonian fluid in a completely wetting system. Phys Rev Lett 90:144501CrossRefGoogle Scholar
  2. Bruin GJM (2000) Recent developments in electrokinetically driven analysis on micro-fabricated devices. Electrophoresis 21:3931–3951CrossRefGoogle Scholar
  3. Bruus H (2008) Theoretical microfluidics. Oxford University Press, OxfordGoogle Scholar
  4. Chakraborty S (2008) Generalization of interfacial electrohydrodynamics in the presence of hydrophobic interactions in narrow fluidic confinements. Phys Rev Lett 100:097801CrossRefGoogle Scholar
  5. Dolnik V, Hutterer KM (2001) Capillary electrophoresis of proteins 1999–2001. Electrophoresis 22:4163–4178CrossRefGoogle Scholar
  6. Duan Z, Muzychka YS (2007) Slip flow in elliptic microchannels. Int J Therm Sci 46:1104–1111CrossRefGoogle Scholar
  7. Figeys D, Pinto D (2001) Proteomics on a chip: promising developments. Electrophoresis 22:208–216CrossRefGoogle Scholar
  8. Galea TM, Attard P (2004) Molecular dynamics study of the effect of atomic roughness on the slip length at the fluid–solid boundary during shear flow. Langmuir 20:3477CrossRefGoogle Scholar
  9. Hunter RJ (2000) Foundations of colloid science, 2nd edn. Oxford University Press, OxfordGoogle Scholar
  10. Joly L, Ybert C, Trizac E, Bocquet L (2004) Hydrodynamics within the electric double layer on slipping surfaces. Phys Rev Lett 93:257805CrossRefGoogle Scholar
  11. Landers JP (2003) Molecular diagnostics on electrophoretic microchips. Anal Chem 75:2919–2927CrossRefGoogle Scholar
  12. Li D (2004) Electrokinetics in microfluidics. Elsevier, AmsterdamGoogle Scholar
  13. Park HM, Kim TW (2009) Extension of the Helmholtz–Smoluchowski velocity to the hydrophobic microchannels with velocity slip. Lab Chip 9:291–296CrossRefGoogle Scholar
  14. Park HM, Lim JY (2009) Streaming potential for microchannels of arbitrary cross-sectional shapes for thin electric double layers. J Colloid Interface Sci 336:834–841CrossRefGoogle Scholar
  15. Reyes DR, Iossifidis D, Auroux PA, Manz A (2002) Micro total analysis systems. 1. Introduction, theory, and technology. Anal Chem 74:2623–2636CrossRefGoogle Scholar
  16. Shih FS (1967) Laminar flow in axisymmetric conduits by a rational approach. Can J Chem Eng 45:285–294CrossRefGoogle Scholar
  17. Stone HA, Kim S (2001) Microfluidics: basic issues, applications, and challenges. AIChE J 47:1250–1254CrossRefGoogle Scholar
  18. Tsao HK (2000) Electroosmotic flow through an annulus. J Colloid Interface Sci 225:247–250CrossRefGoogle Scholar
  19. Whitesides GM, Stroock AD (2001) Flexible methods for microfluidics. Phys Today 54:42–48CrossRefGoogle Scholar
  20. Yang J, Kwok DY (2002) A new method to determine zeta potential and slip coefficient simultaneously. J Phys Chem B106:12851–12855Google Scholar
  21. Yang J, Kwok DY (2003) Microfluid flow in circular microchannel with electrokinetic effects and Navier’s slip condition. Langmuir 19:1047–1053CrossRefGoogle Scholar
  22. Yang J, Kwok DY (2004) Analytical treatment of electrokinetic microfluidics in hydrophobic microchannels. Anal Chim Acta 507:39–53CrossRefGoogle Scholar
  23. Zhang Y, Wong TN, Yang C, Ooi KT (2005) Electroosmotic flow in irregular shape microchannels. Int J Eng Sci 43:1450–1463CrossRefGoogle Scholar
  24. Zhu Y, Granick S (2002) Limits of the hydrodynamic no-slip boundary condition. Phys Rev Lett 88:106102CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology, KharagpurKharagpurIndia
  2. 2.Department of Mechanical EngineeringIndian Institute of Technology, KharagpurKharagpurIndia

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