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Research Papers: Fundamental Issues and Canonical Flows

Electroosmotic Flow Through a Circular Tube With Slip-Stick Striped Wall

[+] Author and Article Information
Henry C. W. Chu

Department of Mechanical Engineering,  The University of Hong Kong, Pokfulam Road, Hong Kong, SAR

Chiu-On Ng1

Department of Mechanical Engineering,  The University of Hong Kong, Pokfulam Road, Hong Kong, SARcong@hku.hk

1

Corresponding author.

J. Fluids Eng 134(11), 111201 (Oct 23, 2012) (13 pages) doi:10.1115/1.4007690 History: Received March 28, 2012; Revised September 18, 2012; Published October 23, 2012

This is an analytical study on electrohydrodynamic flows through a circular tube, of which the wall is micropatterned with a periodic array of longitudinal or transverse slip-stick stripes. One unit of the wall pattern comprises two stripes, one slipping and the other nonslipping, and each with a distinct ζ potential. Using the methods of eigenfunction expansion and point collocation, the electric potential and velocity fields are determined by solving the linearized Poisson–Boltzmann equation and the Stokes equation subject to the mixed electrohydrodynamic boundary conditions. The effective equations for the fluid and current fluxes are deduced as functions of the slipping area fraction of the wall, the intrinsic hydrodynamic slip length, the Debye parameter, and the ζ potentials. The theoretical limits for some particular wall patterns, which are available in the literature only for plane channels, are extended in this paper to the case of a circular channel. We confirm that some remarks made earlier for electroosmotic flow over a plane surface are also applicable to the present problem involving patterns on a circular surface. We pay particular attention to the effects of the pattern pitch on the flow in both the longitudinal and transverse configurations. When the wall is uniformly charged, the adverse effect on the electroosmotic flow enhancement due to a small fraction of area being covered by no-slip slots can be amplified if the pitch decreases. Reducing the pitch will also lead to a greater deviation from the Helmholtz–Smoluchowski limit when the slipping regions are uncharged. With oppositely charged slipping regions, local recirculation or a net reversed flow is possible, even when the wall is on the average electropositive or neutral. The flow morphology is found to be subject to the combined influence of the geometry of the tube and the electrohydrodynamic properties of the wall.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

Electrokinetic flow through a tube of radius R, where the wall is patterned with a periodic array of (a) longitudinal stripes or (b) transverse stripes. One unit of wall pattern consists of a nonslipping stripe of ζ potential ζNS, and a slipping stripe of slip length λ and ζ potential ζS. The radial, angular and axial coordinates are respectively (r,θ,z) and their corresponding velocity components are denoted by (u,v,w). With longitudinal stripes, the flow is unidirectional and purely along the z-direction; and with transverse stripes, the flow is two-dimensional in the (r,z) plane. In (a), one periodic unit is from θ = 0 to θ = 2π/K, where K is the number of periodic units on the wall. In (b), one periodic unit is from z = 0 to z = 2L. In either case, the area fraction of the slipping region is denoted by a.

Grahic Jump Location
Figure 2

The longitudinal and transverse streaming conductance, L12|| and L12⊥, as functions of the Debye parameter κ̂ and the slipping area fraction a where the intrinsic slip length λ̂ = 1 and wall potential ζ̂S = 1. The number of periodic units for longitudinal stripes K = 5 in 2(a) and 2(c), while the period length for transverse stripes L̂ = 1 in 2(b) and 2(d). Dashed lines in 2(c) and 2(d) represent the case where K = 10 and L̂ = 5 respectively. The two insets show the effective slip lengths δ̂|| and δ̂⊥ as functions of a.

Grahic Jump Location
Figure 3

The longitudinal and transverse streaming conductance, L12|| and L12⊥, as functions of the Debye parameter κ̂ and the slipping area fraction a where the intrinsic slip length λ̂ = ∞ and wall potential ζ̂S = 0. The number of periodic units for longitudinal stripes K = 5 in 3(a) and 3(c), while the period length for transverse stripes L̂ = 1 in 3(b) and 3(d). Dashed lines in 3(a) and 3(b) represent the case where λ̂ = 1. Dashed lines in 3(c) and 3(d) represent the case where K = 10 and L̂ = 5 respectively.

Grahic Jump Location
Figure 4

The longitudinal and transverse streaming conductance, L12|| and L12⊥, as functions of the slipping area fraction a, Debye parameter κ̂, intrinsic slip length λ̂, the number of periodic units for longitudinal stripes K, and the period length for transverse stripes L̂, where wall potential ζ̂S = -1. The parameter K = 5 in 4(a) and L̂ = 1 in 4(b), λ̂ = 0.1 in 4(a) and 4(b) while a = 0.1 and κ̂ = 100 in 4(c) and 4(d). The two insets show the effective slip lengths δ̂|| and δ̂⊥ as functions of K and L̂. Diamonds in 4(c) are the results from the analytical solution of Philip [(18),19]. The dotted line in 4(d) is the asymptotic limit obtained by Lauga and Stone [20] when L̂→∞ and a fixed.

Grahic Jump Location
Figure 5

Streamlines of EOF transverse to the slipping stripes where slipping area fraction a = 0.35, period length L̂ = 1 and wall potential ζ̂S = -0.43. From left to right intrinsic slip length is increased λ̂ = 0.001,0.1,100. From top to bottom Debye parameter is increased κ̂ = 10,100,500. Dashed lines represent negative value of stream function, i.e., flow reversal.

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