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# Oscillatory Flow Through a Channel With Stick-Slip Walls: Complex Navier’s Slip Length

[+] Author and Article Information
Chiu-On Ng1

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

C. Y. Wang

Department of Mathematics, Michigan State University, East Lansing, MI 48824

1

Corresponding author.

J. Fluids Eng 133(1), 014502 (Jan 13, 2011) (6 pages) doi:10.1115/1.4003219 History: Received August 28, 2010; Revised December 08, 2010; Published January 13, 2011; Online January 13, 2011

## Abstract

Effective slip lengths for pressure-driven oscillatory flow through a parallel-plate channel with boundary slip are deduced using a semi-analytic method of eigenfunction expansions and point matching. The channel walls are each a superhydrophobic surface micropatterned with no-shear alternating with no-slip stripes, which are aligned either parallel or normal to the flow. The slip lengths are complex quantities that are functions of the oscillation frequency, the channel height, and the no-shear area fraction of the wall. The dependence of the complex nature of the slip length on the oscillation frequency is investigated in particular.

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## Figures

Figure 1

Oscillatory flow through a parallel-plate channel with walls that are patterned with a periodic array of longitudinal or transverse no-shear stripes. The patterns of the two walls are arranged in-phase with each other so that the flow is symmetrical about the centerline of the channel. The x- and z-axes are normal and parallel to the stripes, respectively, while the y-axis is perpendicular to the channel walls. The length dimensions are normalized with respect to half the period of the wall pattern.

Figure 2

For longitudinal flow, (a) the real part of the slip length δ∥, (b) the imaginary part of the slip length δ∥, (c) the magnitude of the mean velocity |W¯| relative to that without wall slip |W¯0|, and (d) the phase ϕ of the mean velocity are plotted as functions of the oscillation parameter Ω, and the no-shear area fraction of the wall a, where the channel height h=2. In (a), the dotted lines are the limiting steady-state values of the slip length δ∥P given by Eq. 1. In (d), the phase is in degrees, and the dashed line is for the case without wall slip or a=0.

Figure 3

For longitudinal flow, (a) the real part of the slip length δ∥, (b) the imaginary part of the slip length δ∥, (c) the magnitude of the mean velocity |W¯| relative to that without wall slip |W¯0|, and (d) the phase ϕ of the mean velocity are plotted as functions of the channel height h, and the no-shear area fraction of the wall a, where the oscillation parameter Ω=5. In (d), the phase is in degrees, and the dashed line is for the case without wall slip or a=0.

Figure 4

For transverse flow, (a) the real part of the slip length δ⊥, (b) the imaginary part of the slip length δ⊥, (c) the magnitude of the mean velocity |U¯| relative to that without wall slip |U¯0|, and (d) the phase ϕ of the mean velocity are plotted as functions of the oscillation parameter Ω, and the no-shear area fraction of the wall a, where the channel height h=2. In (a), the dotted lines are the limiting steady-state values of the slip length δ⊥P given by Eq. 1. In (d), the phase is in degrees, and the dashed line is for the case without wall slip or a=0.

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