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# Discussion: “Slip-Flow Pressure Drop in Microchannels of General Cross Section”OPEN ACCESS

[+] Author and Article Information
C. Y. Wang1

Departments of Mathematics and Mechanical Engineering,  Michigan State University, East Lansing, MI 48824cywang@mth.msu.edu

1

Corresponding author.

J. Fluids Eng 134(5), 055501 (May 22, 2012) (2 pages) doi:10.1115/1.4005724 History: Received April 21, 2010; Revised July 19, 2010; Published May 18, 2012; Online May 22, 2012

## Abstract

Partial slip occurs in a variety of important fluid flow situations. Recently several sources used the constant boundary slip assumption for the flow in a tube. By comparing with the exact solution for the slip flow in a triangular duct, we show the constant slip assumption invokes substantial errors in both local and global fluid dynamic properties.

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## Introduction

Partial slip for fluid flow past a solid boundary occurs not only in rarefied gas flow in tubes at small Knudsen numbers [1] but also in many other practical situations including rough boundary [2], lubricated surfaces, superhydrophobic surfaces [3], and particulate fluids such as blood, foam, emulsion, suspension or polymers [4]. In all cases, especially when the nominal dimension is small, the no slip boundary condition should be replaced by Navier’s (first order) partial slip condition [5] Display Formula

$w′s=Nτ′$
(1)
where $w′s$ is the wall slip velocity, N is the slip constant and $τ′$ is the local shear stress. Normalize lengths by nominal width L and velocity by $GL2/μ$ where G is the pressure gradient, $μ$ is the viscosity, and drop primes. Eq. 1 becomes Display Formula
$ws=λ∂w∂n$
(2)
Here $λ=Nμ/L$ is the normalized slip factor and n is the normal direction. For parallel flow in tubes the (constant density) Navier-Stokes equations reduce to Display Formula
$∇2w=-1$
(3)
How large is $λ$? For superhydrophobic microchannels [3] of tens of microns across, $λ$ could be of order one. For lubricated surfaces the slip factor may be much larger [6].

Recently Bahrami et al.  [7] suggested an approximate method where a constant slip assumption is used along the wall. Such an assumption has also been proposed previously [8-12]. Thus, this Discussion applies to Refs. [7]–[12].

The constant slip assumption involves no error for circular ducts or parallel plates where, due to symmetry, slip is indeed uniform. However, care must be taken for other geometries.

We illustrate by the equilateral triangular duct of normalized height 3. The no-slip velocity is classical Display Formula

$w0=-(x2+y2)4-(x3-3xy2)12+13$
(4)
Using the constant slip assumption, for slip flow let Display Formula
$w=w0+ws$
(5)
where $ws$ is the slip velocity. The average shear stress on the boundary is − 1/2. Using Navier’s slip condition we find $ws=λ/2$. The Poiseuille number (friction factor-Reynolds number) is Display Formula
$Po=403+10λ$
(6)

On the other hand, an exact closed form solution for slip flow in an equilateral triangle exists [13] Display Formula

$w=-(x2+y2)4-(x3-3xy2)12(1+λ)+(2+6λ+3λ2)6(1+λ)$
(7)
The corresponding Poiseuille number is Display Formula
$Po=40(1+λ)3+15λ+10λ2$
(8)
Figure 1 shows a comparison of the constant velocity lines. It is seen that the velocity distributions are very different. Thus, local properties such as stress or heat transfer cannot be predicted by the constant slip approximation. Table 1 shows a comparison of the Poiseuille number. We see the error in the global properties such as Poiseuille number is also large.

We expect the error is even larger for cross sections which deviate much from the circle, such as star shaped ducts or biconcave ducts (a partially collapsed circular tube).

## References

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## Figures

Figure 1

Constant velocity lines for the equilateral triangular duct (λ=1) (a) Exact solution. Curves from inside: w = 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3. (b) Using constant velocity assumption. Curves from inside: w = 0.8, 0.7, 0.6, and 0.5 on boundary.

## Tables

Table 1
Comparison of Poiseuille numbers

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