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Research Papers: Fundamental Issues and Canonical Flows

# Numerical Investigation of Combined Effects of Rarefaction and Compressibility for Gas Flow in Microchannels and Microtubes

[+] Author and Article Information
Xiaohong Yan

State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China

Qiuwang Wang1

State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, Chinawangqw@mail.xjtu.edu.cn

1

Corresponding author.

J. Fluids Eng 131(10), 101201 (Sep 17, 2009) (7 pages) doi:10.1115/1.3215941 History: Received November 14, 2008; Revised July 02, 2009; Published September 17, 2009

## Abstract

In this paper, first, the Navier–Stokes equations for incompressible fully developed flow in microchannels and microtubes with the first-order and second-order slip boundary conditions are analytically solved. Then, the compressible Navier–Stokes equations are numerically solved with slip boundary conditions. The numerical methodology is based on the control volume scheme. Numerical results reveal that the compressibility effect increases the velocity gradient near the wall and the friction factor. On the other hand, the increment of velocity gradient near the wall leads to a much larger slip velocity than that for incompressible flow with the same value of Knudsen number and results in a corresponding decrement of friction factor. General correlations for the Poiseuille number $(fRe)$, the Knudsen number (Kn), and the Mach number (Ma) containing the first-order and second-order slip coefficients are proposed. Correlations are validated with available experimental and numerical results.

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## Figures

Figure 6

Combined effects of rarefaction and compressibility under various slip boundary conditions (Ccomp,slip=Pocomp,slip/Poincomp,noslip, (a) microchannel, and (b) microtube)

Figure 7

Combined effects of rarefaction and compressibility (Ccomp,slip/(Cslip⋅Ccomp) indicates the effect of velocity derivative on the slip velocity, (a) microchannel, and (b) microtube)

Figure 8

Prediction of Poiseuille number for a special case (Ccomp indicates the effect of compressibility, Cslip indicates the effect of rarefaction, and Ccomp,slip indicates the combined effects of rarefaction and compressibility, (a) microchannel, and (b) microtube)

Figure 9

Prediction of Poiseuille number along microchannels with different length-height ratios. (second-order slip boundary condition is used for A1=1.1466 and A2=0.9756—the symbols denote the numerical results and the lines denote the predictions of Eq. 20)

Figure 10

Prediction of Poiseuille number along a microchannel under three second-order slip boundary conditions (the symbols denote the numerical results and the lines denote the predictions of Eq. 20)

Figure 1

Sketch of the microchannel and the microtube

Figure 2

Effect of rarefaction for various slip models (Cslip=Poincomp,slip/Poincomp,noslip)

Figure 3

Comparison of reduced Poiseuille number (Ccomp=Pocomp,noslip/Poincomp,noslip) with available results (effect of compressibility, (a) microchannel, and (b) microtube)

Figure 4

Comparison of dimensionless velocity profile of compressible and incompressible flow

Figure 5

Second-order velocity derivative for compressible flow (Cderi=(∂2u/∂x22)wall,comp/(∂2u/∂x22)wall,incomp, which indicates the enhancement of the second-order velocity derivative due to compressibility)

## Errata

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