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Research Papers: Flows in Complex Systems

Modeling of Knudsen Layer Effects in Micro/Nanoscale Gas Flows

[+] Author and Article Information
Nishanth Dongari, Yonghao Zhang

Jason M Reese

Department of Mechanical Engineering,  University of Strathclyde, Glasgow G1 1XJ, United Kingdomjason.reese@strath.ac.uk

J. Fluids Eng 133(7), 071101 (Jul 05, 2011) (10 pages) doi:10.1115/1.4004364 History: Received December 16, 2010; Accepted June 03, 2011; Published July 05, 2011; Online July 05, 2011

We propose a power-law based effective mean free path (MFP) model so that the Navier-Stokes-Fourier equations can be employed for the transition-regime flows typical of gas micro/nanodevices. The effective MFP model is derived for a system with planar wall confinement by taking into account the boundary limiting effects on the molecular free paths. Our model is validated against molecular dynamics simulation data and compared with other theoretical models. As gas transport properties can be related to the mean free path through kinetic theory, the Navier-Stokes-Fourier constitutive relations are then modified in order to better capture the flow behavior in the Knudsen layers close to surfaces. Our model is applied to fully developed isothermal pressure-driven (Poiseuille) and thermal creep gas flows in microchannels. The results show that our approach greatly improves the near-wall accuracy of the Navier-Stokes-Fourier equations, well beyond the slip-flow regime.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic of the momentum Knudsen layer (KL) close to a planar wall showing the microscopic slip u1 (x, 0); the macroscopic slip u(x, 0) is required when using the classical Navier-Stokes equations with a slip boundary condition. Here S denotes the plane at the outer edge of the KL.

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Figure 2

A molecule confined between two planar walls with spacing H. The molecule has an equal probability to travel in any zenith angle θ− or θ+ or to travel in either the positive or negative y direction. The molecule under consideration is assumed to have just experienced an intermolecular collision at its current position H/2 + y.

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Figure 3

A molecule at a distance H/2 + y from a planar wall; possible trajectories for traveling in the negative y direction in cylindrical coordinates [H/2 + y, (H/2 + y) tan θ]

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Figure 4

Variation of normalized mean free path β with normalized distance from a surface; (a) single-wall case and (b) parallel-wall case. Comparison of our power-law (PL) model with molecular dynamics (MD) simulation data [14], and Arlemark [13] and Stops [11] exponential models for various Knudsen numbers.

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Figure 5

Normalized half-cross-channel velocity profiles for various Knudsen numbers. Comparison of our power-law (PL) model results with the solution of the Boltzmann equation [31], R26 moment equations [32], and conventional NS equations with first- and second-order slip.

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Figure 6

Variation of normalized slip velocity with Knudsen number, comparison of power-law (PL) model with the solution of the Boltzmann equation [31], R26 moment equations [32], and the conventional NS equations with first- and second-order slip

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Figure 7

Normalized mass flow rate (G) variation with inverse Knudsen number (δm ). Comparison of power-law (PL) model results with: (a) experimental data [33] and BGK simulation results (- -) [34]; (b) the solution of the Boltzmann equation [31], R26 moment equations (-·-) [32], and the conventional NS equations with first- and second-order slip.

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Figure 8

Normalized thermal creep component of half-cross-channel velocity profiles for various Knudsen numbers. Comparison of our power-law (PL) model results (thick line) with the solution of the Boltzmann equation (symbols, Ohwada [31]), exponential MFP model (dashed line, Arlemark [13]), and the conventional slip solution (dotted line).

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Figure 9

Variation of normalized thermal creep component (GT ) and Poiseuille component (GP ) of flow rates, and TMPD, with Knudsen number (Kn). Comparison of our power-law (PL) model results with: hard sphere Boltzmann equation [38], BGK simulation data [34], exponential MFP model [13], and the conventional second-order slip solution.

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