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Research Papers

# An Extension to the Navier–Stokes Equations to Incorporate Gas Molecular Collisions With Boundaries

[+] Author and Article Information
Erik J. Arlemark1

Department of Mechanical Engineering, University of Strathclyde, Glasgow G1 1XJ, UKerik.arlemark@strath.ac.uk

S. Kokou Dadzie, Jason M. Reese

Department of Mechanical Engineering, University of Strathclyde, Glasgow G1 1XJ, UK

The viscous tangential stress vector, $τ$, relates to the viscous stress tensor, $Π$, through the expression $τ=(n̂⋅Π)⋅(I−n̂n̂)$, where $n̂$ is the unit vector normal to a surface and $I$ is the identity tensor (6).

In Eqs. 3,4 the “$−$” sign is applied to the first-order velocity gradient if $y$ is oriented in the opposite direction to the wall normal, otherwise “$−$” is replaced with “$+$” in these equations.

The difference in our mass flow results, see below, for 14 and 16 integration intervals is 1.54% for $Kn=1$, indicating that further increase of the number of integration intervals will only marginally affect the results.

The second-order velocity-slip solution is the same as the first-order slip solution because the second gradient of the velocity does not exist in this test case.

1

Corresponding author.

J. Heat Transfer 132(4), 041006 (Feb 18, 2010) (8 pages) doi:10.1115/1.4000877 History: Received October 02, 2008; Revised March 05, 2009; Published February 18, 2010; Online February 18, 2010

## Abstract

We investigate a model for microgas-flows consisting of the Navier–Stokes equations extended to include a description of molecular collisions with solid-boundaries together with first- and second-order velocity-slip boundary conditions. By considering molecular collisions affected by boundaries in gas flows, we capture some of the near-wall effects that the conventional Navier–Stokes equations with a linear stress-/strain-rate relationship are unable to describe. Our model is expressed through a geometry-dependent mean-free-path yielding a new viscosity expression, which makes the stress-/strain-rate constitutive relationship nonlinear. Test cases consisting of Couette and Poiseuille flows are solved using these extended Navier–Stokes equations and we compare the resulting velocity profiles with conventional Navier–Stokes solutions and those from the BGK kinetic model. The Poiseuille mass flow rate results are compared with results from the BGK-model and experimental data for various degrees of rarefaction. We assess the range of applicability of our model and show that it can extend the applicability of conventional fluid dynamic techniques into the early continuum-transition regime. We also discuss the limitations of our model due to its various physical assumptions and we outline ideas for further development.

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## Figures

Figure 1

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.

Figure 2

A molecule at a distance H/2+y from a planar wall; possible trajectories for a molecule traveling in the negative y-direction in cylindrical coordinates (H/2+y,(H/2+y)tan θ), where ∞ denotes infinity.

Figure 3

Comparison of different λeff-models in a half-channel for different Knudsen numbers, where λeff=λK(y) and KnA=0.04, KnB=0.25, KnC=1, and KnD=20

Figure 4

Half-channel Couette flow velocity profiles using conventional NS and our effective viscosity model (NSeff), using first- and second-order BCs, compared with the BGK-results of Sharipov (13). The velocity profiles are for KnE=0.01, KnB=0.04 and KnF=0.08, and y=0 is the channel center. The slip coefficients for our second-order model are A2=0.05 and A3=0.63 and for our first-order model A1=1.

Figure 5

Couette flow velocity profiles using conventional NS and our effective viscosity model (NSeff) with first- and second-order BCs, compared with the BGK-results of Sharipov (13); KnE=0.01, KnG=0.113 and KnH=0.339, and y=0 is the channel center. The coefficients for our second-order slip model are A2=0.05 and A3=0.63 and for our first-order model A1=1.

Figure 6

Half-channel Poiseuille flow velocity profiles using conventional NS and our effective viscosity model NSeff, using first- and second-order BCs compared with the BGK-results of Sharipov (13). The velocity profiles are for KnE=0.01, KnB=0.04 and KnF=0.08, and y=0 is the channel center. The slip coefficients for our second-order model are A2=0.05 and A3=0.63 and for our first-order model A1=1.

Figure 7

Half-channel Poiseuille flow velocity profiles using conventional NS and our effective viscosity model NSeff with first- and second-order BCs, compared with the BGK-results of Sharipov (13); KnG=0.113, KnH=0.339 and KnI=0.903, and y=0 is the channel center. The coefficients for our second-order slip model are A2=0.05 and A3=0.63 and for our first-order model A1=1.

Figure 8

Mass flow results from conventional NS and our effective viscosity model NSeff, using first- and second-order BCs. The results are compared with BGK solutions by Sharipov (13) and experimental results by Ewart (9). The height of the error bars of the experimental data is set to 4.5% of the normalized mass flow rate values, consistent with the data in Ref. 9. The coefficients for our second-order slip model are A2=0.05 and A3=0.63 and for our firs-order slip model A1=1.

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