Extended Navier-Stokes Equations and Treatments of Micro-Channel Gas Flows

Abstract

Gas flows in micro-channels are, in general, theoretically treated with the Maxwell slip velocity as a boundary condition for the convection velocity at the wall. It is pointed out that wall slip is conventionally introduced in theoretical/numerical treatments of gas flows through micro-channels to obtain agreement with experimental results. In the present paper, we provide an alternative by solving the extended Navier-Stokes equations for compressible gas flows in micro-channels using the conventional no-slip velocity boundary condition for the convection velocity. Results obtained with this approach are presented and compared with experiments. It is also shown that the theoretical treatment of micro-channel gas flows using the "extended Navier-Stokes equations" also permits the phenomena such as the Knudsen paradox, to be treated in an analytical manner. Comparison with experimental data suggests that the derived analytical solution has excellent agreement up to Knudsen number of approximately 1, which shows the validity of extended Navier-Stokes equations with the conventional no-slip velocity boundary condition up to the early transition regime of micro-channel gas flows.

Numerical Investigations of Rarefied Gas Flows Using the Continuum Description by the CIP Method

“…The extended Navier-Stokes equation (23) is also a possible and effective approach, and Dongari.et.al have reported that it can permit characteristic phenomena such as the Knudsen paradox in the transition regime (24) . Nevertheless, it is found that this macroscopic approach can give accurate and comparable solutions to kinetic-type approaches for flows within the slip regime.…”

Pressure Drop and Flow Development in the Entrance Region of Microchannels With Second-Order Velocity Slip Condition and the Requirement for Development Length

“…Predictions of flows using the conventional Navier-Stokes (NS) equations along with different velocity slip and temperature jump conditions at the wall (Arkilic et al, 1997;Karniadakis and Beskok, 2002;Colin, 2005;Karniadakis et al, 2005;Barber and Emerson, 2006;Dongari et al, 2007;Tang et al, 2007a,b;Weng and Chen, 2008;Cao et al, 2009;Chen and Bogy, 2010;Colin, 2012;Zhang et al, 2012b), 2. Modelling of flows employing higher order equations than the conventional NS equations (Hadjiconstantinou, 2000;Jin and Slemrod, 2001;Struchtrup and Torrilhon, 2003;Hadjiconstantinou, 2006;Shan et al, 2006;Ansumali et al, 2007;Lilley and Sader, 2008;Struchtrup and Torrilhon, 2008;Weng and Chen, 2008;Dongari et al, 2009;Roohi and Darbandi, 2009;Dongari et al, 2010), 3. Use of i) kinetic theory of gases (Loyalka, 1971;Cercignani, 1990), ii) Molecular Dynamic (MD) simulations (Bird, 1994;Zhang et al, 2012a), iii) Discrete Simulations Monte Carlo (DSMC) methods (Pan et al, 1999;Hadjiconstantinou, 2000), iv) Non-linear and linearised Boltzmann Equation (BE) (Cercignani, 1975(Cercignani, , 1988Li and Kwok, 2003) and v) Lattice Boltzmann Method (LBM) (Cornubert et al, 1991;Sbragaglia and Succi, 2005;Zheng et al, 2006;…”

Diffusive Boltzmann equation, its fluid dynamics, Couette flow and Knudsen layers

“…Some of these works [6][7][8] were based on the idea of introducing the concept of the "volume velocity", which differs from the usual mass velocity by a small flux term derived from Fick's law. Others [16,18] introduced similar additional terms to model the self-diffusion of mass. Among those listed above, the works [15,29,38] appear to employ the closest conceptual approach to what we suggest here, namely, in [15,29,38] an ad hoc diffusion term was introduced directly into the Boltzmann equation (1.1) via the assumption of an additional empirical stochasticity of the molecular motion to complement the already present intermolecule collisions.…”