Self-diffusion in compressible gas flow through a microconduit

Steady viscous compressible gas flow through a microconduit is studied using the classical formulation of compressible Navier-Stokes equations with the no-slip condition. The mathematical theory of Klainerman and Majda for low Mach number flow is employed to derive asymptotic equations in the limit of small Mach number. The leading-order equations give rise to the Hagen-Poiseuille solution, and the equations at a higher order lead to a diffusive velocity field induced by self-diffusion of the gas. An explicit solution for the diffusive field is obtained under the no-slip condition and the overall flow exhibits a sliplike mass flow rate even though the velocity satisfies the no-slip condition. The results indicate that the classical formulation includes the self-diffusion effect and it embeds the extended Navier-Stokes equation (ENSE) theory without the need of introducing an additional constitutive hypothesis or assuming slip on the boundary. The predicted pressure profile agrees well with experiments. Contrary to reports in many ENSE publications, an extensive comparison with 35 experiments shows that in the slip-flow regime the predicted mass flow rate is still significantly below the measured mass flow rate, as the self-diffusion effect is too small to account for the observed mass flow-rate enhancement in this flow regime.

Figure 1

Microflow through a channel. For a microcapillary, the radius of the capillary is $R$.

Figure 2

Comparison of the predicted pressure profiles with the measurements of Pong et al. [65], as well as the incompressible flow linear pressure profiles. The five profiles correspond to five different inlet pressures. The outlet pressure is atmospheric.

Figure 3

(a)–(e) Comparison of the pressure deviations from the incompressible pressure with the measurements of Pong et al. [65].

Figure 4

(a)–(c) Mass flow-rate comparison between theory and the experiment of Ewart et al. [17] for nitrogen flow through a microtube. (a) Dimensional mass flow rates in log-log scale. (b) S plots in linear-linear scale. (c) G plots in linear-log scale.

Figure 5

(a)–(c) Mass flow-rate comparison between theory and the experiment of Ewart et al. [18] for helium flow through a microchannel. (a) Dimensional mass flow rates in log-log scale. (b) S plots in linear-linear scale. (c) G plots in linear-log scale.

Figure 6

(a)–(c) Mass flow-rate comparison between theory and the experiment of Maurer et al. [15] for helium flow through a microchannel. (a) Dimensional mass flow rates in log-log scale. (b) S plots in linear-linear scale. (c) G plots in linear-log scale.

Figure 7

Streamwise velocity profile in a microchannel flow. The streamwise velocity is normalized by the mean velocity. There is no difference between the normalized velocity from the Hagen-Poiseuille flow and the present theory [they differ dimensionally; see Fig. 8].

Figure 8

Streamwise and cross stream velocity profiles in a microchannel flow at the entry, the middle, and the exit sections of the microchannel.