Perturbation analysis on gas flow in a straight microchannel

Abstract

In the present paper, a steady subsonic gas flow either in a circular micropipe or in a planar microchannel driven by pressure within the slip flow regime is studied theoretically by using a perturbation expansion method to solve compressible Navier-Stokes equations. The isothermal flow assumption used in previous theoretical studies is given up. High-order boundary conditions of velocity slip and temperature jump are adopted at the wall. The set of dimensionless governing equations with two small similarity parameters, namely, the ratio of height to length ε, and the Knudsen number Kn, is approximated successively by using the perturbation expansions. The various cases such as ε≪Kn2, ε∼Kn2, and ε∼Kn1.5 are studied in detail. Explicit analytical solutions for pressure, density, velocity, temperature, and mass flow rate are obtained up to order of O(Kn2). It is shown that the solution formulas for long channels (ε≪Kn2) in lower order are in exact agreement with previous theoretical results. In particular, it is proved that the isothermal flow assumption is indeed reliable for relatively lower-order expansions. However, for higher-order expansions, the flow cannot be considered as isothermal, and the higher-order temperature correction is also given. The present high-order perturbation solutions can be applied even to a relatively shorter channel, and the results agree very well with those by the direct simulation Monte Carlo approach.

An analytical solution to the extended Navier–Stokes equations using the Lambert W function

“…The perturbation solution of Arkilic and Schmidt 32 which applies to long channels (( h / L ) 2 ≪ Kn) is retrieved by Qin et al in the appropriate limit. Our analytically derived pressure profiles obtained by utilizing the lubrication approximation (figure 2) also agree with those derived using the perturbation solution in these cited works in the correct limit. However, Qin et al introduce two additional material constants corresponding to slip boundary conditions (slip coefficients), which are obtained from the literature.…”

Comment on “Heat transfer and fluid flow in microchannels and nanochannels at high Knudsen number using thermal lattice-Boltzmann method”

“…The flow through a long microchannel in 2D has been studied extensively [18][19][20][21][22][23]. The analytic solution of the compressible isothermal flow through a long microchannel in 2D can be obtained by solving the steady, isothermal compressible Navier-Stokes equations in 2D perturbatively.…”

Gas flows in microchannels and microtubes

“…(v) Some researchers use a second-order slip model for the wall boundary conditions, Qin et al (2007). For this treatment, an extra term with temperature gradient should be included, since it is not smaller than the second term in the expression.…”

“…Hence, a study of the temperature field inside a microchannel or a microtube is of practical importance with mathematical significance. Therefore, some recent research work has investigated the temperature field or the heat transfer in microchannels or microtubes, for example, Kedzierski (2003), Wang & Li (2004), Hetstroni et al (2005) and Qin, Sun & Yin (2007).…”