Interdisciplinary Applied Mathematics

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This resulted in a modified streamwise momentum equation, which could be reduced to    the    Navier-Stokes    level    only    for    low    subsonic    (M    ^ 1)

isothermal flows in large aspect ratio channels. One important deviation here is in the cross-flow momentum equation, where we have shown that the pressure variation in this direction is balanced by the viscous normal stresses of the Burnett equations (see Section 4.2.1). Since the (Navier-Stokes-based) Reynolds equation neglects the pressure variations in the cross-flow direction, we may expect some deviations between the Burnett-and Navier-Stokes-based models. However, these deviations are proportional    to    the    Mach    number in equation    (4.17b),    and    thus    they    can be

neglected for most low Mach number applications.

The linear Couette flow in the transition flow regime has been investigated in (Schamberg, 1947), using the Burnett equations. Schamberg showed that the streamwise momentum equation can be reduced to the Navier-Stokes level, resulting in linear velocity variation across the channel under negligible heat transfer and compressibility effects (Schamberg, 1947). However, the velocity distribution must be subject to a second-order slip condition (see also Section 3.2). Recalling that

• the flowrate of linear Couette flow is independent of the Knudsen number,

we can    conclude    that    for    low-speed    isothermal    flows    in    long    channels,

the Reynolds equation with second-order slip corrections is similar to the Reynolds equation derived using the Burnett equations. Hence, the agreement of experimental and numerical results presented by Hsia and Domoto in integral quantities can be explained. Fukui and Kaneko (1988) have shown that the second-order (slip based) Reynolds equation is valid for Kn < 1. This is expected, since the Burnett equations and the corresponding second-order slip models are derived via a second-order Chapman-Enskog expansion, which is a perturbation expansion in Kn, and it is valid for Kn < 1.

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