Interdisciplinary Applied Mathematics

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   The high-order boundary conditions proposed include Maxwell’s first-order slip conditions (2.19), (2.19) as the leading-order term. Hence, these results are correct up to O(Kn) in the slip flow regime, irrespective of the formal order of the utilized slip conditions.

   The general boundary condition for slip (equation (2.43)) converges to a finite value for large Kn, unlike the first-order Maxwell’s boundary condition.

2.3.3 Comparison of Slip Models

For isothermal flows with tangential momentum accommodation coefficient av = 1, the general second-order slip condition has the nondimensional form

where (d/dn) denotes gradients normal to the wall surface. The coefficients Ci    and    C2    are    the    slip coefficients.    Typical    values    of the    slip coefficients

developed by different investigators are shown in Table 2.2.

We will apply the second-order slip boundary conditions given above for channel flows in Chapter 4 to examine their accuracy in representing the flow profile, including the velocity slip predictions. According to Sreekanth

(Sreekanth, 1969), Cercignani’s second-order boundary conditions should be used    only    for    evaluating    the    flow    states    far    from the wall,    and these

conditions should not be used to evaluate space integrals in regions extending close to the walls. Sreekanth reports good agreement of second-order slip boundary conditions with his experimental results for Kn as high as Kn = 1.5 (Sreekanth, 1969). However, Sreekanth used a different second-order slip coefficient (C2 = 0.14) than the original ones shown in Table 2.2. He also reports a change of the first slip coefficient (Ci) from 1.00 to 1.1466 as the Knudsen number is increased. First-order boundary conditions cease to be accurate, according to Sreekanth’s study, above Kn > 0.13. More recent studies also show that Maxwell’s slip boundary condition breaks down around Kn = 0.15 (Piekos and Breuer, 1995).

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