# Interdisciplinary Applied Mathematics

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the parameter B(Kn) has a definite value. This value can be used to make equation (2.35) second-order accurate in Kn for finite Kn. For the rest of the Kn values, B(Kn) can be curve-fitted accurately using the solutions of corresponding numerical models (i.e., Navier-Stokes and DSMC models) for the    entire    Kn range    (0.0    <    Kn    <    ж).    Equation    (2.35)    suggests    finite

corrections for slip effects for the entire Kn range, provided that B(Kn) < 0. It is possible to obtain the value of the parameter B(Kn) for small Kn, especially    for    the    slip    flow regime,    by    Taylor series    expansion    of    B(Kn)

about Kn = 0. We thus obtain

B(Kn)

Bo +

dB

dKn

Kn + • ••

0

b + Kn c + • ••.

(2.36)

Assuming that B(Kn) < 1, we expand equation (2.35) in geometric series, including    also    the    expansion    given    in    equation    (2.36)    for    B(Kn).

This results in

Us — Uw — -Kn ——[1 + 6Kn +(62 + c) Kn2 + •••].    (2.37)

ov    on

Next, we    substitute    the    asymptotic    expansion    for    the    velocity    field    (equa

tion (2.32)) to the general slip condition given above, and rearrange the terms as a function of their Knudsen number order. This results in

 O(1) : U0 O(Kn) : Ui O(Kn2) : U2 O(Kn3) : Us

s

s

s

s

Uw

(2.38)

2 О v

O v

2 ov

O v

2 ov

O v

(U0 )s;

(bU0 + U1 )s;

(U2 + bU1 + (b2 + c)U0 )s

Comparing these equations with the conditions obtained from the Taylor series expansion in equation (2.35) order by order, we obtain that the two representations are identical up to the first-order terms in Kn. To match the second-order terms we must choose the parameter b as

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