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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