# Interdisciplinary Applied Mathematics

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us = ^[ux + (1 — av)ux + avuw.    (2.26)

Schaaf and Chambre (1961) have written this expression as an average tangential velocity on a surface adjacent to an isothermal wall. Our derivation results in the same relation with approximately similar assumptions. Notice that instead of obtaining the slip information u one mean free path away from the wall, a fraction of A may be used; see (Thompson and Owens, 1975). Using a Taylor series expansion for u about us, we obtain

1

Us

2

du

us + A ——    H——

dn

s

A2 ( d2

u

2 V dn2

+

s

. du    A2 (d2u

Us+

u

w

where the normal coordinate to the wall is denoted by n. This expansion results in the following slip relation on the boundaries:

us    uw

2    (7 v

Ov

A

du A    A2

dn)s+ 2

d2u dn2

+ •••

s

(2.27)

After nondimensionalization with a reference length and velocity scale (such as free-stream velocity), we obtain

Us- Uw

2 o~v

O V

Kn

dU Kn2 dn)s+ 2

d2U

dn2

+ ••• j

s

(2.28)

where    we    have    denoted    the    nondimensional    quantities    with    capital    let

ters. By neglecting the higher-order terms in the above equation we recover Maxwell’s first-order slip boundary condition (2.19) in nondimensional form. Similarly, if we truncate the above equation to include only up to second-order terms in Kn, we obtain

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