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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