Interdisciplinary Applied Mathematics

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Us    Uw



2 o~v


O V



Kn



dU A Kn2 dn)s+ 2



d2U dn2



s



(2.29)


We will use this equation for comparison of various slip models in Section 2.3.3 and in Section 4.2 .


Equation (2.26) excludes the thermal creep terms of equation (2.19), since isothermal conditions are assumed in its derivation. For nonisothermal


flows, the thermal creep effects are included to equation (2.26) separately, resulting in the following relation:


us



2 [^A H~“ (1



3Pr(7-l) 4 7 pRTw



(-qs )■


For the temperature jump boundary condition, a derivation based on the kinetic theory of gases is given in (Kennard, 1938). We propose the following form for the high-order temperature jump condition by analogy with equation (2.28):


T —T


s —Lw


2(7т


27


1


ат


7 +1


Pr


Kn3


fd3T


+6


у dn3 )


+ • •


s



dT    Kn2


dn)s+ 2



d2T


dn2



s



(2.30)


which can be rearranged by recognizing the Taylor series expansion of T about    Ts    to    give    a bf    temperature    jump    boundary    condition    similar    to


equation (2.26) as


Ts



(2aT) 27


Pr (7 + 1)



T + ат Tu



aT +    27    (2    ) . (2.31)


(7+1) Pr 1


Here T is the temperature at the edge of the Knudsen layer, i.e., one mean free path (A) away from the wall.

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