# Interdisciplinary Applied Mathematics

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passing through s are coming from A away from this surface n = nSl the other half of the molecules passing through s are reflected from the wall (see Figure 2.5), and they bring to surface s a tangential momentum flux of

1

~ flw TflVw Ur ,

where the subscript w indicates wall conditions and the number density nw is equal to ns. The average tangential velocity of the molecules reflected from the wall    is    shown    by    ur.    For    determination    of ur    we    will    use the    def

inition of tangential momentum accommodation coefficient av. Assuming that av (in percentage) of the molecules are reflected from the wall diffusely (i.e., with average tangential velocity corresponding to that of the wall uw), and (1av) (in percentage) of the molecules are reflected from the wall specularly (i.e., conserving their average incoming tangential velocity u), we have

ur(1 Xv )uX + X vuw •

Therefore, the total tangential momentum flux on surface s is written as

1

—nsmvu

s

11

-nmvU + -nw

mvw [(1

Vv )uX + (Xvuw] •

Since we have assumed that the temperatures of the fluid and the surface are the same, the mean thermal speeds are identical (i.e., vsVVw); this is a rather strong assumption in our derivation. The number density ns of molecules passing through the surface is composed of n and nw. We have assumed that n = nw = nSl which is approximately true if there is no accumulation or condensation of gas on the surface. Using the tangential momentum flux relation, the mean tangential velocity of the gas molecules on the surface, called slip velocity, is

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