S1'RFAC.E LAYER IN UNSTAB1.E ('OKDITIONS
I25
the immediate problem that small T is not the same as large z because of the
presence of H, . In addition, of course. if we let 7 approach zero. the Reynolds number approaches zero. Let us proceed, nevertheless, by allowing T
to approach zero. We may write
2/3 - 4/3
(12)
P z = (7
Y 3 ( i 1 R m )
and the similarity argument is that
lim g3(c, R,) = c*
r - 0
where L' is finite, nonzero constant. It is easy to investigate the regime of
weak shear implied by T -+ 0. In fact, for any value oft or Au, we may write
(15)
where
(16)
C'= Au/(vAp)' ' ,
= ~ ( A p ) " ~ / v " ~
It is easy to decide on physical grounds that the following nonzero limits
exist
q = (Ap)413v' I: \ ; ( V )
lim . f , ( V ) = a,,
Comparing (12) and ( 14) and using (IS), we get
lim f 3 ( V , a) =fj(O, a)
v-0
v-0
(17)
lim g 3 ( R , , c) = (q'1'22'iJ!\~'/4u~/4)fJ(0, a)
1 - 0
(18)
The similarity argument is that this is a nonzero constant, i.e.,
(1'))
j3(0, a) = const a- ' I 3
.for crll I ~ I I P . \ of i and this is clearly impossible. On the other hand, (12)
shows that large z and small r are not equivalent limit processes so that we
may investigittc separately the limit as z -+ xi. We get
To obtain a 2-4!-3 law for 3,. we must have
lim j j ( V , a) = f 3 ( V)o: -+ 7(21)
This is a conceivable behavior, but the argument is certainly not rigorous or
even convincing. It leads precisely to the behavior in (11) which may be