Boundary Lyer Theory

Neglecting u12, we obtain
XXIV. Prce twbulcnt flows; jets nnd.wnkos
+ m
n=he U,/u,dy
y- -m
Substit,uting D = 4 c, d h e rJ:, whcrc d denotcs
we obtain
+m
t,hc thiclrncss of the cylinder,
As deduced in,Scc. XXIVb, thc width and the velocity difference vary in a manner
to give 1) N x1I2 and u, N x-'12.
Shearing stress hypothesis from eqn. (24.3): Since the term v8ul8y in eqn.
(24.1) is small, we obtain
- a14! = 2 la 2 au aau
--I.
(24.33)
ax ay ag
It is assumed that thc mixing length 1 is constant over the width h and proportional
to it, i. c. t,lmt 1 = /I b(x). In vicw of the similarity of the velocity profiles the ratio
7 = ?//h is inlrotlurcd as tho intlcpcntlcnt variable. In agrccrnct~t with tho power
laws for the width and for the dopth of depression in the vclority profilc wc makc
the assumptions :
h = B (en d x)'I2 (24.34)
Inserting into eqn. (24.33), wc arc led to the following different.ia1 equation for
the frtnction /(v) :
--
1
(/ -1 7 /') =
21J2
- --- /' /"
2 U
wit,l~ the honndary conditions u1 = 0 and aul/ay - 0 at y = h, i. e. f = /' = 0
at 11 -= 1. I~~tcgrathg oncc, we obt.ain
whrrr thr constrant of intqption Itas been mn?tle ccpl to zero in vicw of the boun-
dary rontli tion. I