20 
LORD RAYLEIGH ON THE CIRCULATION 
si n 2hx f „ . , _ „ (y, — ?A 2 
« =-——|c : ‘(4 Kin fiii -f- 2 cos/3./+<- ')+f — | J( , 
■ ( 91 ). 
2Jm { ? cos 2 lex 
v ~ ^{ e ^(sin /3y+3 cos $y-\-\e ^+fi%i-y)-f/3 (j ^^ !• (92). 
?/f 
Outside the thin film of air immediately influenced by the friction we may put 
e -ft/_ 0 } anc [ then 
= - ?3 ^{ 1 -^}.03). 
3u 0 2 2k cos 2 Z;a? 
16a 
2/i—2/ 
(3/i—2/) 3 ] 
Vi J 
(94). 
From (93) we see that u changes sign as we pass from the boundary y— 0 to the 
plane of symmetry y=y Y the critical value of y being ^(1 — \/a)> or ‘423 y Y 
The principal motion being u— — u 0 cos kx cos nt, the loops correspond to kx=0, it, 
27r, . . . , and the nodes correspond to ^ 7 r, f n, . . . Thus v is positive at the nodes and 
negative at the loops, vanishing of course in either case both at the wall y= 0 , and at 
the plane of symmetry y—y Y 
Plane of symmetry., 
Wall t 
0 ^7T 7T 
loop node loop 
- 1 
|tt 
node 
To obtain the mean velocities of the 'particles parallel to x, we must make an addition 
to u, as in the former problems. 
In the present case the mean value of 
du 
dx 
y , du w 0 2 sin 21cx e Pv 
S + dy 71 - 4a 
so that 
u 
it ( ~ sin 2 lex f 
8a 
1 
e“ ft, (4 sin ySy+3c“^) + 
9(li 
1 2 
■ y ) 2 
Vi 
When /3y is small, 
/ 
u = 
^l 0 2 sin 2 kx 
8 a 
(95). 
( 96 ). 
Inside the frictional layer the motion is in the same direction as just beyond it. 
