Theory of Stability of Laminar Flow 457 
p' (a,aR)¥3 wy! -(ia+e,y) e,y 0 ie,a,y" 
1 
tz » (3.12) 
p wy e,c (Hare, Cy 0 te,a,y" 
This step requires examination. If the matrix has no inverse, if the operation cannot be 
carried. out, then we have a special problem. In general, the boundary layer will be unstable 
if the zeros of the determinant (3.13) have positive real parts: 
i e5€ ology 
A = SCM ap SINE IC Seay 
eC (20 ey) 
= -ia + iaCU,p Mag = yU_ p Ee (3.13) 
Since we assume the compliances are passive, this will generally not be the case, so that 
we will be able actually to divide through by the matrix and obtain (3.12). Our stability 
conditions are then given in 
dp! (a,aR)1/3 y' ¢' den 
i = 0. (3.14) 
P Ww OQ NS, 
As can be seen the first term is the expression which must be set equal to zero in the case 
of the rigid boundary. The second term contains the added conditions imposed by the flex- 
ibility. Developing this determinant, we finally obtain the following equation for the condi- 
tions of the boundary: 
wy db ie, de,¢ ay" 
Y sn) a Wee, a 0. (3.15) 
Teaeny a | Y_(a,aR)/3 y 
We will note that this boundary condition is expressed in terms of three functions depending 
upon the hydrodynamic conditions and on the two compliances. Two of the three functions 
which are involved are identical with the ones which appeared in the solution for the rigid 
body as developed by previous authors. We may rewrite in the form 
HCE) = G(a,C) =" — | ze) -foe, V GCa,€) | a H(A 
Be Se ae eee (Cont) 
