Theory of Stability of Laminar Flow 455 
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RES lis whey Nit (c + iar) + 7009), : (3.5) 
The tangential force will be given in turn by 
(3.6) 
If we substitute Eqs. (3.5) and (3.6) into Eq. (3.3) which expresses the deformation of the 
surface in terms of the surface forces, we will obtain a relationship between the velocity on 
the boundary and its derivatives along the wall. Every term in this new expression will de- 
pend upon the stream function or on its derivatives and on the parameters of the system. 
These two equations, therefore, replace as boundary conditions the vanishing of the compo- 
nents of velocity on the surface of the body. The conditions at infinity will not be changed 
by the flexible wall and will, therefore, remain v, = v, = 0 as in the classical equation. 
It was pointed out previously that the stream function ® could be expressed as a linear 
sum of two functions ¢ and w. This will still be true since the differential equation is un- 
changed by the flexible wall. Consequently, these two functions obey the boundary condi- 
tions at infinity both for the flexible and for the inflexible wall and can be used in linear 
combination to develop the solution of the equation of the flow in the presence of a flexible 
wall. If we substitute into Eq. (3.3). a linear sum of these two functions, multiplying by 
arbitrary coefficients, we will obtain a set of two equations in the two arbitrary coefficients 
which much be satisfied. The condition of existence of solutions to these equations is the 
vanishing of a determinant which is related to the determinant obtained for the rigid wall. 
The set of linear equations which must be satisfied are given in 
mw U 
44 3) ~ Oy0 Vy Eee ae Hates 3" 19" 
BY (aaRy'* v'Cy) -UpY,, 2 yl" + (c + BY cary"? y+ 99 | 
Up 173 2 
a (a,aR) ee = 10) 
: ae mt - U_p 
A-iag = Up ee iz abe ar is) g' + | = oi v.07} 
+ p{-iay - Up io E yn +(c +: A) cau)? y + yw | 
(3.7) 
Up 1/3 
meee) Youd" f = ©. 
