Theory of Stability of Laminar Flow 453 
velocity U,,. We will take Y = y/5 as an independent variable and introduce the following 
parameters: 
8U V 
R=—", a= 48, See @=—. (2.4) 
Equation (2.3) then becomes 
" 2 " i on 2au 4 
GS ac) (2) = ar) = B40 + pyr = 200) it  Oc@)), (2-5) 
The solution of Eq. (2.5) has been extensively discussed in the literature. It has been found 
that it may be expressed in terms of solutions of the left-hand side of Eq. (2.5) set equal to 
zero (proper attention being paid to the behavior at the branch point) and in terms of solu- 
tions of 
= 0 (2.6) 
where 7 =(a,dR)!/3(Y - Y,), Y, is the value of y for which g/y) = C and a, is the deriva- 
tive of g at the point y= Y,. All previous solutions to these equations have assumed that 
v, and v, vanished on all boundaries and at infinity if the flow was not confined. In the 
case of the boundary layer with which we will primarily be concerned, the conditions are 
V, = Vy = 0 for y=0 and y=. This requires four boundary conditions. If we choose an 
appropriate solution @ of the left-hand side of Eq. (2.5) which vanishes with its first deriv- 
ative for large values of y and a similar solution of & of Eq. (2.6), then it can be shown that 
for the familiar solution of the stable equation, we obtain 
e o 
=| e02 (2.7) 
dp! (a,aR)*/3y' 
This determinant is a complex function of the three variables 0, R, and C. If for all real 
values of @ and R the imaginary part of C is negative, then the flow will be stable. If we 
choose C real, we can plot any one of these variables (a, R, and C) as a function of one of 
the others for the condition that satisfies Eq. (2.7). The curves so obtained are known as 
curves of neutral stability and establish the boundary of stable flow conditions. Diagrams 
of this type permit one to predict not only whether the flow will be stable but the wave- 
lengths which occur in the unstable regions of the diagrams. Our objective in this paper is 
to examine how these curves of neutral stability will be influenced by the compliant sur- 
faces. This paper, however, contains only the basic principles which will need to be de- 
veloped somewhat more fully. 
3. BOUNDARY CONDITIONS FOR COMPLIANT SURFACES 
The velocity of flow of a viscous liquid must be identical with the velocity of the wall 
at every point of contact. Suppose that 7 is the position of a point on the undeformed wall 
and that the deformation caused by forces in the flow is designated by the vector €(r). The 
