Theory of Stability of Laminar Flow 465 
and 
a= 5 
vJ/R 
which we know must be mutually tangent and pass through the origin for small values of 
both variables. If now for a fixed value of 4 we describe one of the families of curves, its 
intersection or intersections with the member of the envelope for a corresponding value of @ 
will give points on the boundary line separating stable conditions from unstable conditions. 
In general, there will be two such intersections and usually the stable conditions will be 
those sections between the points of intersection. If the members of the family of curves of 
the compliance do not intersect the corresponding member of the family of envelopes, then 
either the system will always be stable or will always be unstable. Since the envelopes 
corresponding to unstable conditions for a rigid wall always enclose the origin and since the 
compliances must pass through the origin, it follows that there will be at all times at least 
one intersection with the compliance unless the compliance is so small as to be completely 
enclosed by the envelope. Since, however, in this family of curves the portion closest to 
the origin represents unstable conditions, it follows also that in the case where there is no 
intersection between the compliance and the envelope that the boundary layer will always 
be unstable. Consequently, in the case of interest there will always be at least one inter- 
section corresponding to low frequencies between the members of the envelope and the 
family of compliances. These curves considered as a function of R, £, and the velocity of 
flow will form a family from which the extreme values of the Reynolds number can be deduced. 
SUMMARY 
We have given a very brief discussion of the principles on which an analysis of the ef- 
fect of a flexible surface on fluid flow is based. It shows that through the introduction of 
the concept of surface compliance, the conditions of stability of laminar flow may be ana- 
lyzed. A more detailed presentation would allow us to give conditions which will be ful- 
filled for a stable flow. In general, we show that the flow will be stable when certain con- 
tours in the complex plane exclude the origin. The presence of the flexible wall replaces 
the origin by a curve and the points of instability are the intersection of this curve with the 
contours. 
APPENDIX 
Reference 4 gives the relation 
-0 Pé 
v(L) \ | eM? Hy 3[ 3 (iey/"] do dé 
_ ¢+0 J+ 
F eA ee ease Se SE OS ee ee ee 
1 A) e 
pet Hy 3[4 (ipy?/7| dp 
