Theory of Stability of Laminar Flow 463 
In addition, we must specify that we are on the envelope. This will give a relationship 
between @ and C. These two relations together will, therefore, lead to a relationship be- 
tween the Reynolds number and the velocity, from which a family of curves of critical 
Reynolds numbers can be constructed. 
A delineation of the family formed of the envelopes obtained by the elimination of the 
Reynolds numbers through the construction of the envelope requires a more detailed anal- 
ysis of the way the right-hand side of Eq. (5.1) is formed. If we consider the function given 
to the left of Eq. (5.1), it is not difficult formally to construct the envelope. 
The analytic expressions for the functions F, G, and H are inconvenient. We have used 
an approximate expression for the denominator and have constructed the envelope graphically. 
The value of the function in the numerator for a given set of values of © and C is obtained 
by simple graphical subtraction. Holding constant and varying C, we obtain a family of 
curves with the general appearance given in Fig. 1. This family of curves has an envelope 
which is formed by the tangent to the common apices of the curves. There will be one such 
envelope for each value of &. This family of envelopes is given in Fig. 2, which was con- 
structed from existing data. Each member of this family of envelopes separates the plane 
into two domains. Any point in the lower domain will have two members of the family of 
which it is an envelope passing through it, a point on the envelope will have only one, and 
finally a point above the envelope will have none. The locus of the points of intersection 
between the family of envelopes on the one hand and the compliances on the other will form 
the boundary between areas of stability and instability. Points corresponding to the com- 
pliance which are above the envelope cannot lead to unstable solutions. 
In the discussion that follows we will consider the special case when e, vanishes. 
For small values of 8 and @, e.g., moderate speeds and large Reynolds numbers, this 
will be the most important case. 
ted Se a 
Fig. 1. Family of 
curves from Eq. (5.1) 
for constant @ and a 
set of values of C 
4 
0.0 
SSS 
o2 a=0.4 
me) 
a=0.5 
Fig. 2. Envelopes obtained = 506 
as shown in Fig. 1 ; 
a=0.7 
A 
fe 
