On Possible Similarity Solutions for Three-Dimensional Incompressible Laminar Boundary-Layer Flows Over Developable Surfaces and with Proportional Mainstream Velocity Components Page: 70 of 82

                                
NACA TM 1437

Now, differentiating equation (D8) with respect to     x2   and recalling
. Bhl
that    x    0 (as  kI1 has been assumed zero) give (with eq. (D9))
a2 In k2         k2
x2xl= b6hl       - = 0
Ak2
Thus, 62     0  is incompatible with   k2 1 0, k1 = 0, and the only remain-
ing possibility is that            k2  is a nonconstant function of    x1  alone.
Defining   k2 = f7(x1) gives from equation (DM)
f6(x2)
h2 -                                   (D10)
S f7(x1)                           (i)
Finally, from the definition of k2,
Ik2 "
C1
I
!In h2
xl 1f7(xl)
h-=          -                            (D1)
k2        g2(xl)
From these results, Theorem 2 follows directly.
Because of the symmetrical nature of the equations, similar results
can be obtained if it is first assumed that     k2 --0  and   kI1  0.  In
this case,
k 1 = f8(x2)
fg(x1)
hl1 = f8(x2
and
f(2)
f8(x2)

67