34
Substituting into Eq. (3.15) gives
(*)
The first term of the Greens identity using Eq. (3.6) becomes
r f*ndz=- r fvi+dz=-r /v*. (v^* (3.19)
* ho ho ho
Now
Vh = + efVh4> +e4>Vhf (3.20)
and
^h = (/ V/1^) + eVhij> Vhf + zV2hf (3-21)
Retaining only terms of order z and dropping the last term since it is second order in the
slope the integral is equal to
r /Vfc.(/V^)dz- r fVh4>Vhf dz (3.22)
J ho /lo
These two terms are handled using Leibnitzs rule of integration.
vaT /(/v)dz = r vhf(fvj)dz+r fvh(fvht)dz
* ho ho hp
+^hVf2 U + |_a V/,<£ (3.23)
Using this relationship and recognizing that / |