It is easy to see from eq 144 that Tp depends on neither αo nor α1. Differentiating eq 170
with respect to σ, we find that Tp is a decreasing function of σ.
We will introduce a straight line Lp starting from the origin (Fig. 2c) defined as
{
}
[
}]
{
-1
Lp = (α1 , α o ):α o = k1 ko + LK2 (Tp ) / ηeo (Tp )
α1 .
(172)
+
The line Lp divides the region Sp into two regions Sx and Spp defined as
{
}]-1α1}
[
{
Sx = (α1 , α o ):α o < k1 ko + LK2 (Tp ) / ηeo (Tp )
(173)
{
}]-1α1}.
[
{
Spp = (α1 , α o ):α o > k1 ko + LK2 (Tp ) / ηeo (Tp )
(174)
It is clear that fo < 0 in Sx while fo > 0 in Spp
Now we will study Case 2 where Tσ ≤ Tx. In this case a solution T1 may be greater than
Tσ by Proposition 10. First we will search a solution T1 of Problem P such that T1 < Tσ ≤ Tx
~
on L+ under the assumption of Vo > 0. If such a solution exists, then Vo > 0 and T1 > 0 on L+
by the same reasoning as used in the proof of Proposition 7.
Proposition 13
There exists a unique solution T1 of Problem P on L+ such that T1 < Tσ ≤ Tx and fo ≥ 0 if
σ ≥ σx.
Proof
First we assume that Vo > 0. Because of eq 105, 106, and 107 for a unique solution T1 to
exist we must have:
Wo (Tσ -) = g1(Tσ -) - g2 (Tσ -) > 0.
(175)
From eq 99 and 100 we obtain:
(s, Tσ )
0
Wo = ao (Tσ -) fo - (s2 / e1)(Q - Y) ∫
ds.
(176)
Tσ- K1(s)T ′(s, Tσ -)
where ao is defined as
1
0
ao (T ) = (δo / Ko ) + (η / αo ) ∫
ds
(177)
T K1(s)[1 + e4(s, T )]
where fo, e1 and Y are functions of Tσ - , and e4 defined by eq 114 is given as
e4 (Tσ -) = (s2 s3 / α o )[Q - Y(Tσ -)] / e1(Tσ -).
(178)
From eq 177 we find that ao > 0. Since Vo > 0, so (Q Y) > 0. When fo ≥ 0, Wo > 0. Therefore,
a unique solution T1 of Problem P such that fo ≥ 0 exists on L+ if Vo > 0.
We will denote one of such unique solution T1 on L+ such that fo = 0 by Tp. Let Tp be
+
located at (α1p, αo) on L+. Since fo > 0 in a neighborhood of Cs in Sp , the point (α1p, αo) must
+
be in Sp . We will consider a segment lp (α o ) of L+ defined as
[
]
lp (α o ) = (α1 , α o ):α1s < α1 < α1p .
(179)
Since fo > 0 on lp (α o ) , there exists a unique solution T1 < Tσ on lp (α o ) . Since Vo > 0 in a
22
to Contents
Previous Page
