�
�
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�
C 2 f ""
C 2
""
∂ 3 Ψ
∂ ∂ 2 Ψ
∂
2
2 ∂
f
∂η
=
=
√
=
√
·
∂y 3
∂y ∂y 2
∂y C 1 x
C 1 ∂η
x
∂y

C 2
C 3 f """

2 1 f
""" C 2
2
=
√
√ =
C 1 x
x
C 1 x
(
)
∂ 2 Ψ
∂ ∂Ψ
∂ C 2
∂ C 2
∂η
C 2
η
f
"
f
"
f
""
=
=
=
=
−
∂x ∂y
∂x ∂y
∂x C 1
∂η C 1
∂x
C 1
2x
Inserting them into the above stream function momentum equation, we
obtain
ff
""
+ 2υc 1 c 2 f
"""
= 0
(7.7)
This is the similarity momentum equation.
Assume
υc 1 c 2 = 1
(7.8)
at
∂Ψ �
y = ∞, u = U ∞ =
�
= f
" c 2
∂y
c 1
y=∞
Let
c 2 = U ∞ .
(7.9)
c 1
From Equations 7.8 and 7.9, we obtain
U ∞
1
c 2 =
, c 1 = √
υ
υU ∞
Therefore,
)
y
y
U ∞
y
η = C 2 √ = √
=
Re x
(7.10)
x
x
υ
x
Ψ
Ψ
f = C 1 √ = √
(7.11)
x
υU ∞ x
where f and η are similarity functions and similarity variables, respectively.
Finally, we obtain the following:
∂Ψ
C 2
u =
=
f
"
= U ∞ f
"
(7.12)
∂y C 1
∂Ψ
1 1 (
) 1 υU ∞ (
)
"
v = −
= −
√ f − f η =
f
"
η − f
(7.13)
∂x
2 C 1 x
2
x
143
External Forced Convection