�
�
�
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Applying the similarity concept from Figure 7.1,
√
c 2 y
y ∼ x ⇒ η = √
(7.5)
x
√
Ψ
Ψ ∼ x ⇒ f = c 1 √ = f (η)
(7.6)
x
η is the similarity variable.
f is the similarity function.
u(x, y) ⇒ Ψ(x, y) ⇒ Ψ(η) ⇒ f (η)
v(x, y)
√
∂Ψ
∂Ψ ∂η
x f
" c 2
c 2
=
=
√ = f
"
∂y
∂η ∂y
c 1
x
c 1
� √
�
∂Ψ
∂f x
f 1 1
=
+
√
∂x
∂x C 1
C 1 2 x
�
√
�
∂f ∂η x
f 1 1
=
+
√
∂η ∂x C 1
C 1 2 x
1 1 1
f 1 1
= f
"
η − √
+
√
2 x C 1
C 1 2 x
1 1 (
)
=
√ f − f
"
η
2 C 1 x
where
∂f
∂η
C 2 ∂η
1
η
f
"
=
,
= √ ,
= − C 2 yx
−(3/2)
= − ,
∂η
∂y
x ∂x
2
2x
Similarly,
∂ 2 Ψ
∂ ∂Ψ
∂ c 2
c 2 ∂f " ∂η
c 2 f ""
f
"
2
=
=
=
= √
∂y 2
∂y ∂y
∂y c 1
c 1 ∂η ∂y
c 1 x
142
Analytical Heat Transfer
Streamline
U ∞
U ∞
y
x
FIGURE 7.1
Stream lines for flow over a flat plate.