�
�
�
�
�
�
�
�
��
The right-hand side can be expanded as
Substitute the mass conservation equation into and we will obtain the
x-direction momentum equation
∂u
∂u
1 ∂P
∂ 2 u
∂ 2 u
u
+ v
= −
+ υ
+ υ
.
(6.11)
∂x
∂y
ρ ∂x
∂x 2
∂y 2
'
-v
' ' -v ' '
-v
'
pressure
convection
stress
Similarly, we will obtain the y-direction momentum equation from Equation
6.11 by changing u and v, x and y:
∂v
∂v
1 ∂P
∂ 2 v
∂ 2 v
u
+ v
= −
+ υ
+ υ
(6.12)
∂x
∂y
ρ ∂y
∂x 2
∂y 2
Conservation of energy:
Unsteady state:
Perform energy balance shown in Figure 6.7,
∂q x
∂q y
∂ (
)
∂ (
)

−
dx −
dy −
ρu dy · C p · T dx −
ρv dx · C p · T dy

∂x
∂y
∂x
∂y
(
)
∂ ρ dx dy · C p · T
=
(6.13)
∂t
(
)
∂ ρC p T
∂ (
)
∂ (
)
+
ρuC p T +
ρvC p T
∂t
∂x
∂y
' -v '
'
-v
'
unsteady steady
convection
∂
∂T
∂
∂T
q ˙
=
k
+
k
+
+
μΦ
(6.14)
∂x
∂x
∂y
∂y
k
'-v'
'-v'
'
-v
'
heat dissipation
heat
heat diffusion
source due to fricion
generation
For steady-state constant properties,
∂ (uT) ∂ (vT)
∂ 2 T
∂ 2 T
μ
+
= α
+
+
Φ.
(6.15)
∂x
∂y
∂x 2
∂y 2
ρC p
�
� 2
� � 2 � � 2
� � 2
∂u ∂v
∂u
∂v
∂u
μΦ = μ
+
+ 2
+
∼ μ
(6.16)
∂y
∂x
∂x
∂y
∂y
134
Analytical Heat Transfer