� �
� �
� �
�
�
∂ρ + ∇ · (ρV) = 0
(6.1)
∂t
where
V = iu + jv + kw
and
∂
∂
∂
∇ = i
+ j
+ k
∂x
∂y
∂z
are the velocity vector and the del operator for unit vectors, i, j, and k in the
x-, y-, and z-direction, respectively.
Conservation of momentum:
ρ
DV = −∇P + μ∇
2 V + ρg
(6.2)
Dt
Conservation of energy:
Dh
DP
ρ
=
+ ∇ · k∇T + μΦ + q ˙
(6.3)
Dt
Dt
where h = e + (1/2)V · V, and e is the specific internal energy. Φ is often called
the dissipation function with the form
� � 2 � � 2 �
� 2
�
� 2
∂u
∂v
∂w
∂u ∂v
Φ = 2
+
+
+
+
∂x
∂y
∂z
∂y
∂x
�
� 2 �
� 2
∂u ∂w
∂v ∂w
+
+
+
+
(6.4)
∂z
∂x
∂z
∂y
q ˙ is the heat generation in unit volume.
130
Analytical Heat Transfer
6.2 General Heat Convection Equations
For flow moving over a heated or cooled solid body, the general 3-D pressure profiles P(x, y, z, t), velocity profiles u(x, y, z, t), v(x, y, z, t), and w(x, y, z, t),
and temperature profiles T(x, y, z, t) can be obtained by solving the following
continuity, momentum, and energy equations inside the hydrodynamic and
thermal boundary layers over the heated or cooled solid surface [1–3].
Conservation of mass (continuity equation):