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a. Velocity profile
Continuity equation:
∂u
∂υ
∂u
+
= 0 ⇒
= 0
∂x
∂y
∂x
Momentum equation:
∂u
∂u
1 ∂P
∂ 2 u
∂ 2 u
u
+ υ
= −
+ ν
+
∂x
∂y
ρ ∂x
∂x 2
∂y 2
υ = 0
∂u
∂ 2 u
= 0 ⇒
= 0
2
∂x
∂x
1 ∂P
∂ 2 u
= ν
⇒ Governing equation
(8.28)
ρ ∂x
∂y 2
with solution
u =
1 ∂P y 2 + c 1 y + c 2
(8.29)
2μ ∂x
where ν = μ/ρ
Boundary conditions:
At y = 0, ∂u/∂y = 0 (maximum velocity)
At y = H, u = 0
c 1 = 0
1 ∂P
c 2 = −
H 2
2μ ∂x
176
Analytical Heat Transfer
q″ s
x
y
2H
q″ s
FIGURE 8.6
Two parallel plates at a uniform wall heat flux.
�
a. Velocity profile
Continuity equation:
∂u
∂υ
∂u
+
= 0 ⇒
= 0
∂x
∂y
∂x
Momentum equation:
∂u
∂u
1 ∂P
∂ 2 u
∂ 2 u
u
+ υ
= −
+ ν
+
∂x
∂y
ρ ∂x
∂x 2
∂y 2
υ = 0
∂u
∂ 2 u
= 0 ⇒
= 0
2
∂x
∂x
1 ∂P
∂ 2 u
= ν
⇒ Governing equation
(8.28)
ρ ∂x
∂y 2
with solution
u =
1 ∂P y 2 + c 1 y + c 2
(8.29)
2μ ∂x
where ν = μ/ρ
Boundary conditions:
At y = 0, ∂u/∂y = 0 (maximum velocity)
At y = H, u = 0
c 1 = 0
1 ∂P
c 2 = −
H 2
2μ ∂x
176
Analytical Heat Transfer
q″ s
x
y
2H
q″ s
FIGURE 8.6
Two parallel plates at a uniform wall heat flux.