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If the flow is hydrodynamically fully developed, then
v = 0
(8.5)
∂u = 0
(8.6)
∂x
1 ∂P
1 ∂
∂u
0 = −
+ υ
r
(8.7)
ρ ∂x
r ∂r
∂r
and u = u(r) only.
Solve momentum equation
1 dP
1 d
du
= υ
r
ρ dx
r dr
dr
dP
du
r dr = μd r
dx
dr
dP 1
du
r
2
= μr
+ C 1
dx 2
dr
at r = 0, du/dr = 0, C 1 = 0
dP 1 r dr = μ du
dx 2
dP 1 r
2
= μu + C 2
dx 4
at r = R, u = 0, C 2 = (1/4)R 2 (dP/dx)
1 2 dP
1 R
2 dP
r
= μu +
4 dx
4 dx
(
)
1 dP 2 − R
2
4μ dx
and r = 0, u = u max = −(1/4μ)R 2 (dP/dx).
We obtain u = −u max (r 2 − R 2 )/R 2
( ) 2
u =
r
u
r
= 1 −
(8.8)
U max
R
170
Analytical Heat Transfer
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If the flow is hydrodynamically fully developed, then
v = 0
(8.5)
∂u = 0
(8.6)
∂x
1 ∂P
1 ∂
∂u
0 = −
+ υ
r
(8.7)
ρ ∂x
r ∂r
∂r
and u = u(r) only.
Solve momentum equation
1 dP
1 d
du
= υ
r
ρ dx
r dr
dr
dP
du
r dr = μd r
dx
dr
dP 1
du
r
2
= μr
+ C 1
dx 2
dr
at r = 0, du/dr = 0, C 1 = 0
dP 1 r dr = μ du
dx 2
dP 1 r
2
= μu + C 2
dx 4
at r = R, u = 0, C 2 = (1/4)R 2 (dP/dx)
1 2 dP
1 R
2 dP
r
= μu +
4 dx
4 dx
(
)
1 dP 2 − R
2
4μ dx
and r = 0, u = u max = −(1/4μ)R 2 (dP/dx).
We obtain u = −u max (r 2 − R 2 )/R 2
( ) 2
u =
r
u
r
= 1 −
(8.8)
U max
R
170
Analytical Heat Transfer