HOMEWORK #7 SOLUTIONS
Number 1
Find: Velocity and temperature profiles within the fluid, location and value of maximum
temperature, and the heat flux going into each plate.
Known:
T
O
= 18°C (temperature at both top and bottom plates)
Velocity of top plate = v
T
= 12 m/s
Velocity of bottom plate = 0 m/s
From Table A.5, properties of oil at 18°C are:
ρ = 890 kg/m
3
μ
= 0.999 Ns/m
2
k = 0.145 W/mK
To find the velocity profile, start with the momentum equation in the x direction (eq.
6.26):
2
2
2
2
u
u
P
u
u
u
v
x
y
x
x
y
ρ
μ
�
�
�
�
L
Κ
+
= 
+
+
�
�
�
�
L
Κ
•♠
�
�
�
�
(1)
From continuity (eq. 6.25) we know that
0
u
x
P
=
.
Given the nature of the system, there is
no vertical velocity, v = 0.
Equation (1) reduced to:
2
2
0
d u
dy
=
(2)
Integrating twice yields:
(
29
1
2
u
y
C y
C
=
+
(3)
The two boundary conditions are:
(
29
0
0
(
)
T
u
y
u y
L
v
=
=
=
=
(4)
Solving for C
1
and C
2
and plugging into equation (3) yields:
(
29
(12)
8000
.0015
T
v y
y
u
y
y
L
=
=
=
(5)
This equation represents the velocity profile between the two plates.
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To find the temperature distribution between the two plates, start with the energy
equation (6.28).
This equation reduces to:
2
2
2
0
T
u
k
x
y
μ
�
�
°
=
+
�
�
°
�
�
(6)
From equation (5), we see that
T
v
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 Spring '08
 HERMAN
 Fluid Dynamics, Thermodynamics, Heat, Heat Transfer

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