CE 3500 Fluid Mechanics – Fall 2014
Exercises
1 Heated Pipe (4.56)
Air flows steadily through a long pipe with a speed of u=50 [ m s−1 ] +0.5 [ s−1 ] x , where x is the
distance along the pipe in ft and u is in ft s−1 . Due to heat transfer into the pipe, the air
temperature T within the pipe is T =300 [ ˚ F ] +10 [ ˚ F ft −1] x . Determine the rate of change of
the temperature of air particles
DT ∂ T
∂T
as they flow past the section at x=5 [ ft ] .
=
+u
Dt ∂ t
∂x
u=50 [ ft s−1 ] +0.5 [ s−1 ] x=50+0.5 [ s−1 ] 5 [ ft ] =52.5 [ ft s−1 ]
DT ∂ T
∂T DT
,
=
+u
=u 10 [ ˚ F ft −1 ] =525 [ ˚ F s−1 ]
Dt ∂ t
∂x
Dt
1
CE 3500 Fluid Mechanics – Fall 2014
Exercises
2 Oil Layer
( )(
A layer of oil flows down a vertical plate with a velocity of U⃗ =
U0
h
2
2hx−x 2 ) ^j , where U 0 and h
are constants.
a) Show that the fluid sticks to the plate and that the shear stress τ=μ
dv
at the edge of the layer
dx
is zero.
b) Determine the volumetric flowrate across surface AB. Assume the width of the plate is b .
(Note: The velocity profile for laminar flow in a pipe has a similar shape).
( )(
U⃗ =
U0
h
2
2hx−x 2 ) ^j , v ( x ) =
U0
h
2
U
dv ( h )
( 2 hx− x 2 ) → dv = 20 ( 2 h−2 x ) →
=0
dx
dx
h
[
]
h
dV out
U0 h
U
x3 2
⃗
dA=b dx ,
=∫ U⋅^n dA= 2 ∫ ( 2 hx− x 2 ) b dx=b 20 h x 2− = bU 0 h
dt A
3 0 3
h x=0
h
2
CE 3500 Fluid Mechanics – Fall 2014
Exercises
3 Bicycle ride to the beach
A bicyclist leaves from her home at 9 A.M. and rides to a beach 40 mi away. Because of a breeze
off the ocean, the temperature at the beach remains T beach =60 [ ˚ F ] throughout the day. At the
cyclist’s home the temperature increases linearly with time, going from T home , 9 am =60 [ ˚ F ] at
9am. to T home , 1 pm =80 [ ˚ F ] by 1pm. The temperature is assumed to vary linearly as a function of
position between the cyclist’s home and the beach. Determine the rate of change of temperature
observed by the cyclist for the following conditions:
a) as she pedals 10 mph through a town 10 mi from her home at 10am
b) as she eats lunch at a rest stop 30 mi from her home at noon;
c) as she arrives enthusiastically at the beach at 1 P.M., pedaling 20 mph.
T =T 0 +a x t → a=
5
[ ˚ F mi−1 hr −1 ]
40
DT ∂ T
∂T
DT
=
+u
→
=ax+u at
Dt ∂ t
∂x
Dt
x=30 [ mi ] , t=1 [ hr ] , u=−10 [ mph ] →
x=10 [ mi ] , t=3 [ hr ] , u=0 [ mph ] →
DT 15
5
= [ ˚ F hr −1] − [ ˚ F hr−1 ]=2.5 [ ˚ F hr −1 ]
Dt
4
4
DT 5
= [ ˚ F hr −1 ]=1.2 5 [ ˚ F hr −1 ]
Dt 4
x=0 [ mi ] , t=0 [ hr ] , u=−20 [ mph ] →
DT
=−10 [ ˚ F hr −1]
Dt
3
CE 3500 Fluid Mechanics – Fall 2014
Exercises
4 Moving Container (20 pts)
Water flows from a nozzle with a speed of U =10 [ m s−1 ] and is collected in a container that
moves toward the nozzle with a speed of U cv =2 [ m s−1] . The moving control surface consists of
the inner surface of the container. The system consists of the water in the container at time
t=0 [ s ] and the water between the nozzle and the tank. The jet from the nozzle to the tank has
constant diameter stream d=0.1 [ m ] .
a) At time t 0=0 [ s ] , what volume of the system is outside of the control volume?
b) At time t 1=0.1 [ s ] what volume of the system remains outside of the control volume?
c) How much water enters the control volume during this time period?
d) Repeat the problem for t 2=0.3 [ s ] .
D b m sys D b mcv
DV sys ∂ V cv
∂ V cv
⃗ n
⃗ n^ dA=0 →
⃗ n^ A
^ dA →
=
+∫ ρ b U⋅
=
+∫ U⋅
=−U⋅
Dt
∂t
Dt
∂
t
∂
t
cs
cs
2
V sys ,t −V cv , t =
0
0
( 0.1 [ m ] )
d2
3
π l=
π 3 [ m ]=.02355 [ m ]
4
4
2
( 0.1 [ m ] )
d2
3
V sys ,t −V cv , t = π [ l −( U +U cv ) t 1 ]=
π 3 [ m ]=.01413 [ m ]
4
4
1
1
t1
2
∂ V cv
∂ V cv
d2
⃗ n^ A= d π (U +U cv ) →V cv , t −V cv , t = ∫
=−U⋅
dt= π (U +U cv ) t
∂t
4
∂t
4
t =t
0
0
4