University of California, Berkeley
Mechanical Engineering
Prof S. Morris
ME106 Fluid Mechanics, Problem Set 4
1.
The ¯gure shows the cross{section of a large{diameter pipe from which you are permitted to withdraw
water through a hole of ¯xed area
A
i
. You have attached a di®user with exit area
A
e
. The °ow is incompress
ible and inviscid. Find (a) the volume °owrate
Q
as a function of the pipe pressure
p
o
, atmospheric pressure
p
a
, °uid density
½
and the exit area
A
e
; and (b) the pressure
p
i
in the hole of area
A
i
. Your analysis will
predict that
Q
can be made arbitrarily large by making
A
e
large enough, but what e®ect already discussed
in this class would limit
Q
?
air
p
a
free jet
p
o
A
i
A
e
2.
The ¯gure shows a piston of cross{sectional area
A
p
sinking under its own weight into an air{¯lled cylinder.
The air thus expelled leaves the cylinder as a free jet of total cross{sectional area
A
j
. The gap between the
piston and cylinder is narrow, so that
A
j
¿
A
p
. (a) Sketch the stagnation streamline for a reference frame
¯xed in the cylinder wall, and hence sketch the streamlines. (b) Then ¯nd the pressure acting on the base
on the piston. (Assume the °ow is quasi{steady, and that the gravitational potential energy of the air is
negligible compared with the kinetic energy of the free jet.) (c) Hence show that the piston sinks at a speed
V
given approximately by
V
=
q
2
mgA
2
j
=
(
½A
3
p
). (A similar device is used as a hydraulic bu®er.)
V
m
air
Piston area
A
p
Jet area
A
j
g
3.
Flow in a sink vortex can be approximated by
V
=
K
^
Á=r
; here
K
is a constant,
r
and
Á
are plane polar
coordinates, and
^
Á
is a unit vector in the circumferential direction. (This simpli¯ed model does not include
the downward °ow into the drain; that °ow is small compared with the swirling motion.)
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 Spring '08
 Morris
 Mechanical Engineering, AP, Velocity, Trigraph, th e acceleration, Aj Piston area Ap

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