fluid_mechanics

Problems 325
ω
a
b
C
B
Δθ
D
r
A
6.54 Water flows over a flat surface at 4 fts, as shown in
Fig. P6.54. A pump draws off water through a narrow slit at a volume
rate of 0.1 ft 3 s per foot length of the slit. Assume that the fluid
is incompressible and inviscid and can be represented by the combination
of a uniform flow and a sink. Locate the stagnation point
on the wall 1point A2 and determine the equation for the stagnation
streamline. How far above the surface, H, must the fluid be so that
it does not get sucked into the slit?
4 ft/s
F I G U R E P6.51
6.52 The motion of a liquid in an open tank is that of a combined
vortex consisting of a forced vortex for 0 r 2 ft and a free vortex
for r 7 2 ft. The velocity profile and the corresponding shape of
the free surface are shown in Fig. P6.52. The free surface at the center
of the tank is a depth h below the free surface at r q. Determine
the value of h. Note that h h forced h free , where h forced
and h free are the corresponding depths for the forced vortex and the
free vortex, respectively. 1See Section 2.12.2 for further discussion
regarding the forced vortex.2
10
v θ , ft/s
H
0.1 ft 3 /s
(per foot of length of slit)
F I G U R E P6.54
6.55 Two sources, one of strength m and the other with strength 3m,
are located on the x axis as shown in Fig. P6.55. Determine the location
of the stagnation point in the flow produced by these sources.
A
0 2
z
2
h
r, ft
r, ft
y
2
3
x
+m +3m
F I G U R E P6.55
6.56 The velocity potential for a spiral vortex flow is given by
f 12p2 u 1m2p2 ln r, where and m are constants. Show
that the angle, a, between the velocity vector and the radial direction
is constant throughout the flow field 1see Fig. P6.562.
F I G U R E P6.52
y
6.53 When water discharges from a tank through an opening in its
bottom, a vortex may form with a curved surface profile, as shown
in Fig. P6.53 and Video V6.4. Assume that the velocity distribution
in the vortex is the same as that for a free vortex. At the same time
the water is being discharged from the tank at point A, it is desired
to discharge a small quantity of water through the pipe B. As the
discharge through A is increased, the strength of the vortex, as indicated
by its circulation, is increased. Determine the maximum
strength that the vortex can have in order that no air is sucked in at
B. Express your answer in terms of the circulation. Assume that the
fluid level in the tank at a large distance from the opening at A remains
constant and viscous effects are negligible.
V α
r
θ
F I G U R E P6.56
x
B
1 ft
6.57 For a free vortex (see Video V6.4) determine an expression
for the pressure gradient (a) along a streamline, and (b) normal to a
streamline. Assume that the streamline is in a horizontal plane, and
express your answer in terms of the circulation.
F I G U R E P6.53
A
2 ft
6.58 (See Fluids in the News article titled “Some hurricanes
facts,” Section 6.5.3.) Consider a category five hurricane that has a
maximum wind speed of 160 mph at the eye wall, 10 miles from the
center of the hurricane. If the flow in the hurricane outside of
the hurricane’s eye is approximated as a free vortex, determine the
wind speeds at locations 20 mi, 30 mi, and 40 mi from the center of
the storm.