fluid_mechanics

180 Chapter 4 ■ Fluid Kinematics
polar coordinates by v r 10r, and v u 10r. This flow approximates
a fluid swirling into a sink as shown in Fig. P4.7. Plot the velocity
field at locations given by r 1, 2, and 3 with u 0, 30, 60,
and 90°.
vθ
r
θ
v r
x (in.) y (in.)
0 0
0.25 0.13
0.50 0.16
0.75 0.13
1.0 0.00
1.25 0.20
1.50 0.53
1.75 0.90
2.00 1.43
y
y
θ
θ
V 0
V 0
(a)
x
x
(b)
F I G U R E P4.14
F I G U R E P4.7
4.8 The velocity field of a flow is given by V
20y1x 2 y 2 2 12 î 20x1x 2 y 2 2 1 2
ĵ fts, where x and y are in
feet. Determine the fluid speed at points along the x axis; along the
y axis. What is the angle between the velocity vector and the x axis
at points 1x, y2 15, 02, 15, 52, and 10, 52?
4.9 The components of a velocity field are given by u x y,
v xy 3 16, and w 0. Determine the location of any stagnation
points 1V 02 in the flow field.
4.10 The x and y components of velocity for a two-dimensional
flow are u 6y fts and v 3 fts, where y is in feet. Determine
the equation for the streamlines and sketch representative streamlines
in the upper half plane.
4.11 Show that the streamlines for a flow whose velocity components
are u c1x 2 y 2 2 and v 2cxy, where c is a constant, are
given by the equation x 2 y y 3 3 constant. At which point
1points2 is the flow parallel to the y axis? At which point 1points2 is
the fluid stationary?
4.12 A velocity field is given by V xî x1x 121y 12ĵ,
where u and v are in fts and x and y are in feet. Plot the streamline
that passes through x 0 and y 0. Compare this streamline with
the streakline through the origin.
4.13 From time t 0 to t 5 hr radioactive steam is released from
a nuclear power plant accident located at x 1 mile and y
3 miles. The following wind conditions are expected: V 10î 5ĵ
mph for 0 6 t 6 3 hr, V 15î 8ĵ mph for 3 6 t 6 10 hr, and
V 5î mph for t 7 10 hr. Draw to scale the expected streakline of
the steam for t 3, 10, and 15 hr.
*4.14 Consider a ball thrown with initial speed V 0 at an angle
of u as shown in Fig. P4.14a. As discussed in beginning physics, if
friction is negligible the path that the ball takes is given by
y 1tan u2x 3g12 V 2 0 cos 2 u24x 2
That is, y c 1 x c 2 x 2 , where c 1 and c 2 are constants. The path
is a parabola. The pathline for a stream of water leaving a small
nozzle is shown in Fig. P4.14b and Video V4.12. The coordinates
for this water stream are given in the following table. (a) Use the
given data to determine appropriate values for c 1 and c 2 in the above
equation and, thus, show that these water particles also follow a
parabolic pathline. (b) Use your values of c 1 and c 2 to determine
the speed of the water, V 0 , leaving the nozzle.
4.15 The x and y components of a velocity field are given by
u x 2 y and v xy 2 . Determine the equation for the streamlines
of this flow and compare it with those in Example 4.2. Is the flow
in this problem the same as that in Example 4.2? Explain.
4.16 A flow in the x–y plane is given by the following velocity
field: u 3 and v 6 ms for 0 6 t 6 20 s; u 4 and
v 0 ms for 20 6 t 6 40 s. Dye is released at the origin
1x y 02 for t 0. (a) Draw the pathlines at t 30 s for two
particles that were released from the origin—one released at t 0
and the other released at t 20 s. (b) On the same graph draw the
streamlines at times t 10 s and t 30 s.
4.17 In addition to the customary horizontal velocity components of
the air in the atmosphere 1the “wind”2, there often are vertical air currents
1thermals2 caused by buoyant effects due to uneven heating of the
air as indicated in Fig. P4.17. Assume that the velocity field in a certain
region is approximated by u u 0 , v v 0 11 yh2 for 0 6 y 6 h,
and u u 0 , v 0 for y 7 h. Plot the shape of the streamline that
passes through the origin for values of u 0v 0 0.5, 1, and 2.
u 0
F I G U R E P4.17
y
0
*4.18 Repeat Problem 4.17 using the same information except
that u u 0 yh for 0 y h rather than u u 0 . Use values of
u 0v 0 0, 0.1, 0.2, 0.4, 0.6, 0.8, and 1.0.
4.19 As shown in Video V4.6 and Fig. P4.19, a flying airplane
produces swirling flow near the end of its wings. In certain circumstances
this flow can be approximated by the velocity field
u Ky1x 2 y 2 2 and v Kx1x 2 y 2 2, where K is a constant
depending on various parameters associated with the airplane (i.e.,
its weight, speed) and x and y are measured from the center of the
swirl. (a) Show that for this flow the velocity is inversely proportional
to the distance from the origin. That is, V K1x 2 y 2 2 12 .
(b) Show that the streamlines are circles.
x