fluid_mechanics

324 Chapter 6 ■ Differential Analysis of Fluid Flow
6.44 The velocity potential
f k1x 2 y 2 2 1k constant2
may be used to represent the flow against an infinite plane boundary,
as illustrated in Fig. P6.44. For flow in the vicinity of a stagnation
point, it is frequently assumed that the pressure gradient along
the surface is of the form
0p
0x Ax
where A is a constant. Use the given velocity potential to show that
this is true.
y
6.47 It is suggested that the velocity potential for the incompressible,
nonviscous, two-dimensional flow along the wall shown in
Fig. P6.47 is
f r 4 3 cos 4 3 u
Is this a suitable velocity potential for flow along the wall? Explain.
3 π/4
θ
r
F I G U R E P6.47
x
F I G U R E P6.44
6.45 Water is flowing between wedge-shaped walls into a small
opening as shown in Fig. P6.45. The velocity potential with units
m 2 s for this flow is f 2 ln r with r in meters. Determine the
pressure differential between points A and B.
Section 6.5 Some Basic, Plane Potential Flows
6.48 Obtain a photograph/image of a situation which approximates
one of the basic, plane potential flows. Print this photo and
write a brief paragraph that describes the situation involved.
6.49 As illustrated in Fig. P6.49, a tornado can be approximated
by a free vortex of strength for r 7 R c , where R c is the radius of
the core. Velocity measurements at points A and B indicate that
V A 125 fts and V B 60 fts. Determine the distance from point
A to the center of the tornado. Why can the free vortex model not be
used to approximate the tornado throughout the flow field 1r 02?
y
r
__ π
6
r
A
θ
B
R c
A
B
x
0.5 m 1.0 m
F I G U R E P6.45
100 ft
6.46 An ideal fluid flows between the inclined walls of a twodimensional
channel into a sink located at the origin 1Fig. P6.462.
The velocity potential for this flow field is
where m is a constant. (a) Determine the corresponding stream
function. Note that the value of the stream function along the wall
OA is zero. (b) Determine the equation of the streamline passing
through the point B, located at x 1, y 4.
ψ = 0
f m 2p ln r
A
y
F I G U R E P6.49
6.50 If the velocity field is given by V axî ay ĵ, and a is a constant,
find the circulation around the closed curve shown in Fig. P6.50.
(1, 2) (2, 2)
y
B (1, 4)
(1, 1) (2, 1)
O
r
θ
π__
3
F I G U R E P6.46
x
x
F I G U R E P6.50
6.51 The streamlines in a particular two-dimensional flow field are
all concentric circles, as shown in Fig. P6.51. The velocity is given by
the equation v u vr where v is the angular velocity of the rotating
mass of fluid. Determine the circulation around the path ABCD.