fluid_mechanics

6.5 Some Basic, Plane Potential Flows 289
ψ = constant
y
φ = constant
v r
r
θ
x
F I G U R E 6.17
source.
The streamline pattern for a
A source or sink
represents a purely
radial flow.
Also, since the flow is a purely radial flow, v u 0, the corresponding velocity potential can be
obtained by integrating the equations
It follows that
0f
0r m
2pr
1 0f
r 0u 0
f m 2p ln r
(6.82)
v r
~ v r
r
1__ r
If m is positive, the flow is radially outward, and the flow is considered to be a source flow. If m
is negative, the flow is toward the origin, and the flow is considered to be a sink flow. The flowrate,
m, is the strength of the source or sink.
As shown by the figure in the margin, at the origin where r 0 the velocity becomes infinite,
which is of course physically impossible. Thus, sources and sinks do not really exist in real flow
fields, and the line representing the source or sink is a mathematical singularity in the flow field.
However, some real flows can be approximated at points away from the origin by using sources or
sinks. Also, the velocity potential representing this hypothetical flow can be combined with other
basic velocity potentials to approximately describe some real flow fields. This idea is further discussed
in Section 6.6.
The stream function for the source can be obtained by integrating the relationships
to yield
v r 1 r
0c
0u m
2pr
v u 0c
0r 0
c m 2p u
(6.83)
It is apparent from Eq. 6.83 that the streamlines 1lines of c constant2 are radial lines, and from Eq.
6.82 the equipotential lines 1lines of f constant2 are concentric circles centered at the origin.
E XAMPLE 6.5
Potential Flow—Sink
GIVEN A nonviscous, incompressible fluid flows between
wedge-shaped walls into a small opening as shown in Fig. E6.5.
The velocity potential 1in ft 2 s 2, which approximately describes
this flow is
f 2 ln r
y
v r
π_
6
FIND Determine the volume rate of flow 1per unit length2 into
the opening.
r
R
θ
F I G U R E E6.5
x