fluid_mechanics

290 Chapter 6 ■ Differential Analysis of Fluid Flow
SOLUTION
The components of velocity are
v r 0f
0r 2 r
which indicates we have a purely radial flow. The flowrate per
unit width, q, crossing the arc of length Rp6 can thus be obtained
by integrating the expression
COMMENT Note that the radius R is arbitrary since the
v u 1 flowrate crossing any curve between the two walls must be the
0f
r 0u 0 same. The negative sign indicates that the flow is toward the opening,
that is, in the negative radial direction.
p6
p6
q v r R du a 2 R b R du
0
0
p 3 1.05 ft2 s
(Ans)
6.5.3 Vortex
We next consider a flow field in which the streamlines are concentric circles—that is, we interchange
the velocity potential and stream function for the source. Thus, let
f Ku
(6.84)
A vortex represents
a flow in which the
streamlines are concentric
circles.
and
(6.85)
where K is a constant. In this case the streamlines are concentric circles as are illustrated in Fig.
6.18, with v r 0 and
v u 1 r
c K ln r
0f
0u 0c 0r K r
(6.86)
v θ
v θ ~ 1__ r
r
This result indicates that the tangential velocity varies inversely with the distance from the origin,
as shown by the figure in the margin, with a singularity occurring at r 0 1where the velocity
becomes infinite2.
It may seem strange that this vortex motion is irrotational 1and it is since the flow field
is described by a velocity potential2. However, it must be recalled that rotation refers to the
orientation of a fluid element and not the path followed by the element. Thus, for an irrotational
vortex, if a pair of small sticks were placed in the flow field at location A, as indicated in Fig.
6.19a, the sticks would rotate as they move to location B. One of the sticks, the one that is
aligned along the streamline, would follow a circular path and rotate in a counterclockwise
y
ψ = constant
v θ
r
θ
x
φ = constant
F I G U R E 6.18
pattern for a vortex.
The streamline