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 Demonstrate that a line source of strength
(running along the
axis) situated in a uniform flow of (unperturbed) velocity
(lying in the

plane) and density
experiences a force per unit length
 Demonstrate that a vortex filament of intensity
(running along the
axis) situated in a uniform flow of (unperturbed) velocity
(lying in the

plane) and density
experiences a force per unit length
 Show that two parallel line sources of strengths
and
, located a perpendicular distance
apart, exert a radial
force per unit length
on one another, the force being attractive if
, and
repulsive if
.
 Show that two parallel vortex filaments of intensities
and
, located a perpendicular distance
apart, exert a radial
force per unit length
on one another, the force being repulsive if
, and
attractive if
.
 A vortex filament of intensity
runs parallel to, and lies a perpendicular distance
from, a rigid planar boundary. Demonstrate that
the boundary experiences a net force per unit length
directed toward the
filament.
 Two rigid planar boundaries meet at rightangles. A line source of strength
runs parallel to the line of intersection of the planes, and is situated a
perpendicular distance
from each. Demonstrate that the source is subject to a force per unit length
directed towards the line of intersection of the planes.
 A line source of strength
is located a distance
from an impenetrable circular cylinder of radius
(the axis
of the cylinder being parallel to the source). Demonstrate that the cylinder experiences a net force per unit length
directed toward the source.
 A dipole line source consists of a line source of strength
, running parallel to the
axis, and intersecting the

plane at
,
, and
a parallel source of strength
that intersects the

plane at
,
. Show
that, in the limit
, and
, the complex velocity potential of the source is
Here,
is termed the complex dipole strength.
 A dipole line source of complex strength
is placed in a uniformly flowing fluid
of speed
whose direction of motion subtends a (counterclockwise) angle
with the
axis.
Show that, while no net force acts on the source, it is subject to a moment (per unit length)
about the
axis.
 Consider a dipole line source of complex strength
running along the
axis,
and a second parallel source of complex strength
that intersects the

plane
at
,
.
Demonstrate that the first source is subject to a moment (per unit length) about the
axis of
as well as a force (per unit length) whose
 and
components are
respectively. Show that the second source is subject to the same moment, but an equal and opposite force.
 A dipole line source of complex strength
runs parallel to, and is located a
perpendicular distance
from, a rigid planar boundary. Show that the
boundary experiences a force per unit length
acting toward the source.
 Demonstrate that a conformal map converts a line source into a line source of the same strength, and
a vortex filament into a vortex filament of the same intensity.
 Consider the conformal map
where
,
, and
is real and positive. Show that
Demonstrate that
, where
, maps to a circular arc of center
,
, and
radius
, that connects the points
,
, and lies in the region
. Demonstrate that
maps to the continuation of this arc in the region
. In particular, show that
maps to the region
on the
axis, whereas
maps to the region
. Finally, show that
maps to a circle of center
,
, and radius
.
 Consider the complex velocity potential
where
Here,
,
, and
are real and positive. Show that
Hence, deduce that the flow at
is uniform,
parallel to the
axis, and of speed
. Demonstrate that
Hence, deduce that the streamline
runs along the
axis for
, but along a
circular arc connecting the points
,
for
. Furthermore, show that if
then this arc lies above the
axis, and is of maximum height
but if
then the arc lies below the
axis, and is of maximum depth
Hence, deduce that if
then the complex velocity potential under investigation corresponds to uniform flow of speed
,
parallel to a planar boundary that possesses
a cylindrical bump (whose axis is normal to the flow) of height
and width
, but if
then the potential corresponds to flow parallel to a planar
boundary that possesses a cylindrical depression of depth
and width
. Show, in particular, that if
then
the bump is a halfcylinder, and if
then the depression is a halfcylinder. Finally, demonstrate that the
flow speed at the top of the bump (in the case
), or the bottom of the depression (in the case
) is
 Show that
maps the semiinfinite strip
,
in the
plane
onto the upper half (
) of the
plane. Hence, show that the stream function due to a line source of strength
placed at
,
, in the rectangular region
,
bounded by the rigid planes
,
, and
, is
 Show that the complex velocity potential
can be interpreted as that due to uniform flow of speed
over a cylindrical log of radius
lying on the flat bed of a deep
stream (the axis of the log being normal to the flow).
Demonstrate that the flow speed at the top of the log is
. Finally, show that the pressure
difference between the top and bottom of the log is
.
 Show that the complex potential
where
(
) represents uniform flow of unperturbed speed
, whose direction subtends a (counterclockwise) angle
with
the
axis, around an impenetrable elliptic cylinder of major radius
, aligned along the
axis, and
minor radius
, aligned along the
axis. Demonstrate that the moment per unit length (about the
axis)
exerted on the cylinder by the flow is
Hence, deduce that the moment acts to turn the cylinder broadsideon to the flow (i.e.,
is a dynamically stable
equilibrium state), and that the equilibrium state in which the cylinder is aligned with the flow (i.e.,
) is
dynamically unstable.
 Consider a simplyconnected region of a twodimensional flow pattern bounded on the inside by the closed curve
(lying in the

plane), and on the
outside by the closed curve
. Here,
and
do not necessarily correspond to streamlines of the flow. Demonstrate that the kinetic energy per unit
length (in the
direction) of the fluid lying between the two curves is
where
is the fluid mass density,
the velocity potential, and
the stream function. Here,
is a curve that runs from
to
, and
denotes the amount by which the velocity potential increases as the
argument of
increases by
.
 Show that the complex potential
where
(
) represents the flow pattern around an impenetrable elliptic cylinder of major radius
, aligned along the
axis, and
minor radius
, aligned along the
axis, moving with speed
, in a direction that makes a counterclockwise angle
with the
axis, through a fluid that is at rest far from the cylinder. Demonstrate that the kinetic energy per unit length of the
flow pattern is
where
is the fluid mass density.
Hence, show that the cylinder's added mass per unit length is
 Demonstrate from Equation (6.110) that the equation of the free streamline
, in the case of a liquid jet emerging from
a twodimensional orifice of semiwidth
formed by a gap between two semiinfinite plane walls that subtend an angle
, can be written parametrically
as:
where
. Here, the orifice corresponds to the plane
, and the flow a long way from the orifice is in the
direction.
Show that for the case of a twodimensional Borda mouthpiece,
, the previous equations reduce to
Finally, show that the previous equations predict that the free streamline is reentrant, with
attaining its minimum value
when
.
Next: Axisymmetric Incompressible Inviscid Flow
Up: TwoDimensional Potential Flow
Previous: Blasius Theorem
Richard Fitzpatrick
20160331