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- For the case of the two-dimensional motion of an incompressible fluid, determine the condition that the velocity components

satisfy the equation of continuity. Show that the magnitude of the vorticity is
.

- For the case of the two-dimensional motion of an incompressible fluid, show that

are the velocity components of a possible flow pattern. Determine the stream function and sketch the streamlines.
Prove that the motion is irrotational, and find the velocity potential.

- A cylindrical vortex in an incompressible fluid is co-axial with the
-axis, and such that
takes the constant value
for
, and is zero for
, where
is a cylindrical coordinate. Show that

where
is the pressure at radius
inside the vortex, and the circulation of the fluid outside the vortex is
.
Deduce that

where
is the pressure at the center of the vortex.

- Consider the cylindrical vortex discussed in Exercise 3. If
is the pressure at radius
external to the
vortex, demonstrate that

where
is the pressure at infinity.

- Show that the stream function for the cylindrical vortex discussed in Exercises 3 and 4 is
for
, and
for
.

- Prove that in the two-dimensional motion of a liquid the mean tangential
fluid velocity around any small circle of radius
is
, where
is the value of

at the center of the circle. Neglect terms of order
.

- Show that the equation of continuity for the two-dimensional motion of an
incompressible fluid can be written

where
,
are cylindrical coordinates. Demonstrate that this
equation is satisfied when
and
. Determine the stream function, and
show that the fluid speed at any point is

where
is the stream function at that point (defined such that
at
).

- Demonstrate that streamlines cross at right-angles at a stagnation point in two-dimensional, incompressible,
irrotational flow.

- Consider two-dimensional, incompressible, inviscid flow. Demonstrate that the fluid motion
is governed by the following equations:

where
,
, and
.

- For irrotational, incompressible, inviscid motion in two dimensions show that

where
.

** Next:** Two-Dimensional Potential Flow
** Up:** Two-Dimensional Incompressible Inviscid Flow
** Previous:** Two-Dimensional Jets
Richard Fitzpatrick
2016-03-31