- Liquid is led steadily through a pipeline that passes over a hill of height
into the valley below, the
speed at the crest being
. Show that, by properly adjusting the ratio of the cross-sectional areas of the pipe
at the crest and in the valley, the pressure may be equalized at these two places. (Milne-Thomson 1958.)
- Water of mass density
and pressure
flows through a curved pipe of
uniform cross-sectional area
, whose radius of curvature is
, at the
uniform speed
. Demonstrate that there is a net force per unit length
acting on the
pipe, and that this force is everywhere directed away from the pipe's local center of curvature. Here,
is atmospheric pressure.
- Water is held in a right circular conical tank whose apex lies vertically below the center of its
base. The water initially fills the tank to a height
above the vertex. Let
be the initial
radius of the surface of the water inside the tank. A small hole of area
(that is much less than
)
is made at the bottom of the tank. Demonstrate that the time required to empty the tank is at least
- Water is held in a spherical tank of radius
, and initially fills the tank to a height
above
its lowest point. A small hole of area
(that is much less than
) is made
at the bottom of the tank. Demonstrate that the time required to empty the tank is at least
- Water is held in two contiguous tanks whose cross-sectional areas,
and
, are independent of height. A
small hole of area
(where
,
) is made in the wall connecting the tanks. Assuming that the initial
difference in water level between the two tanks is
, show that the time
required for the water levels to equilibrate is at least
- For a channel of width
, having a discharge rate
, show that there is
a critical depth
, where
- Show that for a stationary hydraulic jump in a rectangular channel, the
upstream Froude number
, and the downstream Froude number
,
are related by
- Consider a simply-connected volume
whose boundary is the surface
. Suppose that
contains an
incompressible fluid whose motion is irrotational. Let the velocity potential
be constant over
. Prove that
has the same constant value throughout
. [Hint: Consider the identity
.]
- In Exercise 8, suppose that, instead of
taking a constant value on the boundary, the
normal velocity is everywhere zero on the boundary. Show that
is constant throughout
.
- An incompressible fluid flows in a simply-connected volume
bounded by a surface
.
The normal flow at the boundary is prescribed. Show that the flow pattern with the lowest kinetic
energy is irrotational. This result is known as the
*Kelvin minimum energy theorem*. [Hint: Try writing , where is the velocity potential of the irrotational flow pattern. Let throughout , and on . Show that the kinetic energy is lowest when throughout .]