Mechanics of Fluids

and frictional effects to be negligible, calculate the net axial
force exerted by the air on the cone.
4.4 Two adjacent parallel and horizontal rectangular conduits A
and B, of cross-sectional areas 0.2 m 2 and 0.4 m 2 respectively,
discharge water axially into another conduit C of crosssectional
area 0.6 m 2 and of sufficient length for the individual
streams to become thoroughly mixed. The rates of flow through
A and B are 0.6 m 3 · s −1 and 0.8 m 3 · s −1 respectively and the
pressures there are 31 kPa and 30 kPa respectively. Neglecting
friction at the boundaries, determine the energy lost by each
entry stream (divided by mass) and the total power dissipated.
4.5 A boat is driven at constant velocity c (relative to the undisturbed
water) by a jet-propulsion unit which takes in water
at the bow and pumps it astern, beneath the water surface,
at velocity u relative to the boat. Show that the efficiency of
the propulsion, if friction and other losses are neglected, is
2c/(c + u).
Such a boat moves steadily up a wide river at 8 m · s −1 (relative
to the land). The river flows at 1.3 m · s −1 . The resistance to
motion of the boat is 1500 N. If the velocity of the jet relative to
the boat is 17.5 m · s −1 , and the overall efficiency of the pump
is 65%, determine the total area of the outlet nozzles, and the
engine power required.
4.6 A toy balloon of mass 86 g is filled with air of density
1.29 kg · m −3 . The small filling tube of 6 mm bore is pointed
vertically downwards and the balloon is released. Neglecting
frictional effects calculate the rate at which the air escapes if
the initial acceleration of the balloon is 15 m · s −2 .
4.7 A rocket sled of 2.5 Mg (tare) burns 90 kg of fuel a second
and the uniform exit velocity of the exhaust gases relative to
the rocket is 2.6 km · s −1 . The total resistance to motion at the
track on which the sled rides and in the air equals KV, where
K = 1450 N · m −1 · sandV represents the velocity of the sled.
Assuming that the exhaust gases leave the rocket at atmospheric
pressure, calculate the quantity of fuel required if the sled is to
reach a maximum velocity of 150 m · s −1 .
4.8 A boat travelling at 12 m · s −1 in fresh water has a 600 mm
diameter propeller which takes water at 4.25 m 3 · s −1 between
its blades. Assuming that the effects of the propeller hub and the
boat hull on flow conditions are negligible, calculate the thrust
on the boat, the efficiency of the propulsion, and the power
input to the propeller.
4.9 To propel a light aircraft at an absolute velocity of 240 km · h −1
against a head wind of 48 km · h −1 a thrust of 10.3 kN is
required. Assuming an efficiency of 90% and a constant air
density of 1.2 kg · m −3 determine the diameter of ideal propeller
required and the power needed to drive it.
Problems 157