Mechanics of Fluids

2.5 Two small vessels are connected to a U-tube manometer
containing mercury (relative density 13.56) and the connecting
tubes are filled with alcohol (relative density 0.82). The
vessel at the higher pressure is 2 m lower in elevation than the
other. What is the pressure difference between the vessels when
the steady difference in level of the mercury menisci is 225 mm?
What is the difference of piezometric head? If an inverted U-tube
manometer containing a liquid of relative density 0.74 were
used instead, what would be the manometer reading for the
same pressure difference?
2.6 A manometer consists of two tubes A and B, with vertical
axes and uniform cross-sectional areas 500 mm 2 and 800 mm 2
respectively, connected by a U-tube C of cross-sectional area
70 mm 2 throughout. Tube A contains a liquid of relative density
0.8; tube B contains one of relative density 0.9. The surface
of separation between the two liquids is in the vertical side of C
connected to tube A. What additional pressure, applied to the
tube B, would cause the surface of separation to rise 60 mm in
the tube C?
2.7 Assuming that atmospheric temperature decreases with increasing
altitude at a uniform rate of 0.0065 K · m −1 , determine the
atmospheric pressure at an altitude of 7.5 km if the temperature
and pressure at sea level are 15 ◦ C and 101.5 kPa respectively.
(R = 287 J · kg −1 · K −1 .)
2.8 At the top a mountain the temperature is −5 ◦ C and a mercury
barometer reads 566 mm, whereas the reading at the foot of
the mountain is 749 mm. Assuming a temperature lapse rate of
0.0065 K · m −1 and R = 287 J · kg −1 · K −1 , calculate the height
of the mountain. (Neglect thermal expansion of mercury.)
2.9 A rectangular plane, 1.2 m by 1.8 m is submerged in water and
makes an angle of 30 ◦ with the horizontal, the 1.2 m sides being
horizontal. Calculate the magnitude of the net force on one face
and the position of the centre of pressure when the top edge of
the plane is (a) at the free surface, (b) 500 mm below the free
surface, (c) 30 m below the free surface.
2.10 What is the position of the centre of pressure for a vertical semicircular
plane submerged in a homogeneous liquid and with its
diameter d at the free surface?
2.11 An open channel has a cross-section in the form of an equilateral
triangle with 2.5 m sides and a vertical axis of symmetry. Its
end is closed by a triangular vertical gate, also with 2.5 m sides,
supported at each corner. Calculate the horizontal thrust on
each support when the channel is brim-full of water.
2.12 A circular opening 1.2 m in diameter in the vertical side of a
reservoir is closed by a disc which just fits the opening and is
pivoted on a shaft along its horizontal diameter. Show that, if
the water level in the reservoir is above the top of the disc, the
Problems 85