Mechanics of Fluids

Solution
(a) From the equation of state
ρ1 = p1
=
RT1
(1.5 × 105 ) N · m−2 287 J · kg −1 · K −1 = 1.93 kg · m−3
× 271 K
(b) From eqn 11.13, and using the value for cp from Appendix 2,
T0 = T1 + u2 1
= 271 K +
2cp
(c) Since T0 is constant in adiabatic flow:
T2 = T0 − u2 2
= 307.3 K −
2cp
and
ρ2 = p2
=
RT2
11.4 THE SPEED OF SOUND
(270 m · s −1 ) 2
2 × 1005 J · kg −1 = 307.3 K
−1
· K
(320 m · s −1 ) 2
2 × 1005 J · kg −1 = 256.4 K
−1
· K
(1.2 × 105 ) N · m−2 287 J · kg −1 · K −1 = 1.63 kg · m−3
× 256.4 K
If the pressure at a point in a fluid is altered, the density is also altered –
even if only slightly – and in consequence individual particles undergo small
changes in position. To maintain a continuum, adjacent particles also change
position and thus the new pressure is progressively, yet rapidly, transmitted
throughout the rest of the fluid. Indeed, in a completely incompressible
fluid any disturbances would be propagated with infinite velocity because all
particles would have to change position simultaneously. Even in an actual
fluid, changes of pressure are transmitted so rapidly that the time necessary
for them to be spread throughout the fluid may often be negligible
compared with the time taken for the original change. Thus, in previous
chapters of this book we have assumed that pressure adjustments occur simultaneously
throughout the fluid. But if the pressure at a point is suddenly
altered, or the fluid is moving with high velocity relative to some solid body,
then the exact speed with which pressure changes are transmitted is of great
importance. This speed is determined by the relation between changes of
pressure and changes of density, that is, by the elastic properties of the
fluid.
Consider an instant after a small change of pressure has been caused at
some point in a fluid. The change may have resulted from the movement
of a solid body, such as a piston, the breaking of a thin membrane across
which a pressure difference existed, or an electrical discharge such as lightning.
Not all the fluid has yet experienced the pressure change and so at a
certain distance from the place where the change originated there is a more
or less abrupt discontinuity of pressure. This discontinuity is known as a
The speed of sound 493
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