Page 426 - Introduction to Continuum Mechanics

410 Irrotational, Barotropic Flows of Inviscid Compressible Fluid

Example 6.28.1

Show that for steady isentropic irrotational flows of an inviscid compressible fluid (body
force neglected)

Solution. For an isentropic flow

so that

Thus, the Bernoulli equation [Eq. (6.28.11)] becomes

We note that this is the same result as that obtained in Example 26.1, Eq. (6.26.8), by the use
of the energy equation. In other words, under the conditions stated (inviscid, non-heat
conducting, initial homogeneous state), the Bernoulli equation and the energy equation are
the same.

Example 6.28.2

Let p 0 denote the pressure at zero speed (called stagnation pressure .) Show that for
Y
isentropic steady flow (p/p = constant) of an ideal gas,

where c is the local speed of sound.
Solution. Since (see the previous example)

therefore,