Page 428 - Introduction to Continuum Mechanics

412 One-Dimensional Flow of a Compressible Fluid

Example 6.28.3

For steady, barotropic irrotational flow, derive the equation for the velocity potential <p.
Neglect body forces.
Solution. For steady flow, the equation of continuity is, with v,- = -dp/dx;,

and the equation of motion is

2
Let c = dp/ dp (the local sound speed), then

Substituting Eq. (iii) into Eq. (i), we obtain

or,

In long form, Eq. (v) reads

6.29 One-Dimensional Flow of a Compressible Fluid

In this section, we discuss some internal flow problems of a compressible fluid. The fluid
will be assumed to be an ideal gas. The flow will be assumed to be one-dimensional in the
sense that the pressure, temperature, density, velocity, etc. are uniform over any cross-section
of the channel or duct in which the fluid is flowing. The flow will also be assumed to be steady
and adiabatic.