Page 425 - Introduction to Continuum Mechanics

Newtonian Viscous Fluid 409

To satisfy the mass conservation principle, we must have

In terms of <p this equation becomes

The equations of motion for an inviscid fluid are the Euler equations

We assume that the flow is barotropic, that is, the pressure is an explicit function of density
only (such as in isentropic or isothermal flow). Thus, in a barotropic flow,

Now,

Therefore, for barotropic flows of an inviscid fluid under conservative body forces (i.e.,
<3Q
». - —__^ the equations of motion can be written
oX{

Comparing Eq. (6.28.7) with (6.21.6), we see immediately that under the conditions stated,
irrotational flows are again always dynamically possible. In fact, the integration of Eq. (6.28.7)
(in exactly the same way as was done in Section 6.21) gives the following Bernoulli equation

which for steady flow, becomes

For most problems in gas dynamics, the body force is small compared with other forces and
often neglected. We then have