Bernoulli and Euler Equations

• 
  For inviscid flows, the steady form of the momentum equation is the Euler equation,

(39.11)

Integrating along a streamline, we get the Bernoulli's equation for a compressible flow as

(39.12)

• For adiabatic frictionless flows the Bernoulli's equation is identical to the energy equation. Recall, that this is an isentropic flow, so that the Tds equation is given by

For isentropic flow, ds=0

Therefore,

Hence, the Euler equation (39.11) can also be written as

This is identical to the adiabatic form of the energy Eq. (39.10).

Momentum Principle for a Control Volume

For a finite control volume between Sections 1 and 2 (Fig. 39.3), the momentum principle is

(39.13)

where F is the x-component of resultant force exerted on the fluid by the walls. Note that the momentum principle, Eq. (39.13), is applicable even when there are frictional dissipative processes within the control volume.