§3.5 Integration of Euler differential equation

Euler equation in differential form given is used in the situations that the ideal fluid flows along a streamline in steady.The Euler equation are multiplied by dx, dy and dz, and then added:

3.5.1 Integration under steady potential flow, gravity, incompressible condition

1．Bernoulli equation

Because the ideal fluid only have gravity, fx=fy=0, fz=-g, then

Because the ideal fluid is incompressible, then

Because the ideal fluid is potential flow,,then

So

Integration

or

The Bernoulli equation can be obtained with an integral along a streamline as formula (3-24).

2．The physical meaning of Bernoulli equation

The physical meaning and geometric meaning of Bernoulli equation is shown in Tab.3-1.

3.5.2 Integration along the stream under steady, gravity, incompressible condition

Integration result in the potential flow only apply to potential flow.Along with a stream line is also available for integration to rotational flow.

Energy equation is

Formula(3-36)and(3-34)are exactly same in formation.However, integration constant are different.