Derive eulers equation

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Problem:

1- Derive. From first principles Euler's equation from the rate of change of momentum
2- Show by integration how Euler's equation derives Bernoulli's equation
3- Explain the use of Euler's equation as applied to compressible fluids.

Solution:

1. The Euler's equation can be derived by applying the Newton's 2-nd law on a volume of an inviscid fluid (force = rate of change of momentum):

( 1)
where sum of all forces acting on the considered volume of fluid
m = mass of the considered volume of fluid:
( 2)
acceleration, ( 3)

The nature of the forces can be:
- pressure
- external field (like gravity, electromagnetic etc.)

The integral of pressure on the surface [] bounding the volume is given by:
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