## Derivation of Bernoulli’s Equation

**The Bernoulli’s equation** for incompressible fluids can be derived from the **Euler’s equations** of motion under rather severe restrictions.

- The velocity must be derivable from a
**velocity potential**. - External forces must be conservative. That is, derivable from a potential.
- The density must either be constant, or a function of the pressure alone.
- Thermal effects, such as natural convection, are ignored.

In fluid dynamics, the Euler equations are a set of quasilinear hyperbolic equations governing adiabatic and inviscid flow. Euler equations can be obtained by linearization of these Navier–Stokes equations.

## Bernoulli’s Equation

**The Bernoulli’s equation** can be considered to be a statement of the **conservation of energy principle** appropriate for flowing fluids. It is one of the most important/useful equations in **fluid mechanics**. It puts into a relation **pressure and velocity** in an **inviscid incompressible flow**. **Bernoulli’s equation** has some restrictions in its applicability, they summarized in following points:

- steady flow system,
- density is constant (which also means the fluid is incompressible),
- no work is done on or by the fluid,
- no heat is transferred to or from the fluid,
- no change occurs in the internal energy,
- the equation relates the states at two points along a single streamline (not conditions on two different streamlines)

Under these conditions, the general energy equation is simplified to:

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