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.
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: