Bernoulli's equation is one of the cornerstones of classical

fluid
mechanics. The equation describes the relationship between

kinetic
energy,

pressure and

potential energy in an

inviscid fluid (

ideal fluid). This
requires an

fluid in which there is no internal

friction, and thus no
conversion of mechanical energy to

heat.

The equation was derived by Daniel Bernoulli, who investigated the
forces present in a moving fluid:

P + ½ ρ v^{2} + ρ g h = constant

where P is pressure, ρ is the fluid density, v is the
velocity, h is elevation, and g is the gravitational acceleration.
The equation is only valid under the following conditions:

Despite these seemingly severe restrictions, Bernoulli's equation is
used very commonly, since it gives a good insight in the interconversion
between pressure (P), kinetic energy (½ρv^{2}), and
potential energy (ρgh). For many applications, the equation gives
a good quantitative estimate, or a qualitative argument for its
behavior.

It is easy to see that the Bernoulli effect (an increase in fluid velocity results in a
drop in total pressure) follows directly from Bernoulli's Equation.
Another effect is directly related to this is the Magnus effect (as
described nicely by bigmouth_strikes).
Some other applications:

- Airfoil calculations: The aerodynamic forces on wings (lift and
drag) can be described using Bernoulli's equation.
- Pitot tube: This device measures the difference between static
and dynamic pressure in a fluid. It can be used to calculate fluid
velocities;
*e.g.*airplanes are fitted with these devices to
measure air speed.
- General flow phenomena: pressure drop of flow through
orifices, free falling liquid flow (explains why the flow
from a tap contracts and accelerates as it falls), flow from open or
pressurized containers.