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 v2 + 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 (½ρv2), 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.

For the form of Bernoulli's equation given in Professor Pi's writeup, it's important to remember that ρ g h is only the gravitational potential energy in a uniform gravitational field, such as you might find in most engineering problems. In astrophysical problems it is usually the case that the gravitational field is nonuniform, in which case the ρ g h term must be replaced with the appropriate expression for the gravitational potential energy for the gravitational field in the problem. One particularly hairy case is the case of a self-gravitating fluid, in which the fluid itself has sufficient mass to contribute to the overall gravitational field. In that case, calculating the gravitational potential, using Poisson's Equation can be far more difficult than actually solving the fluid dynamics equations.

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