This entry contributed by Dana Romero

Bernoulli's law describes the behavior of a fluid under varying conditions of flow and height. It states

(1)

where P is the static pressure (in Newtons per square meter), is the fluid density (in kg per cubic meter), v is the velocity of fluid flow (in meters per second) and h is the height above a reference surface. The second term in this equation is known as the dynamic pressure. The effect described by this law is called the Bernoulli effect, and (1) is sometimes known as Bernoulli's equation.

For a heuristic derivation of the law, picture a pipe through which and ideal fluid is flowing at a steady rate. Let W denote the work done by applying a pressure P over an area A, producing an offset of , or volume change of . Let a subscript 1 denote fluid parcels at an initial point down the pipe, and a subscript 2 denote fluid parcels further down the pipe. Then the work done by pressure force

(2)

at points 1 and 2 is

			(3)
			(4)

(5)

Equating this with the change in total energy (written as the sum of kinetic and potential energies gives

			(6)

(7)

which, upon rearranging, gives

(8)

so writing the density as then gives

(9)

This quantity is constant for all points along the streamline, and this is Bernoulli's theorem, first formulated by Daniel Bernoulli in 1738. Although it is not a new principle, it is an expression of the law of conservation of mechanical energy in a form more convenient for fluid mechanics.

A more rigorous derivation proceeds using the one-dimensional Euler's equation of inviscid motion,

(10)

along a streamline, where u is used for speed instead of v (a common convention in fluid mechanics). Integrating gives

(11)

(12)

In a gravitational field, this becomes

(13)

However, if the flow has zero vorticity, then

(14)

but

(15)

so, for incompressible flow,

(16)

(17)

throughout the entire fluid.

Bernoulli Effect, d'Alembert's Paradox, Dynamic Pressure, Kutta-Zhukovski Theorem, Lift, Lift Coefficient, Lift Force, Static Pressure