# Bernoulli's theorem

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## Bernoulli's theorem

An idealized algebraic relation between pressure, velocity, and elevation for flow of an inviscid fluid. Its most commonly used form is for steady flow of an incompressible fluid, and is given by the equation below,

where*p*is pressure, &rgr; is fluid density (assumed constant),

*V*is flow velocity,

*g*is the acceleration of gravity, and

*z*is the elevation of the fluid particle. The relation applies along any particular streamline of the flow. The constant may vary across streamlines unless it can be further shown that the fluid has zero local angular velocity.

The above equation may be extended to steady compressible flow (where changes in &rgr; are important) by adding the internal energy per unit mass, *e*, to the left-hand side. *See* Compressible flow

The equation is limited to inviscid flows with no heat transfer, shaft work, or shear work. Although no real fluid truly meets these conditions, the relation is quite accurate in free-flow or “core” regions away from solid boundaries or wavy interfaces, especially for gases and light liquids. Thus Bernoulli's theorem is commonly used to analyze flow outside the boundary layer, flow in supersonic nozzles, flow over airfoils, and many other practical problems. *See* Aerodynamics, Boundary-layer flow

## Bernoulli’s Theorem

one of the major theorems of the theory of probability; it is the simplest case of the so-called law of large numbers. Bernoulli’s theorem was first published in Jakob Bernoulli’s treatise *Ars Conjectandi,* published in 1713. The first proofs of Bernoulli’s theorem required complex mathematical methods, and only in the mid-19th century did P. L. Chebyshev find an unusually elegant and short proof of it. The exact formulation of Bernoulli’s theorem is the following: if for each of *n* independent trials the probability of a certain event is equal to *p,* then the probability that the frequency *m\n* of occurrence of the event satisfies the inequality *ǀm/n - pǀ< ε(ε* is an arbitrarily small positive number) becomes as close to 1 as desired for a sufficiently large number *n* of trials. The following simple quantitative estimate of this probability results from Chebyshev’s proof:

*P{ǀm/n - p \ <*ε} > 1 - *p(1 - p)/n*ε^{2}

V. I. BITIUTSKOV

## Bernoulli's theorem

*P*+ ¼ ρV

^{2}=

*K*where

*P*is the static pressure, ρ is the density,

*V*is the air speed, and

*K*is a constant. From this law, it follows that where there is a velocity increase in a fluid flow, there must be a corresponding pressure decrease. Thus an airfoil, by increasing the velocity of the flow over its upper surface, derives lift from the decreased pressure. Swiss physicist and mathematician Daniel Bernoulli formulated this theorem in 1738.