# Bernoulli’s principle

# Bernoulli’s principle

Applying Bernoulli’s principle

Bernoulli’s principle states that the pressure exerted by a flowing fluid decreases quadratically with speed, that is, as a function of the speed squared.

## History

Over two hundred years ago, Daniel Bernoulli (1700–1782) pioneered the use of kinetic theory that molecules moved and bumped things. He also knew that flowing fluids pressed less, but he did not connect these ideas logically. In *Hydrodynamica,* Daniel’s logic that flow reduced pressure was obscure, and his formula was awkward. Daniel’s father Johann, amid controversy, improved his son’s insight and presentation in *Hydraulica.* This research was centered in St. Petersburg where Leonhard Euler, a colleague of Daniel and a student of Johann, generalized a rate-of-change dependence of pressure and density on speed of flow. Bernoulli’s principle for liquids was then formulated in modern form for the first time.

In this same group of scientists was d’Alembert, who found paradoxically that fluids stopped ahead of obstacles, so frictionless flow did not push.

Progress then seems to have halted for about a century and a half until Ludwig Prandtl or one of his students solved Euler’s equation for smooth streams of air in order to have a mathematical model of flowing air for designing wings. Here, speed lowers pressure more than it lowers density because expanding air cools, and the ratio of density times degrees-kelvin divided by pressure is constant for an ideal gas.

More turbulent flow, as in atmospheric winds, requires an alternative solution of Euler’s equation because mixing keeps air-temperature fixed.

## Applying Bernoulli’s principle

Bernoulli’s principle acts in concert with the phenomenon known as molecular entrainment. Molecules in the lower pressure of faster flow aspirate and whisk away molecules from the higher pressure of slower flow. Solid obstacles such as airfoils carry a very thin stagnant layer of air with them. A swift low-pressure airstream takes some molecules from this boundary layer and reduces molecular impacts on that surface of the wing across which the airstream moves faster.

For Bernoulli’s principle to dominate a dynamic situation, friction must be non-dominant. Elastic molecular impacts are frictionless. Molecules of dry air, even more than those of water, collide elastically, so Bernoulli’s principle with its molecular-entrainment agent is dominate for windy air.

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**Bernoulli’s principle**