﻿ Bernoulli's principle

# Bernoulli's principle

As flow Bernoulli and Venturi refers developed by Giovanni Battista Venturi and Daniel Bernoulli in the 18th century theories of fluid mechanics that build on each other and form the basis for important aero- and hydrodynamic calculations.

## Venturi effect

The Italian Giovanni Battista Venturi discovered that the flow rate of a flowing incompressible fluid through a tube is inversely proportional to a changing pipe diameter. That is, the velocity of the fluid is at its greatest where the cross-section of the tube is smallest. According to the continuity equation for incompressible fluids occurs from any pipe section of the same amount of fluid that has been introduced into it. The fluid must be the bottleneck that is, with the same flow rate ( volume / time ) happen as the rest of the pipe. Therefore, the velocity of the fluid must increase mandatory.

This alone results in the following two questions:

## Bernoulli's law

Daniel Bernoulli discovered the relationship between the flow rate of a fluid and its pressure. He found out that, in a flowing fluid ( gas or liquid), a speed increase is accompanied by a pressure drop.

### Derivation of the energy equation

#### The mass point in the potential field

Considering a mass point of constant mass m, which moves only on the basis of a stationary force field which can be written as the gradient of a scalar potential, it satisfies the equation of motion:

Here, the speed and the acceleration. In classical physics many processes are described by an equation of this type. Examples are:

• Satellite movements, here is the Gravitionspotential and the acceleration due to gravity
• Harmonic oscillations, here is the potential of the restoring force
• Electrically charged particles in the electrostatic field. the electric potential of the field.

And linear combinations of these processes are described by the equation of motion.

Multiplying the equation of motion with scalar obtained after integrations with the substitution rule, the equations for the specific energy:

The sum of kinetic () and potential () of specific energy is constant.

#### The Navier -Stokes equations

The equation of motion of fluid mechanics, the Navier -Stokes equation. For stationarity and friction -free, they have the same shape as general above. The pressure here is the scalar potential of the force field. The above requirement of a constant grounding point mass is achieved by a constant density of the flow.

The energy equation is derived as above, and is the law of Bernoulli:

The term is also called dynamic pressure or back pressure. Its sum with the hydrostatic pressure is constant. is equivalent as the potentials discussed in the previous section as a potential. The flow is, for example, in the gravitational field of the earth, is added to the Bernoulli law Geopotential.

From the derivation it follows that the Bernoulli's law is valid for currents which meet all the following criteria:

• Incompressibility
• Valid only along a trajectory ( ground point )
• Stationarity (the pressure is only a function of position )
• No friction and no other forces (eg Coriolis force )

### Direct observation of the equation of motion

The Impulsadvektion in the Navier -Stokes equation writes itself based on the Weber transformation:

With the additional assumption that the flow is irrotational, the Navier -Stokes equation for steady frictionless flow of constant density is:

This equation can now be directly along any path integrate the Bernoulli's law:

Conclusion: Is the flow irrotational, the Bernoulli's law applies not only along a trajectory as from the energetics derived, but in the entire flow.

### Response to the open questions of paragraph Venturi effect

With the above derivations the open questions to be answered:

### Application

This principle can be found in everyday life in many things again. Here are some examples:

• The water jet pump.
• The velocity stacks of a carburetor. After its inventor, also the venturi flow meter and the venturi are named.
• The Prandtl Pitot tube, which, inter alia, is used to measure the speed of an aircraft. Due to the incompressibility requirement, it provides reliable results only in subsonic flight ( eg sport aircraft ).
• The flow around an airfoil is enough with the same limitation to the Bernoulli law. The equation describes sufficiently well the lift of aircraft wings up to speeds of 300 km / h (see chart).
• Venting of vessels by sea bream and cowl vent fan
• Ecological energy supply by vertical wind turbines in the Pearl River Tower ( a skyscraper in Guangzhou).

For these applications:

• As they describe no change in pressure along a trajectory, this eddy freedom of the entire flow should be required.
• The Bernoulli's law describes no causality, but a relation between velocity and pressure field.
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