Chapter 7

Bernoulli's equation

Fondamental : In the case of a perfect, steady, irrotational, incompressible and homogeneous flow

It is assumed in the following that the only volume force (other than the pression forces) is the weight.

We study the case of a perfect, steady, irrotational, incompressible and homogeneous flow.

The Euler equation becomes :

Noting : (the (Oz) axis is upward)

So :

Where :

A scalar field whose gradient is zero is independent of the point ; this is only a function of time .

As the flow is stationary, this function is constant :

This is Bernoulli's theorem, which states that the quantity remains in every point of the fluid equal to the same constant.

Remarque :

It is noted that  and denote the kinetic and potential energies density (gravity), homogeneous to a pressure.

Remarque :

In the particular case , there is the hydrostatic law of fluid :

Attention : Bernoulli's equation for a perfect, steady, irrotational, incompressible and homogeneous flow

The constant is the same for all points of the fluid.

Fondamental : For a perfect, stationary, incompressible and homogeneous flow

The Euler equation becomes now : ( )

Bernouilli's equation around a current line

In order to eliminate the curl term, we do a dot product this term with and we integrate it along the current line between points A and B :

At any point, is parallel to the velocity field and the term is null.

So :

Either :

Thus, the abandonment of the "irrotational flow" hypothesis restricted Bernoulli's theorem at points and in the same current line.

Attention : Bernoulli's equation for a perfect, stationary, incompressible and homogeneous flow

The abandonment of the "irrotational flow" hypothesis restricted Bernoulli's theorem at points and in the same current line :

Remarque : Energetical interpretation of Bernoulli's equation

We are in the case of a perfect, steady, irrotational, incompressible and homogeneous flow. So :

By multiplying with the volume of a fluid particle :

We recognize :

•  : gravitational potential energy of the fluid particle.

•    : kinetic energy of the fluid particle.

•  : potential energy associated with the pressure forces.

Bernoulli's theorem, more general than the simple conservation of mechanical energy of a material point in a non-continuous medium, is a special case of the first law of thermodynamics :

With and : the implicit assumptions of the application of Bernoulli's theorem are isothermal and adiabatic aspects of evolution.

Indeed, for a perfect fluid, the internal energy depends only on the temperature.

Furthermore, the aspect of adiabatic transformation is not problematic ; in the absence of heat transfer from the outside of the fluid, there 's no heat within a non-viscous fluid.

Complément : The phenomenon of Venturi

A homogeneous, steady, incompressible flow and subject only to pressure forces, is limited by a duct with variable section.

The problem is one-dimensional : all quantities have a uniform value on a straight section of the pipe.

The phenomenon of Venturi

The conservation of volume flow rate between the two cross-sections and gives :

Uncompressible flow and fluid velocity

The application of the Bernoulli's theorem between two points and located on the same horizontal gives :

We deduce that  : the small section parts, so high speed, are also areas of low pressure (Venturi effect).

In the vertical tubes, the fluid is immobile and liquid heights measure pressures and  :

We deduce the pressure difference :

Can then deduce the volume flow rate in the pipe, by calculating beforehand :

Where :

And the volumetric flow rate is :

The Venturi tube may also serve as a flow rate meter.

Remark :

The Venturi effect remains verified by a compressible gas such as air, as long as the speed remains below the velocity of sound propagation.

Venturi effect (Vidéo d'Alain Le Rille, enseignant en CPGE au lycée Janson de Sailly, Paris)
A few applications of Bernoulli's equation

Exemple : Some videos of experiments

A recalcitrant ping-pong ball

A recalcitrant ping-pong ball.

Levitate a ping-pong ball

Levitate a ping-pong ball
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