# Fluid Mechanics

# Bernoulli's equation

### Complément : A video (in french) about Bernoulli's equation (with some fun science experiments)

("Unisciel" and "Culture Sciences Physiques")

### 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 : ( )

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

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.

### Exemple : Some videos of experiments

*A recalcitrant ping-pong ball*

*Levitate a ping-pong ball*