Draupnir

The hydrostatic is the study of the motionless Fluide S. Founded by Archimedes, it is by far the simplest case of the Mécanique of the fluids, but it is however rich lesson.

Pressure in a fluid

In experiments, it is noted that the pressure in water depends only the depth and not on the direction. Indeed, if one takes a small open rigid box on a side and that one tightens an elastic membrane, this box locking up of the air with Atmospheric pressure, and that one plunges this box in water, the deformation of the membrane makes it possible to visualize the difference in pressure between the air and water, and this one depends only on the depth, not of the orientation of the box nor of its position in the horizontal plane.

Convention : in the example which follows, we direct the vertical axis to the bottom ( Z grows when one goes down).

Case of an incompressible fluid at rest in a uniform field of gravity

The fluid being incompressible, it transmits the efforts completely. The pressure with a depth Z thus results from the pressure P ₀ which the air on the surface exerts, and of the weight p of the water column above the membrane.

Let us suppose that the membrane horizontal and is directed upwards, and that its surface is S . The water column located above has as a volume S · Z , therefore for mass ρ· S · Z if ρ is the Density of water. The weight p of water is thus

$p\; =\; \backslash \; rho\; \backslash \; cdot\; G\; \backslash \; cdot\; (S\; \backslash \; cdot\; Z)$

where G is the acceleration of the Gravité. The membrane is then subjected to a force F

$F\; =\; P\_0\; \backslash \; cdot\; S\; +\; \backslash \; rho\; \backslash \; cdot\; G\; \backslash \; cdot\; (S\; \backslash \; cdot\; Z)$

that is to say a pressure

$P\; =\; \backslash \; frac\; \{F\}\; \{S\}\; =\; P\_0\; +\; \backslash \; rho\; \backslash \; cdot\; G\; \backslash \; cdot\; z$

It is this variation of the pressure according to the depth which creates the Poussée of Archimedes.

Barometer

Let us put a liquid in a closed tube on a side; this tube is immersed in a basin of liquid (it fills completely), then one places it vertically, the side closed in top, the open side soaking in the basin.

The atmospheric pressure being exerted on the surface of the liquid in the basin makes assemble the liquid in the tube. If the tube is sufficiently large, from the vacuum is obtained above the column of liquid in the tube (in fact, it is there in small quantity of the vapor of liquid to a very low pressure, the saturating Steam pressure).

By measuring the height H of the column, one can determine the atmospheric pressure:

$P\_0\; =\; \backslash \; rho\; \backslash \; cdot\; G\; \backslash \; cdot\; h$

This height is equivalent approximately to 10  m if the liquid is water, and with 76  cm if it is mercury. There is thus a Baromètre.

Now let us take a U-shaped tube of which each one of the end open and is connected to a tight enclosure. The tube contains a liquid. If the pressure reigning in the two enclosures is identical, the height of liquid is identical in the two branches. If the height differs from a value δ H , then the difference in pressure is worth:

$\backslash \; delta\; P\; =\; \backslash \; rho\; \backslash \; cdot\; G\; \backslash \; cdot\; \backslash \; delta\; h$

One expresses thus sometimes a low overpressure in height of the water column (mm EC).

Variations with this ideal case

When great variations of altitude are considered, one cannot regard the field of gravity any more as constant, G thus depends on Z . And when the fluid is a Gaz, one cannot regard this one any more as incompressible, therefore ρ also depends on Z ; but this is sensitive only for significant variations of pressure, therefore ρ being a weak in the case of gas, this intervenes only for rather large variations of Z .

Locally, for small variations dz of Z , one can always write:

$P\; (z+dz)\; =\; P\; (Z)\; +\; \backslash \; rho\; (Z)\; \backslash \; cdot\; G\; (Z)\; \backslash \; cdot\; dz$

This law should then be integrated:

$P\; (Z)\; =\; P\; (z\_0)\; +\; \backslash \; int\_0^\; \{z\_0\}\; \backslash \; rho\; (Z)\; \backslash \; cdot\; G\; (Z)\; \backslash \; cdot\; dz$

if one knows the law of reaction of gas, for example if it is about a Perfect gas, then for a mass m of gas given, one can connect volume V to the pressure P , therefore the density ρ with the pressure P :

$\backslash \; rho\; =\; \backslash \; rho\_0\; \backslash \; cdot\; \backslash \; frac\; \{P\}\; \{P\_0\}$

if ρ₀ and P ₀ are values at an altitude Z ₀ of reference.

In addition, the variation of gravity is calculated with the law of Newton.

In the case of the atmosphere, it is necessary moreover take into account the temperature variation and the variation of composition.

Application

Measure pressure

Barometer of Torriccelli, barometer out of U.

Pressure of water

Pumping by aspiration, Plunged underwater, Osmose reverses.

Meteorology

In Meteorology, the hydrostatic approximation stipulates that the vertical component of the compressive force is in exact balance with the gravitational force: the hydrostatic balance. It makes it possible to neglect, in the calculation of the pressure along the vertical axis, the forces due:

• with the horizontal or vertical movement of the air;

• with the Force of Coriolis.

It follows that the pressure, in any point of atmospheric volume, is only and directly proportional to the weight of the airstream above this point. This approximation is valid in particular with a great degree of accuracy in a very great number of the natural states of the atmosphere for the movements of large scales. It ceases being valid with small scales (< 10 km) and for intense systems like the tornadoes and the lines of grains.

See too

• a whole in closed circuit of a Hydraulic pump and a Moteur hydraulics is called by the professionals a transmission Hydrostatique
• hydraulic Pompe
• Moteur hydraulics

Related articles

• lithostatic Pressure

External bonds

• a video explanatory on the vases communicating

Simple: Fluid statics

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