# Hydrostatic pressure

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The hydrostatic pressure ( ancient Greek ὕδωρ hýdor , German 'water' ), also gravitational pressure or gravity pressure , is the pressure that is created within a stationary fluid , that is a liquid or a gas, through the influence of gravity . In closed vessels, constant hydrostatic pressure can also occur in weightless spaces. Contrary to the meaning of the word “water”, the term is also used for other liquids and even for gases, as it represents the type of stress tensor that also occurs in water at rest (i.e. free from shear stress).

Dynamic pressure from fluid flows such as B. the dynamic pressure is not recorded by the hydrostatic pressure .

## Incompressible liquids in a homogeneous gravity field

### Pascal's law

The pressure increases with the depth of the water. In addition to the hydrostatic pressure, there is also the air pressure on the water surface. The various scales on the y-axis must be observed : the pressure in the water column rises much faster than in the air column .
The hydrostatic pressure on the bottom is the same in all three vessels despite the different filling quantities.

The hydrostatic pressure for fluids with constant density in a homogeneous gravitational field (=  incompressible fluids , especially liquids ) is calculated according to Pascal's (or Pascal's ) law (named after Blaise Pascal ):

${\ displaystyle p (h) = \ rho gh + p_ {0}}$
${\ displaystyle \ rho}$= Density [for water : ≈ 1,000 kg / m³]${\ displaystyle \ rho}$
${\ displaystyle g}$= Acceleration due to gravity [for Germany: ≈ 9.81 m / s²]${\ displaystyle g}$
${\ displaystyle h}$= Height of the liquid level above the point under consideration
${\ displaystyle p_ {0}}$= Air pressure on the surface of the liquid
${\ displaystyle p (h)}$ = hydrostatic pressure as a function of the height of the liquid level.

#### units

The physical units for hydrostatic pressure are:

• internationally the SI unit
Pascal (Pa): 1 Pa = 1 N / m²;
• also in Germany and Austria the "legal unit"
Bar (bar): 1 bar = 100,000 Pa or N / m² (= 100  kPa )

To describe the hydrostatic pressure, the non- SI -compliant, outdated unit of measurement, the meter water column (mWS), is sometimes used.

#### Example of the hydrostatic paradox

• Water column , homogeneous water temperature: 3.98 ° C, height: 50 meters:
1,000 kg / m³ × 9.81 m / s² × 50 m ≈ 490,500 N / m² ≈ 4.90 bar
At a temperature of 20 ° C, water has a density of 998.203 kg / m³. The hydrostatic pressure changes minimally to
998.203 kg / m³ × 9.81 m / s² × 50 m ≈ 489,618.57 N / m² ≈ 4.90 bar

The hydrostatic pressure does not depend on the shape of a vessel; critical to the pressure at the bottom of which is itself the height of the fluid - or liquid level and the density of which (depending on the temperature), but not the absolute amount of the fluid in the vessel. This phenomenon came to be known as the Hydrostatic (or Pascal's ) paradox .

#### Total pressure (absolute pressure) at the bottom of the liquid

For a complete description of the pressure at the bottom of an incompressible fluid at rest, however, the ambient pressure has to be added to the hydrostatic pressure. For example, the water pressure acting on a diver in a body of calm water corresponds to the sum

from the air pressure that acts on the surface of the water plus the hydrostatic pressure of the water itself.

### Examples

• It is important for divers to know what pressure their body gases ( nitrogen ) are exposed to in order to avoid diving illness .
• A bathyscaphe has to withstand particularly high hydrostatic pressure.
• Water towers use the hydrostatic pressure to generate the line pressure necessary to supply end users.
• In hydrogeology , according to Darcy's law, a flow between two points can only be established if the pressure difference is different from the difference in hydrostatic pressures at the two points.
• A siphon is a device or a device with which a liquid can be transferred from a container over the edge of the container to a lower container or emptied into the open without tipping the container over and without a hole or an outlet under the liquid level.

## Continuum mechanics

The hydrostatic axis in the Haight-Westergaard coordinate system is times the mean stress in 3D :${\ displaystyle {\ sqrt {3}}}$${\ displaystyle \ xi = - {\ sqrt {3}} \, p}$

In every point (whether in a fluid, a solid or in a vacuum) there is a stress tensor # pressure

${\ displaystyle {\ boldsymbol {\ sigma}} = {\ begin {pmatrix} \ sigma _ {11} & \ sigma _ {12} & \ sigma _ {13} \\\ sigma _ {21} & \ sigma _ {22} & \ sigma _ {23} \\\ sigma _ {31} & \ sigma _ {32} & \ sigma _ {33} \ end {pmatrix}} \ ,,}$

this consists of a hydrostatic part

${\ displaystyle {\ boldsymbol {\ sigma}} _ {\ mathrm {hydro}} = - p \, {\ boldsymbol {I}} = {\ begin {pmatrix} -p & 0 & 0 \\ 0 & -p & 0 \\ 0 & 0 & -p \ end {pmatrix}} \ ,,}$

with the hydrostatic pressure and a deviatoric part . ${\ displaystyle p = {\ frac {\ sigma _ {11} + \ sigma _ {22} + \ sigma _ {33}} {3}}}$${\ displaystyle {\ boldsymbol {\ sigma}} _ {\ mathrm {dev}} = {\ boldsymbol {\ sigma}} - {\ boldsymbol {\ sigma}} _ {\ mathrm {hydro}}}$

In the case of isotropic (= direction-independent) materials, the failure area is usually given as a function of the hydrostatic and deviatoric component (for example the Mises stress or the Drucker-Prague failure criterion ); the Haight-Westergaard coordinate system is often used for this , where the hydrostatic axis represents a line and the deviatic plane spans the three-dimensional main stress space orthonally to it.

## Gravitational pressure in planets, moons, asteroids and meteorites

### Dependence of g

With increasing depth, it can no longer be regarded as constant. If the shape of the celestial body is described by a sphere with a radius and the density is considered to be constant, the pressure can be calculated as follows: ${\ displaystyle g}$${\ displaystyle R}$

${\ displaystyle p (h) = \ int _ {0} ^ {h} \ rho \, g (Rr) \, \ mathrm {d} r}$ .

The spatial factor follows from Newton's law of gravitation : ${\ displaystyle g (r)}$

${\ displaystyle g (r) = G {\ frac {M (r)} {r ^ {2}}} = G {\ frac {Mr} {R ^ {3}}} = \ rho r {\ frac { 4 \ pi G} {3}}}$,

where indicates the mass within a concentric sphere within the celestial body and its total mass. The formula for the spherical volume gives the pressure in the center: ${\ displaystyle M (r)}$${\ displaystyle M = M (R)}$ ${\ displaystyle V = {\ tfrac {4} {3}} \ pi R ^ {3}}$

${\ displaystyle p _ {\ text {Z}} = p (R) = {\ frac {3G} {8 \ pi}} {\ frac {M ^ {2}} {R ^ {4}}} = \ rho ^ {2} R ^ {2} {\ frac {2 \ pi G} {3}}}$.

## Gravitational pressure in stars

### Stars in balance

Gravitational pressure in stars is a special case of hydrostatic pressure . This results from the force of gravity contracting the star . In contrast, z. B. the radiation pressure as the star expanding force. In the case of a stable star, a balance of all forces is established and the star has a stable shape. This is approximately the state of stars on the main sequence of the Hertzsprung-Russell diagram .

### Examples of stars in imbalance

In the case of emerging stars that contract, the gravitational pressure outweighs the sum of all forces that build up counter pressure. Examples of back pressure are the kinetic gas pressure of the gas itself and, when the fusion reaction starts, the radiation pressure caused by all types of radiation. This changes the hydrostatic pressure within the emerging star.

In some classes of variable stars , periodic or transient changes in star density occur, causing the star's amount of matter inside or outside a sphere with a fixed radius to change, and with it the hydrostatic pressure at a given radius from the star's center.

Due to the stellar wind , stars steadily lose mass to their surroundings. This also changes the hydrostatic pressure. For main sequence stars, however, this change is very slow.

In the later stages of star life, changes in the star structure also occur , which affect the hydrostatic pressure in the star.

## Individual evidence

1. ^ Lew Dawidowitsch Landau , Jewgeni Michailowitsch Lifschitz : Statistical Physics. Part I. Akademie Verlag , Berlin 1979/1987, ISBN 3-05-500069-2 , p. 70.