Editing Pressure (section)

==Types==

===Fluid pressure===
''Fluid pressure'' is most often the compressive stress at some point within a [[fluid]].  (The term ''fluid'' refers to both liquids and gases – for more information specifically about liquid pressure, see [[#Liquid pressure|section below]].)
[[File:Defekter unterflurhydrant goettingen.jpg|thumb|upright=1.2|Water escapes at high speed from a damaged hydrant that contains water at high pressure]]
Fluid pressure occurs in one of two situations:
* An open condition, called "open channel flow", e.g. the ocean, a swimming pool, or the atmosphere.
* A closed condition, called "closed conduit", e.g. a water line or gas line.

Pressure in open conditions usually can be approximated as the pressure in "static" or non-moving conditions (even in the ocean where there are waves and currents), because the motions create only negligible changes in the pressure. Such conditions conform with principles of [[fluid statics]]. The pressure at any given point of a non-moving (static) fluid is called the ''hydrostatic pressure''.

Closed bodies of fluid are either "static", when the fluid is not moving, or "dynamic", when the fluid can move as in either a pipe or by compressing an air gap in a closed container. The pressure in closed conditions conforms with the principles of [[fluid dynamics]].

The concepts of fluid pressure are predominantly attributed to the discoveries of [[Blaise Pascal]] and [[Daniel Bernoulli]]. [[Bernoulli's equation]] can be used in almost any situation to determine the pressure at any point in a fluid. The equation makes some assumptions about the fluid, such as the fluid being ideal<ref name=Finnemore>{{cite book |last=Finnemore, John, E. and Joseph B. Franzini |title=Fluid Mechanics: With Engineering Applications |year=2002 |publisher=McGraw Hill, Inc. |location=New York |isbn=978-0-07-243202-2 |pages=14–29}}</ref> and incompressible.<ref name=Finnemore/> An ideal fluid is a fluid in which there is no friction, it is [[inviscid]]<ref name=Finnemore/> (zero [[viscosity]]).<ref name=Finnemore/> The equation for all points of a system filled with a constant-density fluid is<ref name=NCEES>{{cite book |title=Fundamentals of Engineering: Supplied Reference Handbook|year=2011|publisher=NCEES|location=Clemson, South Carolina|isbn=978-1-932613-59-9|page=64}}</ref>
<math display="block">\frac{p}{\gamma} + \frac{v^2}{2g} + z = \mathrm{const},</math>

where:
*''p'', pressure of the fluid,
*''<math>{\gamma}</math>'' = ''ρg'', density × acceleration of gravity is the (volume-) [[specific weight]] of the fluid,<ref name=Finnemore/>
*''v'', velocity of the fluid,
*''g'', [[gravitational acceleration|acceleration of gravity]],
*''z'', elevation,
*<math>\frac{p}{\gamma}</math>, pressure head,
*<math>\frac{v^2}{2g}</math>, velocity head.

====Applications====
* [[Hydraulic brakes]]
* [[Artesian well]]
* [[Blood pressure]]
* [[Hydraulic head]]
* [[Turgor pressure|Plant cell turgidity]]
* [[Pythagorean cup]]
* [[Pressure washing]]

===Explosion or deflagration pressures===
[[Explosion]] or [[deflagration]] pressures are the result of the ignition of explosive [[gas]]es, mists, dust/air suspensions, in unconfined and confined spaces.

=== Negative pressures ===
[[File:13-07-23-kienbaum-unterdruckkammer-33.jpg|thumb|Low-pressure chamber in [[Bundesleistungszentrum Kienbaum]], Germany]]
While ''pressures'' are, in general, positive, there are several situations in which negative pressures may be encountered:
*When dealing in relative (gauge<!--Editors are asked to PLEASE check the discussion page for this article before making changes regarding "gauge" vs. "gage" spelling issues. Much debate has transpired on this issue.-->) pressures. For instance, an absolute pressure of 80&nbsp;kPa may be described as a gauge<!--Editors are asked to PLEASE check the discussion page for this article before making changes regarding "gauge" vs. "gage" spelling issues. Much debate has transpired on this issue.--> pressure of −21&nbsp;kPa (i.e., 21&nbsp;kPa below an atmospheric pressure of 101&nbsp;kPa). For example, [[abdominal decompression]] is an [[obstetric]] procedure during which negative gauge pressure is applied intermittently to a pregnant woman's abdomen.
*Negative absolute pressures are possible. They are effectively [[tension (physics)|tension]], and both bulk solids and bulk liquids can be put under negative absolute pressure by pulling on them.<ref name="Imre2007">{{cite book|last1=Imre|first1=A. R.|title=Soft Matter under Exogenic Impacts|chapter=How to generate and measure negative pressure in liquids?|series=NATO Science Series II: Mathematics, Physics and Chemistry | volume=242|year=2007|pages=379–388|issn=1568-2609|doi=10.1007/978-1-4020-5872-1_24|isbn=978-1-4020-5871-4}}</ref> Microscopically, the molecules in solids and liquids have attractive interactions that overpower the thermal kinetic energy, so some tension can be sustained. Thermodynamically, however, a bulk material under negative pressure is in a [[metastable]] state, and it is especially fragile in the case of liquids where the negative pressure state is similar to [[superheating]] and is easily susceptible to [[cavitation]].<ref name=liqneg>{{cite book |title=Liquids Under Negative Pressure (Nato Science Series II) |date=2002 |publisher=Springer | isbn=978-1-4020-0895-5 | doi=10.1007/978-94-010-0498-5|editor1-last=Imre |editor1-first=A. R |editor2-last=Maris |editor2-first=H. J |editor3-last=Williams |editor3-first=P. R }}</ref> In certain situations, the cavitation can be avoided and negative pressures sustained indefinitely,<ref name=liqneg/> for example, liquid mercury has been observed to sustain up to {{val|-425|u=atm}} in clean glass containers.<ref name="Briggs1953">{{cite journal|last1=Briggs|first1=Lyman J.|title=The Limiting Negative Pressure of Mercury in Pyrex Glass|journal=Journal of Applied Physics|volume=24|issue=4|year=1953|pages=488–490|issn=0021-8979|doi=10.1063/1.1721307|bibcode=1953JAP....24..488B}}</ref> Negative liquid pressures are thought to be involved in the [[ascent of sap]] in plants taller than 10&nbsp;m (the atmospheric [[pressure head]] of water).<ref>{{cite web | title=The Physics of Negative Pressure | url=http://discovermagazine.com/2003/mar/featscienceof | publisher=[[Discover (magazine)|Discover]] | author=Karen Wright | access-date=31 January 2015 | date=March 2003 | url-status=live | archive-url=https://web.archive.org/web/20150108182004/http://discovermagazine.com/2003/mar/featscienceof | archive-date=8 January 2015 }}</ref>
*The [[Casimir effect]] can create a small attractive force due to interactions with [[vacuum energy]]; this force is sometimes termed "vacuum pressure" (not to be confused with the negative ''gauge<!--Editors are asked to PLEASE check the discussion page for this article before making changes regarding "gauge" vs. "gage" spelling issues. Much debate has transpired on this issue.--> pressure'' of a vacuum).
*For non-isotropic stresses in rigid bodies, depending on how the orientation of a surface is chosen, the same distribution of forces may have a component of positive stress along one [[Normal (geometry)|surface normal]], with a component of negative stress acting along another surface normal. The pressure is then defined as the average of the three principal stresses.
**The stresses in an [[electromagnetic field]] are generally non-isotropic, with the stress normal to one surface element (the [[normal stress]]) being negative, and positive for surface elements perpendicular to this.
*In [[cosmology]], [[dark energy]] creates a very small yet cosmically significant amount of negative pressure, which accelerates the [[expansion of the universe]].

===Stagnation pressure===<!--This section is linked from [[Drag equation]]-->
{{Main|Stagnation pressure}}
[[Stagnation pressure]] is the pressure a fluid exerts when it is forced to stop moving. Consequently, although a fluid moving at higher speed will have a lower [[static pressure]], it may have a higher stagnation pressure when forced to a standstill. Static pressure and stagnation pressure are related by:
<math display="block">p_{0} = \frac{1}{2}\rho v^2 + p</math>
where 
*<math>p_0</math> is the [[stagnation pressure]],
*<math>\rho</math> is the density,
*<math>v</math> is the flow velocity,
*<math>p</math> is the static pressure.

The pressure of a moving fluid can be measured using a [[Pitot tube]], or one of its variations such as a [[Kiel probe]] or [[Cobra probe]], connected to a [[manometer]]. Depending on where the inlet holes are located on the probe, it can measure static pressures or stagnation pressures.

===Surface pressure and surface tension===
There is a two-dimensional analog of pressure – the lateral force per unit length applied on a line perpendicular to the force.

Surface pressure is denoted by π:
<math display="block">\pi = \frac{F}{l}</math>
and shares many similar properties with three-dimensional pressure. Properties of surface chemicals can be investigated by measuring pressure/area isotherms, as the two-dimensional analog of [[Boyle's law]], {{nowrap|''πA'' {{=}} ''k''}}, at constant temperature.

[[Surface tension]] is another example of surface pressure, but with a reversed sign, because "tension" is the opposite to "pressure".

=== Pressure of an ideal gas ===
{{main|Ideal gas law}}
In an [[ideal gas]], molecules have no volume and do not interact. According to the [[ideal gas law]], pressure varies linearly with temperature and quantity, and inversely with volume:
<math display="block">p = \frac{nRT}{V},</math>
where:
*''p'' is the absolute pressure of the gas,
*''n'' is the [[amount of substance]],
*''T'' is the absolute temperature,
*''V'' is the volume,
*''R'' is the [[ideal gas constant]].

[[Real gas]]es exhibit a more complex dependence on the variables of state.<ref>P. Atkins, J. de Paula ''Elements of Physical Chemistry'', 4th Ed, W. H. Freeman, 2006. {{ISBN|0-7167-7329-5}}.</ref>

===Vapour pressure===
{{main|Vapour pressure}}
Vapour pressure is the pressure of a [[vapour]] in [[thermodynamic equilibrium]] with its condensed [[Phase (matter)|phase]]s in a closed system. All liquids and [[solid]]s have a tendency to [[evaporate]] into a gaseous form, and all [[gas]]es have a tendency to [[condense]] back to their liquid or solid form.

The [[atmospheric pressure]] [[boiling point]] of a liquid (also known as the [[normal boiling point]]) is the temperature at which the vapor pressure equals the ambient atmospheric pressure. With any incremental increase in that temperature, the vapor pressure becomes sufficient to overcome atmospheric pressure and lift the liquid to form vapour bubbles inside the bulk of the substance. [[liquid bubble|Bubble]] formation deeper in the liquid requires a higher pressure, and therefore higher temperature, because the fluid pressure increases above the atmospheric pressure as the depth increases.

The vapor pressure that a single component in a mixture contributes to the total pressure in the system is called [[partial pressure|partial vapor pressure]].

===Liquid pressure===
{{See also|Fluid statics#Pressure in fluids at rest}}
{{Continuum mechanics}}
When a person swims under the water, water pressure is felt acting on the person's eardrums. The deeper that person swims, the greater the pressure. The pressure felt is due to the weight of the water above the person. As someone swims deeper, there is more water above the person and therefore greater pressure. The pressure a liquid exerts depends on its depth.

Liquid pressure also depends on the density of the liquid. If someone was submerged in a liquid more dense than water, the pressure would be correspondingly greater. Thus, we can say that the depth, density and liquid pressure are directly proportionate. The pressure due to a liquid in liquid columns of constant density and gravity at a depth within a substance is represented by the following formula:
<math display="block">p = \rho gh,</math>
where:
*''p'' is liquid pressure,
*''g'' is gravity at the surface of overlaying material,
*''ρ'' is [[density]] of liquid,
*''h'' is height of liquid column or depth within a substance.

Another way of saying the same formula is the following:
<math display="block">p = \text{weight density} \times \text{depth}.</math>

{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!Derivation of this equation
|-
|This is derived from the definitions of pressure and weight density. Consider an area at the bottom of a vessel of liquid. The weight of the column of liquid directly above this area produces pressure. From the definition
<math display="block">\text{weight density} = \frac{\text{weight}}{\text{volume}}</math>
we can express this weight of liquid as
<math display="block">\text{weight} = \text{weight density} \times \text{volume},</math>
where the volume of the column is simply the area multiplied by the depth. Then we have
<math display="block">\text{pressure} = \frac{\text{force}}{\text{area}} = \frac{\text{weight}}{\text{area}} = \frac{\text{weight density} \times \text{volume}}{\text{area}},</math>
<math display="block">\text{pressure} = \frac{\text{weight density} \times \text{(area} \times \text{depth)}}{\text{area}}.</math>

With the "area" in the numerator and the "area" in the denominator canceling each other out, we are left with
<math display="block">\text{pressure} = \text{weight density} \times \text{depth}.</math>

Written with symbols, this is our original equation:
<math display="block">p = \rho gh.</math>
|}

The pressure a liquid exerts against the sides and bottom of a container depends on the density and the depth of the liquid. If atmospheric pressure is neglected, liquid pressure against the bottom is twice as great at twice the depth; at three times the depth, the liquid pressure is threefold; etc. Or, if the liquid is two or three times as dense, the liquid pressure is correspondingly two or three times as great for any given depth. Liquids are practically incompressible – that is, their volume can hardly be changed by pressure (water volume decreases by only 50 millionths of its original volume for each atmospheric increase in pressure). Thus, except for small changes produced by temperature, the density of a particular liquid is practically the same at all depths.

Atmospheric pressure pressing on the surface of a liquid must be taken into account when trying to discover the ''total'' pressure acting on a liquid. The total pressure of a liquid, then, is ''ρgh'' plus the pressure of the atmosphere. When this distinction is important, the term ''total pressure'' is used. Otherwise, discussions of liquid pressure refer to pressure without regard to the normally ever-present atmospheric pressure.

The pressure does not depend on the ''amount'' of liquid present. Volume is not the important factor – depth is. The average water pressure acting against a dam depends on the average depth of the water and not on the volume of water held back. For example, a wide but shallow lake with a depth of {{convert|3|m|0|abbr=on}} exerts only half the average pressure that a small {{convert|6|m|abbr=on}} deep pond does. (The ''total force'' applied to the longer dam will be greater, due to the greater total surface area for the pressure to act upon. But for a given {{convert|5|ft|m|adj=on}}-wide section of each dam, the {{convert|10|ft|m|abbr=on}} deep water will apply one quarter the force of {{convert|20|ft|m|abbr=on}} deep water). A person will feel the same pressure whether their head is dunked a metre beneath the surface of the water in a small pool or to the same depth in the middle of a large lake.

If four interconnected vases contain different amounts of water but are all filled to equal depths, then a fish with its head dunked a few centimetres under the surface will be acted on by water pressure that is the same in any of the vases. If the fish swims a few centimetres deeper, the pressure on the fish will increase with depth and be the same no matter which vase the fish is in. If the fish swims to the bottom, the pressure will be greater, but it makes no difference which vase it is in. All vases are filled to equal depths, so the water pressure is the same at the bottom of each vase, regardless of its shape or volume. If water pressure at the bottom of a vase were greater than water pressure at the bottom of a neighboring vase, the greater pressure would force water sideways and then up the neighboring vase to a higher level until the pressures at the bottom were equalized. Pressure is depth dependent, not volume dependent, so there is a reason that water seeks its own level.

Restating this as an energy equation, the energy per unit volume in an ideal, incompressible liquid is constant throughout its vessel. At the surface, [[gravitational energy|gravitational potential energy]] is large but liquid pressure energy is low. At the bottom of the vessel, all the gravitational potential energy is converted to pressure energy. The sum of pressure energy and gravitational potential energy per unit volume is constant throughout the volume of the fluid and the two energy components change linearly with the depth.<ref>Streeter, V. L., ''Fluid Mechanics'', Example&nbsp;3.5, McGraw&ndash;Hill Inc. (1966), New York.</ref>  Mathematically, it is described by [[Bernoulli's equation]], where velocity head is zero and comparisons per unit volume in the vessel are
<math display="block">\frac{p}{\gamma} + z = \mathrm{const}.</math>

Terms have the same meaning as in [[#Fluid pressure|section Fluid pressure]].

===Direction of liquid pressure===
An experimentally determined fact about liquid pressure is that it is exerted equally in all directions.<ref name="Hewitt">Hewitt 251 (2006){{full citation needed|date=July 2020}}</ref> If someone is submerged in water, no matter which way that person tilts their head, the person will feel the same amount of water pressure on their ears. Because a liquid can flow, this pressure is not only downward. Pressure is seen acting sideways when water spurts sideways from a leak in the side of an upright can. Pressure also acts upward, as demonstrated when someone tries to push a beach ball beneath the surface of the water. The bottom of a ball is pushed upward by water pressure ([[buoyancy]]).

When a liquid presses against a surface, there is a net force that is perpendicular to the surface. Although pressure does not have a specific direction, force does. A submerged triangular block has water forced against each point from many directions, but components of the force that are not perpendicular to the surface cancel each other out, leaving only a net perpendicular point.<ref name="Hewitt" /> This is why liquid particles' velocity only alters in a [[normal (geometry)|normal]] component after they are collided to the container's wall. Likewise, if the collision site is a hole, water spurting from the hole in a bucket initially exits the bucket in a direction at right angles to the surface of the bucket in which the hole is located. Then it curves downward due to gravity. If there are three holes in a bucket (top, bottom, and middle), then the force vectors perpendicular to the inner container surface will increase with increasing depth – that is, a greater pressure at the bottom makes it so that the bottom hole will shoot water out the farthest. The force exerted by a fluid on a smooth surface is always at right angles to the surface. The speed of liquid out of the hole is <math>\scriptstyle \sqrt{2gh}</math>, where ''h'' is the depth below the free surface.<ref name="Hewitt" /> As predicted by [[Torricelli's law]] this is the same speed the water (or anything else) would have if freely falling the same vertical distance ''h''.

===Kinematic pressure===
<math display="block">P = \frac{p}{\rho_0}</math>
is the kinematic pressure, where <math>p</math> is the pressure and <math>\rho_0</math> constant mass density. The SI unit of ''P'' is m<sup>2</sup>/s<sup>2</sup>. Kinematic pressure is used in the same manner as [[kinematic viscosity]] <math>\nu</math> in order to compute the [[Navier–Stokes equations|Navier–Stokes equation]] without explicitly showing the density <math>\rho_0</math>.

==== Navier–Stokes equation with kinematic quantities ====
<math display="block"> \frac{\partial u}{\partial t} + (u \nabla) u = - \nabla P + \nu \nabla^2 u. </math>