View source for Physics:Pressure

{{short description|Force distributed over an area}}
{{Infobox physical quantity
| name = Pressure
| unit = [[Pascal (unit)|pascal]] (Pa)
| baseunits = [[Kilogram|kg]]&sdot;[[Metre|m]]{{sup|−1}}&sdot;[[Second|s]]{{sup|−2}}
| symbols = {{mvar|p}}, {{mvar|P}}
| derivations = {{math|1=''p'' = [[Physics:Force|''F'']] / [[Area|''A'']]}}
| dimension = <math>\mathsf{M} \mathsf{L}^{-1} \mathsf{T}^{-2}</math>
}}

[[File:Pressure exerted by collisions.svg|thumb|right|250px|alt=A figure showing pressure exerted by particle collisions inside a closed container. The collisions that exert the pressure are highlighted in red.|Pressure as exerted by particle collisions inside a closed container]]
{{Thermodynamics|cTopic=[[Physics:List of thermodynamic properties|System properties]]}}

'''Pressure''' (symbol: ''p'' or ''P'') is the [[Physics:Force|force]] applied [[Perpendicular|perpendicular]] to the surface of an object per unit [[Area|area]] over which that force is distributed.<ref>{{cite book |last1=Knight, PhD |first1=Randall D. |title=Physics for Scientists and Engineers: A Strategic Approach |date=2007 |publisher=Pearson Addison Wesley |location=San Francisco |isbn=978-0-321-51671-8 |pages=1183 |edition=2nd |url=https://books.google.com/books?id=32yzygAACAAJ&q=Physics+for+Scientists+and+Engineers:+A+Strategic+Approach |access-date=6 April 2020 |language=en-us |format=google books |chapter=Fluid Mechanics |quote={{small|''Pressure itself is not a Force, even though we sometimes talk "informally" about the "force exerted by the pressure. The correct statement is that the Fluid exerts a force on a surface. In addition, Pressure is a scalar, not a vector.'' }}}}</ref>{{rp|445}} Gauge pressure (also spelled ''gage'' pressure)<ref group=lower-alpha>The preferred spelling varies by country and even by industry. Further, both spellings are often used ''within'' a particular industry or country. Industries in British English-speaking countries typically use the "gauge" spelling.</ref> is the pressure relative to the ambient pressure.

Various units are used to express pressure. Some of these derive from a unit of force divided by a unit of area; the [[International System of Units|SI]] unit of pressure, the [[Pascal (unit)|pascal]] (Pa), for example, is one [[Newton (unit)|newton]] per [[Square metre|square metre]] (N/m<sup>2</sup>); similarly, the [[Pound (force)|pound-force]] per [[Square inch|square inch]] ([[Pound per square inch|psi]], symbol lbf/in<sup>2</sup>) is the traditional unit of pressure in the [[Imperial units|imperial]] and US customary systems. Pressure may also be expressed in terms of [[Physics:Standard atmosphere (unit)|standard atmospheric pressure]]; the [[Physics:Atmosphere (unit)|atmosphere]] (atm) is equal to this pressure, and the [[Torr|torr]] is defined as {{frac|760}} of this. Manometric units such as the [[Physics:Centimetre of water|centimetre of water]], [[Physics:Millimetre of mercury|millimetre of mercury]], and [[Inch of mercury|inch of mercury]] are used to express pressures in terms of the height of [[Physics:Pressure measurement#Liquid column (manometer)|column of a particular fluid]] in a manometer.

==Definition==
''Pressure'' is the amount of force applied [[Perpendicular|perpendicular]] to the surface of an object per unit area. The symbol for it is "p" or ''P''.<ref>
{{Cite book |last=Giancoli |first=Douglas G. |title=Physics: principles with applications |year=2004 |publisher=Pearson Education |location=Upper Saddle River, N.J. |isbn=978-0-13-060620-4 |url-access=registration |url=https://archive.org/details/physics00doug }}</ref>
The IUPAC recommendation for pressure is a lower-case ''p''.<ref name="IUPACGoldPressure">
{{cite book|url = http://goldbook.iupac.org/P04819.html|title = IUPAC. Compendium of Chemical Terminology, 2nd ed. (the "Gold Book")|publisher = Blackwell Scientific Publications|author1 = McNaught, A. D.|author2 = Wilkinson, A.|author3 = Nic, M.|author4 = Jirat, J.|author5 = Kosata, B.|author6 = Jenkins, A.|year = 2014|location = Oxford|isbn = 978-0-9678550-9-7|doi = 10.1351/goldbook.P04819|version = 2.3.3|url-status = live|archive-url = https://web.archive.org/web/20160304071856/http://goldbook.iupac.org/E02281.html|archive-date = 2016-03-04}}</ref>
However, upper-case ''P'' is widely used. The usage of ''P'' vs ''p'' depends upon the field in which one is working, on the nearby presence of other symbols for quantities such as [[Physics:Power|power]] and [[Physics:Momentum|momentum]], and on writing style.

===Formula===
{{Conjugate variables (thermodynamics)}}
[[File:Pressure force area.svg|200px|right]]
Mathematically:<ref name="Georgia State University, Physics & Astronomy">{{cite web |url=http://hyperphysics.phy-astr.gsu.edu/hbase/press.html |title=Pressure |author=R Nave |website=Hyperphysics |publisher=Georgia State University, Dept. of Physics and Astronomy |access-date=2022-03-05 }}</ref>
<math display="block">p = \frac{F}{A},</math>
where:
*<math>p</math> is the pressure,
*<math>F</math> is the magnitude of the [[Physics:Normal force|normal force]],
*<math>A</math> is the area of the surface on contact.

Pressure is a [[Physics:Scalar|scalar]] quantity. It relates the [[Vector area|vector area]] element (a vector normal to the surface) with the [[Physics:Normal force|normal force]] acting on it. The pressure is the scalar proportionality constant that relates the two normal vectors:
<math display="block">d\mathbf{F}_n = -p\,d\mathbf{A} = -p\,\mathbf{n}\,dA.</math>

The minus sign comes from the convention  that the force is considered towards the surface element, while the normal vector points outward. The equation has meaning in that, for any surface ''S'' in contact with the fluid, the total force exerted by the fluid on that surface is the [[Surface integral|surface integral]] over ''S'' of the right-hand side of the above equation.

It is incorrect (although rather usual) to say "the pressure is directed in such or such direction". The pressure, as a scalar, has no direction. The force given by the previous relationship to the quantity has a direction, but the pressure does not. If we change the orientation of the surface element, the direction of the normal force changes accordingly, but the pressure remains the same.{{citation needed|date=October 2021}}

Pressure is distributed to solid boundaries or across arbitrary sections of fluid ''normal to'' these boundaries or sections at every point. It is a fundamental parameter in [[Physics:Thermodynamics|thermodynamics]], and it is [[Physics:Conjugate variables (thermodynamics)|conjugate]] to [[Physics:Volume (thermodynamics)|volume]].<ref>{{cite journal |last1=Alberty |first1=Robert A. |title=USE OF LEGENDRE TRANSFORMS IN CHEMICAL THERMODYNAMICS (IUPAC Technical Report) |journal=Pure Appl. Chem. |date=2001 |volume=73 |issue=8 |pages=1349–1380 |doi=10.1351/pac200173081349 |s2cid=98264934 |url=http://publications.iupac.org/pac/2001/pdf/7308x1349.pdf |access-date=1 November 2021 |quote=See Table 1 Conjugate pairs of variables ... (p.1357)}}</ref>

===Units===
[[File:Barometer mercury column hg.jpg|thumb|right|Mercury column]]
The SI unit for pressure is the [[Pascal (unit)|pascal]] (Pa), equal to one [[Newton (unit)|newton]] per [[Square metre|square metre]] (N/m<sup>2</sup>, or kg·m<sup>−1</sup>·s<sup>−2</sup>). This name for the unit was added in 1971;<ref>{{cite web |url=http://www.bipm.fr/en/convention/cgpm/14/pascal-siemens.html |title=14th Conference of the International Bureau of Weights and Measures |publisher=Bipm.fr |access-date=2012-03-27 |url-status=dead |archive-url=https://web.archive.org/web/20070630020548/http://www.bipm.fr/en/convention/cgpm/14/pascal-siemens.html |archive-date=2007-06-30 }}</ref> before that, pressure in SI was expressed simply in newtons per square metre.

Other units of pressure, such as pounds per square inch (lbf/in<sup>2</sup>) and [[Bar (unit)|bar]], are also in common use. The [[Centimetre–gram–second system of units|CGS]] unit of pressure is the [[Physics:Barye|barye]] (Ba), equal to 1&nbsp;dyn·cm<sup>−2</sup>, or 0.1&nbsp;Pa. Pressure is sometimes expressed in grams-force or kilograms-force per square centimetre (g/cm<sup>2</sup> or kg/cm<sup>2</sup><!--don't add an f to kg, this is making the point about usage without it-->) and the like without properly identifying the force units. But using the names kilogram, gram, kilogram-force, or gram-force (or their symbols) as units of force is expressly forbidden in SI. The [[Technical atmosphere|technical atmosphere]] (symbol: at) is 1&nbsp;kgf/cm<sup>2</sup> (98.0665&nbsp;kPa, or 14.223&nbsp;psi).

Pressure is related to [[Physics:Energy density|energy density]] and may be expressed in units such as [[Joule|joule]]s per cubic metre (J/m<sup>3</sup>, which is equal to Pa).
Mathematically:
<math display="block">p = \frac{F \cdot \text{distance}}{A \cdot \text{distance}} = \frac{\text{Work}}{\text{Volume}} = \frac{\text{Energy (J)}}{\text{Volume }(\text{m}^3)}.</math>

Some [[Meteorologist|meteorologist]]s prefer the hectopascal (hPa) for atmospheric air pressure, which is equivalent to the older unit millibar (mbar). Similar pressures are given in kilopascals (kPa) in most other fields, except aviation where the hecto- prefix is commonly used. The inch of mercury is still used in the United States. Oceanographers usually measure underwater pressure in decibars (dbar) because pressure in the ocean increases by approximately one decibar per metre depth.

The [[Physics:Atmosphere (unit)|standard atmosphere]] (atm) is an established constant. It is approximately equal to typical air pressure at Earth mean sea level and is defined as {{val|101325|u=Pa}}.

Because pressure is commonly measured by its ability to displace a column of liquid in a manometer, pressures are often expressed as a depth of a particular fluid (e.g., centimetres of water, millimetres of mercury or inches of mercury). The most common choices are [[Chemistry:Mercury (element)|mercury]] (Hg) and water; water is nontoxic and readily available, while mercury's high density allows a shorter column (and so a smaller manometer) to be used to measure a given pressure. The pressure exerted by a column of liquid of height ''h'' and density ''ρ'' is given by the hydrostatic pressure equation {{nowrap|1=''p'' = ''ρgh''}}, where ''g'' is the [[Physics:Gravitational acceleration|gravitational acceleration]]. Fluid density and local gravity can vary from one reading to another depending on local factors, so the height of a fluid column does not define pressure precisely. 

When millimetres of mercury (or inches of mercury) are quoted today, these units are not based on a physical column of mercury; rather, they have been given precise definitions that can be expressed in terms of SI units.<ref>{{SIbrochure8th|page=127}}</ref> One millimetre of mercury is approximately equal to one [[Torr|torr]]. The water-based units still depend on the density of water, a measured, rather than defined, quantity. These ''manometric units'' are still encountered in many fields. [[Physics:Blood pressure|Blood pressure]] is measured in millimetres (or centimetres) of mercury in most of the world, and lung pressures in centimetres of water are still common.{{citation needed|date=October 2021}}

Underwater divers use the [[Physics:Metre sea water|metre sea water]] (msw or MSW) and foot sea water (fsw or FSW) units of pressure, and these are the units for pressure gauges used to measure pressure exposure in [[Engineering:Diving chamber|diving chamber]]s and personal decompression computers. A msw is defined as 0.1&nbsp;bar (= 10,000&nbsp;Pa), is not the same as a linear metre of depth. 33.066&nbsp;fsw = 1&nbsp;atm{{cn|date=February 2023}} (1&nbsp;atm = 101,325&nbsp;Pa / 33.066 = 3,064.326 Pa).  The pressure conversion from msw to fsw is different from the length conversion: 10&nbsp;msw = 32.6336&nbsp;fsw, while 10&nbsp;m = 32.8083&nbsp;ft.{{cn|date=February 2023}}

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 is often given in units with "g" appended, e.g. "kPag", "barg" or "psig", and units for measurements of absolute pressure are sometimes given a suffix of "a", to avoid confusion, for example "kPaa", "psia". However, the US [[Organization:National Institute of Standards and Technology|National Institute of Standards and Technology]] recommends that, to avoid confusion, any modifiers be instead applied to the quantity being measured rather than the unit of measure.<ref name = pubs>{{cite journal |access-date= 2009-07-07 |journal=NIST |url=http://physics.nist.gov/Pubs/SP811/sec07.html#7.4 |title=Rules and Style Conventions for Expressing Values of Quantities |date=2 July 2009 |url-status=live |archive-url=https://web.archive.org/web/20090710192735/http://physics.nist.gov/Pubs/SP811/sec07.html#7.4 |archive-date=2009-07-10 }}</ref> For example, {{nowrap|1="''p''<sub>g</sub> = 100 psi"}} rather than {{nowrap|1="''p'' = 100 psig"}}.

Differential pressure is expressed in units with "d" appended; this type of measurement is useful when considering sealing performance or whether a valve will open or close.

Presently or formerly popular pressure units include the following:
*[[Physics:Atmosphere (unit)|atmosphere]] (atm)
*manometric units:
**centimetre, inch, millimetre (torr) and micrometre (mTorr, micron) of mercury,
**{{anchor|H2O}}height of equivalent column of water, including [[Physics:Millimeters, water gauge|millimetre]] (mm {{chem|H|2|O}}), [[Physics:Centimetre of water|centimetre]] (cm {{chem|H|2|O}}), metre, [[Physics:Inch of water|inch]], and foot of water;
*imperial and customary units:
**[[Kip (unit)|kip]], [[Physics:Ton-force#Short ton-force|short ton-force]], [[Physics:Ton-force#Long ton-force|long ton-force]], pound-force, ounce-force, and [[Poundal|poundal]] per square inch,
**short ton-force and long ton-force per square inch,
**fsw (feet sea water) used in underwater diving, particularly in connection with diving pressure exposure and decompression;
*non-SI metric units:
**[[Bar (unit)|bar]], decibar, millibar,
***msw (metres sea water), used in underwater diving, particularly in connection with diving pressure exposure and decompression,
**kilogram-force, or kilopond, per square centimetre ([[Technical atmosphere|technical atmosphere]]),
**gram-force and tonne-force (metric ton-force) per square centimetre,
**[[Physics:Barye|barye]] ([[Physics:Dyne|dyne]] per square centimetre),
**kilogram-force and tonne-force per square metre,
**sthene per square metre (pieze).
<!--

{{Pressure Units}}
[[File:Pressure units.png|alt=A vizualization of pressure units as circles in log scale. A circle is to imagine a bubble, that such pressure would occupy.|thumb|Pressure units as circles in log scale. 4 graphs show scaled P units to unitary mTorr, Pa, mbar, bar. A circle is to imagine a bubble, that such pressure would occupy.]]
-->

===Examples===
[[File:Aluminium cylinder.jpg|120px|thumbnail|right|The effects of an external pressure of 700&nbsp;bar on an aluminum cylinder with {{convert|5|mm|in|3|abbr=on}} wall thickness]]
As an example of varying pressures, a finger can be pressed against a wall without making any lasting impression; however, the same finger pushing a thumbtack can easily damage the wall. Although the force applied to the surface is the same, the thumbtack applies more pressure because the point concentrates that force into a smaller area. Pressure is transmitted to solid boundaries or across arbitrary sections of fluid ''normal to'' these boundaries or sections at every point. Unlike [[Physics:Stress|stress]], pressure is defined as a [[Physics:Scalar|scalar quantity]]. The negative [[Gradient|gradient]] of pressure is called the [[Physics:Force density|force density]].<ref>{{cite book |last1=Lautrup |first1=Benny |title=Physics of continuous matter : exotic and everyday phenomena in the macroscopic world |date=2005 |publisher=Institute of Physics |location=Bristol |isbn=9780750307529 |page=50}}</ref>

Another example is a knife. If the flat edge is used, force is distributed over a larger surface area resulting in less pressure, and it will not cut. Whereas using the sharp edge, which has less surface area, results in greater pressure, and so the knife cuts smoothly. This is one example of a practical application of pressure<ref>{{cite book |last1=Breithaupt |first1=Jim |title=Physics |date=2015 |location=Basingstoke |isbn=9781137443243 |page=106 |edition=Fourth |publisher=Palgrave Macmillan }}</ref>

For gases, pressure is sometimes measured not as an ''absolute pressure'', but relative to [[Physics:Atmospheric pressure|atmospheric pressure]]; such measurements are called ''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''. An example of this is the air pressure in an automobile [[Engineering:Tire|tire]], which might be said to be "{{convert|220 |kPa |psi|abbr=on|lk=in}}", but is actually 220&nbsp;kPa (32&nbsp;psi) above atmospheric pressure. Since atmospheric pressure at sea level is about 100&nbsp;kPa (14.7&nbsp;psi), the absolute pressure in the tire is therefore about {{convert|320|kPa|psi|abbr=on}}. In technical work, this is written "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 {{convert|220|kPa|psi|abbr=on}}". 

Where space is limited, such as on pressure 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.-->s, name plates, graph labels, and table headings, the use of a modifier in parentheses, such as "kPa (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.-->)" or "kPa (absolute)", is permitted. In non-SI technical work, 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 {{convert|32 |psi|kPa|abbr=on}} is sometimes written as "32&nbsp;psig", and an absolute pressure as "32&nbsp;psia", though the other methods explained above that avoid attaching characters to the unit of pressure are preferred.<ref name = pubs/>

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 is the relevant measure of pressure wherever one is interested in the stress on [[Physics:Pressure vessel|storage vessels]] and the plumbing components of fluidics systems. However, whenever equation-of-state properties, such as densities or changes in densities, must be calculated, pressures must be expressed in terms of their absolute values. For instance, if the atmospheric pressure is {{convert|100|kPa|psi|abbr=on}}, a gas (such as helium) at {{convert|200|kPa|psi|abbr=on}} (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.-->) ({{convert|300|kPa|psi|abbr=on|disp=or}} [absolute]) is 50% denser than the same gas at {{convert|100|kPa|psi|abbr=on}} (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.-->) ({{convert|200|kPa|psi|abbr=on|disp=or}} [absolute]). Focusing on 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.--> values, one might erroneously conclude the first sample had twice the density of the second one.{{citation needed|date=October 2021}}

===Scalar nature===
In a static [[Physics:Gas|gas]], the gas as a whole does not appear to move. The individual molecules of the gas, however, are in constant [[Brownian motion|random motion]]. Because there are an extremely large number of molecules and because the motion of the individual molecules is random in every direction, no motion is detected. When the gas is at least partially confined (that is, not free to expand rapidly), the gas will exhibit a hydrostatic pressure. This confinement can be achieved with either a physical container of some sort, or in a gravitational well such as a planet, otherwise known as ''[[Physics:Atmospheric pressure|atmospheric pressure]]''. 

In the case of planetary [[Astronomy:Atmosphere|atmosphere]]s, the [[Physics:Pressure-gradient force|pressure-gradient force]] of the gas pushing outwards from higher pressure, lower altitudes to lower pressure, higher altitudes is balanced by the gravitational force, preventing the gas from diffusing into [[Astronomy:Outer space|outer space]] and maintaining [[Astronomy:Hydrostatic equilibrium|hydrostatic equilibrium]].

In a physical container, the pressure of the gas originates from the molecules colliding with the walls of the container. The walls of the container can be anywhere inside the gas, and the force per unit area (the pressure) is the same. If the "container" is shrunk down to a very small point (becoming less true as the atomic scale is approached), the pressure will still have a single value at that point. Therefore, pressure is a scalar quantity, not a vector quantity. It has magnitude but no direction sense associated with it. Pressure force acts in all directions at a point inside a gas. At the surface of a gas, the pressure force acts perpendicular (at right angle) to the surface.{{citation needed|date=October 2021}}

A closely related quantity is the [[Physics:Stress|stress]] tensor ''σ'', which relates the vector force <math>\mathbf{F}</math> to the 
[[Vector area|vector area]] <math>\mathbf{A}</math> via the linear relation <math>\mathbf{F} = \sigma\mathbf{A}</math>.

This [[Tensor|tensor]] may be expressed as the sum of the [[Physics:Viscous stress tensor|viscous stress tensor]] minus the hydrostatic pressure. The negative of the stress tensor is sometimes called the pressure tensor, but in the following, the term "pressure" will refer only to the scalar pressure.{{citation needed|date=October 2021}}

According to the theory of [[Physics:General relativity|general relativity]], pressure increases the strength of a gravitational field (see [[Stress–energy tensor|stress–energy tensor]]) and so adds to the mass-energy cause of [[Physics:Gravity|gravity]]. This effect is unnoticeable at everyday pressures but is significant in [[Astronomy:Neutron star|neutron star]]s, although it has not been experimentally tested.<ref>{{cite journal|title=Einstein's gravity under pressure |doi=10.1007/s10509-009-0016-8 |volume=321 |issue=2 |journal=Astrophysics and Space Science |pages=151–156|arxiv=0705.0825 |bibcode=2009Ap&SS.321..151V |year=2009 |last1=Vishwakarma |first1=Ram Gopal |s2cid=218673952 }}</ref>

==Types==

===Fluid pressure===
''Fluid pressure'' is most often the compressive stress at some point within a [[Physics:Fluid|fluid]].  (The term ''fluid'' refers to both liquids and gases – for more information specifically about liquid pressure, see 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 [[Physics:Fluid dynamics|fluid dynamics]].

The concepts of fluid pressure are predominantly attributed to the discoveries of [[Biography:Blaise Pascal|Blaise Pascal]] and [[Biography:Daniel Bernoulli|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 [[Physics:Viscosity|viscosity]]).<ref name=Finnemore/> The equation for all points of a system filled with a constant-density fluid is<ref name=NCEES>{{cite book|last=NCEES|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-) [[Physics:Specific weight|specific weight]] of the fluid,<ref name=Finnemore/>
*''v'', velocity of the fluid,
*''g'', [[Physics: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
* [[Earth:Artesian well|Artesian well]]
* [[Physics:Blood pressure|Blood pressure]]
* [[Physics:Hydraulic head|Hydraulic head]]
* [[Biology:Turgor pressure|Plant cell turgidity]]
* [[Physics:Pythagorean cup|Pythagorean cup]]
* [[Engineering:Pressure washing|Pressure washing]]

===Explosion or deflagration pressures===
Explosion or [[Physics:Deflagration|deflagration]] pressures are the result of the ignition of explosive [[Physics:Gas|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 [[Physics:Tension|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 [[Physics:Superheating|superheating]] and is easily susceptible to [[Physics:Cavitation|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 [[Biology:Ascent of sap|ascent of sap]] in plants taller than 10&nbsp;m (the atmospheric [[Physics:Pressure head|pressure head]] of water).<ref>{{cite web | title=The Physics of Negative Pressure | url=http://discovermagazine.com/2003/mar/featscienceof | publisher=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 [[Physics:Casimir effect|Casimir effect]] can create a small attractive force due to interactions with [[Physics:Vacuum energy|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 [[Physics:Electromagnetic field|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 [[Astronomy:Cosmology|cosmology]], [[Astronomy:Dark energy|dark energy]] creates a very small yet cosmically significant amount of negative pressure, which accelerates the [[Astronomy:Expansion of the universe|expansion of the universe]].

===Stagnation pressure===<!--This section is linked from [[Drag equation]]-->
{{Main|Physics:Stagnation pressure}}
[[Physics: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 [[Physics:Static pressure|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 [[Physics:Stagnation pressure|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 [[Engineering:Pitot tube|Pitot tube]], or one of its variations such as a [[Engineering:Kiel probe|Kiel probe]] or [[Engineering:Cobra probe|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.

[[Physics:Surface tension|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|Physics:Ideal gas law}}
In an [[Physics:Ideal gas|ideal gas]], molecules have no volume and do not interact. According to the [[Physics:Ideal gas law|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 [[Physics:Amount of substance|amount of substance]],
*''T'' is the absolute temperature,
*''V'' is the volume,
*''R'' is the [[Physics:Ideal gas constant|ideal gas constant]].

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

The [[Physics:Atmospheric pressure|atmospheric pressure]] [[Physics:Boiling point|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. 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 [[Physics:Partial pressure|partial vapor pressure]].

===Liquid pressure===
{{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 or 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 [[Physics:Density|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 narrower 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, [[Physics: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 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 boat is pushed upward by water pressure ([[Physics:Buoyancy|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 water spurting from a 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" /> 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 = 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 [[Physics:Kinematic viscosity|kinematic viscosity]] <math>\nu</math> in order to compute the [[Physics: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>

==See also==
{{cmn|
*{{annotated link|Boyle's law}}
*{{annotated link|Physics:Combined gas law}}
*{{annotated link|Conversion of units}}
*{{annotated link|Critical point (thermodynamics)}}
*{{annotated link|Dimensional analysis}}
*{{annotated link|Physics:Dynamic pressure}}
*{{annotated link|Physics:Electric potential}}
*{{annotated link|Astronomy:Electron degeneracy pressure}}
*{{annotated link|Physics:High pressure}}
*{{annotated link|Physics:Hydraulics}}
*{{annotated link|Physics:Internal pressure}}
*{{annotated link|Microphone}}
*{{annotated link|Orders of magnitude (pressure)}}
*{{annotated link|Physics:Partial pressure}}
*{{annotated link|Physics:Pressure measurement}}
*{{annotated link|Physics:Pressure sensor}}
*{{annotated link|Physics:Sound pressure}}
*{{annotated link|Physics:Static pressure}}
*{{annotated link|Engineering:Timeline of temperature and pressure measurement technology}}
*{{annotated link|Physics:Torricelli's law}}
*{{annotated link|Physics:Vacuum pump}}
*{{annotated link|Vertical pressure variation}}
}}

==Notes==
{{Reflist|group=lower-alpha}}

==References==
{{Reflist|30em}}

==External links==
*[https://web.archive.org/web/20190929053038/http://www.physnet.org/modules/pdf_modules/m48.pdf ''Introduction to Fluid Statics and Dynamics''] on [http://www.physnet.org/ Project PHYSNET]
*[https://web.archive.org/web/20100106203418/http://www.grc.nasa.gov/WWW/K-12/airplane/pressure.html Pressure being a scalar quantity]
*[https://web.archive.org/web/20200311121248/http://www.wikiunits.org/ wikiUnits.org - Convert units of pressure]


[[Category:Pressure| ]]
[[Category:Atmospheric thermodynamics]]
[[Category:Underwater diving physics]]
[[Category:Fluid dynamics]]
[[Category:Fluid mechanics]]
[[Category:Hydraulics]]
[[Category:Thermodynamic properties]]
[[Category:State functions]]

{{Sourceattribution|Pressure}}