TheInfoList

Pressure (symbol: ''p'' or ''P'') is the
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

applied perpendicular to the surface of an object per unit
area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

over which that force is distributed.
Gauge pressure Pressure measurement is the analysis of an applied force In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that st ...
(also spelled ''gage'' pressure)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. is the pressure relative to the ambient pressure. Various
units Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in ...
are used to express pressure. Some of these derive from a unit of force divided by a unit of area; the SI unit of pressure, the
pascal Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, French ...
(Pa), for example, is one
newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * Newton (film), ''Newton'' (film), a 2017 Indian fil ...
per
square metre The square metre ( international spelling as used by the International Bureau of Weights and Measures The International Bureau of Weights and Measures (french: Bureau international des poids et mesures, BIPM) is an intergovernmental organis ...
(N/m2); similarly, the
pound-force The pound of force or pound-force (symbol: lbf, sometimes lbf,) is a Units of measurement, unit of force used in some System of measurement, systems of measurement, including English Engineering units and the foot–pound–second system. Pound- ...
per
square inch A square inch (plural: square inches) is a of , equal to the area of a with sides of one . The following symbols are used to denote square inches: *square in *sq inches, sq inch, sq in *inches/-2, inch/-2, in/-2 *inches^2, inch^2, in^2 *inches2 ...
( psi) is the traditional unit of pressure in the
imperial Imperial is that which relates to an empire, emperor, or imperialism. Imperial or The Imperial may also refer to: Places United States * Imperial, California * Imperial, Missouri * Imperial, Nebraska * Imperial, Pennsylvania * Imperial, Texas * ...
and U.S. customary systems. Pressure may also be expressed in terms of
standard atmospheric pressure The standard atmosphere (symbol: atm) is a unit Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit ...
; the
atmosphere An atmosphere (from the greek words ἀτμός ''(atmos)'', meaning 'vapour', and σφαῖρα ''(sphaira)'', meaning 'ball' or 'sphere') is a layer or a set of layers of gases surrounding a planet or other material body, that is held in ...
(atm) is equal to this pressure, and the
torr The torr (symbol: Torr) is a unit of pressure based on an absolute scale An absolute scale is a system of measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attri ...
is defined as of this. Manometric units such as the
centimetre of water A centimetre or millimetre of water (US spelling ''centimeter'' or ''millimeter of water'') are less commonly used measures of pressure Pressure (symbol: ''p'' or ''P'') is the force In physics Physics (from grc, φυσική ...
,
millimetre of mercury A millimetre of mercury is a manometric unit of pressure Pressure (symbol: ''p'' or ''P'') is the force In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''p ...
, and
inch of mercury Inch of mercury (inHg and ″Hg) is a unit of measurement A unit of measurement is a definite magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (math ...
are used to express pressures in terms of the height of
column of a particular fluid A column or pillar in architecture and structural engineering is a structural element that transmits, through compression (physical), compression, the weight of the structure above to other structural elements below. In other words, a column is ...
in a manometer.

# Definition

Pressure is the amount of force applied at
right angle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

s to the surface of an object per unit area. The symbol for it is "p" or ''P''. The
IUPAC The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering OrganizationsNational Adhering Organizations in chemistry are the organizations that work as the authoritative power over chemist ...
recommendation for pressure is a lower-case ''p''. 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
power Power typically refers to: * Power (physics) In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older works, p ...
and
momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinui ...

, and on writing style.

## Formula

Mathematically: :$p = \frac,$ where: :$p$ is the pressure, :$F$ is the magnitude of the
normal force In mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces applied to objects result in Displacement (ve ...

, :$A$ is the area of the surface on contact. Pressure is a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...
quantity. It relates the
vector areaIn 3-dimensional geometry, for a finite planar surface of scalar area and unit normal , the vector area is defined as the unit normal scaled by the area: :\mathbf = \mathbfS For an orientable is non-orientable In mathematics, orientability ...

element (a vector normal to the surface) with the
normal force In mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces applied to objects result in Displacement (ve ...

acting on it. The pressure is the scalar
proportionality constant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
that relates the two normal vectors: :$d\mathbf_n = -p\,d\mathbf = -p\,\mathbf\,dA.$ The minus sign comes from the fact 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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. 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
thermodynamics Thermodynamics is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in ot ...

, and it is
conjugate Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the change ...
to
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...
.

## Units

The unit for pressure is the
pascal Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, French ...
(Pa), equal to one
newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * Newton (film), ''Newton'' (film), a 2017 Indian fil ...
per
square metre The square metre ( international spelling as used by the International Bureau of Weights and Measures The International Bureau of Weights and Measures (french: Bureau international des poids et mesures, BIPM) is an intergovernmental organis ...
(N/m2, or kg·m−1·s−2). This name for the unit was added in 1971; before that, pressure in SI was expressed simply in newtons per square metre. Other units of pressure, such as
pounds per square inch The pound per square inch or, more accurately, pound-force per square inch (symbol: lbf/in2; abbreviation: psi) is a unit of pressure Pressure (symbol: ''p'' or ''P'') is the force In physics, a force is an influence that can change ...
(lbf/in2) and
bar Bar or BAR may refer to: Food *Bar (establishment) A bar is a long raised narrow table or bench designed for dispensing beer or other alcoholic beverage, alcoholic drinks. They were originally chest high, and a bar, often brass, ran the len ...
, are also in common use. The CGS unit of pressure is the
barye The barye (symbol: Ba), or sometimes barad, barrie, bary, baryd, baryed, or barie, is the centimetre–gram–second (CGS) unit of pressure Pressure (symbol: ''p'' or ''P'') is the force In physics, a force is an influence that can ...
(Ba), equal to 1 dyn·cm−2, or 0.1 Pa. Pressure is sometimes expressed in grams-force or kilograms-force per square centimetre (g/cm2 or kg/cm2) 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 may refer to: * Technical (vehicle) A technical, in professional military parlance often called a non-standard tactical vehicle (NSTV), is a light improvised fighting vehicle, typically an open-backed civilian pickup truck or four-whe ...
(symbol: at) is 1 kgf/cm2 (98.0665 kPa, or 14.223 psi). Since a system under pressure has the potential to perform work on its surroundings, pressure is a measure of potential energy stored per unit volume. It is therefore related to energy density and may be expressed in units such as
joule The joule ( ; symbol: J) is a derived unit of energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates ...

s per cubic metre (J/m3, which is equal to Pa). Mathematically: :$p = \frac = \frac = \frac.$ Some
meteorologist A meteorologist is a scientist who studies and works in the field of meteorology Meteorology is a branch of the (which include and ), with a major focus on . The study of meteorology dates back , though significant progress in meteorology ...
s prefer the hectopascal (hPa) for atmospheric air pressure, which is equivalent to the older unit
millibar The bar is a metric unit of pressure Pressure (symbol: ''p'' or ''P'') is the force In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), ...
(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
decibar The bar is a metric units, metric unit of pressure, but not part of the International System of Units (SI). It is defined as exactly equal to 100,000 Pascal (unit), Pa (100 kPa), or slightly less than the current average atmospheric pr ...
s (dbar) because pressure in the ocean increases by approximately one decibar per metre depth. The standard atmosphere (atm) is an established constant. It is approximately equal to typical air pressure at Earth
mean sea level There are several kinds of mean in mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contain ...
and is defined as . Because pressure is commonly measured by its ability to displace a column of liquid in a
manometer Pressure measurement is the analysis of an applied force In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that st ...

, pressures are often expressed as a depth of a particular fluid (e.g.,
centimetres of water A centimetre or millimetre of water (US spelling ''centimeter'' or ''millimeter of water'') are less commonly used measures of pressure Pressure (symbol: ''p'' or ''P'') is the force In physics Physics (from grc, φυσική ...
, millimetres of mercury or
inches of mercury Inch of mercury (inHg and ″Hg) is a unit of measurement for pressure. It is used for barometric pressure in Weather forecasting, weather reports, refrigeration and aviation in the United States customary units, United States. It is the pressur ...
). The most common choices are
mercury Mercury usually refers to: * Mercury (planet) Mercury is the smallest planet in the Solar System and the closest to the Sun. Its orbit around the Sun takes 87.97 Earth days, the shortest of all the Sun's planets. It is named after the Roman g ...

(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 , where ''g'' is the
gravitational acceleration In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
. 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. One millimetre of mercury is approximately equal to one
torr The torr (symbol: Torr) is a unit of pressure based on an absolute scale An absolute scale is a system of measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attri ...
. 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.
Blood pressure Blood pressure (BP) is the pressure Pressure (symbol: ''p'' or ''P'') is the force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motio ...

is measured in millimetres of mercury in most of the world, and lung pressures in centimetres of water are still common.
Underwater divers Underwater divers Underwater divers are people who take part in underwater diving Action (philosophy), activities – Underwater diving is practiced as part of an occupation, or for recreation, where the practitioner submerges belo ...
use the
metre sea water The metre (or meter) sea water (msw) is a of used in . It is defined as one tenth of a . The unit used in the US is the foot sea water (fsw), based on and a sea-water density of 64 lb/ft3. According to the US Navy Diving Manual, one fs ...
(msw or MSW) and
foot sea water The metre (or meter) sea water (msw) is a Metric units, metric unit of pressure used in underwater diving. It is defined as one tenth of a Bar (unit), bar. The unit used in the US is the foot sea water (fsw), based on standard gravity and a sea- ...
(fsw or FSW) units of pressure, and these are the standard units for pressure gauges used to measure pressure exposure in
diving chamber A diving chamber is a vessel for human occupation, which may have an entrance that can be sealed to hold an internal pressure significantly higher than ambient pressure Ambient or Ambiance or Ambience may refer to: Music and sound * Ambience ...
s and personal decompression computers. A msw is defined as 0.1 bar (= 100000 Pa = 10000 Pa), is not the same as a linear metre of depth. 33.066 fsw = 1 atm (1 atm = 101325 Pa / 33.066 = 3064.326 Pa). Note that the pressure conversion from msw to fsw is different from the length conversion: 10 msw = 32.6336 fsw, while 10 m = 32.8083 ft. Gauge 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
National Institute of Standards and Technology The National Institute of Standards and Technology (NIST) is a physical sciences Physical science is a branch of natural science that studies abiotic component, non-living systems, in contrast to life science. It in turn has many branches, e ...
recommends that, to avoid confusion, any modifiers be instead applied to the quantity being measured rather than the unit of measure. For example, rather than . 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: *
atmosphere An atmosphere (from the greek words ἀτμός ''(atmos)'', meaning 'vapour', and σφαῖρα ''(sphaira)'', meaning 'ball' or 'sphere') is a layer or a set of layers of gases surrounding a planet or other material body, that is held in ...
(atm) *manometric units: **centimetre, inch, millimetre (torr) and micrometre (mTorr, micron) of mercury, **height of equivalent column of water, including
millimetre 330px, Different lengths as in respect to the Electromagnetic spectrum, measured by the Metre and its deriveds scales. The Microwave are in-between 1 meter to 1 millimeter. The millimetre (American and British English spelling differences#-re, ...
(mm ),
centimetre 330px, Different lengths as in respect to the Electromagnetic spectrum, measured by the Metre and its deriveds scales. The Microwave are in-between 1 meter to 1 millimeter. A centimetre (international spelling) or centimeter (American spellin ...
(cm ), metre, inch, and foot of water; *imperial and customary units: **
kip Kip, KIP or kips may refer to: Athletics * Kip (artistic gymnastics) In artistic gymnastics, a kip is a technique that involves flexing or piking at the hips, and then rapidly extending the hip joints to impart momentum. It may be performed in ...
, short ton-force, long ton-force,
pound-force The pound of force or pound-force (symbol: lbf, sometimes lbf,) is a Units of measurement, unit of force used in some System of measurement, systems of measurement, including English Engineering units and the foot–pound–second system. Pound ...
,
ounce-force The pound of force or pound-force (symbol: lbf, sometimes lbf,) is a unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete p ...
, and
poundal The poundal (symbol: pdl) is a unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical ...
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 Bar or BAR may refer to: Food *Bar (establishment) A bar is a long raised narrow table or bench designed for dispensing beer or other alcoholic beverage, alcoholic drinks. They were originally chest high, and a bar, often brass, ran the len ...
, decibar,
millibar The bar is a metric unit of pressure Pressure (symbol: ''p'' or ''P'') is the force In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), ...
, ***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 may refer to: * Technical (vehicle) A technical, in professional military parlance often called a non-standard tactical vehicle (NSTV), is a light improvised fighting vehicle, typically an open-backed civilian pickup truck or four-whe ...
), **gram-force and tonne-force (metric ton-force) per square centimetre, **
barye The barye (symbol: Ba), or sometimes barad, barrie, bary, baryd, baryed, or barie, is the centimetre–gram–second (CGS) unit of pressure Pressure (symbol: ''p'' or ''P'') is the force In physics, a force is an influence that can ...
(
dyne The dyne (symbol dyn, from grc, δύναμις, dynamis, power, force) is a derived unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of acti ...
per square centimetre), **kilogram-force and tonne-force per square metre, ** sthene per square metre ().

## Examples

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 A drawing pin (British English) or thumb tack (North American English) is a short nail Nail or Nails may refer to: In biology * Nail (anatomy), toughened protective protein-keratin (known as alpha-keratin, also found in hair) at the end of ...

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 stress, pressure is defined as a scalar quantity. The negative
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

of pressure is called the
force densityIn fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force ...
. Another example is a knife. If we try to cut with the flat edge, 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. For gases, pressure is sometimes measured not as an ''absolute pressure'', but relative to
atmospheric pressure Atmospheric pressure, also known as barometric pressure (after the barometer A barometer is a scientific instrument that is used to measure air pressure Atmospheric pressure, also known as barometric pressure (after the barometer), is the ...
; such measurements are called ''gauge pressure''. An example of this is the air pressure in an
automobile A car (or automobile) is a wheeled motor vehicle A motor vehicle, also known as motorized vehicle or automotive vehicle, is a self-propelled vehicle, commonly wheeled, that does not operate on Track (rail transport), rails (such as trains o ...

tire A tire (American English American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of variety (linguistics), varieties of the English language native to the United States ...

, which might be said to be "", but is actually 220 kPa (32 psi) above atmospheric pressure. Since atmospheric pressure at sea level is about 100 kPa (14.7 psi), the absolute pressure in the tire is therefore about . In technical work, this is written "a gauge pressure of ". Where space is limited, such as on
pressure gauge Pressure measurement is the analysis of an applied force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. m ...

s, name plates, graph labels, and table headings, the use of a modifier in parentheses, such as "kPa (gauge)" or "kPa (absolute)", is permitted. In non- technical work, a gauge pressure of is sometimes written as "32 psig", and an absolute pressure as "32 psia", though the other methods explained above that avoid attaching characters to the unit of pressure are preferred. Gauge pressure is the relevant measure of pressure wherever one is interested in the stress on
storage vessels Storage may refer to: Storage of goods Storage containers * Dry cask storage, storing high-level radioactive waste * Food storage * Intermodal container, cargo shipping * Storage tank Storage facilities * Mail storage, storage by mail or delivery s ...
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 , a gas (such as helium) at (gauge) ( bsolute is 50% denser than the same gas at (gauge) ( bsolute. Focusing on gauge values, one might erroneously conclude the first sample had twice the density of the second one.

## Scalar nature

In a static
gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...

, the gas as a whole does not appear to move. The individual molecules of the gas, however, are in constant . Because we are dealing with an extremely large number of molecules and because the motion of the individual molecules is random in every direction, we do not detect any motion. If we enclose the gas within a container, we detect a pressure in the gas from the molecules colliding with the walls of our container. We can put the walls of our container anywhere inside the gas, and the force per unit area (the pressure) is the same. We can shrink the size of our "container" down to a very small point (becoming less true as we approach the atomic scale), and 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. A closely related quantity is the stress tensor ''σ'', which relates the vector force $\mathbf$ to the
vector areaIn 3-dimensional geometry, for a finite planar surface of scalar area and unit normal , the vector area is defined as the unit normal scaled by the area: :\mathbf = \mathbfS For an orientable is non-orientable In mathematics, orientability ...

$\mathbf$ via the linear relation $\mathbf = \sigma\mathbf$. This
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

may be expressed as the sum of the 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. According to the theory of
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
, pressure increases the strength of a gravitational field (see
stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the Cauchy str ...
) and so adds to the mass-energy cause of
gravity Gravity (), or gravitation, is a by which all things with or —including s, s, , and even —are attracted to (or ''gravitate'' toward) one another. , gravity gives to s, and the causes the s of the oceans. The gravitational attracti ...

. This effect is unnoticeable at everyday pressures but is significant in
neutron star A neutron star is the collapsed core Core or cores may refer to: Science and technology * Core (anatomy) In common parlance, the core of the body is broadly considered to be the torso. Functional movements are highly dependent on this par ...
s, although it has not been experimentally tested.

# Types

## Fluid pressure

Fluid pressure is most often the compressive stress at some point within a
fluid In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
. (The term ''fluid'' refers to both liquids and gases – for more information specifically about liquid pressure, see section below.) 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 Fluid statics or hydrostatics is the branch of fluid mechanics that studies "fluids at hydrostatic equilibrium and the pressure in a fluid or exerted by a fluid on an immersed body". It encompasses the study of the conditions under which fluid ...
. 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 In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
. The concepts of fluid pressure are predominantly attributed to the discoveries of
Blaise Pascal Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, writer and Catholic Church, Catholic theologian. He was a child prodigy who was educated by his father, a tax collector i ...

and
Daniel Bernoulli Daniel Bernoulli Fellows of the Royal Society, FRS (; – 27 March 1782) was a Swiss people, Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for ...
.
Bernoulli's equation Video of a venturi meter used in a lab experiment In fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, inc ...
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 and incompressible. An ideal fluid is a fluid in which there is no friction, it is
inviscid The viscosity of a fluid is a measure of its drag (physics), resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity can be ...
(zero
viscosity The viscosity of a fluid In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, ...

). The equation for all points of a system filled with a constant-density fluid is :$\frac + \frac + z = \mathrm,$ where: :''p'', pressure of the fluid, :'''' = ''ρg'', density × acceleration of gravity is the (volume-)
specific weightThe specific weight, also known as the unit weight, is the weight In science and engineering, the weight of an object is the force acting on the object due to gravity. Some standard textbooks define weight as a Euclidean vector, vector quantity, t ...
of the fluid, :''v'', velocity of the fluid, :''g'', acceleration of gravity, :''z'', elevation, :$\frac$, pressure head, :$\frac$, velocity head.

### Applications

*
Hydraulic brakes A hydraulic brake is an arrangement of braking A brake is a machine, mechanical device that inhibits motion by absorbing energy from a moving system. It is used for Acceleration, slowing or stopping a moving vehicle, wheel, axle, or to prevent it ...
*
Artesian well :''See Great Artesian Basin for the water source in Australia.'' scheme: 1. Aquifer 2. Impervious strata 3. Infiltration area 4. Artesian well 5. Saturation level 6. Subartesian well 7. Artesian spring Image:Artesian Well (PSF).png, 300px, Schem ...

*
Blood pressure Blood pressure (BP) is the pressure Pressure (symbol: ''p'' or ''P'') is the force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motio ...

*
Hydraulic head Hydraulic head or piezometric head is a specific measurement of liquid pressure above a vertical datum A vertical datum, altimetric datum, or height datum is a reference surface for vertical position Vertical position or vertical location ...

* Plant cell turgidity *
Pythagorean cup A Pythagorean cup (also known as a Pythagoras cup, Greedy Cup, Cup of Justice, Tantalus cup or ''i koupa tis dikaiosynis'') is a practical joke device in a form of a Drinkware, drinking cup, credited to Pythagoras, Pythagoras of Samos. When it is ...

## Explosion or deflagration pressures

Explosion An explosion is a rapid expansion in volume Volume is a expressing the of enclosed by a . For example, the space that a substance (, , , or ) or occupies or contains. Volume is often quantified numerically using the , the . The volum ...

or
deflagration Deflagration (Lat: ''de + flagrare'', "to burn down") is subsonic Subsonic may refer to: Motion through a medium * Any speed lower than the speed of sound within a sound-propagating medium * Subsonic aircraft, a flying machine that flies at air ...
pressures are the result of the ignition of explosive
gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...

es, mists, dust/air suspensions, in unconfined and confined spaces.

## Negative pressures

While pressures are, in general, positive, there are several situations in which negative pressures may be encountered: *When dealing in relative (gauge) pressures. For instance, an absolute pressure of 80 kPa may be described as a gauge pressure of −21 kPa (i.e., 21 kPa below an atmospheric pressure of 101 kPa). For example, abdominal decompression is an
obstetric Obstetrics is the field of study concentrated on pregnancy, childbirth and the postpartum period. As a medical specialty, obstetrics is combined with gynecology under the discipline known as obstetrics and gynecology (OB/GYN), which is a surgical ...
procedure during which negative gauge pressure is applied intermittently to a pregnant woman's abdomen. *Negative absolute pressures are possible. They are effectively
tension Tension may refer to: Science * Psychological stress * Tension (physics), a force related to the stretching of an object (the opposite of compression) * Tension (geology), a stress which stretches rocks in two opposite directions * Voltage or elect ...
, and both bulk solids and bulk liquids can be put under negative absolute pressure by pulling on them. 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 In chemistry and physics, metastability denotes an intermediate energetic state within a dynamical system other than the system's ground state, state of least energy. A ball resting in a hollow on a slope is a simple example of metastability. I ...

state, and it is especially fragile in the case of liquids where the negative pressure state is similar to
superheating In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...

and is easily susceptible to
cavitation Cavitation is a phenomenon in which the static pressure of the liquid reduces to below the liquid's vapour pressure, leading to the formation of small vapor-filled cavities in the liquid. When subjected to higher pressure, these cavities, ...

. In certain situations, the cavitation can be avoided and negative pressures sustained indefinitely, for example, liquid mercury has been observed to sustain up to in clean glass containers. Negative liquid pressures are thought to be involved in the ascent of sap in plants taller than 10 m (the atmospheric
pressure head In fluid mechanics Fluid mechanics is the branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the orde ...
of water). *The
Casimir effect In quantum field theory, the Casimir effect is a physical force (physics), force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of the field. It is named after the Dutch physicist Hendrik Cas ...
can create a small attractive force due to interactions with
vacuum energy Vacuum energy is an underlying background energy In physics, energy is the physical quantity, quantitative physical property, property that must be #Energy transfer, transferred to a physical body, body or physical system to perform Work ...
; this force is sometimes termed "vacuum pressure" (not to be confused with the negative ''gauge 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 pressure along one
surface normal In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
, with a component of negative pressure acting along another surface normal. **The stresses in an
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the in ...
are generally non-isotropic, with the pressure normal to one surface element (the
normal stress In continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as point particle, discrete particles. The French mathematician Augustin-Louis ...

) being negative, and positive for surface elements perpendicular to this. *In
cosmology Cosmology (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...
,
dark energy In physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the unive ...

creates a very small yet cosmically significant amount of negative pressure, which accelerates the
expansion of the universe The expansion of the universe is the increase in distance between any two given gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion whereby ''the scale of space itself changes''. The universe does n ...
.

## Stagnation pressure

Stagnation pressure In fluid dynamics In and , fluid dynamics is a subdiscipline of that describes the flow of s—s and es. It has several subdisciplines, including ' (the study of air and other gases in motion) and hydrodynamics (the study of liquids in mot ...
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 In fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among for ...
, it may have a higher stagnation pressure when forced to a standstill. Static pressure and stagnation pressure are related by: :$p_ = \frac\rho v^2 + p$ where :$p_0$ is the
stagnation pressure In fluid dynamics In and , fluid dynamics is a subdiscipline of that describes the flow of s—s and es. It has several subdisciplines, including ' (the study of air and other gases in motion) and hydrodynamics (the study of liquids in mot ...
, :$\rho$ is the density, :$v$ is the flow velocity, :$p$ is the static pressure. The pressure of a moving fluid can be measured using a
Pitot tube , combines a pitot tube (right) with a static port and an angle-of-attack vane (left). Air-flow is right to left. helicopter A pitot ( ) tube, also known as pitot probe, is a flow measurement Flow measurement is the quantification of bulk ...

, or one of its variations such as a or Cobra probe, connected to a
manometer Pressure measurement is the analysis of an applied force In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that st ...

. 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: :$\pi = \frac$ 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, , at constant temperature.
Surface tension Surface tension is the tendency of liquid A liquid is a nearly incompressible In fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics Mechanics (Ancient Greek, Greek: ) is the area of physi ...

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

## Pressure of an ideal gas

In an
ideal gas An ideal gas is a theoretical gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion ...
, molecules have no volume and do not interact. According to the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the na ...

, pressure varies linearly with temperature and quantity, and inversely with volume: :$p = \frac,$ where: :''p'' is the absolute pressure of the gas, :''n'' is the
amount of substance In chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a ...
, :''T'' is the absolute temperature, :''V'' is the volume, :''R'' is the
ideal gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant The Boltzmann constant ( or ) is the proportionality fa ...
.
Real gas Real gases are nonideal gases whose molecules occupy space and have interactions; consequently, they do not adhere to the ideal gas law. To understand the behaviour of real gases, the following must be taken into account: * compressibility effects; ...
es exhibit a more complex dependence on the variables of state.

## Vapour pressure

Vapour pressure is the pressure of a
vapour In physics, a vapor (American English American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of varieties of the English language native to the United States. Currently, America ...
in
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic An axiom, postulate or assumption is a statement that is taken to be true True most commonly refers to truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Onlin ...
with its condensed
phase Phase or phases may refer to: Science * State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter) In the physical sciences, a phase is a region of space (a thermodynamic system A thermodynamic system is a ...
s in a closed system. All liquids and
solid Solid is one of the four fundamental states of matter 4 (four) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is an ...

s have a tendency to
evaporate Evaporation is a type of vaporization Vaporization (or vaporisation) of an element or compound is a phase transition from the liquid phase to vapor. There are two types of vaporization: evaporation and boiling. Evaporation is a surface phe ...
into a gaseous form, and all
gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...

es have a tendency to
condense Condensation is the change of the state of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...
back to their liquid or solid form. The
atmospheric pressure Atmospheric pressure, also known as barometric pressure (after the barometer A barometer is a scientific instrument that is used to measure air pressure Atmospheric pressure, also known as barometric pressure (after the barometer), is the ...
boiling point The boiling point of a substance is the temperature at which the vapor pressure 280px, The ''pistol test tube'' experiment. The tube contains alcohol and is closed with a piece of cork. By heating the alcohol, the vapors fill in the space, inc ...
of a liquid (also known as the
normal boiling point The boiling point of a substance is the temperature at which the vapor pressure of a liquid A liquid is a nearly incompressible fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an ...
) 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 Bubble or Bubbles may refer to: Physical bubbles * Bubble (physics), a globule of one substance in another, usually gas in a liquid ** Soap bubble, commonly referred to as a "bubble" People * Bubbles, a contestant on ''Real Chance of Love ( ...
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 vapor pressure.

## Liquid pressure

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: :$p = \rho gh,$ where: :''p'' is liquid pressure, :''g'' is gravity at the surface of overlaying material, :''ρ'' is
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

of liquid, :''h'' is height of liquid column or depth within a substance. Another way of saying the same formula is the following: :$p = \text \times \text.$ 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 exerts only half the average pressure that a small 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 -wide section of each dam, the deep water will apply one quarter the force of deep water). A person will feel the same pressure whether his/her 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 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 what 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 energy equation, the energy per unit volume in an ideal, incompressible liquid is constant throughout its vessel. At the surface, 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.Streeter, V. L., ''Fluid Mechanics'', Example 3.5, McGraw–Hill Inc. (1966), New York. Mathematically, it is described by
Bernoulli's equation Video of a venturi meter used in a lab experiment In fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, inc ...
, where velocity head is zero and comparisons per unit volume in the vessel are :$\frac + z = \mathrm.$ 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.Hewitt 251 (2006) If someone is submerged in water, no matter which way that person tilts his/her head, the person will feel the same amount of water pressure on his/her ears. Because a liquid can flow, this pressure isn't 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 (
buoyancy Buoyancy (), or upthrust, is an upward exerted by a that opposes the of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pressure at the bo ...

). When a liquid presses against a surface, there is a net force that is perpendicular to the surface. Although pressure doesn't 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. 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 $\scriptstyle \sqrt$, where ''h'' is the depth below the free surface. This is the same speed the water (or anything else) would have if freely falling the same vertical distance ''h''.

## Kinematic pressure

:$P=p/\rho_0$ is the kinematic pressure, where $p$ is the pressure and $\rho_0$ constant mass density. The SI unit of ''P'' is m2/s2. Kinematic pressure is used in the same manner as
kinematic viscosity The viscosity of a fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external force. Fluids are a Phase (matter), phase of matter and include liquids, Gas, ...
$\nu$ in order to compute the Navier–Stokes equation without explicitly showing the density $\rho_0$. ;Navier–Stokes equation with kinematic quantities :$\frac + \left(u \nabla\right) u = - \nabla P + \nu \nabla^2 u.$