Pressure

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Pressure (symbol: ''p'' or ''P'') is the
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
applied
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
to the surface of an object per unit
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...
over which that force is distributed.
Gauge pressure Pressure measurement is the measurement of an applied force by a fluid ( liquid or gas) on a surface. Pressure is typically measured in units of force per unit of surface area. Many techniques have been developed for the measurement of pre ...
(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, 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 presentation Music * Unit (album), ...
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 (Pa), for example, is one newton per square metre (N/m2); similarly, the pound-force per
square inch A square inch (plural: square inches) is a unit of area, equal to the area of a square with sides of one inch. 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, inc ...
(
psi Psi, PSI or Ψ may refer to: Alphabetic letters * Psi (Greek) (Ψ, ψ), the 23rd letter of the Greek alphabet * Psi (Cyrillic) (Ѱ, ѱ), letter of the early Cyrillic alphabet, adopted from Greek Arts and entertainment * "Psi" as an abbreviatio ...
) 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, Texa ...
and U.S. customary systems. Pressure may also be expressed in terms of
standard atmospheric pressure The standard atmosphere (symbol: atm) is a unit of pressure defined as Pa. It is sometimes used as a ''reference pressure'' or ''standard pressure''. It is approximately equal to Earth's average atmospheric pressure at sea level. History The s ...
; the
atmosphere An atmosphere () is a layer of gas or layers of gases that envelop a planet, and is held in place by the gravity of the planetary body. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A s ...
(atm) is equal to this pressure, and the
torr The torr (symbol: Torr) is a unit of pressure based on an absolute scale, defined as exactly of a standard atmosphere (). Thus one torr is exactly (≈ ). Historically, one torr was intended to be the same as one " millimeter of merc ...
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 derived from pressure head. Centimetre of water A ''centimetre of water'' (US spelling ''centimeter of water ...
,
millimetre of mercury A millimetre of mercury is a manometric unit of pressure, formerly defined as the extra pressure generated by a column of mercury one millimetre high, and currently defined as exactly pascals. It is denoted mmHg or mm Hg. Although not an ...
, and inch of mercury are used to express pressures in terms of the height of column of a particular fluid in a manometer.

Definition

Pressure is the amount of force applied
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
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 Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
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 most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...
and
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
, and on writing style.

Formula

Mathematically: :$p = \frac,$ where: :$p$ is the pressure, :$F$ is the magnitude of the
normal force In mechanics, the normal force F_n is the component of a contact force that is perpendicular to the surface that an object contacts, as in Figure 1. In this instance '' normal'' is used in the geometric sense and means perpendicular, as oppos ...
, :$A$ 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 ...
quantity. It relates the
vector area In 3-dimensional geometry and vector calculus, an area vector is a vector combining an area quantity with a direction, thus representing an ''oriented area'' in three dimensions. Every bounded surface in three dimensions can be associated with ...
element (a vector normal to the surface) with the
normal force In mechanics, the normal force F_n is the component of a contact force that is perpendicular to the surface that an object contacts, as in Figure 1. In this instance '' normal'' is used in the geometric sense and means perpendicular, as oppos ...
acting on it. The pressure is the scalar
proportionality constant In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...
that relates the two normal vectors: :$d\mathbf_n = -p\,d\mathbf = -p\,\mathbf\,dA.$ 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 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 that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
, and it is conjugate to
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
.

Units

The SI unit for pressure is the pascal (Pa), equal to one newton per square metre (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 (lbf/in2) and
bar Bar or BAR may refer to: Food and drink * Bar (establishment), selling alcoholic beverages * Candy bar * Chocolate bar Science and technology * Bar (river morphology), a deposit of sediment * Bar (tropical cyclone), a layer of cloud * Bar (u ...
, 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. It is equal to 1 dyne per square centimetre. : =  =  = =  = See also *Pasca ...
(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), an improvised fighting vehicle * Technical analysis, a discipline for forecasting the future direction of prices through the study of past market data * Technical drawing, showing how something is co ...
(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 the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force applie ...
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 aiming to understand or predict Earth's atmospheric phenomena including the weather. Those who study meteorological phenomena are meteorologists in research, while t ...
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, but not part of the International System of Units (SI). It is defined as exactly equal to 100,000  Pa (100 kPa), or slightly less than the current average atmospheric pressure on Earth at sea leve ...
(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 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, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the '' ari ...
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 measurement of an applied force by a fluid (liquid or gas) on a surface. Pressure is typically measured in units of force per unit of surface area. Many techniques have been developed for the measurement of pressu ...
, 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 derived from pressure head. Centimetre of water A ''centimetre of water'' (US spelling ''centimeter of water ...
, millimetres of mercury or
inches of mercury Inch of mercury (inHg and ″Hg) is a non- SI unit of measurement for pressure. It is used for barometric pressure in weather reports, refrigeration and aviation in the United States. It is the pressure exerted by a column of mercury in heigh ...
). The most common choices are 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 , where ''g'' is the
gravitational acceleration In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodi ...
. 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, defined as exactly of a standard atmosphere (). Thus one torr is exactly (≈ ). Historically, one torr was intended to be the same as one " millimeter of merc ...
. 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 is measured in millimetres of mercury in most of the world, and lung pressures in centimetres of water are still common.
Underwater divers This is a list of underwater divers whose exploits have made them notable. Underwater divers are people who take part in underwater diving activities – Underwater diving is practiced as part of an occupation, or for recreation, where t ...
use the
metre sea water The metre (or meter) sea water (msw) is a metric unit of pressure used in underwater diving. It is defined as one tenth of a bar. The unit used in the US is the foot sea water (fsw), based on standard gravity and a sea-water density of 64&nb ...
(msw or MSW) and
foot sea water The metre (or meter) sea water (msw) is a metric unit of pressure used in underwater diving. It is defined as one tenth of a bar. The unit used in the US is the foot sea water (fsw), based on standard gravity and a sea-water density of 64  ...
(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, a pressurised gas system to control the internal pressure, and a supply of ...
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 an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical s ...
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 () is a layer of gas or layers of gases that envelop a planet, and is held in place by the gravity of the planetary body. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A s ...
(atm) *manometric units: **centimetre, inch, millimetre (torr) and micrometre (mTorr, micron) of mercury, **height of equivalent column of water, including millimetre (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 Measuring tape with inches The inch (symbol: in or ″) is a unit of length in the British imperial and the United States customary systems of measurement. It is equal to yard or of a foot. Derived from the Roman uncia ("twelfth") ...
, and foot of water; *imperial and customary units: **
kip Kip, KIP or kips may refer to: Athletics * Kip (artistic gymnastics), a basic skill on the women's uneven bars * Kip (trampolining), a coaching skill used in trampolining * Kip-up, an acrobatic manoeuvre used in martial arts and gymnastics People ...
, short ton-force, long ton-force, pound-force,
ounce-force The pound of force or pound-force (symbol: lbf, sometimes lbf,) is a unit of force used in some systems of measurement, including English Engineering units and the foot–pound–second system. Pound-force should not be confused with pound-m ...
, and
poundal The poundal (symbol: pdl) is a unit of force, introduced in 1877, that is part of the Absolute English system of units, which itself is a coherent subsystem of the foot–pound–second system. :1\,\text = 1\,\text\text/\text^2 The poundal is de ...
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 and drink * Bar (establishment), selling alcoholic beverages * Candy bar * Chocolate bar Science and technology * Bar (river morphology), a deposit of sediment * Bar (tropical cyclone), a layer of cloud * Bar (u ...
, decibar,
millibar The bar is a metric unit of pressure, but not part of the International System of Units (SI). It is defined as exactly equal to 100,000  Pa (100 kPa), or slightly less than the current average atmospheric pressure on Earth at sea leve ...
, ***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), an improvised fighting vehicle * Technical analysis, a discipline for forecasting the future direction of prices through the study of past market data * Technical drawing, showing how something is co ...
), **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. It is equal to 1 dyne per square centimetre. : =  =  = =  = See also *Pasca ...
(
dyne The dyne (symbol: dyn; ) is a derived unit of force specified in the centimetre–gram–second (CGS) system of units, a predecessor of the modern SI. History The name dyne was first proposed as a CGS unit of force in 1873 by a Committee of ...
per square centimetre), **kilogram-force and tonne-force per square metre, ** sthene per square metre ( pieze).

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 (in British English) or thumb tack (in North American English) is a short nail or pin used to fasten items to a wall or board for display and intended to be inserted by hand, usually using the thumb. A variety of names is used ...
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 Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
, pressure is defined as a scalar quantity. The negative
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of pressure is called the force density. 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), is the pressure within the atmosphere of Earth. The standard atmosphere (symbol: atm) is a unit of pressure defined as , which is equivalent to 1013.25 millibars, ...
; such measurements are called ''gauge pressure''. An example of this is the air pressure in an
automobile A car or automobile is a motor vehicle with Wheel, wheels. Most definitions of ''cars'' say that they run primarily on roads, Car seat, seat one to eight people, have four wheels, and mainly transport private transport#Personal transport, pe ...
tire A tire (American English) or tyre (British English) is a ring-shaped component that surrounds a wheel's rim to transfer a vehicle's load from the axle through the wheel to the ground and to provide traction on the surface over which t ...
, 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 measurement of an applied force by a fluid ( liquid or gas) on a surface. Pressure is typically measured in units of force per unit of surface area. Many techniques have been developed for the measurement of pres ...
s,
name plate A nameplate identifies and displays a person or product's name. Nameplates are usually shaped as rectangles but are also seen in other shapes, sometimes taking on the shape of someone's written name. Nameplates primarily serve an informat ...
s, graph labels, and table headings, the use of a modifier in parentheses, such as "kPa (gauge)" or "kPa (absolute)", is permitted. In non- SI 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 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 (the others being solid, liquid, and plasma). A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
, the gas as a whole does not appear to move. The individual molecules of the gas, however, are in constant
random motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
. 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 Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
tensor ''σ'', which relates the vector force $\mathbf$ to the
vector area In 3-dimensional geometry and vector calculus, an area vector is a vector combining an area quantity with a direction, thus representing an ''oriented area'' in three dimensions. Every bounded surface in three dimensions can be associated with ...
$\mathbf$ via the linear relation $\mathbf = \sigma\mathbf$. This
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
may be expressed as the sum of the
viscous stress tensor The viscous stress tensor is a tensor used in continuum mechanics to model the part of the stress at a point within some material that can be attributed to the strain rate, the rate at which it is deforming around that point. The viscous stress ...
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 and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, pressure increases the strength of a gravitational field (see stress–energy tensor) and so adds to the mass-energy cause of
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
. This effect is unnoticeable at everyday pressures but is significant in
neutron star A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. w ...
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. (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 the condition of the equilibrium of a floating body and submerged body " fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an imm ...
. 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) an ...
. 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, and Catholic Church, Catholic writer. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pa ...
and Daniel Bernoulli.
Bernoulli's equation In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after the Swiss mathematic ...
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 (zero
viscosity The viscosity of a fluid is a measure of its 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 quantifies the inte ...
). 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 weight The specific weight, also known as the unit weight, is the weight per unit volume of a material. A commonly used value is the specific weight of water on Earth at , which is .National Council of Examiners for Engineering and Surveying (2005). ''Fu ...
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 mechanism which uses brake fluid, typically containing glycol ethers or diethylene glycol, to transfer pressure from the controlling mechanism to the braking mechanism. History During 1904, Frederi ...
* Artesian well * Blood pressure * Hydraulic head * Plant cell turgidity *
Pythagorean cup A Pythagorean cup (also known as a Pythagoras cup, Greedy Cup, Cup of Justice or Tantalus cup) is a practical joke device in a form of a drinking cup, credited to Pythagoras of Samos. When it is filled beyond a certain point, a siphoning effect ...
*
Pressure washing Pressure washing or power washing is the use of high-pressure water spray to remove loose paint, mold, grime, dust, mud, and dirt from surfaces and objects such as buildings, vehicles and concrete surfaces. The volume of a mechanical pressure w ...

Explosion or deflagration pressures

Explosion or deflagration pressures are the result of the ignition of explosive
gas Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma). A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
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 surg ...
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 el ...
, 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 Energy level, 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 me ...
state, and it is especially fragile in the case of liquids where the negative pressure state is similar to superheating and is easily susceptible to cavitation. 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 The ascent of sap in the xylem tissue of plants is the upward movement of water and minerals from the root to the aerial parts of the plant. The conducting cells in xylem are typically non-living and include, in various groups of plants, vessel me ...
in plants taller than 10 m (the atmospheric
pressure head In fluid mechanics, pressure head is the height of a liquid column that corresponds to a particular pressure exerted by the liquid column on the base of its container. It may also be called static pressure head or simply static head (but not ''sta ...
of water). *The
Casimir effect In quantum field theory, the Casimir effect is a physical 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 Casimir, who pr ...
can create a small attractive force due to interactions with
vacuum energy Vacuum energy is an underlying background energy that exists in space throughout the entire Universe. The vacuum energy is a special case of zero-point energy that relates to the quantum vacuum. The effects of vacuum energy can be experiment ...
; 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, with a component of negative pressure acting along another surface normal. **The stresses in an electromagnetic field are generally non-isotropic, with the pressure normal to one surface element (the
normal stress In continuum mechanics, stress is a physical quantity. It is a quantity that describes the magnitude of forces that cause deformation. Stress is defined as ''force per unit area''. When an object is pulled apart by a force it will cause elonga ...
) being negative, and positive for surface elements perpendicular to this. *In
cosmology Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', and in 1731 taken up in Latin by German philosopher ...
,
dark energy In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the univ ...
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 not exp ...
.

Stagnation pressure

Stagnation pressure In fluid dynamics, stagnation pressure is the static pressure at a stagnation point in a fluid flow.Clancy, L.J., ''Aerodynamics'', Section 3.5 At a stagnation point the fluid velocity is zero. In an incompressible flow, stagnation pressure is equ ...
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 the term static pressure has several uses: * In the design and operation of aircraft, ''static pressure'' is the air pressure in the aircraft's static pressure system. * In fluid dynamics, many authors use the term ''static pres ...
, 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, stagnation pressure is the static pressure at a stagnation point in a fluid flow.Clancy, L.J., ''Aerodynamics'', Section 3.5 At a stagnation point the fluid velocity is zero. In an incompressible flow, stagnation pressure is equ ...
, :$\rho$ 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 A pitot ( ) tube (pitot probe) measures fluid flow velocity. It was invented by a French engineer, Henri Pitot, in the early 18th century, and was modified to its modern form in the mid-19th century by a French scientist, Henry Darcy. It ...
, or one of its variations such as a Kiel probe or
Cobra probe A Cobra probe is a device to measure the pressure and velocity components of a moving fluid. It is a multi-holed pressure probe with rotational axis of the probe shaft coplanar with the measurement plane of the instrument. Because of this geomet ...
, connected to a
manometer Pressure measurement is the measurement of an applied force by a fluid (liquid or gas) on a surface. Pressure is typically measured in units of force per unit of surface area. Many techniques have been developed for the measurement of pressu ...
. 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 Boyle's law, also referred to as the Boyle–Mariotte law, or Mariotte's law (especially in France), is an experimental gas law that describes the relationship between pressure and volume of a confined gas. Boyle's law has been stated as: The ...
, , at constant temperature. Surface tension is another example of surface pressure, but with a reversed sign, because "tension" is the opposite to "pressure".

Pressure of an ideal gas

In an
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
, 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 of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
, 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, the amount of substance ''n'' in a given sample of matter is defined as the quantity or number of discrete atomic-scale particles in it divided by the Avogadro constant ''N''A. The particles or entities may be molecules, atoms, ions, ...
, :''T'' is the absolute temperature, :''V'' is the volume, :''R'' is the ideal gas constant.
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 effect ...
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) or vapour (British English and Canadian English; see spelling differences) is a substance in the gas phase at a temperature lower than its critical temperature,R. H. Petrucci, W. S. Harwood, and F. G. Herr ...
in
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In the ...
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), a region of space throughout which all physical properties are essentially uniform * Phase space, a mathematic ...
s in a closed system. All liquids and
solid Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structural ...
s have a tendency to evaporate into a gaseous form, and all
gas Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma). A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
es have a tendency to
condense Condensation is the change of the state of matter from the gas phase into the liquid phase, and is the reverse of vaporization. The word most often refers to the water cycle. It can also be defined as the change in the state of water vapor to ...
back to their liquid or solid form. The
atmospheric pressure Atmospheric pressure, also known as barometric pressure (after the barometer), is the pressure within the atmosphere of Earth. The standard atmosphere (symbol: atm) is a unit of pressure defined as , which is equivalent to 1013.25 millibars, ...
boiling point 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 equals the pressure surrounding the liquid and the liquid changes into a vapor. The boiling point of a liquid varies depending upon the surrounding envir ...
) 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, Bubbles or The Bubble may refer to: Common uses * Bubble (physics), a globule of one substance in another, usually gas in a liquid ** Soap bubble * Economic bubble, a situation where asset prices are much higher than underlying funda ...
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 Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
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 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 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 Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released (conver ...
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 In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after the Swiss mathematic ...
, 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 their head, the person will feel the same amount of water pressure on their 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 force exerted by a fluid that opposes the weight 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 ...
). 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 is a measure of its 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 quantifies the int ...
$\nu$ in order to compute the Navier–Stokes equation without explicitly showing the density $\rho_0$. ;Navier–Stokes equation with kinematic quantities :$\frac + \left(u \nabla\right) u = - \nabla P + \nu \nabla^2 u.$