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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 ...
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 ca ...
to the surface of an object per unit area over which that force is distributed. Gauge pressure (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 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 The square metre ( international spelling as used by the International Bureau of Weights and Measures) or square meter ( American spelling) is the unit of area in the International System of Units (SI) with symbol m2. It is the area of a square ...
(N/m2); similarly, the pound-force per square inch ( psi) is the traditional unit of pressure in the imperial and U.S. customary systems. Pressure may also be expressed in terms of standard atmospheric pressure; 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 is defined as of this. Manometric units such as the centimetre of water, millimetre of mercury, 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 ca ...
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 ...
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 a ...
, 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 oppo ...
, :$A$ is the area of the surface on contact. Pressure is a scalar 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 oppo ...
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 In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one m ...
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 o ...
, and it is conjugate to volume.

## Units

The SI unit for pressure is the pascal (Pa), equal to one newton per
square metre The square metre ( international spelling as used by the International Bureau of Weights and Measures) or square meter ( American spelling) is the unit of area in the International System of Units (SI) with symbol m2. It is the area of a square ...
(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 or of stress based on avoirdupois units. It is the pressure resulting from a force of one pound-force applied t ...
(lbf/in2) and bar, are also in common use. The CGS unit of pressure is the barye (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 (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 applied ...
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 (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 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 ...
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, 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 '' a ...
and is defined as . Because pressure is commonly measured by its ability to displace a column of liquid in a manometer, pressures are often expressed as a depth of a particular fluid (e.g., centimetres of water, millimetres of mercury or inches of mercury). The most common choices are 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 bo ...
. 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 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 of circulating blood against the walls of blood vessels. Most of this pressure results from the heart pumping blood through the circulatory system. When used without qualification, the term "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 use the metre sea water (msw or MSW) and foot sea water (fsw or FSW) units of pressure, and these are the standard units for pressure gauges used to measure pressure exposure in diving chambers 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 sc ...
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 330px, Different lengths as in respect to the electromagnetic spectrum, measured by the metre and its derived scales. The microwave is between 1 meter to 1 millimeter. The millimetre (American and British English spelling differences#-re, -er, ...
(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, 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-ma ...
, and poundal per square inch, **short ton-force and long ton-force per square inch, **fsw (feet sea water) used in underwater diving, particularly in connection with diving pressure exposure and decompression; *non-SI metric units: ** bar, decibar, millibar, ***msw (metres sea water), used in underwater diving, particularly in connection with diving pressure exposure and decompression, **kilogram-force, or kilopond, per square centimetre ( technical atmosphere), **gram-force and tonne-force (metric ton-force) per square centimetre, ** barye ( dyne 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 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, 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 gradi ...
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 wheels. Most definitions of ''cars'' say that they run primarily on roads, seat one to eight people, have four wheels, and mainly transport people instead of goods. The year 1886 is regarded ...
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 gauges,
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), o ...
, the gas as a whole does not appear to move. The individual molecules of the gas, however, are in constant random motion. 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 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 tenso ...
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 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 physic ...
, 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 str ...
. This effect is unnoticeable at everyday pressures but is significant in neutron stars, 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. 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 and Daniel Bernoulli. Bernoulli's equation can be used in almost any situation to determine the pressure at any point in a fluid. The equation makes some assumptions about the fluid, such as the fluid being ideal 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 in ...
). 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). ...
of the fluid, :''v'', velocity of the fluid, :''g'', acceleration of gravity, :''z'', elevation, :$\frac$, pressure head, :$\frac$, velocity head.

### Applications

* Hydraulic brakes * Artesian well *
Blood pressure Blood pressure (BP) is the pressure of circulating blood against the walls of blood vessels. Most of this pressure results from the heart pumping blood through the circulatory system. When used without qualification, the term "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

## Explosion or deflagration pressures

Explosion or
deflagration Deflagration (Lat: ''de + flagrare'', "to burn down") is subsonic combustion in which a pre-mixed flame propagates through a mixture of fuel and oxidizer. Deflagrations can only occur in pre-mixed fuels. Most fires found in daily life are dif ...
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), o ...
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 procedure during which negative gauge pressure is applied intermittently to a pregnant woman's abdomen. *Negative absolute pressures are possible. They are effectively tension, 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 state of least energy. A ball resting in a hollow on a slope is a simple example of metastability. If the ball ...
state, and it is especially fragile in the case of liquids where the negative pressure state is similar to
superheating In thermodynamics, superheating (sometimes referred to as boiling retardation, or boiling delay) is the phenomenon in which a liquid is heated to a temperature higher than its boiling point, without boiling. This is a so-called ''metastable st ...
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 in plants taller than 10 m (the atmospheric pressure head of water). *The Casimir effect 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 experim ...
; 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, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve ...
, 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 produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classica ...
are generally non-isotropic, with the pressure normal to one surface element (the normal stress) being negative, and positive for surface elements perpendicular to this. *In cosmology,
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 uni ...
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 ex ...
.

## 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 e ...
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 pr ...
, 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 e ...
, :$\rho$ 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, or one of its variations such as a Kiel probe or Cobra probe, connected to a manometer. Depending on where the inlet holes are located on the probe, it can measure static pressures or stagnation pressures.

## Surface pressure and surface tension

There is a two-dimensional analog of pressure – the lateral force per unit length applied on a line perpendicular to the force. Surface pressure is denoted by π: :$\pi = \frac$ 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 Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) t ...
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, molecules have no volume and do not interact. According to the ideal gas law, pressure varies linearly with temperature and quantity, and inversely with volume: :$p = \frac,$ 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 gases exhibit a more complex dependence on the variables of state.

## Vapour pressure

Vapour pressure is the pressure of a vapour 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 ther ...
with its condensed phases in a closed system. All liquids and solids 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), o ...
es have a tendency to condense 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 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 en ...
of a liquid (also known as the normal boiling point) is the temperature at which the vapor pressure equals the ambient atmospheric pressure. With any incremental increase in that temperature, the vapor pressure becomes sufficient to overcome atmospheric pressure and lift the liquid to form vapour bubbles inside the bulk of the substance. Bubble formation deeper in the liquid requires a higher pressure, and therefore higher temperature, because the fluid pressure increases above the atmospheric pressure as the depth increases. The vapor pressure that a single component in a mixture contributes to the total pressure in the system is called 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. Mathematica ...
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 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, 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 $\nu$ in order to compute the Navier–Stokes equation without explicitly showing the density $\rho_0$. ;Navier–Stokes equation with kinematic quantities :$\frac + \left(u \nabla\right) u = - \nabla P + \nu \nabla^2 u.$