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Fluid statics or hydrostatics is the branch of
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...
that studies the condition of the equilibrium of a floating body and submerged body "
fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
s at
hydrostatic equilibrium
In fluid mechanics, hydrostatic equilibrium (hydrostatic balance, hydrostasy) is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetary ...
and the pressure in a fluid, or exerted by a fluid, on an immersed body".
It encompasses the study of the conditions under which fluids are at rest in
stable equilibrium as opposed to
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 study of fluids in motion. Hydrostatics is a subcategory of fluid statics, which is the study of all fluids, both compressible or incompressible, at rest.
Hydrostatics is fundamental to
hydraulics
Hydraulics (from Greek: Υδραυλική) is a technology and applied science using engineering, chemistry, and other sciences involving the mechanical properties and use of liquids. At a very basic level, hydraulics is the liquid counter ...
, the
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
of equipment for storing, transporting and using fluids. It is also relevant to
geophysics
Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' som ...
and
astrophysics
Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the h ...
(for example, in understanding
plate tectonics
Plate tectonics (from the la, label=Late Latin, tectonicus, from the grc, τεκτονικός, lit=pertaining to building) is the generally accepted scientific theory that considers the Earth's lithosphere to comprise a number of large ...
and the anomalies of the
Earth's gravitational field), to
meteorology
Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did not ...
, to
medicine
Medicine is the science and practice of caring for a patient, managing the diagnosis, prognosis, prevention, treatment, palliation of their injury or disease, and promoting their health. Medicine encompasses a variety of health care pract ...
(in the context of
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" r ...
), and many other fields.
Hydrostatics offers physical explanations for many phenomena of everyday life, such as why
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, 7 ...
changes with
altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
, why wood and oil float on water, and why the surface of still water is always level according to the curvature of the earth.
History
Some principles of hydrostatics have been known in an empirical and intuitive sense since antiquity, by the builders of boats,
cistern
A cistern (Middle English ', from Latin ', from ', "box", from Greek ', "basket") is a waterproof receptacle for holding liquids, usually water. Cisterns are often built to catch and store rainwater. Cisterns are distinguished from wells by t ...
s,
aqueducts and
fountain
A fountain, from the Latin "fons" (genitive "fontis"), meaning source or Spring (hydrology), spring, is a decorative reservoir used for discharging water. It is also a structure that jets water into the air for a decorative or dramatic effect. ...
s.
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
is credited with the discovery of
Archimedes' Principle
Archimedes' principle (also spelled Archimedes's principle) states that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially, is equal to the weight of the fluid that the body displaces. Archimede ...
, which relates the
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 p ...
force on an object that is submerged in a fluid to the weight of fluid displaced by the object. The
Roman
Roman or Romans most often refers to:
*Rome, the capital city of Italy
*Ancient Rome, Roman civilization from 8th century BC to 5th century AD
*Roman people, the people of ancient Rome
*''Epistle to the Romans'', shortened to ''Romans'', a letter ...
engineer
Vitruvius
Vitruvius (; c. 80–70 BC – after c. 15 BC) was a Roman architect and engineer during the 1st century BC, known for his multi-volume work entitled ''De architectura''. He originated the idea that all buildings should have three attribute ...
warned readers about
lead
Lead is a chemical element with the symbol Pb (from the Latin ) and atomic number 82. It is a heavy metal that is denser than most common materials. Lead is soft and malleable, and also has a relatively low melting point. When freshly cu ...
pipes bursting under hydrostatic pressure.
[Marcus Vitruvius Pollio (ca. 15 BCE)]
"The Ten Books of Architecture"
Book VIII, Chapter 6. At the University of Chicago's Penelope site. Accessed on 2013-02-25.
The concept of pressure and the way it is transmitted by fluids was formulated by the
French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
and
philosopher
A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...
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 ...
in 1647.
Hydrostatics in ancient Greece and Rome
Pythagorean Cup
The "fair cup" or
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 ...
, which dates from about the 6th century BC, is a hydraulic technology whose invention is credited to the Greek mathematician and geometer Pythagoras. It was used as a learning tool.
The cup consists of a line carved into the interior of the cup, and a small vertical pipe in the center of the cup that leads to the bottom. The height of this pipe is the same as the line carved into the interior of the cup. The cup may be filled to the line without any fluid passing into the pipe in the center of the cup. However, when the amount of fluid exceeds this fill line, fluid will overflow into the pipe in the center of the cup. Due to the drag that molecules exert on one another, the cup will be emptied.
Heron's fountain
Heron's fountain
Heron's fountain is a hydraulic machine invented by the 1st century AD inventor, mathematician, and physicist Heron of Alexandria (also known as Hero of Alexandria).
Heron studied the pressure of air and steam, described the first steam engin ...
is a device invented by
Heron of Alexandria
Hero of Alexandria (; grc-gre, Ἥρων ὁ Ἀλεξανδρεύς, ''Heron ho Alexandreus'', also known as Heron of Alexandria ; 60 AD) was a Greek mathematician and engineer who was active in his native city of Alexandria, Roman Egypt. He ...
that consists of a jet of fluid being fed by a reservoir of fluid. The fountain is constructed in such a way that the height of the jet exceeds the height of the fluid in the reservoir, apparently in violation of principles of hydrostatic pressure. The device consisted of an opening and two containers arranged one above the other. The intermediate pot, which was sealed, was filled with fluid, and several
cannula
A cannula (; Latin meaning 'little reed'; plural or ) is a tube that can be inserted into the body, often for the delivery or removal of fluid or for the gathering of samples. In simple terms, a cannula can surround the inner or outer surfaces ...
(a small tube for transferring fluid between vessels) connecting the various vessels. Trapped air inside the vessels induces a jet of water out of a nozzle, emptying all water from the intermediate reservoir.
Pascal's contribution in hydrostatics
Pascal made contributions to developments in both hydrostatics and hydrodynamics.
Pascal's Law
Pascal's law (also Pascal's principle or the principle of transmission of fluid-pressure) is a principle in fluid mechanics given by Blaise Pascal that states that a pressure change at any point in a confined incompressible fluid is transmitted ...
is a fundamental principle of fluid mechanics that states that any pressure applied to the surface of a fluid is transmitted uniformly throughout the fluid in all directions, in such a way that initial variations in pressure are not changed.
Pressure in fluids at rest
Due to the fundamental nature of fluids, a fluid cannot remain at rest under the presence of a
shear stress
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...
. However, fluids can exert
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular 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 e ...
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
to any contacting surface. If a point in the fluid is thought of as an infinitesimally small cube, then it follows from the principles of equilibrium that the pressure on every side of this unit of fluid must be equal. If this were not the case, the fluid would move in the direction of the resulting force. Thus, the
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular 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 e ...
on a fluid at rest is
isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
; i.e., it acts with equal magnitude in all directions. This characteristic allows fluids to transmit force through the length of pipes or tubes; i.e., a force applied to a fluid in a pipe is transmitted, via the fluid, to the other end of the pipe. This principle was first formulated, in a slightly extended form, by Blaise Pascal, and is now called
Pascal's law
Pascal's law (also Pascal's principle or the principle of transmission of fluid-pressure) is a principle in fluid mechanics given by Blaise Pascal that states that a pressure change at any point in a confined incompressible fluid is transmitted ...
.
Hydrostatic pressure
In a fluid at rest, all frictional and inertial stresses vanish and the state of stress of the system is called ''hydrostatic''. When this condition of is applied to the
Navier–Stokes equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
, the gradient of pressure becomes a function of body forces only. For a
barotropic fluid
In fluid dynamics, a barotropic fluid is a fluid whose density is a function of pressure only. The barotropic fluid is a useful model of fluid behavior in a wide variety of scientific fields, from meteorology to astrophysics.
The density of mos ...
in a conservative force field like a gravitational force field, the pressure exerted by a fluid at equilibrium becomes a function of force exerted by gravity.
The hydrostatic pressure can be determined from a control volume analysis of an infinitesimally small cube of fluid. Since
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular 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 e ...
is defined as the force exerted on a test area (, with : pressure, : force normal to area , : area), and the only force acting on any such small cube of fluid is the weight of the fluid column above it, hydrostatic pressure can be calculated according to the following formula:
:
where:
* is the hydrostatic pressure (Pa),
* is the fluid
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 ...
(kg/m
3),
* is
gravitational
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 strong ...
acceleration (m/s
2),
* is the test area (m
2),
* is the height (parallel to the direction of gravity) of the test area (m),
* is the height of the
zero reference point of the pressure (m).
For water and other liquids, this integral can be simplified significantly for many practical applications, based on the following two assumptions: Since many liquids can be considered
incompressible
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An eq ...
, a reasonable good estimation can be made from assuming a constant density throughout the liquid. (The same assumption cannot be made within a gaseous environment.) Also, since the height of the fluid column between and is often reasonably small compared to the radius of the Earth, one can neglect the variation of . Under these circumstances, the integral is simplified into the formula:
:
where is the height of the liquid column between the test volume and the zero reference point of the pressure. This formula is often called
Stevin's law. Note that this reference point should lie at or below the surface of the liquid. Otherwise, one has to split the integral into two (or more) terms with the constant and . For example, the
absolute 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 pressur ...
compared to vacuum is:
:
where is the total height of the liquid column above the test area to the surface, and is 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, 7 ...
, i.e., the pressure calculated from the remaining integral over the air column from the liquid surface to infinity. This can easily be visualized using a
pressure prism.
Hydrostatic pressure has been used in the preservation of foods in a process called
pascalization
Pascalization, bridgmanization, high pressure processing (HPP) or high hydrostatic pressure (HHP) processing is a method of preserving and sterilizing food, in which a product is processed under very high pressure, leading to the inactivation of ...
.
Medicine
In medicine, hydrostatic pressure in
blood vessel
The blood vessels are the components of the circulatory system that transport blood throughout the human body. These vessels transport blood cells, nutrients, and oxygen to the tissues of the body. They also take waste and carbon dioxide away ...
s is the pressure of the blood against the wall. It is the opposing force to
oncotic pressure
Oncotic pressure, or colloid osmotic-pressure, is a form of osmotic pressure induced by the proteins, notably albumin, in a blood vessel's plasma (blood/liquid) that causes a pull on fluid back into the capillary. Participating colloids displace ...
.
Atmospheric pressure
Statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
shows that, for a pure
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 ...
of constant temperature in a gravitational field, ''T'', its pressure, ''p'' will vary with height, ''h'', as:
:
where:
* is the
acceleration due to gravity
* is the
absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
* is
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
* is the mass of a single
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
of gas
* is the pressure
* is the height
This is known as the
barometric formula
The barometric formula, sometimes called the ''exponential atmosphere'' or ''isothermal atmosphere'', is a formula used to model how the pressure (or density) of the air changes with altitude. The pressure drops approximately by 11.3 pascals pe ...
, and maybe derived from assuming the pressure is
hydrostatic
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 imme ...
.
If there are multiple types of molecules in the gas, the
partial pressure
In a mixture of gases, each constituent gas has a partial pressure which is the notional pressure of that constituent gas as if it alone occupied the entire volume of the original mixture at the same temperature. The total pressure of an ideal gas ...
of each type will be given by this equation. Under most conditions, the distribution of each species of gas is independent of the other species.
Buoyancy
Any body of arbitrary shape which is immersed, partly or fully, in a fluid will experience the action of a net force in the opposite direction of the local pressure gradient. If this pressure gradient arises from gravity, the net force is in the vertical direction opposite that of the gravitational force. This vertical force is termed buoyancy or buoyant force and is equal in magnitude, but opposite in direction, to the weight of the displaced fluid. Mathematically,
:
where is the density of the fluid, is the acceleration due to gravity, and is the volume of fluid directly above the curved surface.
In the case of a
ship
A ship is a large watercraft that travels the world's oceans and other sufficiently deep waterways, carrying cargo or passengers, or in support of specialized missions, such as defense, research, and fishing. Ships are generally distinguished ...
, for instance, its weight is balanced by pressure forces from the surrounding water, allowing it to float. If more cargo is loaded onto the ship, it would sink more into the water – displacing more water and thus receive a higher buoyant force to balance the increased weight.
Discovery of the principle of buoyancy is attributed to
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
.
Hydrostatic force on submerged surfaces
The horizontal and vertical components of the hydrostatic force acting on a submerged surface are given by the following:
:
where:
* is the pressure at the centroid of the vertical projection of the submerged surface
* is the area of the same vertical projection of the surface
* is the density of the fluid
* is the acceleration due to gravity
* is the volume of fluid directly above the curved surface
Liquids (fluids with free surfaces)
Liquids can have
free surface
In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress,
such as the interface between two homogeneous fluids.
An example of two such homogeneous fluids would be a body of water (liquid) and the air in ...
s at which they interface with gases, or with a
vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
. In general, the lack of the ability to sustain a
shear stress
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...
entails that free surfaces rapidly adjust towards an equilibrium. However, on small length scales, there is an important balancing force from
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) to f ...
.
Capillary action
When liquids are constrained in vessels whose dimensions are small, compared to the relevant length scales,
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) to f ...
effects become important leading to the formation of a
meniscus
Meniscus may refer to:
*Meniscus (anatomy), crescent-shaped fibrocartilaginous structure that partly divides a joint cavity
*Meniscus (liquid)
The meniscus (plural: ''menisci'', from the Greek for "crescent") is the curve in the upper surface ...
through
capillary action
Capillary action (sometimes called capillarity, capillary motion, capillary rise, capillary effect, or wicking) is the process of a liquid flowing in a narrow space without the assistance of, or even in opposition to, any external forces li ...
. This capillary action has profound consequences for biological systems as it is part of one of the two driving mechanisms of the flow of water in
plant
Plants are predominantly photosynthetic eukaryotes of the kingdom Plantae. Historically, the plant kingdom encompassed all living things that were not animals, and included algae and fungi; however, all current definitions of Plantae exclud ...
xylem
Xylem is one of the two types of transport tissue in vascular plants, the other being phloem. The basic function of xylem is to transport water from roots to stems and leaves, but it also transports nutrients. The word ''xylem'' is derived from ...
, the
transpirational pull
Xylem is one of the two types of transport tissue in vascular plants, the other being phloem. The basic function of xylem is to transport water from roots to stems and leaves, but it also transports nutrients. The word ''xylem'' is derived from ...
.
Hanging drops
Without surface tension,
drops would not be able to form. The dimensions and stability of drops are determined by surface tension. The drop's surface tension is directly proportional to the cohesion property of the fluid.
See also
*
*
*
References
Further reading
*
*
*
*
*
*
External links
*
{{DEFAULTSORT:Fluid Statics
Pressure
Underwater diving physics