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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 immersed body". It encompasses the study of the conditions under which fluids are at rest in
stable equilibrium Stable equilibrium can refer to: *Homeostasis, a state of equilibrium used to describe organisms *Mechanical equilibrium, a state in which all particles in a system are at rest, and total force on each particle is permanently zero *Balance of natur ...
as opposed to fluid dynamics, 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, 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 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 he ...
(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 t ...
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 no ...
, to
medicine Medicine is the science and Praxis (process), practice of caring for a patient, managing the diagnosis, prognosis, Preventive medicine, prevention, therapy, treatment, Palliative care, palliation of their injury or disease, and Health promotion ...
(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 pressur ...
), 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 millibar ...
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 ...
s, aqueducts and
fountain A fountain, from the Latin "fons" (genitive "fontis"), meaning source or 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. Fountains were or ...
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 scienti ...
is credited with the discovery of Archimedes' Principle, 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 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 attribut ...
warned readers about
lead Lead is a chemical element with the Symbol (chemistry), symbol Pb (from the Latin ) and atomic number 82. It is a heavy metals, heavy metal that is density, denser than most common materials. Lead is Mohs scale of mineral hardness#Intermediate ...
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 French (french: français(e), link=no) may refer to: * Something of, from, or related to France ** French language, which originated in France, and its various dialects and accents ** French people, a nation and ethnic group identified with Franc ...
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, mathematical structure, structure, space, Mathematica ...
and philosopher
Blaise Pascal Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic writer. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pascal's earlies ...
in 1647.


Hydrostatics in ancient Greece and Rome


Pythagorean Cup

The "fair cup" or Pythagorean cup, 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 is a device invented by Heron of Alexandria 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 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. 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 a ...
normal 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 a ...
on a fluid at rest is isotropic; 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.


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 G ...
, the gradient of pressure becomes a function of body forces only. For a barotropic fluid 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 a ...
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: :p(z)-p(z_0)=\frac\int_^z dz' \iint_A dx' dy'\, \rho (z') g(z') = \int_^z dz'\, \rho (z') g(z') , 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. Mathematicall ...
(kg/m3), * is gravitational acceleration (m/s2), * is the test area (m2), * 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, 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: :p - p_0 = \rho g h, 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 compared to vacuum is: :p = \rho g H + p_\mathrm, 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 millibar ...
, 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 A pressure prism is a way of visually describing the variation of hydrostatic pressure within a volume of fluid. When variables of fluid density, depth, gravity, and other forces such as atmospheric pressure are charted, the resulting figure some ...
. Hydrostatic pressure has been used in the preservation of foods in a process called pascalization.


Medicine

In medicine, hydrostatic pressure in
blood vessel Blood vessels are the structures 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 from ...
s is the pressure of the blood against the wall. It is the opposing force to oncotic pressure.


Atmospheric pressure

Statistical mechanics 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 ...
of constant temperature in a gravitational field, ''T'', its pressure, ''p'' will vary with height, ''h'', as: :p (h)=p (0) e^ 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 ...
* 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 consta ...
* 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 bio ...
of gas * is the pressure * is the height This is known as the barometric formula, and maybe derived from assuming the pressure is hydrostatic. If there are multiple types of molecules in the gas, the partial pressure 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, :F = \rho g V 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 distinguishe ...
, 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 scienti ...
.


Hydrostatic force on submerged surfaces

The horizontal and vertical components of the hydrostatic force acting on a submerged surface are given by the following: :\begin F_\mathrm &= p_\mathrmA \\ F_\mathrm &= \rho g V \end 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 surfaces 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 di ...
. In general, the lack of the ability to sustain a shear stress entails that free surfaces rapidly adjust towards an equilibrium. However, on small length scales, there is an important balancing force from surface tension.


Capillary action

When liquids are constrained in vessels whose dimensions are small, compared to the relevant length scales, surface tension 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), a curve in the upper surface of liquid contained in an object *Meniscus (optics) A lens is a ...
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 Photosynthesis, photosynthetic eukaryotes of the Kingdom (biology), kingdom Plantae. Historically, the plant kingdom encompassed all living things that were not animals, and included algae and fungi; however, all curr ...
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 fr ...
, 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 ...
.


Hanging drops

Without surface tension,
drop Drop, DROP, drops or DROPS may refer to: * Drop (liquid) or droplet, a small volume of liquid ** Eye drops, saline (sometimes mydriatic) drops used as medication for the eyes * Drop (unit), a unit of measure of volume * Falling (physics), allowi ...
s 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