Editing Hydrostatics

{{short description|Branch of fluid mechanics that studies fluids at rest}}
{{about|the discipline branch|the concept|Hydrostatic equilibrium}}
[[File:Table of Hydraulics and Hydrostatics, Cyclopaedia, Volume 1.jpg|thumb|right|250px|Table of Hydraulics and Hydrostatics, from the 1728 ''[[Cyclopædia, or an Universal Dictionary of Arts and Sciences|Cyclopædia]]'']]
{{Continuum mechanics|cTopic=fluid}}

'''Hydrostatics''' is the branch of [[fluid mechanics]] that studies [[fluid]]s at [[hydrostatic equilibrium]]<ref>{{Cite web|title=Fluid Mechanics/Fluid Statics/Fundamentals of Fluid Statics - Wikibooks, open books for an open world|url=https://en.wikibooks.org/wiki/Fluid_Mechanics/Fluid_Statics/Fundamentals_of_Fluid_Statics#Hydrostatic_Equilibrium|access-date=2021-04-01|website=en.wikibooks.org|language=en}}</ref> and "the pressure in a fluid or exerted by a fluid on an immersed body".<ref name="MW dictionary def">{{cite web |title=Hydrostatics |url=https://www.merriam-webster.com/dictionary/hydrostatics |website=Merriam-Webster |access-date=11 September 2018}}</ref> The word "hydrostatics" is sometimes used to refer specifically to water and other liquids, but more often it includes both gases and liquids, whether [[Compressible flow|compressible]] or [[Incompressible flow|incompressible]].

It encompasses the study of the conditions under which fluids are at rest in [[Mechanical equilibrium|stable equilibrium]]. It is opposed to ''[[fluid dynamics]]'', the study of fluids in motion. 

Hydrostatics is fundamental to ''[[hydraulics]]'', the [[engineering]] of equipment for storing, transporting and using fluids.  It is also relevant to [[geophysics]] and [[astrophysics]] (for example, in understanding [[plate tectonics]] and the anomalies of the [[Gravity of Earth|Earth's gravitational field]]), to [[meteorology]], to [[medicine]] (in the context of [[blood pressure]]), and many other fields.

Hydrostatics offers physical explanations for many phenomena of everyday life, such as why [[atmospheric pressure]] changes with [[altitude]], why wood and oil float on water, and why the surface of still water is always [[Water level|level]] according to the [[spherical Earth|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]]s, [[Aqueduct (water supply)|aqueduct]]s and [[fountain]]s.  [[Archimedes]] is credited with the discovery of [[Archimedes' Principle]], which relates the [[buoyancy]] force on an object that is submerged in a fluid to the weight of fluid displaced by the object. The [[Roman Empire|Roman]] engineer [[Vitruvius]] warned readers about [[lead]] pipes bursting under hydrostatic pressure.<ref name=VitruviusVIII.6>Marcus Vitruvius Pollio (ca. 15 BCE), [https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Vitruvius/8*.html "The Ten Books of Architecture"], Book VIII, Chapter 6.  At the University of Chicago's Penelope site. Accessed on 2013-02-25.</ref>

The concept of pressure and the way it is transmitted by fluids was formulated by the [[France|French]] [[mathematician]] and [[philosopher]] [[Blaise Pascal]] in 1647.{{cn|date=July 2022}}

===Hydrostatics in ancient Greece and Rome===

====Pythagorean Cup====
{{Main|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.{{cn|date=July 2022}}

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====

{{Main|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 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.{{cn|date=July 2022}}

===Pascal's contribution in hydrostatics===

{{Main|Pascal's law}}

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]] [[surface normal|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 [[fluid pressure|pressure]] 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]].{{cn|date=July 2022}}

===Hydrostatic pressure===<!-- This section is linked from [[Water tower]] --><!-- target for redirect [[hHydrostic pressure]] -->
{{See also|Vertical pressure variation}}

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 {{math|''V'' {{=}} 0}} is applied to the [[Navier–Stokes equations]] for viscous fluids or [[Euler equations (fluid dynamics)]] for ideal inviscid fluid, the gradient of pressure becomes a function of body forces only. 
The Navier-Stokes momentum equations are:
{{Equation box 1
|indent=:
|title='''Navier–Stokes momentum equation''' (''convective form'')
|equation=<math> \rho \frac{\mathrm{D} \mathbf{u}}{\mathrm{D} t} = - \nabla [p - \zeta (\nabla\cdot\mathbf{u})] + \nabla \cdot \left\{ \mu \left[\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T} - \tfrac23 (\nabla\cdot\mathbf{u})\mathbf I\right] \right\} + \rho\mathbf{g} .</math>
|cellpadding
|border
|border colour = #FF0000
|background colour = #DCDCDC
}}

By setting the [[flow velocity]] <math> \mathbf u = \mathbf 0</math>, they become simply:

<math> \mathbf 0 = - \nabla p + \rho\mathbf{g}</math>

or:

<math> \nabla p = \rho \mathbf{g}</math>

This is the general form of Stevin's law: the [[pressure gradient]] equals the [[body force]] [[force density]] field.

Let us now consider two particular cases of this law. In case of a [[conservative force|conservative]] body force with [[scalar potential]] <math> \phi</math>:

<math> \rho \mathbf{g} = - \nabla \phi</math>

the Stevin equation becomes:
<math> \nabla p = - \nabla \phi</math>

That can be integrated to give:

<math> \Delta p = - \Delta \phi</math>

So in this case the pressure difference is the opposite of the difference of the scalar potential associated to the body force.
In the other particular case of a body force of constant direction along z:

<math> \mathbf{g} = - g(x ,y,z) \hat k</math>

the generalised Stevin's law above becomes:

<math> \frac {\partial p}{\partial z}= - \rho(x,y,z) g(x, y, z)</math>

That can be integrated to give another (less-) generalised Stevin's law:

<math> p (x,y,z) - p_0(x,y) = - \int_0^z \rho(x,y,z') g(x, y, z') dz'</math>

where:
* <math>p</math> is the hydrostatic pressure (Pa),
* <math>\rho</math> is the fluid [[density]] (kg/m<sup>3</sup>),
* <math>g</math> is [[gravity|gravitational]] acceleration (m/s<sup>2</sup>),
* <math>z</math> is the height (parallel to the direction of gravity) of the test area (m),
* <math>0</math> is the height of the [[Pressure measurement#Absolute, gauge and differential pressures — zero reference|zero reference point of the pressure]] (m)
* <math>p_0</math> is the hydrostatic pressure field (Pa) along x and y at the zero reference point 

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 flow|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 <math>\Delta z</math> of the fluid column between {{mvar|z}} and {{math|''z''<sub>0</sub>}} is often reasonably small compared to the radius of the Earth, one can neglect the variation of {{mvar|[[Gravity|g]]}}. Under these circumstances, one can transport out of the integral the density and the gravity acceleration and the law is simplified into the formula
:<math>\Delta p (z) = \rho g \Delta z,</math>

where <math>\Delta z</math> is the height {{math|''z'' − ''z''<sub>0</sub>}} of the liquid column between the test volume and the zero reference point of the pressure. This formula is often called [[Simon Stevin|Stevin's]] law.<ref>{{cite book|last1=Bettini|first1=Alessandro|title=A Course in Classical Physics 2—Fluids and Thermodynamics|date=2016|publisher=Springer|isbn=978-3-319-30685-8|page=8}}</ref><ref>{{cite book|last1=Mauri|first1=Roberto|title=Transport Phenomena in Multiphase Flow|date=8 April 2015|publisher=Springer|isbn=978-3-319-15792-4|page=24|url=https://books.google.com/books?id=S3L0BwAAQBAJ&pg=PA24|access-date=3 February 2017}}</ref> 
One could arrive to the above formula also by considering the first particular case of the equation for a conservative body force field: in fact the body force field of uniform intensity and direction:

<math> \rho \mathbf{g}(x,y,z) = - \rho g \hat k</math>

is conservative, so one can write the body force density as:

<math> \rho \mathbf{g} = \nabla (- \rho g z) </math>

Then the body force density has a simple scalar potential:

<math> \phi(z) = - \rho g z</math>

And the pressure difference follows another time the Stevin's law:

<math> \Delta p = - \Delta \phi = \rho g \Delta z</math>

The 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 {{math|''ρ''<sub>liquid</sub>}} and {{math|''ρ''(''z''′)<sub>above</sub>}}. For example, the [[Pressure measurement#Absolute, gauge and differential pressures - zero reference|absolute pressure]] compared to vacuum is

:<math>p = \rho g \Delta z + p_\mathrm{0},</math>

where <math>\Delta z</math> is the total height of the liquid column above the test area to the surface, and {{math|''p''<sub>0</sub>}} is the [[atmospheric pressure]], 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]].<ref>{{cite book|url=https://books.google.com/books?id=edPzm5KSMmYC|title=Understanding Food: Principles and Preparation|last=Brown|first=Amy Christian|publisher=Cengage Learning|year=2007|edition=3|isbn=978-0-495-10745-3|page=546}}</ref>

===Medicine===
In medicine, hydrostatic pressure in [[blood vessel]]s is the pressure of the blood against the wall. It is the opposing force to [[oncotic pressure]]. In capillaries, hydrostatic pressure (also known as capillary blood pressure) is higher than the opposing “colloid osmotic pressure” in blood—a “constant” pressure primarily produced by circulating albumin—at the arteriolar end of the capillary. This pressure forces plasma and nutrients out of the capillaries and into surrounding tissues. Fluid and the cellular wastes in the tissues enter the capillaries at the venule end, where the hydrostatic pressure is less than the osmotic pressure in the vessel.<ref name="Openstax Anatomy & Physiology attribution">{{CC-notice|cc=by4|url=https://openstax.org/books/anatomy-and-physiology/pages/26-1-body-fluids-and-fluid-compartments}} {{cite book|last1=Betts|first1=J Gordon|last2=Desaix|first2=Peter|last3=Johnson|first3=Eddie|last4=Johnson|first4=Jody E|last5=Korol|first5=Oksana|last6=Kruse|first6=Dean|last7=Poe|first7=Brandon|last8=Wise|first8=James|last9=Womble|first9=Mark D|last10=Young|first10=Kelly A|title=Anatomy & Physiology|location=Houston|publisher=OpenStax CNX|isbn=978-1-947172-04-3|date=September 16, 2023|at=26.1 Body fluids and fluid compartments}}</ref>

===Atmospheric pressure===
[[Statistical mechanics]] shows that, for a pure [[ideal gas]] of constant temperature in a gravitational field, ''T'', its pressure, ''p'' will vary with height, ''h'', as

:<math>p (h)=p (0) e^{-\frac{Mgh}{kT}}</math>

where
* {{mvar|g}} is the [[standard gravity|acceleration due to gravity]]
* {{mvar|T}} is the  [[absolute temperature]]
* {{mvar|k}} is [[Boltzmann constant]]
* {{mvar|M}} is the [[molecular mass]] of the gas
* {{mvar|p}} is the pressure
* {{mvar|h}} is the height

This is known as the [[barometric formula]], and may be derived from assuming the pressure is [[Hydrostatic pressure|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===
{{Main|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,

:<math>F = \rho g V </math>

where {{mvar|ρ}} is the density of the fluid, {{mvar|g}} is the acceleration due to gravity, and {{mvar|V}} is the volume of fluid directly above the curved surface.<ref name="F-M">{{cite book|last1=Fox|first1=Robert|last2=McDonald|first2=Alan|last3=Pritchard|first3=Philip|title=Fluid Mechanics|edition=8|year=2012|publisher=[[John Wiley & Sons]]|isbn=978-1-118-02641-0|pages=76–83}}</ref>  In the case of a [[ship]], 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&nbsp;– displacing more water and thus receive a higher buoyant force to balance the increased weight.{{cn|date=July 2022}}

Discovery of the principle of buoyancy is attributed to [[Archimedes]].
<!-- this section is poorly translated and hard to understand; needs complete rewriting.

The principle of Archimedes proved experimentally in the following manner: taking a body hanging in the small dynamometer, read the indication of weight. Then, add a deep dish and keep the body hanging on the dynamometer immersed in glass completely overflowing with water.  The indication of the immersed dynamometer will be smaller than the pre-immersion reading.  At the same time, some of the water from the overflowing glass will be poured on the plate; the weight of the overflowed water will equal the difference in body weight between dynamometer readings. This test is more accurate when a special "weir tank" is used properly.
After the above experiment, the principle of Archimedes can be simplified:
: Each body is immersed in a liquid loses both by weight, as the weight of the liquid that displaces.

Accordingly, when a body is found inside a liquid, two main forces ("resultants") will be observed: the body weight and the force applied to this buoyancy.  A further three cases can be distinguished, depending on the values of the resultants:
# The body weight is greater than the buoyancy, causing the body to sink
# The body weight is equal to the buoyancy, causing the body to be suspended in the liquid
# The body weight is less than the buoyancy, causing the body to float

The actual case depends on the [[specific weight]] of the body (solid or liquid) and its relationship to the specific gravity of the liquid. For example, wood, cork, and oil float on water, while iron, aluminum, and mercury sink.

The principle finds very wide application in daily life, particularly in engineering. Anything that floats, such as ships, all lighter water bodies, the human body, floats, amphibious vehicles, etc. obey the principle.  The principle is particularly applicable in shipbuilding, which implements the principle in extensive detail.
-->

===Hydrostatic force on submerged surfaces===
The horizontal and vertical components of the hydrostatic force acting on a submerged surface are given by the following formula:<ref name=" F-M" />

:<math>\begin{align} F_\mathrm{h} &= p_\mathrm{c}A \\ F_\mathrm{v} &= \rho g V \end{align}</math>

where
* {{math|''p''<sub>c</sub>}} is the pressure at the centroid of the vertical projection of the submerged surface
* {{mvar|A}} is the area of the same vertical projection of the surface
* {{mvar|ρ}} is the density of the fluid
* {{mvar|g}} is the acceleration due to gravity
* {{mvar|V}} is the volume of fluid directly above the curved surface

==Liquids (fluids with free surfaces)==
Liquids can have [[free surface]]s at which they interface with gases, or with a [[vacuum]]. 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 (liquid)|meniscus]] through [[capillary action]]. 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]] [[xylem]], the [[transpirational pull]].

===Hanging drops===
Without surface tension, [[Drop (liquid)|drop]]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==
* {{annotated link|Communicating vessels}}
* {{annotated link|Hydrostatic test}}
* {{annotated link|D-DIA}}

==References==
{{Reflist}}

== Further reading ==
* {{cite book|last=Batchelor|first=George K.|date=1967|title=An Introduction to Fluid Dynamics|publisher=Cambridge University Press|isbn=0-521-66396-2}}
* {{cite book|last=Falkovich|first=Gregory|date=2011|title=Fluid Mechanics (A short course for physicists)|publisher=Cambridge University Press|isbn=978-1-107-00575-4}}
* {{cite book|last1=Kundu|first1=Pijush K.|last2=Cohen|first2=Ira M.|date=2008|title=Fluid Mechanics|edition=4th rev.|publisher=Academic Press|isbn=978-0-12-373735-9}}
* {{cite book|last=Currie|first=I. G.|date=1974|title=Fundamental Mechanics of Fluids|publisher=McGraw-Hill|isbn=0-07-015000-1}}
* {{cite book|last1=Massey|first1=B.|last2=Ward-Smith|first2=J.|date=2005|title=Mechanics of Fluids|edition=8th|publisher=Taylor & Francis|isbn=978-0-415-36206-1}}
* {{cite book|last=White|first=Frank M.|date=2003|title=Fluid Mechanics|publisher=McGraw–Hill|isbn=0-07-240217-2}}

==External links==
{{Wiktionary|hydrostatics}}
*[https://www.feynmanlectures.caltech.edu/II_40.html The Flow of Dry Water - The Feynman Lectures on Physics]

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