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Mathematical example: a little price stickiness can go a long way.

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en>JimJJewett: /* Significance in macroeconomics */ slight rewording

2014-01-24T20:49:24Z

Significance in macroeconomics: slight rewording

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[[File:Table of Hydraulics and Hydrostatics, Cyclopaedia, Volume 1.jpg|thumb|right|250px|Table of Hydraulics and Hydrostatics, from the 1728 ''[[Cyclopaedia, or an Universal Dictionary of Arts and Sciences|Cyclopaedia]]'']]

'''Fluid statics''' or '''hydrostatics''' is the branch of [[fluid mechanics]] that studies [[fluid]]s at rest. It embraces the study of the conditions under which fluids are at rest in [[mechanical equilibrium|stable]] [[hydrostatic equilibrium|equilibrium]]; and is contrasted with [[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 water is always flat and horizontal whatever the shape of its container.

==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 the mathematical [[Archimedes's principle|law that bears his name]], that relates the [[buoyancy]] force to the volume and [[density]] of the displaced fluid. 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), [http://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 were formulated by the [[France|French]] [[mathematician]] and [[philosopher]] [[Blaise Pascal]] in 1647.

==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]].

===Hydrostatic pressure===
{{Also|Vertical pressure variation}}

In a fluid at rest, all frictional stresses vanish and the state of stress of the system is called ''hydrostatic''. When this condition of (''V=0'') is applied to [[Navier-Stokes equation]], gradient of pressure becomes a function of body forces only. For a [[Barotropic fluid]] in a conservative force field like a gravitational force field, 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]] is defined as the force exerted on a test area (''p'' = ''F''/''A'', with ''p'': pressure, ''F'': force normal to area ''A'', ''A'': 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:

:<math>p(z)=\frac{1}{A}\int_{z_0}^z dz' \iint\limits_A dx' dy'\, \rho (z') g(z') = \int_{z_0}^z dz'\, \rho (z') g(z') </math>,

where:
* ''p'' is the hydrostatic pressure (Pa),
* ''ρ'' is the fluid [[density]] (kg/m3),
* ''g'' is [[gravity|gravitational]] acceleration (m/s2),
* ''A'' is the test area (m2),
* ''z'' is the height (parallel to the direction of gravity) of the test area (m),
* ''z0'' is the height of the [[Pressure measurement#Absolute, gauge and differential pressures - zero reference|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 reasonably 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 ''h'' of the fluid column between ''z'' and ''z''0 is often reasonably small compared to the radius of the Earth, one can neglect the variation of [[Gravity|''g'']]. Under these circumstances, the integral boils down to the simple formula:
:<math>\ p = \rho g h,</math>

where ''h'' is the height ''z'' − ''z''0 of the liquid column between the test volume and the zero reference point of the pressure. 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 ''ρ''liquid and ''ρ''(''z''<nowiki>'</nowiki>)above. For example, the [[Pressure measurement#Absolute, gauge and differential pressures - zero reference|absolute pressure]] compared to vacuum is:

:<math>\ p = \rho g H + p_\mathrm{atm},</math>

where ''H'' is the total height of the liquid column above the test area to the surface, and ''p''atm is the [[atmospheric pressure]], i.e., the pressure calculated from the remaining integral over the air column from the liquid surface to infinity.

Hydrostatic pressure has been used in the preservation of foods in a process called [[pascalization]].<ref>{{cite book|url=http://books.google.com/books?id=edPzm5KSMmYC&printsec=frontcover&cad=0#v=onepage&q&f=false|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>

In medicine, hydrostatic pressure in [[blood vessels]] 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 gas of constant temperature, ''T'', its pressure, ''p'' will vary with height, ''h'', as:

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

where:
*''g'' is the [[standard gravity|acceleration due to gravity]]
*''T'' is the [[absolute temperature]]
*''k'' is [[Boltzmann constant]]
*''M'' is the mass of a single [[molecule]] of gas
*''p'' is the pressure
*''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 ''ρ'' is the density of the fluid, ''g'' is the acceleration due to gravity, and ''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 – 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]].

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

:<math>F_H = p_cA</math>
:<math>F_V = \rho g V</math>

where:
*''p''c is the pressure at the center of the vertical projection of the submerged surface
*''A'' is the area of the same vertical projection of the surface
*''ρ'' is the density of the fluid
*''g'' is the acceleration due to gravity
*''V'' 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]]. 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]] 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==
{{Continuum mechanics|cTopic=[[Fluid mechanics]]}}
{{Wiktionary|hydrostatics}}
{{portal|Underwater diving}}
* [[Communicating vessels]]
* [[Hydrostatic test]]

==References==
{{Reflist}}

==External links==
* J.B. Calvert (2003) [http://mysite.du.edu/~jcalvert/tech/fluids/hydstat.htm Hydrostatics] from [[University of Denver]]. Accessed on 2013-05-22.

{{Physics-footer}}
{{Diving medicine, physiology and physics}}
{{DEFAULTSORT:Fluid Statics}}
[[Category:Pressure]]
[[Category:Hydrostatics]]
[[Category:Underwater diving physics]]

Nominal rigidity - Revision history

en>Buldri: /* Further reading */

165.123.217.42: /* Mathematical example: a little price stickiness can go a long way. */

en>JimJJewett: /* Significance in macroeconomics */ slight rewording