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1. Shard Calculation

Query:

Response:

Calculated Shard: 152 (from laksa158)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa152.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Hydrostatic_pressure\')), getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Hydrostatic_pressure\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure","crawl_time":1773846763,"first_indexed_time":1428303606,"http_code":200,"src_unparsed":"org,wikipedia!en,\/wiki\/Hydrostatic_pressure s443","src_root_hash":"17790707453426894952","history_drop_reason":null,"meta_title":"Hydrostatic pressure - Wikipedia","meta_descriptions":[],"attrs_boilerpipe_text":"From Wikipedia, the free encyclopedia\nHydrostatic pressure\nis the\nstatic pressure\nexerted at a point of interest by the\nweight\nof a\nfluid\ncolumn above the point.\nDue to the fundamental nature of fluids, a fluid cannot remain at rest under the presence of a\nshear stress\n.  However, fluids can exert\npressure\nnormal\nto 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\npressure\non a fluid at rest is\nisotropic\n; 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\nBlaise Pascal\n, and is now called\nPascal's law\n.\n[\ncitation needed\n]\nIn a fluid at rest, all frictional and inertial stresses vanish and the state of stress of the system is called\nhydrostatic\n. When this condition of\nV\n= 0\nis applied to the\nNavier–Stokes equations\nfor viscous fluids or\nEuler equations (fluid dynamics)\nfor ideal inviscid fluid, the gradient of pressure becomes a function of body forces only. \nThe Navier-Stokes momentum equations are:\nNavier–Stokes momentum equation\n(\nconvective form\n)\nBy setting the\nflow velocity\n :\n, they become simply:\nor:\nThis is the general form of Stevin's law: the\npressure gradient\nequals the\nbody force\nforce density\nfield.\nLet us now consider two particular cases of this law. In case of a\nconservative\nbody force with\nscalar potential\n :\n:\nthe Stevin equation becomes:\nThat can be integrated to give:\nSo in this case the pressure difference is the opposite of the difference of the scalar potential associated to the body force.\nIn the other particular case of a body force of constant direction along z:\nthe generalised Stevin's law above becomes:\nThat can be integrated to give another (less-) generalised Stevin's law:\nwhere:\nSimplification for liquids\n[\nedit\n]\nFor 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\nincompressible\n, 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\nof the fluid column between\nz\nand\nz\n0\nis often reasonably small compared to the radius of the Earth, one can neglect the variation of\ng\n. Under these circumstances, one can transport out of the integral the density and the gravity acceleration and the law is simplified into the formula\nwhere\nis the height\nz\n−\nz\n0\nof the liquid column between the test volume and the zero reference point of the pressure. This formula is often called\nStevin's\nlaw.\n[\n1\n]\n[\n2\n]\nOne 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:\nis conservative, so one can write the body force density as:\nThen the body force density has a simple\nscalar potential\n:\nAnd the pressure difference follows another time the Stevin's law:\nThe 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\nρ\nliquid\nand\nρ\n(\nz\n′)\nabove\n. For example, the\nabsolute pressure\ncompared to vacuum is\nwhere\nis the total height of the liquid column above the test area to the surface, and\np\n0\nis the\natmospheric pressure\n, 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\npressure prism\n.\nHydrostatic pressure has been used in the preservation of foods in a process called\npascalization\n.\n[\n3\n]\nIn medicine, hydrostatic pressure in\nblood vessels\nis the pressure of the blood against the wall. It is the opposing force to\noncotic pressure\n. In capillaries, hydrostatic pressure (also known as capillary blood pressure) is higher than the opposing “colloid\nosmotic pressure\n” 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.\n[\n4\n]\nAtmospheric pressure\n[\nedit\n]\nStatistical mechanics\nshows that, for a pure\nideal gas\nof constant temperature\nT\nin the earth gravitational field, its pressure,\np\nwill vary with height,\nh\n, as\nwhere\ng\nis the\nacceleration due to gravity\nT\nis the\nabsolute temperature\nk\nis\nBoltzmann constant\nM\nis the\nmolecular mass\nof the gas\np\nis the pressure\nh\nis the height\nThis is known as the\nbarometric formula\n, and may be derived from assuming the pressure is\nhydrostatic\n.\nIf there are multiple types of molecules in the gas, the\npartial pressure\nof each type will be given by this equation. Under most conditions, the distribution of each species of gas is independent of the other species.\nAny 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,\nwhere\nρ\nis the density of the fluid,\ng\nis the acceleration due to gravity, and\nV\nis the volume of fluid directly above the curved surface.\n[\n5\n]\nIn the case of a\nship\n, 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.\n[\ncitation needed\n]\nDiscovery of the principle of buoyancy is attributed to\nArchimedes\n.\nHydrostatic force on submerged surfaces\n[\nedit\n]\nThe horizontal and vertical components of the hydrostatic force acting on a submerged surface are given by the following formula:\n[\n5\n]\nwhere\np\nc\nis the pressure at the centroid of the vertical projection of the submerged surface\nA\nis the area of the same vertical projection of the surface\nρ\nis the density of the fluid\ng\nis the acceleration due to gravity\nV\nis the volume of fluid directly above the curved surface\nHydrostatics\nVertical pressure variation\n^\nBettini, Alessandro (2016).\nA Course in Classical Physics 2—Fluids and Thermodynamics\n. Springer. p. 8.\nISBN\n \n978-3-319-30685-8\n.\n^\nMauri, Roberto (8 April 2015).\nTransport Phenomena in Multiphase Flow\n. Springer. p. 24.\nISBN\n \n978-3-319-15792-4\n. Retrieved\n3 February\n2017\n.\n^\nBrown, Amy Christian (2007).\nUnderstanding Food: Principles and Preparation\n(3 ed.). Cengage Learning. p. 546.\nISBN\n \n978-0-495-10745-3\n.\n^\n This article incorporates\ntext\navailable under the\nCC BY 4.0\nlicense.\nBetts, J Gordon; Desaix, Peter; Johnson, Eddie; Johnson, Jody E; Korol, Oksana; Kruse, Dean; Poe, Brandon; Wise, James; Womble, Mark D; Young, Kelly A (September 16, 2023).\nAnatomy & Physiology\n. Houston: OpenStax CNX. 26.1 Body fluids and fluid compartments.\nISBN\n \n978-1-947172-04-3\n.\n^\na\nb\nFox, Robert; McDonald, Alan; Pritchard, Philip (2012).\nFluid Mechanics\n(8 ed.).\nJohn Wiley & Sons\n. pp. \n76–\n83.\nISBN\n \n978-1-118-02641-0\n.","attrs_markdown":"[Jump to content](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#bodyContent)\n\nMain menu\n\nMain menu\n\nmove to sidebar\n\nhide\n\nNavigation\n\n- [Main page](https:\/\/en.wikipedia.org\/wiki\/Main_Page \"Visit the main page [z]\")\n- [Contents](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contents \"Guides to browsing Wikipedia\")\n- [Current events](https:\/\/en.wikipedia.org\/wiki\/Portal:Current_events \"Articles related to current events\")\n- [Random article](https:\/\/en.wikipedia.org\/wiki\/Special:Random \"Visit a randomly selected article [x]\")\n- [About Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:About \"Learn about Wikipedia and how it works\")\n- [Contact us](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contact_us \"How to contact Wikipedia\")\n\nContribute\n\n- [Help](https:\/\/en.wikipedia.org\/wiki\/Help:Contents \"Guidance on how to use and edit Wikipedia\")\n- [Learn to edit](https:\/\/en.wikipedia.org\/wiki\/Help:Introduction \"Learn how to edit Wikipedia\")\n- [Community portal](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Community_portal \"The hub for editors\")\n- [Recent changes](https:\/\/en.wikipedia.org\/wiki\/Special:RecentChanges \"A list of recent changes to Wikipedia [r]\")\n- [Upload file](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:File_upload_wizard \"Add images or other media for use on Wikipedia\")\n- [Special pages](https:\/\/en.wikipedia.org\/wiki\/Special:SpecialPages \"A list of all special pages [q]\")\n\n[![](https:\/\/en.wikipedia.org\/static\/images\/icons\/enwiki-25.svg) ![Wikipedia](https:\/\/en.wikipedia.org\/static\/images\/mobile\/copyright\/wikipedia-wordmark-en-25.svg) ![The Free Encyclopedia](https:\/\/en.wikipedia.org\/static\/images\/mobile\/copyright\/wikipedia-tagline-en-25.svg)](https:\/\/en.wikipedia.org\/wiki\/Main_Page)\n\n[Search](https:\/\/en.wikipedia.org\/wiki\/Special:Search \"Search Wikipedia [f]\")\n\nAppearance\n\n- [Donate](https:\/\/donate.wikimedia.org\/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en)\n- [Create account](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:CreateAccount&returnto=Hydrostatic+pressure \"You are encouraged to create an account and log in; however, it is not mandatory\")\n- [Log in](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:UserLogin&returnto=Hydrostatic+pressure \"You're encouraged to log in; however, it's not mandatory. [o]\")\n\nPersonal tools\n\n- [Donate](https:\/\/donate.wikimedia.org\/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en)\n- [Create account](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:CreateAccount&returnto=Hydrostatic+pressure \"You are encouraged to create an account and log in; however, it is not mandatory\")\n- [Log in](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:UserLogin&returnto=Hydrostatic+pressure \"You're encouraged to log in; however, it's not mandatory. [o]\")\n\n## Contents\nmove to sidebar\n\nhide\n\n- [(Top)](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure)\n- [1 Background](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#Background)\n- [2 Formulation](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#Formulation)\n  Toggle Formulation subsection\n  - [2\\.1 Simplification for liquids](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#Simplification_for_liquids)\n- [3 Applications](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#Applications)\n  Toggle Applications subsection\n  - [3\\.1 Medicine](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#Medicine)\n  - [3\\.2 Atmospheric pressure](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#Atmospheric_pressure)\n- [4 Buoyancy](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#Buoyancy)\n  Toggle Buoyancy subsection\n  - [4\\.1 Hydrostatic force on submerged surfaces](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#Hydrostatic_force_on_submerged_surfaces)\n- [5 See also](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#See_also)\n- [6 References](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#References)\n\nToggle the table of contents\n\n# Hydrostatic pressure\n29 languages\n\n- [Asturianu](https:\/\/ast.wikipedia.org\/wiki\/Presi%C3%B3n_nun_fluy%C3%ADu \"Presión nun fluyíu – Asturian\")\n- [Azərbaycanca](https:\/\/az.wikipedia.org\/wiki\/Hidrostatik_t%C9%99zyiq \"Hidrostatik təzyiq – Azerbaijani\")\n- [Bosanski](https:\/\/bs.wikipedia.org\/wiki\/Pritisak_fluida \"Pritisak fluida – Bosnian\")\n- [Català](https:\/\/ca.wikipedia.org\/wiki\/Pressi%C3%B3_hidroest%C3%A0tica \"Pressió hidroestàtica – Catalan\")\n- [Čeština](https:\/\/cs.wikipedia.org\/wiki\/Hydrostatick%C3%BD_tlak \"Hydrostatický tlak – Czech\")\n- [Deutsch](https:\/\/de.wikipedia.org\/wiki\/Hydrostatischer_Druck \"Hydrostatischer Druck – German\")\n- [Ελληνικά](https:\/\/el.wikipedia.org\/wiki\/%CE%A5%CE%B4%CF%81%CE%BF%CF%83%CF%84%CE%B1%CF%84%CE%B9%CE%BA%CE%AE_%CF%80%CE%AF%CE%B5%CF%83%CE%B7 \"Υδροστατική πίεση – Greek\")\n- [Esperanto](https:\/\/eo.wikipedia.org\/wiki\/Premo_de_fluido \"Premo de fluido – Esperanto\")\n- [Español](https:\/\/es.wikipedia.org\/wiki\/Presi%C3%B3n_en_un_fluido \"Presión en un fluido – Spanish\")\n- [Euskara](https:\/\/eu.wikipedia.org\/wiki\/Presio_hidrostatiko \"Presio hidrostatiko – Basque\")\n- [Suomi](https:\/\/fi.wikipedia.org\/wiki\/Hydrostaattinen_paine \"Hydrostaattinen paine – Finnish\")\n- [Magyar](https:\/\/hu.wikipedia.org\/wiki\/Hidrosztatikai_nyom%C3%A1s \"Hidrosztatikai nyomás – Hungarian\")\n- [Հայերեն](https:\/\/hy.wikipedia.org\/wiki\/%D5%80%D5%AB%D5%A4%D6%80%D5%B8%D5%BD%D5%BF%D5%A1%D5%BF%D5%AB%D5%AF_%D5%B3%D5%B6%D5%B7%D5%B8%D6%82%D5%B4 \"Հիդրոստատիկ ճնշում – Armenian\")\n- [Italiano](https:\/\/it.wikipedia.org\/wiki\/Pressione_idrostatica \"Pressione idrostatica – Italian\")\n- [日本語](https:\/\/ja.wikipedia.org\/wiki\/%E6%B0%B4%E5%9C%A7 \"水圧 – Japanese\")\n- [Қазақша](https:\/\/kk.wikipedia.org\/wiki\/%D0%93%D0%B8%D0%B4%D1%80%D0%BE%D1%81%D1%82%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D2%9B%D1%8B%D1%81%D1%8B%D0%BC \"Гидростатикалық қысым – Kazakh\")\n- [한국어](https:\/\/ko.wikipedia.org\/wiki\/%EC%88%98%EC%95%95 \"수압 – Korean\")\n- [Latviešu](https:\/\/lv.wikipedia.org\/wiki\/Hidrostatiskais_spiediens \"Hidrostatiskais spiediens – Latvian\")\n- [Bahasa Melayu](https:\/\/ms.wikipedia.org\/wiki\/Tekanan_bendalir \"Tekanan bendalir – Malay\")\n- [Nederlands](https:\/\/nl.wikipedia.org\/wiki\/Hydrostatische_druk \"Hydrostatische druk – Dutch\")\n- [Polski](https:\/\/pl.wikipedia.org\/wiki\/Ci%C5%9Bnienie_hydrostatyczne \"Ciśnienie hydrostatyczne – Polish\")\n- [Русский](https:\/\/ru.wikipedia.org\/wiki\/%D0%93%D0%B8%D0%B4%D1%80%D0%BE%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%B4%D0%B0%D0%B2%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5 \"Гидростатическое давление – Russian\")\n- [Simple English](https:\/\/simple.wikipedia.org\/wiki\/Fluid_pressure \"Fluid pressure – Simple English\")\n- [Slovenčina](https:\/\/sk.wikipedia.org\/wiki\/Hydrostatick%C3%BD_tlak \"Hydrostatický tlak – Slovak\")\n- [Slovenščina](https:\/\/sl.wikipedia.org\/wiki\/Hidrostati%C4%8Dni_tlak \"Hidrostatični tlak – Slovenian\")\n- [Српски \/ srpski](https:\/\/sr.wikipedia.org\/wiki\/%D0%A5%D0%B8%D0%B4%D1%80%D0%BE%D1%81%D1%82%D0%B0%D1%82%D0%B8%D0%BA%D0%B0 \"Хидростатика – Serbian\")\n- [Українська](https:\/\/uk.wikipedia.org\/wiki\/%D0%93%D1%96%D0%B4%D1%80%D0%BE%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%87%D0%BD%D0%B8%D0%B9_%D1%82%D0%B8%D1%81%D0%BA \"Гідростатичний тиск – Ukrainian\")\n- [Tiếng Việt](https:\/\/vi.wikipedia.org\/wiki\/%C3%81p_su%E1%BA%A5t_ch%E1%BA%A5t_l%E1%BB%8Fng \"Áp suất chất lỏng – Vietnamese\")\n- [中文](https:\/\/zh.wikipedia.org\/wiki\/%E6%B5%81%E9%AB%94%E5%A3%93%E5%8A%9B \"流體壓力 – Chinese\")\n\n[Edit links](https:\/\/www.wikidata.org\/wiki\/Special:EntityPage\/Q1149672#sitelinks-wikipedia \"Edit interlanguage links\")\n\n- [Article](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure \"View the content page [c]\")\n- [Talk](https:\/\/en.wikipedia.org\/wiki\/Talk:Hydrostatic_pressure \"Discuss improvements to the content page [t]\")\n\nEnglish\n\n- [Read](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure)\n- [Edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&action=edit \"Edit this page [e]\")\n- [View history](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&action=history \"Past revisions of this page [h]\")\n\nTools\n\nTools\n\nmove to sidebar\n\nhide\n\nActions\n\n- [Read](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure)\n- [Edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&action=edit \"Edit this page [e]\")\n- [View history](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&action=history)\n\nGeneral\n\n- [What links here](https:\/\/en.wikipedia.org\/wiki\/Special:WhatLinksHere\/Hydrostatic_pressure \"List of all English Wikipedia pages containing links to this page [j]\")\n- [Related changes](https:\/\/en.wikipedia.org\/wiki\/Special:RecentChangesLinked\/Hydrostatic_pressure \"Recent changes in pages linked from this page [k]\")\n- [Upload file](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:File_Upload_Wizard \"Upload files [u]\")\n- [Permanent link](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&oldid=1341733806 \"Permanent link to this revision of this page\")\n- [Page information](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&action=info \"More information about this page\")\n- [Cite this page](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:CiteThisPage&page=Hydrostatic_pressure&id=1341733806&wpFormIdentifier=titleform \"Information on how to cite this page\")\n- [Get shortened URL](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FHydrostatic_pressure)\n\nPrint\/export\n\n- [Download as PDF](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:DownloadAsPdf&page=Hydrostatic_pressure&action=show-download-screen \"Download this page as a PDF file\")\n- [Printable version](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&printable=yes \"Printable version of this page [p]\")\n\nIn other projects\n\n- [Wikidata item](https:\/\/www.wikidata.org\/wiki\/Special:EntityPage\/Q1149672 \"Structured data on this page hosted by Wikidata [g]\")\n\nAppearance\n\nmove to sidebar\n\nhide\n\nFrom Wikipedia, the free encyclopedia\n\nPhysical quantity\n\n**Hydrostatic pressure** is the [static pressure](https:\/\/en.wikipedia.org\/wiki\/Static_pressure \"Static pressure\") exerted at a point of interest by the [weight](https:\/\/en.wikipedia.org\/wiki\/Weight \"Weight\") of a [fluid](https:\/\/en.wikipedia.org\/wiki\/Fluid \"Fluid\") column above the point.\n\n## Background\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&action=edit&section=1 \"Edit section: Background\")\\]\n\nDue to the fundamental nature of fluids, a fluid cannot remain at rest under the presence of a [shear stress](https:\/\/en.wikipedia.org\/wiki\/Shear_stress \"Shear stress\"). However, fluids can exert [pressure](https:\/\/en.wikipedia.org\/wiki\/Pressure \"Pressure\") [normal](https:\/\/en.wikipedia.org\/wiki\/Surface_normal \"Surface 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](https:\/\/en.wikipedia.org\/wiki\/Fluid_pressure \"Fluid pressure\") on a fluid at rest is [isotropic](https:\/\/en.wikipedia.org\/wiki\/Isotropic \"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](https:\/\/en.wikipedia.org\/wiki\/Blaise_Pascal \"Blaise Pascal\"), and is now called [Pascal's law](https:\/\/en.wikipedia.org\/wiki\/Pascal%27s_law \"Pascal's law\").\\[*[citation needed](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Citation_needed \"Wikipedia:Citation needed\")*\\]\n\n## Formulation\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&action=edit&section=2 \"Edit section: Formulation\")\\]\n\nIn 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 *V* = 0 is applied to the [Navier–Stokes equations](https:\/\/en.wikipedia.org\/wiki\/Navier%E2%80%93Stokes_equations \"Navier–Stokes equations\") for viscous fluids or [Euler equations (fluid dynamics)](https:\/\/en.wikipedia.org\/wiki\/Euler_equations_\\(fluid_dynamics\\) \"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:\n\n**Navier–Stokes momentum equation** (*convective form*)\n\nρ\n\nD\n\nu\n\nD\n\nt\n\n\\=\n\n−\n\n∇\n\n\\[\n\np\n\n−\n\nζ\n\n(\n\n∇\n\n⋅\n\nu\n\n)\n\n\\]\n\n\\+\n\n∇\n\n⋅\n\n{\n\nμ\n\n\\[\n\n∇\n\nu\n\n\\+\n\n(\n\n∇\n\nu\n\n)\n\nT\n\n−\n\n2\n\n3\n\n(\n\n∇\n\n⋅\n\nu\n\n)\n\nI\n\n\\]\n\n}\n\n\\+\n\nρ\n\ng\n\n.\n\n{\\\\displaystyle \\\\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} }-{\\\\tfrac {2}{3}}(\\\\nabla \\\\cdot \\\\mathbf {u} )\\\\mathbf {I} \\\\right\\]\\\\right\\\\}+\\\\rho \\\\mathbf {g} .}\n\n![{\\\\displaystyle \\\\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} }-{\\\\tfrac {2}{3}}(\\\\nabla \\\\cdot \\\\mathbf {u} )\\\\mathbf {I} \\\\right\\]\\\\right\\\\}+\\\\rho \\\\mathbf {g} .}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6fd98d1bcacc93a84fa4fc981a05a200e680be04)\n\nBy setting the [flow velocity](https:\/\/en.wikipedia.org\/wiki\/Flow_velocity \"Flow velocity\") :u \\= 0 {\\\\displaystyle \\\\mathbf {u} =\\\\mathbf {0} } ![{\\\\displaystyle \\\\mathbf {u} =\\\\mathbf {0} }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/92a3de96f1315a7b1aa90a60f2322d7e5012b419), they become simply:\n\n0\n\n\\=\n\n−\n\n∇\n\np\n\n\\+\n\nρ\n\ng\n\n{\\\\displaystyle \\\\mathbf {0} =-\\\\nabla p+\\\\rho \\\\mathbf {g} }\n\n![{\\\\displaystyle \\\\mathbf {0} =-\\\\nabla p+\\\\rho \\\\mathbf {g} }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fadede56aa656a73a1903ef4ef33c276184ecac1)\n\nor:\n\n∇\n\np\n\n\\=\n\nρ\n\ng\n\n{\\\\displaystyle \\\\nabla p=\\\\rho \\\\mathbf {g} }\n\n![{\\\\displaystyle \\\\nabla p=\\\\rho \\\\mathbf {g} }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d5d9f468c7d46b3471f2b1534701b7bbed1fc9e4)\n\nThis is the general form of Stevin's law: the [pressure gradient](https:\/\/en.wikipedia.org\/wiki\/Pressure_gradient \"Pressure gradient\") equals the [body force](https:\/\/en.wikipedia.org\/wiki\/Body_force \"Body force\") [force density](https:\/\/en.wikipedia.org\/wiki\/Force_density \"Force density\") field.\n\nLet us now consider two particular cases of this law. In case of a [conservative](https:\/\/en.wikipedia.org\/wiki\/Conservative_force \"Conservative force\") body force with [scalar potential](https:\/\/en.wikipedia.org\/wiki\/Scalar_potential \"Scalar potential\") :ϕ {\\\\displaystyle \\\\phi } ![{\\\\displaystyle \\\\phi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/72b1f30316670aee6270a28334bdf4f5072cdde4):\n\nρ\n\ng\n\n\\=\n\n−\n\n∇\n\nϕ\n\n{\\\\displaystyle \\\\rho \\\\mathbf {g} =-\\\\nabla \\\\phi }\n\n![{\\\\displaystyle \\\\rho \\\\mathbf {g} =-\\\\nabla \\\\phi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bb954c118e31f5c60fef95bbfe11eaa52a62b9b9)\n\nthe Stevin equation becomes:\n\n∇\n\np\n\n\\=\n\n−\n\n∇\n\nϕ\n\n{\\\\displaystyle \\\\nabla p=-\\\\nabla \\\\phi }\n\n![{\\\\displaystyle \\\\nabla p=-\\\\nabla \\\\phi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5810d8e16907c3b14f06f080fa7406811594ee27)\n\nThat can be integrated to give:\n\nΔ\n\np\n\n\\=\n\n−\n\nΔ\n\nϕ\n\n{\\\\displaystyle \\\\Delta p=-\\\\Delta \\\\phi }\n\n![{\\\\displaystyle \\\\Delta p=-\\\\Delta \\\\phi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bc17ceebfece38c57a6b1ccf34447dbe8b824f92)\n\nSo 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:\n\ng\n\n\\=\n\n−\n\ng\n\n(\n\nx\n\n,\n\ny\n\n,\n\nz\n\n)\n\nk\n\n^\n\n{\\\\displaystyle \\\\mathbf {g} =-g(x,y,z){\\\\hat {k}}}\n\n![{\\\\displaystyle \\\\mathbf {g} =-g(x,y,z){\\\\hat {k}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a192db8a1f1df66661ab6991e2a9ef35311d56b8)\n\nthe generalised Stevin's law above becomes:\n\n∂\n\np\n\n∂\n\nz\n\n\\=\n\n−\n\nρ\n\n(\n\nx\n\n,\n\ny\n\n,\n\nz\n\n)\n\ng\n\n(\n\nx\n\n,\n\ny\n\n,\n\nz\n\n)\n\n{\\\\displaystyle {\\\\frac {\\\\partial p}{\\\\partial z}}=-\\\\rho (x,y,z)g(x,y,z)}\n\n![{\\\\displaystyle {\\\\frac {\\\\partial p}{\\\\partial z}}=-\\\\rho (x,y,z)g(x,y,z)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/836715bf560634b1be927d83a788bd92964b1c2f)\n\nThat can be integrated to give another (less-) generalised Stevin's law:\n\np\n\n(\n\nx\n\n,\n\ny\n\n,\n\nz\n\n)\n\n−\n\np\n\n0\n\n(\n\nx\n\n,\n\ny\n\n)\n\n\\=\n\n−\n\n∫\n\n0\n\nz\n\nρ\n\n(\n\nx\n\n,\n\ny\n\n,\n\nz\n\n′\n\n)\n\ng\n\n(\n\nx\n\n,\n\ny\n\n,\n\nz\n\n′\n\n)\n\nd\n\nz\n\n′\n\n{\\\\displaystyle p(x,y,z)-p\\_{0}(x,y)=-\\\\int \\_{0}^{z}\\\\rho (x,y,z')g(x,y,z')dz'}\n\n![{\\\\displaystyle p(x,y,z)-p\\_{0}(x,y)=-\\\\int \\_{0}^{z}\\\\rho (x,y,z')g(x,y,z')dz'}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/78bcc6804783378024effbb00548d25e28827d18)\n\nwhere:\n\n- p\n  \n  {\\\\displaystyle p}\n  \n  ![{\\\\displaystyle p}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/81eac1e205430d1f40810df36a0edffdc367af36)\n  is the hydrostatic pressure (Pa),\n- ρ\n  \n  {\\\\displaystyle \\\\rho }\n  \n  ![{\\\\displaystyle \\\\rho }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1f7d439671d1289b6a816e6af7a304be40608d64)\n  is the fluid [density](https:\/\/en.wikipedia.org\/wiki\/Density \"Density\") (kg\/m3),\n- g\n  \n  {\\\\displaystyle g}\n  \n  ![{\\\\displaystyle g}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d3556280e66fe2c0d0140df20935a6f057381d77)\n  is [gravitational](https:\/\/en.wikipedia.org\/wiki\/Gravity \"Gravity\") acceleration (m\/s2),\n- z\n  \n  {\\\\displaystyle z}\n  \n  ![{\\\\displaystyle z}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bf368e72c009decd9b6686ee84a375632e11de98)\n  is the height (parallel to the direction of gravity) of the test area (m),\n- 0\n  \n  {\\\\displaystyle 0}\n  \n  ![{\\\\displaystyle 0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2aae8864a3c1fec9585261791a809ddec1489950)\n  is the height of the [zero reference point of the pressure](https:\/\/en.wikipedia.org\/wiki\/Pressure_measurement#Absolute,_gauge_and_differential_pressures_%E2%80%94_zero_reference \"Pressure measurement\") (m)\n- p\n  \n  0\n  \n  {\\\\displaystyle p\\_{0}}\n  \n  ![{\\\\displaystyle p\\_{0}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2b969ada68a88e2aeba9a2d2096abaf1fd53c21d)\n  is the hydrostatic pressure field (Pa) along x and y at the zero reference point\n\n### Simplification for liquids\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&action=edit&section=3 \"Edit section: Simplification for liquids\")\\]\n\nFor 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](https:\/\/en.wikipedia.org\/wiki\/Incompressible_flow \"Incompressible flow\"), 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 Δ z {\\\\displaystyle \\\\Delta z} ![{\\\\displaystyle \\\\Delta z}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/cf59b5e4dc02e6b2b89b426a27e02c84861c0caa) 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 [g](https:\/\/en.wikipedia.org\/wiki\/Gravity \"Gravity\"). Under these circumstances, one can transport out of the integral the density and the gravity acceleration and the law is simplified into the formula\n\nΔ\n\np\n\n(\n\nz\n\n)\n\n\\=\n\nρ\n\ng\n\nΔ\n\nz\n\n,\n\n{\\\\displaystyle \\\\Delta p(z)=\\\\rho g\\\\Delta z,}\n\n![{\\\\displaystyle \\\\Delta p(z)=\\\\rho g\\\\Delta z,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9afa9a1da855d0dde3239d2e26ad69a22c9354c6)\n\nwhere Δ z {\\\\displaystyle \\\\Delta z} ![{\\\\displaystyle \\\\Delta z}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/cf59b5e4dc02e6b2b89b426a27e02c84861c0caa) is the height *z* − *z*0 of the liquid column between the test volume and the zero reference point of the pressure. This formula is often called [Stevin's](https:\/\/en.wikipedia.org\/wiki\/Simon_Stevin \"Simon Stevin\") law.[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_note-1)[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_note-2) 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:\n\nρ\n\ng\n\n(\n\nx\n\n,\n\ny\n\n,\n\nz\n\n)\n\n\\=\n\n−\n\nρ\n\ng\n\nk\n\n^\n\n{\\\\displaystyle \\\\rho \\\\mathbf {g} (x,y,z)=-\\\\rho g{\\\\hat {k}}}\n\n![{\\\\displaystyle \\\\rho \\\\mathbf {g} (x,y,z)=-\\\\rho g{\\\\hat {k}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/286c83b8ad7070465180a3bda0f08ac24ca1b7a7)\n\nis conservative, so one can write the body force density as:\n\nρ\n\ng\n\n\\=\n\n∇\n\n(\n\n−\n\nρ\n\ng\n\nz\n\n)\n\n{\\\\displaystyle \\\\rho \\\\mathbf {g} =\\\\nabla (-\\\\rho gz)}\n\n![{\\\\displaystyle \\\\rho \\\\mathbf {g} =\\\\nabla (-\\\\rho gz)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a4f7a68af5b40d5e2bb679d68fe7d38ad381d1fc)\n\nThen the body force density has a simple [scalar potential](https:\/\/en.wikipedia.org\/wiki\/Scalar_potential \"Scalar potential\"):\n\nϕ\n\n(\n\nz\n\n)\n\n\\=\n\n−\n\nρ\n\ng\n\nz\n\n{\\\\displaystyle \\\\phi (z)=-\\\\rho gz}\n\n![{\\\\displaystyle \\\\phi (z)=-\\\\rho gz}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ad0108c644e333a75e7b0ae3c46a697b51d08b03)\n\nAnd the pressure difference follows another time the Stevin's law:\n\nΔ\n\np\n\n\\=\n\n−\n\nΔ\n\nϕ\n\n\\=\n\nρ\n\ng\n\nΔ\n\nz\n\n{\\\\displaystyle \\\\Delta p=-\\\\Delta \\\\phi =\\\\rho g\\\\Delta z}\n\n![{\\\\displaystyle \\\\Delta p=-\\\\Delta \\\\phi =\\\\rho g\\\\Delta z}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/787b041c209cf18ccbb6b1aa4368c9aee99b2066)\n\nThe 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*′)above. For example, the [absolute pressure](https:\/\/en.wikipedia.org\/wiki\/Pressure_measurement#Absolute,_gauge_and_differential_pressures_-_zero_reference \"Pressure measurement\") compared to vacuum is\n\np\n\n\\=\n\nρ\n\ng\n\nΔ\n\nz\n\n\\+\n\np\n\n0\n\n,\n\n{\\\\displaystyle p=\\\\rho g\\\\Delta z+p\\_{\\\\mathrm {0} },}\n\n![{\\\\displaystyle p=\\\\rho g\\\\Delta z+p\\_{\\\\mathrm {0} },}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/65dc7e8253c8fa7fade48f906aef2b45e821c398)\n\nwhere Δ z {\\\\displaystyle \\\\Delta z} ![{\\\\displaystyle \\\\Delta z}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/cf59b5e4dc02e6b2b89b426a27e02c84861c0caa) is the total height of the liquid column above the test area to the surface, and *p*0 is the [atmospheric pressure](https:\/\/en.wikipedia.org\/wiki\/Atmospheric_pressure \"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](https:\/\/en.wikipedia.org\/wiki\/Pressure_prism \"Pressure prism\").\n\n## Applications\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&action=edit&section=4 \"Edit section: Applications\")\\]\n\nHydrostatic pressure has been used in the preservation of foods in a process called [pascalization](https:\/\/en.wikipedia.org\/wiki\/Pascalization \"Pascalization\").[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_note-3)\n\n### Medicine\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&action=edit&section=5 \"Edit section: Medicine\")\\]\n\nIn medicine, hydrostatic pressure in [blood vessels](https:\/\/en.wikipedia.org\/wiki\/Blood_vessel \"Blood vessel\") is the pressure of the blood against the wall. It is the opposing force to [oncotic pressure](https:\/\/en.wikipedia.org\/wiki\/Oncotic_pressure \"Oncotic pressure\"). In capillaries, hydrostatic pressure (also known as capillary blood pressure) is higher than the opposing “colloid [osmotic pressure](https:\/\/en.wikipedia.org\/wiki\/Osmotic_pressure \"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.[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_note-Openstax_Anatomy_&_Physiology_attribution-4)\n\n### Atmospheric pressure\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&action=edit&section=6 \"Edit section: Atmospheric pressure\")\\]\n\n[Statistical mechanics](https:\/\/en.wikipedia.org\/wiki\/Statistical_mechanics \"Statistical mechanics\") shows that, for a pure [ideal gas](https:\/\/en.wikipedia.org\/wiki\/Ideal_gas \"Ideal gas\") of constant temperature *T* in the earth gravitational field, its pressure, *p* will vary with height, *h*, as\n\np\n\n(\n\nh\n\n)\n\n\\=\n\np\n\n(\n\n0\n\n)\n\ne\n\n−\n\nM\n\ng\n\nh\n\nk\n\nT\n\n{\\\\displaystyle p(h)=p(0)e^{-{\\\\frac {Mgh}{kT}}}}\n\n![{\\\\displaystyle p(h)=p(0)e^{-{\\\\frac {Mgh}{kT}}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/aa359443bd71846a0f795d8af50a298f02fc900b)\n\nwhere\n\n- g is the [acceleration due to gravity](https:\/\/en.wikipedia.org\/wiki\/Standard_gravity \"Standard gravity\")\n- T is the [absolute temperature](https:\/\/en.wikipedia.org\/wiki\/Absolute_temperature \"Absolute temperature\")\n- k is [Boltzmann constant](https:\/\/en.wikipedia.org\/wiki\/Boltzmann_constant \"Boltzmann constant\")\n- M is the [molecular mass](https:\/\/en.wikipedia.org\/wiki\/Molecular_mass \"Molecular mass\") of the gas\n- p is the pressure\n- h is the height\n\nThis is known as the [barometric formula](https:\/\/en.wikipedia.org\/wiki\/Barometric_formula \"Barometric formula\"), and may be derived from assuming the pressure is [hydrostatic]().\n\nIf there are multiple types of molecules in the gas, the [partial pressure](https:\/\/en.wikipedia.org\/wiki\/Partial_pressure \"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.\n\n## Buoyancy\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&action=edit&section=7 \"Edit section: Buoyancy\")\\]\n\nMain article: [Buoyancy](https:\/\/en.wikipedia.org\/wiki\/Buoyancy \"Buoyancy\")\n\nAny 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,\n\nF\n\n\\=\n\nρ\n\ng\n\nV\n\n{\\\\displaystyle F=\\\\rho gV}\n\n![{\\\\displaystyle F=\\\\rho gV}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2b1b65f31d86a2c06763a4f93131781af9867fba)\n\nwhere 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.[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_note-F-M-5) In the case of a [ship](https:\/\/en.wikipedia.org\/wiki\/Ship \"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.\\[*[citation needed](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Citation_needed \"Wikipedia:Citation needed\")*\\]\n\nDiscovery of the principle of buoyancy is attributed to [Archimedes](https:\/\/en.wikipedia.org\/wiki\/Archimedes \"Archimedes\").\n\n### Hydrostatic force on submerged surfaces\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&action=edit&section=8 \"Edit section: Hydrostatic force on submerged surfaces\")\\]\n\nThe horizontal and vertical components of the hydrostatic force acting on a submerged surface are given by the following formula:[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_note-F-M-5)\n\nF\n\nh\n\n\\=\n\np\n\nc\n\nA\n\nF\n\nv\n\n\\=\n\nρ\n\ng\n\nV\n\n{\\\\displaystyle {\\\\begin{aligned}F\\_{\\\\mathrm {h} }&=p\\_{\\\\mathrm {c} }A\\\\\\\\F\\_{\\\\mathrm {v} }&=\\\\rho gV\\\\end{aligned}}}\n\n![{\\\\displaystyle {\\\\begin{aligned}F\\_{\\\\mathrm {h} }&=p\\_{\\\\mathrm {c} }A\\\\\\\\F\\_{\\\\mathrm {v} }&=\\\\rho gV\\\\end{aligned}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8812be401acd9d7effb3ae8d7e6e4da2e9ee95a6)\n\nwhere\n\n- *p*c is the pressure at the centroid of the vertical projection of the submerged surface\n- A is the area of the same vertical projection of the surface\n- ρ is the density of the fluid\n- g is the acceleration due to gravity\n- V is the volume of fluid directly above the curved surface\n\n## See also\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&action=edit&section=9 \"Edit section: See also\")\\]\n\n- [Hydrostatics](https:\/\/en.wikipedia.org\/wiki\/Hydrostatics \"Hydrostatics\")\n- [Vertical pressure variation](https:\/\/en.wikipedia.org\/wiki\/Vertical_pressure_variation \"Vertical pressure variation\")\n\n## References\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&action=edit&section=10 \"Edit section: References\")\\]\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_ref-1)**\n   Bettini, Alessandro (2016). *A Course in Classical Physics 2—Fluids and Thermodynamics*. Springer. p. 8. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-3-319-30685-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-3-319-30685-8 \"Special:BookSources\/978-3-319-30685-8\")\n   \n   .\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_ref-2)**\n   Mauri, Roberto (8 April 2015). [*Transport Phenomena in Multiphase Flow*](https:\/\/books.google.com\/books?id=S3L0BwAAQBAJ&pg=PA24). Springer. p. 24. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-3-319-15792-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-3-319-15792-4 \"Special:BookSources\/978-3-319-15792-4\")\n   \n   . Retrieved 3 February 2017.\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_ref-3)**\n   Brown, Amy Christian (2007). [*Understanding Food: Principles and Preparation*](https:\/\/books.google.com\/books?id=edPzm5KSMmYC) (3 ed.). Cengage Learning. p. 546. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-495-10745-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-495-10745-3 \"Special:BookSources\/978-0-495-10745-3\")\n   \n   .\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_ref-Openstax_Anatomy_&_Physiology_attribution_4-0)**\n   *[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/6\/65\/Creative_Commons_by_small.svg\/120px-Creative_Commons_by_small.svg.png)](https:\/\/creativecommons.org\/licenses\/by\/4.0\/ \"creativecommons:by\/4.0\/\") This article incorporates [text](https:\/\/openstax.org\/books\/anatomy-and-physiology\/pages\/26-1-body-fluids-and-fluid-compartments) available under the [CC BY 4.0](https:\/\/creativecommons.org\/licenses\/by\/4.0\/ \"creativecommons:by\/4.0\/\") license.*\n   \n   Betts, J Gordon; Desaix, Peter; Johnson, Eddie; Johnson, Jody E; Korol, Oksana; Kruse, Dean; Poe, Brandon; Wise, James; Womble, Mark D; Young, Kelly A (September 16, 2023). *Anatomy & Physiology*. Houston: OpenStax CNX. 26.1 Body fluids and fluid compartments. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-1-947172-04-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-947172-04-3 \"Special:BookSources\/978-1-947172-04-3\")\n   \n   .\n5. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_ref-F-M_5-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_ref-F-M_5-1)\n   Fox, Robert; McDonald, Alan; Pritchard, Philip (2012). *Fluid Mechanics* (8 ed.). [John Wiley & Sons](https:\/\/en.wikipedia.org\/wiki\/John_Wiley_%26_Sons \"John Wiley & Sons\"). pp. 76–83\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-1-118-02641-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-118-02641-0 \"Special:BookSources\/978-1-118-02641-0\")\n   \n   .\n\n![](https:\/\/en.wikipedia.org\/wiki\/Special:CentralAutoLogin\/start?useformat=desktop&type=1x1&usesul3=1)\n\nRetrieved from \"<https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&oldid=1341733806>\"\n\n[Categories](https:\/\/en.wikipedia.org\/wiki\/Help:Category \"Help:Category\"):\n\n- [Hydrostatics](https:\/\/en.wikipedia.org\/wiki\/Category:Hydrostatics \"Category:Hydrostatics\")\n- [Pressure](https:\/\/en.wikipedia.org\/wiki\/Category:Pressure \"Category:Pressure\")\n\nHidden categories:\n\n- [Articles with imported Creative Commons Attribution 4.0 text](https:\/\/en.wikipedia.org\/wiki\/Category:Articles_with_imported_Creative_Commons_Attribution_4.0_text \"Category:Articles with imported Creative Commons Attribution 4.0 text\")\n- [Articles with short description](https:\/\/en.wikipedia.org\/wiki\/Category:Articles_with_short_description \"Category:Articles with short description\")\n- [Short description is different from Wikidata](https:\/\/en.wikipedia.org\/wiki\/Category:Short_description_is_different_from_Wikidata \"Category:Short description is different from Wikidata\")\n- [All articles with unsourced statements](https:\/\/en.wikipedia.org\/wiki\/Category:All_articles_with_unsourced_statements \"Category:All articles with unsourced statements\")\n- [Articles with unsourced statements from July 2022](https:\/\/en.wikipedia.org\/wiki\/Category:Articles_with_unsourced_statements_from_July_2022 \"Category:Articles with unsourced statements from July 2022\")\n\n- This page was last edited on 4 March 2026, at 19:25 (UTC).\n- Text is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License \"Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License\"); additional terms may apply. By using this site, you agree to the [Terms of Use](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Terms_of_Use \"foundation:Special:MyLanguage\/Policy:Terms of Use\") and [Privacy Policy](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Privacy_policy \"foundation:Special:MyLanguage\/Policy:Privacy policy\"). Wikipedia® is a registered trademark of the [Wikimedia Foundation, Inc.](https:\/\/wikimediafoundation.org\/), a non-profit organization.\n\n- [Privacy policy](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Privacy_policy)\n- [About Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:About)\n- [Disclaimers](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:General_disclaimer)\n- [Contact Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contact_us)\n- [Legal & safety contacts](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Legal:Wikimedia_Foundation_Legal_and_Safety_Contact_Information)\n- [Code of Conduct](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Universal_Code_of_Conduct)\n- [Developers](https:\/\/developer.wikimedia.org\/)\n- [Statistics](https:\/\/stats.wikimedia.org\/#\/en.wikipedia.org)\n- [Cookie statement](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Cookie_statement)\n- [Mobile view](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&mobileaction=toggle_view_mobile)\n\n- [![Wikimedia Foundation](https:\/\/en.wikipedia.org\/static\/images\/footer\/wikimedia.svg)](https:\/\/www.wikimedia.org\/)\n- [![Powered by MediaWiki](https:\/\/en.wikipedia.org\/w\/resources\/assets\/mediawiki_compact.svg)](https:\/\/www.mediawiki.org\/)\n\nSearch\n\nToggle the table of contents\n\nHydrostatic pressure\n\n29 languages\n\n[Add topic](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure)","attrs_readable_markdown":"From Wikipedia, the free encyclopedia\n\n**Hydrostatic pressure** is the [static pressure](https:\/\/en.wikipedia.org\/wiki\/Static_pressure \"Static pressure\") exerted at a point of interest by the [weight](https:\/\/en.wikipedia.org\/wiki\/Weight \"Weight\") of a [fluid](https:\/\/en.wikipedia.org\/wiki\/Fluid \"Fluid\") column above the point.\n\nDue to the fundamental nature of fluids, a fluid cannot remain at rest under the presence of a [shear stress](https:\/\/en.wikipedia.org\/wiki\/Shear_stress \"Shear stress\"). However, fluids can exert [pressure](https:\/\/en.wikipedia.org\/wiki\/Pressure \"Pressure\") [normal](https:\/\/en.wikipedia.org\/wiki\/Surface_normal \"Surface 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](https:\/\/en.wikipedia.org\/wiki\/Fluid_pressure \"Fluid pressure\") on a fluid at rest is [isotropic](https:\/\/en.wikipedia.org\/wiki\/Isotropic \"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](https:\/\/en.wikipedia.org\/wiki\/Blaise_Pascal \"Blaise Pascal\"), and is now called [Pascal's law](https:\/\/en.wikipedia.org\/wiki\/Pascal%27s_law \"Pascal's law\").\\[*[citation needed](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Citation_needed \"Wikipedia:Citation needed\")*\\]\n\nIn 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 *V* = 0 is applied to the [Navier–Stokes equations](https:\/\/en.wikipedia.org\/wiki\/Navier%E2%80%93Stokes_equations \"Navier–Stokes equations\") for viscous fluids or [Euler equations (fluid dynamics)](https:\/\/en.wikipedia.org\/wiki\/Euler_equations_\\(fluid_dynamics\\) \"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:\n\n**Navier–Stokes momentum equation** (*convective form*)\n\n![{\\\\displaystyle \\\\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} }-{\\\\tfrac {2}{3}}(\\\\nabla \\\\cdot \\\\mathbf {u} )\\\\mathbf {I} \\\\right\\]\\\\right\\\\}+\\\\rho \\\\mathbf {g} .}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6fd98d1bcacc93a84fa4fc981a05a200e680be04)\n\nBy setting the [flow velocity](https:\/\/en.wikipedia.org\/wiki\/Flow_velocity \"Flow velocity\") :![{\\\\displaystyle \\\\mathbf {u} =\\\\mathbf {0} }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/92a3de96f1315a7b1aa90a60f2322d7e5012b419), they become simply:\n\n![{\\\\displaystyle \\\\mathbf {0} =-\\\\nabla p+\\\\rho \\\\mathbf {g} }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fadede56aa656a73a1903ef4ef33c276184ecac1)\n\nor:\n\n![{\\\\displaystyle \\\\nabla p=\\\\rho \\\\mathbf {g} }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d5d9f468c7d46b3471f2b1534701b7bbed1fc9e4)\n\nThis is the general form of Stevin's law: the [pressure gradient](https:\/\/en.wikipedia.org\/wiki\/Pressure_gradient \"Pressure gradient\") equals the [body force](https:\/\/en.wikipedia.org\/wiki\/Body_force \"Body force\") [force density](https:\/\/en.wikipedia.org\/wiki\/Force_density \"Force density\") field.\n\nLet us now consider two particular cases of this law. In case of a [conservative](https:\/\/en.wikipedia.org\/wiki\/Conservative_force \"Conservative force\") body force with [scalar potential](https:\/\/en.wikipedia.org\/wiki\/Scalar_potential \"Scalar potential\") :![{\\\\displaystyle \\\\phi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/72b1f30316670aee6270a28334bdf4f5072cdde4):\n\n![{\\\\displaystyle \\\\rho \\\\mathbf {g} =-\\\\nabla \\\\phi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bb954c118e31f5c60fef95bbfe11eaa52a62b9b9)\n\nthe Stevin equation becomes:\n\n![{\\\\displaystyle \\\\nabla p=-\\\\nabla \\\\phi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5810d8e16907c3b14f06f080fa7406811594ee27)\n\nThat can be integrated to give:\n\n![{\\\\displaystyle \\\\Delta p=-\\\\Delta \\\\phi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bc17ceebfece38c57a6b1ccf34447dbe8b824f92)\n\nSo 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:\n\n![{\\\\displaystyle \\\\mathbf {g} =-g(x,y,z){\\\\hat {k}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a192db8a1f1df66661ab6991e2a9ef35311d56b8)\n\nthe generalised Stevin's law above becomes:\n\n![{\\\\displaystyle {\\\\frac {\\\\partial p}{\\\\partial z}}=-\\\\rho (x,y,z)g(x,y,z)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/836715bf560634b1be927d83a788bd92964b1c2f)\n\nThat can be integrated to give another (less-) generalised Stevin's law:\n\n![{\\\\displaystyle p(x,y,z)-p\\_{0}(x,y)=-\\\\int \\_{0}^{z}\\\\rho (x,y,z')g(x,y,z')dz'}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/78bcc6804783378024effbb00548d25e28827d18)\n\nwhere:\n\n### Simplification for liquids\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&action=edit&section=3 \"Edit section: Simplification for liquids\")\\]\n\nFor 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](https:\/\/en.wikipedia.org\/wiki\/Incompressible_flow \"Incompressible flow\"), 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 ![{\\\\displaystyle \\\\Delta z}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/cf59b5e4dc02e6b2b89b426a27e02c84861c0caa) 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 [g](https:\/\/en.wikipedia.org\/wiki\/Gravity \"Gravity\"). Under these circumstances, one can transport out of the integral the density and the gravity acceleration and the law is simplified into the formula\n\n![{\\\\displaystyle \\\\Delta p(z)=\\\\rho g\\\\Delta z,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9afa9a1da855d0dde3239d2e26ad69a22c9354c6)\n\nwhere ![{\\\\displaystyle \\\\Delta z}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/cf59b5e4dc02e6b2b89b426a27e02c84861c0caa) is the height *z* − *z*0 of the liquid column between the test volume and the zero reference point of the pressure. This formula is often called [Stevin's](https:\/\/en.wikipedia.org\/wiki\/Simon_Stevin \"Simon Stevin\") law.[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_note-1)[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_note-2) 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:\n\n![{\\\\displaystyle \\\\rho \\\\mathbf {g} (x,y,z)=-\\\\rho g{\\\\hat {k}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/286c83b8ad7070465180a3bda0f08ac24ca1b7a7)\n\nis conservative, so one can write the body force density as:\n\n![{\\\\displaystyle \\\\rho \\\\mathbf {g} =\\\\nabla (-\\\\rho gz)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a4f7a68af5b40d5e2bb679d68fe7d38ad381d1fc)\n\nThen the body force density has a simple [scalar potential](https:\/\/en.wikipedia.org\/wiki\/Scalar_potential \"Scalar potential\"):\n\n![{\\\\displaystyle \\\\phi (z)=-\\\\rho gz}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ad0108c644e333a75e7b0ae3c46a697b51d08b03)\n\nAnd the pressure difference follows another time the Stevin's law:\n\n![{\\\\displaystyle \\\\Delta p=-\\\\Delta \\\\phi =\\\\rho g\\\\Delta z}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/787b041c209cf18ccbb6b1aa4368c9aee99b2066)\n\nThe 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*′)above. For example, the [absolute pressure](https:\/\/en.wikipedia.org\/wiki\/Pressure_measurement#Absolute,_gauge_and_differential_pressures_-_zero_reference \"Pressure measurement\") compared to vacuum is\n\n![{\\\\displaystyle p=\\\\rho g\\\\Delta z+p\\_{\\\\mathrm {0} },}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/65dc7e8253c8fa7fade48f906aef2b45e821c398)\n\nwhere ![{\\\\displaystyle \\\\Delta z}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/cf59b5e4dc02e6b2b89b426a27e02c84861c0caa) is the total height of the liquid column above the test area to the surface, and *p*0 is the [atmospheric pressure](https:\/\/en.wikipedia.org\/wiki\/Atmospheric_pressure \"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](https:\/\/en.wikipedia.org\/wiki\/Pressure_prism \"Pressure prism\").\n\nHydrostatic pressure has been used in the preservation of foods in a process called [pascalization](https:\/\/en.wikipedia.org\/wiki\/Pascalization \"Pascalization\").[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_note-3)\n\nIn medicine, hydrostatic pressure in [blood vessels](https:\/\/en.wikipedia.org\/wiki\/Blood_vessel \"Blood vessel\") is the pressure of the blood against the wall. It is the opposing force to [oncotic pressure](https:\/\/en.wikipedia.org\/wiki\/Oncotic_pressure \"Oncotic pressure\"). In capillaries, hydrostatic pressure (also known as capillary blood pressure) is higher than the opposing “colloid [osmotic pressure](https:\/\/en.wikipedia.org\/wiki\/Osmotic_pressure \"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.[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_note-Openstax_Anatomy_&_Physiology_attribution-4)\n\n### Atmospheric pressure\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&action=edit&section=6 \"Edit section: Atmospheric pressure\")\\]\n\n[Statistical mechanics](https:\/\/en.wikipedia.org\/wiki\/Statistical_mechanics \"Statistical mechanics\") shows that, for a pure [ideal gas](https:\/\/en.wikipedia.org\/wiki\/Ideal_gas \"Ideal gas\") of constant temperature *T* in the earth gravitational field, its pressure, *p* will vary with height, *h*, as\n\n![{\\\\displaystyle p(h)=p(0)e^{-{\\\\frac {Mgh}{kT}}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/aa359443bd71846a0f795d8af50a298f02fc900b)\n\nwhere\n\n- g is the [acceleration due to gravity](https:\/\/en.wikipedia.org\/wiki\/Standard_gravity \"Standard gravity\")\n- T is the [absolute temperature](https:\/\/en.wikipedia.org\/wiki\/Absolute_temperature \"Absolute temperature\")\n- k is [Boltzmann constant](https:\/\/en.wikipedia.org\/wiki\/Boltzmann_constant \"Boltzmann constant\")\n- M is the [molecular mass](https:\/\/en.wikipedia.org\/wiki\/Molecular_mass \"Molecular mass\") of the gas\n- p is the pressure\n- h is the height\n\nThis is known as the [barometric formula](https:\/\/en.wikipedia.org\/wiki\/Barometric_formula \"Barometric formula\"), and may be derived from assuming the pressure is [hydrostatic]().\n\nIf there are multiple types of molecules in the gas, the [partial pressure](https:\/\/en.wikipedia.org\/wiki\/Partial_pressure \"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.\n\nAny 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,\n\n![{\\\\displaystyle F=\\\\rho gV}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2b1b65f31d86a2c06763a4f93131781af9867fba)\n\nwhere 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.[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_note-F-M-5) In the case of a [ship](https:\/\/en.wikipedia.org\/wiki\/Ship \"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.\\[*[citation needed](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Citation_needed \"Wikipedia:Citation needed\")*\\]\n\nDiscovery of the principle of buoyancy is attributed to [Archimedes](https:\/\/en.wikipedia.org\/wiki\/Archimedes \"Archimedes\").\n\n### Hydrostatic force on submerged surfaces\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Hydrostatic_pressure&action=edit&section=8 \"Edit section: Hydrostatic force on submerged surfaces\")\\]\n\nThe horizontal and vertical components of the hydrostatic force acting on a submerged surface are given by the following formula:[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_note-F-M-5)\n\n![{\\\\displaystyle {\\\\begin{aligned}F\\_{\\\\mathrm {h} }&=p\\_{\\\\mathrm {c} }A\\\\\\\\F\\_{\\\\mathrm {v} }&=\\\\rho gV\\\\end{aligned}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8812be401acd9d7effb3ae8d7e6e4da2e9ee95a6)\n\nwhere\n\n- *p*c is the pressure at the centroid of the vertical projection of the submerged surface\n- A is the area of the same vertical projection of the surface\n- ρ is the density of the fluid\n- g is the acceleration due to gravity\n- V is the volume of fluid directly above the curved surface\n\n- [Hydrostatics](https:\/\/en.wikipedia.org\/wiki\/Hydrostatics \"Hydrostatics\")\n- [Vertical pressure variation](https:\/\/en.wikipedia.org\/wiki\/Vertical_pressure_variation \"Vertical pressure variation\")\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_ref-1)**\n   Bettini, Alessandro (2016). *A Course in Classical Physics 2—Fluids and Thermodynamics*. Springer. p. 8. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-3-319-30685-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-3-319-30685-8 \"Special:BookSources\/978-3-319-30685-8\")\n   \n   .\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_ref-2)**\n   Mauri, Roberto (8 April 2015). [*Transport Phenomena in Multiphase Flow*](https:\/\/books.google.com\/books?id=S3L0BwAAQBAJ&pg=PA24). Springer. p. 24. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-3-319-15792-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-3-319-15792-4 \"Special:BookSources\/978-3-319-15792-4\")\n   \n   . Retrieved 3 February 2017.\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_ref-3)**\n   Brown, Amy Christian (2007). [*Understanding Food: Principles and Preparation*](https:\/\/books.google.com\/books?id=edPzm5KSMmYC) (3 ed.). Cengage Learning. p. 546. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-495-10745-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-495-10745-3 \"Special:BookSources\/978-0-495-10745-3\")\n   \n   .\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_ref-Openstax_Anatomy_&_Physiology_attribution_4-0)**\n   *[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/6\/65\/Creative_Commons_by_small.svg\/120px-Creative_Commons_by_small.svg.png)](https:\/\/creativecommons.org\/licenses\/by\/4.0\/ \"creativecommons:by\/4.0\/\") This article incorporates [text](https:\/\/openstax.org\/books\/anatomy-and-physiology\/pages\/26-1-body-fluids-and-fluid-compartments) available under the [CC BY 4.0](https:\/\/creativecommons.org\/licenses\/by\/4.0\/ \"creativecommons:by\/4.0\/\") license.*\n   \n   Betts, J Gordon; Desaix, Peter; Johnson, Eddie; Johnson, Jody E; Korol, Oksana; Kruse, Dean; Poe, Brandon; Wise, James; Womble, Mark D; Young, Kelly A (September 16, 2023). *Anatomy & Physiology*. Houston: OpenStax CNX. 26.1 Body fluids and fluid compartments. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-1-947172-04-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-947172-04-3 \"Special:BookSources\/978-1-947172-04-3\")\n   \n   .\n5. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_ref-F-M_5-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Hydrostatic_pressure#cite_ref-F-M_5-1)\n   Fox, Robert; McDonald, Alan; Pritchard, Philip (2012). *Fluid Mechanics* (8 ed.). [John Wiley & Sons](https:\/\/en.wikipedia.org\/wiki\/John_Wiley_%26_Sons \"John Wiley & Sons\"). pp. 76–83\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-1-118-02641-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-118-02641-0 \"Special:BookSources\/978-1-118-02641-0\")\n   \n   .","meta_canonical":null}

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Boilerpipe Text	From Wikipedia, the free encyclopedia Hydrostatic pressure is the static pressure exerted at a point of interest by the weight of a fluid column above the point. 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 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 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 . [ citation needed ] 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 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: Navier–Stokes momentum equation ( convective form ) By setting the flow velocity : , they become simply: or: 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 body force with scalar potential : : the Stevin equation becomes: That can be integrated to give: 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: the generalised Stevin's law above becomes: That can be integrated to give another (less-) generalised Stevin's law: where: Simplification for liquids [ edit ] 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 z and z 0 is often reasonably small compared to the radius of the Earth, one can neglect the variation of 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 where is the height z − z 0 of the liquid column between the test volume and the zero reference point of the pressure. This formula is often called Stevin's law. [ 1 ] [ 2 ] 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: is conservative, so one can write the body force density as: Then the body force density has a simple scalar potential : And the pressure difference follows another time the Stevin's law: 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 ρ liquid and ρ ( z ′) above . For example, the absolute pressure compared to vacuum is where is the total height of the liquid column above the test area to the surface, and p 0 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 . [ 3 ] In medicine, hydrostatic pressure in blood vessels 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. [ 4 ] Atmospheric pressure [ edit ] Statistical mechanics shows that, for a pure ideal gas of constant temperature T in the earth gravitational field, its pressure, p will vary with height, h , as where g is the acceleration due to gravity T is the absolute temperature k is Boltzmann constant M is the molecular mass of the 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 . 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. 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 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. [ 5 ] 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. [ citation needed ] Discovery of the principle of buoyancy is attributed to Archimedes . Hydrostatic force on submerged surfaces [ edit ] The horizontal and vertical components of the hydrostatic force acting on a submerged surface are given by the following formula: [ 5 ] where p c is the pressure at the centroid 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 Hydrostatics Vertical pressure variation ^ Bettini, Alessandro (2016). A Course in Classical Physics 2—Fluids and Thermodynamics . Springer. p. 8. ISBN 978-3-319-30685-8 . ^ Mauri, Roberto (8 April 2015). Transport Phenomena in Multiphase Flow . Springer. p. 24. ISBN 978-3-319-15792-4 . Retrieved 3 February 2017 . ^ Brown, Amy Christian (2007). Understanding Food: Principles and Preparation (3 ed.). Cengage Learning. p. 546. ISBN 978-0-495-10745-3 . ^ This article incorporates text available under the CC BY 4.0 license. Betts, J Gordon; Desaix, Peter; Johnson, Eddie; Johnson, Jody E; Korol, Oksana; Kruse, Dean; Poe, Brandon; Wise, James; Womble, Mark D; Young, Kelly A (September 16, 2023). Anatomy & Physiology . Houston: OpenStax CNX. 26.1 Body fluids and fluid compartments. ISBN 978-1-947172-04-3 . ^ a b Fox, Robert; McDonald, Alan; Pritchard, Philip (2012). Fluid Mechanics (8 ed.). John Wiley & Sons . pp. 76– 83. ISBN 978-1-118-02641-0 .
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[o]") ## Contents move to sidebar hide - [(Top)](https://en.wikipedia.org/wiki/Hydrostatic_pressure) - [1 Background](https://en.wikipedia.org/wiki/Hydrostatic_pressure#Background) - [2 Formulation](https://en.wikipedia.org/wiki/Hydrostatic_pressure#Formulation) Toggle Formulation subsection - [2\.1 Simplification for liquids](https://en.wikipedia.org/wiki/Hydrostatic_pressure#Simplification_for_liquids) - [3 Applications](https://en.wikipedia.org/wiki/Hydrostatic_pressure#Applications) Toggle Applications subsection - [3\.1 Medicine](https://en.wikipedia.org/wiki/Hydrostatic_pressure#Medicine) - [3\.2 Atmospheric pressure](https://en.wikipedia.org/wiki/Hydrostatic_pressure#Atmospheric_pressure) - [4 Buoyancy](https://en.wikipedia.org/wiki/Hydrostatic_pressure#Buoyancy) Toggle Buoyancy subsection - [4\.1 Hydrostatic force on submerged surfaces](https://en.wikipedia.org/wiki/Hydrostatic_pressure#Hydrostatic_force_on_submerged_surfaces) - [5 See also](https://en.wikipedia.org/wiki/Hydrostatic_pressure#See_also) - [6 References](https://en.wikipedia.org/wiki/Hydrostatic_pressure#References) Toggle the table of contents # Hydrostatic pressure 29 languages - [Asturianu](https://ast.wikipedia.org/wiki/Presi%C3%B3n_nun_fluy%C3%ADu "Presión nun fluyíu – Asturian") - [Azərbaycanca](https://az.wikipedia.org/wiki/Hidrostatik_t%C9%99zyiq "Hidrostatik təzyiq – Azerbaijani") - [Bosanski](https://bs.wikipedia.org/wiki/Pritisak_fluida "Pritisak fluida – Bosnian") - [Català](https://ca.wikipedia.org/wiki/Pressi%C3%B3_hidroest%C3%A0tica "Pressió hidroestàtica – Catalan") - [Čeština](https://cs.wikipedia.org/wiki/Hydrostatick%C3%BD_tlak "Hydrostatický tlak – Czech") - [Deutsch](https://de.wikipedia.org/wiki/Hydrostatischer_Druck "Hydrostatischer Druck – German") - [Ελληνικά](https://el.wikipedia.org/wiki/%CE%A5%CE%B4%CF%81%CE%BF%CF%83%CF%84%CE%B1%CF%84%CE%B9%CE%BA%CE%AE_%CF%80%CE%AF%CE%B5%CF%83%CE%B7 "Υδροστατική πίεση – Greek") - [Esperanto](https://eo.wikipedia.org/wiki/Premo_de_fluido "Premo de fluido – Esperanto") - [Español](https://es.wikipedia.org/wiki/Presi%C3%B3n_en_un_fluido "Presión en un fluido – Spanish") - [Euskara](https://eu.wikipedia.org/wiki/Presio_hidrostatiko "Presio hidrostatiko – Basque") - [Suomi](https://fi.wikipedia.org/wiki/Hydrostaattinen_paine "Hydrostaattinen paine – Finnish") - [Magyar](https://hu.wikipedia.org/wiki/Hidrosztatikai_nyom%C3%A1s "Hidrosztatikai nyomás – Hungarian") - [Հայերեն](https://hy.wikipedia.org/wiki/%D5%80%D5%AB%D5%A4%D6%80%D5%B8%D5%BD%D5%BF%D5%A1%D5%BF%D5%AB%D5%AF_%D5%B3%D5%B6%D5%B7%D5%B8%D6%82%D5%B4 "Հիդրոստատիկ ճնշում – Armenian") - [Italiano](https://it.wikipedia.org/wiki/Pressione_idrostatica "Pressione idrostatica – Italian") - [日本語](https://ja.wikipedia.org/wiki/%E6%B0%B4%E5%9C%A7 "水圧 – Japanese") - [Қазақша](https://kk.wikipedia.org/wiki/%D0%93%D0%B8%D0%B4%D1%80%D0%BE%D1%81%D1%82%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D2%9B%D1%8B%D1%81%D1%8B%D0%BC "Гидростатикалық қысым – Kazakh") - [한국어](https://ko.wikipedia.org/wiki/%EC%88%98%EC%95%95 "수압 – Korean") - 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[Get shortened URL](https://en.wikipedia.org/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FHydrostatic_pressure) Print/export - [Download as PDF](https://en.wikipedia.org/w/index.php?title=Special:DownloadAsPdf&page=Hydrostatic_pressure&action=show-download-screen "Download this page as a PDF file") - [Printable version](https://en.wikipedia.org/w/index.php?title=Hydrostatic_pressure&printable=yes "Printable version of this page [p]") In other projects - [Wikidata item](https://www.wikidata.org/wiki/Special:EntityPage/Q1149672 "Structured data on this page hosted by Wikidata [g]") Appearance move to sidebar hide From Wikipedia, the free encyclopedia Physical quantity Hydrostatic pressure is the [static pressure](https://en.wikipedia.org/wiki/Static_pressure "Static pressure") exerted at a point of interest by the [weight](https://en.wikipedia.org/wiki/Weight "Weight") of a [fluid](https://en.wikipedia.org/wiki/Fluid "Fluid") column above the point. ## Background \[[edit](https://en.wikipedia.org/w/index.php?title=Hydrostatic_pressure&action=edit&section=1 "Edit section: Background")\] Due to the fundamental nature of fluids, a fluid cannot remain at rest under the presence of a [shear stress](https://en.wikipedia.org/wiki/Shear_stress "Shear stress"). However, fluids can exert [pressure](https://en.wikipedia.org/wiki/Pressure "Pressure") [normal](https://en.wikipedia.org/wiki/Surface_normal "Surface 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](https://en.wikipedia.org/wiki/Fluid_pressure "Fluid pressure") on a fluid at rest is [isotropic](https://en.wikipedia.org/wiki/Isotropic "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](https://en.wikipedia.org/wiki/Blaise_Pascal "Blaise Pascal"), and is now called [Pascal's law](https://en.wikipedia.org/wiki/Pascal%27s_law "Pascal's law").\[[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")\] ## Formulation \[[edit](https://en.wikipedia.org/w/index.php?title=Hydrostatic_pressure&action=edit&section=2 "Edit section: Formulation")\] 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 V = 0 is applied to the [Navier–Stokes equations](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations "Navier–Stokes equations") for viscous fluids or [Euler equations (fluid dynamics)](https://en.wikipedia.org/wiki/Euler_equations_\(fluid_dynamics\) "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: Navier–Stokes momentum equation (convective form) ρ D u D t \= − ∇ \[ p − ζ ( ∇ ⋅ u ) \] \+ ∇ ⋅ { μ \[ ∇ u \+ ( ∇ u ) T − 2 3 ( ∇ ⋅ u ) I \] } \+ ρ g . {\\displaystyle \\rho {\\frac {\\mathrm {D} \\mathbf {u} }{\\mathrm {D} t}}=-\\nabla \[p-\\zeta (\\nabla \\cdot \\mathbf {u} )\]+\\nabla \\cdot \\left\\{\\mu \\left\[\\nabla \\mathbf {u} +(\\nabla \\mathbf {u} )^{\\mathrm {T} }-{\\tfrac {2}{3}}(\\nabla \\cdot \\mathbf {u} )\\mathbf {I} \\right\]\\right\\}+\\rho \\mathbf {g} .} ![{\\displaystyle \\rho {\\frac {\\mathrm {D} \\mathbf {u} }{\\mathrm {D} t}}=-\\nabla \[p-\\zeta (\\nabla \\cdot \\mathbf {u} )\]+\\nabla \\cdot \\left\\{\\mu \\left\[\\nabla \\mathbf {u} +(\\nabla \\mathbf {u} )^{\\mathrm {T} }-{\\tfrac {2}{3}}(\\nabla \\cdot \\mathbf {u} )\\mathbf {I} \\right\]\\right\\}+\\rho \\mathbf {g} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fd98d1bcacc93a84fa4fc981a05a200e680be04) By setting the [flow velocity](https://en.wikipedia.org/wiki/Flow_velocity "Flow velocity") :u \= 0 {\\displaystyle \\mathbf {u} =\\mathbf {0} } ![{\\displaystyle \\mathbf {u} =\\mathbf {0} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/92a3de96f1315a7b1aa90a60f2322d7e5012b419), they become simply: 0 \= − ∇ p \+ ρ g {\\displaystyle \\mathbf {0} =-\\nabla p+\\rho \\mathbf {g} } ![{\\displaystyle \\mathbf {0} =-\\nabla p+\\rho \\mathbf {g} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fadede56aa656a73a1903ef4ef33c276184ecac1) or: ∇ p \= ρ g {\\displaystyle \\nabla p=\\rho \\mathbf {g} } ![{\\displaystyle \\nabla p=\\rho \\mathbf {g} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5d9f468c7d46b3471f2b1534701b7bbed1fc9e4) This is the general form of Stevin's law: the [pressure gradient](https://en.wikipedia.org/wiki/Pressure_gradient "Pressure gradient") equals the [body force](https://en.wikipedia.org/wiki/Body_force "Body force") [force density](https://en.wikipedia.org/wiki/Force_density "Force density") field. Let us now consider two particular cases of this law. In case of a [conservative](https://en.wikipedia.org/wiki/Conservative_force "Conservative force") body force with [scalar potential](https://en.wikipedia.org/wiki/Scalar_potential "Scalar potential") :ϕ {\\displaystyle \\phi } ![{\\displaystyle \\phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4): ρ g \= − ∇ ϕ {\\displaystyle \\rho \\mathbf {g} =-\\nabla \\phi } ![{\\displaystyle \\rho \\mathbf {g} =-\\nabla \\phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb954c118e31f5c60fef95bbfe11eaa52a62b9b9) the Stevin equation becomes: ∇ p \= − ∇ ϕ {\\displaystyle \\nabla p=-\\nabla \\phi } ![{\\displaystyle \\nabla p=-\\nabla \\phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5810d8e16907c3b14f06f080fa7406811594ee27) That can be integrated to give: Δ p \= − Δ ϕ {\\displaystyle \\Delta p=-\\Delta \\phi } ![{\\displaystyle \\Delta p=-\\Delta \\phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc17ceebfece38c57a6b1ccf34447dbe8b824f92) 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: g \= − g ( x , y , z ) k ^ {\\displaystyle \\mathbf {g} =-g(x,y,z){\\hat {k}}} ![{\\displaystyle \\mathbf {g} =-g(x,y,z){\\hat {k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a192db8a1f1df66661ab6991e2a9ef35311d56b8) the generalised Stevin's law above becomes: ∂ p ∂ z \= − ρ ( x , y , z ) g ( x , y , z ) {\\displaystyle {\\frac {\\partial p}{\\partial z}}=-\\rho (x,y,z)g(x,y,z)} ![{\\displaystyle {\\frac {\\partial p}{\\partial z}}=-\\rho (x,y,z)g(x,y,z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/836715bf560634b1be927d83a788bd92964b1c2f) That can be integrated to give another (less-) generalised Stevin's law: p ( x , y , z ) − p 0 ( x , y ) \= − ∫ 0 z ρ ( x , y , z ′ ) g ( x , y , z ′ ) d z ′ {\\displaystyle p(x,y,z)-p\_{0}(x,y)=-\\int \_{0}^{z}\\rho (x,y,z')g(x,y,z')dz'} ![{\\displaystyle p(x,y,z)-p\_{0}(x,y)=-\\int \_{0}^{z}\\rho (x,y,z')g(x,y,z')dz'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78bcc6804783378024effbb00548d25e28827d18) where: - p {\\displaystyle p} ![{\\displaystyle p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36) is the hydrostatic pressure (Pa), - ρ {\\displaystyle \\rho } ![{\\displaystyle \\rho }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64) is the fluid [density](https://en.wikipedia.org/wiki/Density "Density") (kg/m3), - g {\\displaystyle g} ![{\\displaystyle g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77) is [gravitational](https://en.wikipedia.org/wiki/Gravity "Gravity") acceleration (m/s2), - z {\\displaystyle z} ![{\\displaystyle z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98) is the height (parallel to the direction of gravity) of the test area (m), - 0 {\\displaystyle 0} ![{\\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) is the height of the [zero reference point of the pressure](https://en.wikipedia.org/wiki/Pressure_measurement#Absolute,_gauge_and_differential_pressures_%E2%80%94_zero_reference "Pressure measurement") (m) - p 0 {\\displaystyle p\_{0}} ![{\\displaystyle p\_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b969ada68a88e2aeba9a2d2096abaf1fd53c21d) is the hydrostatic pressure field (Pa) along x and y at the zero reference point ### Simplification for liquids \[[edit](https://en.wikipedia.org/w/index.php?title=Hydrostatic_pressure&action=edit&section=3 "Edit section: Simplification for liquids")\] 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](https://en.wikipedia.org/wiki/Incompressible_flow "Incompressible flow"), 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 Δ z {\\displaystyle \\Delta z} ![{\\displaystyle \\Delta z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf59b5e4dc02e6b2b89b426a27e02c84861c0caa) of the fluid column between z and z0 is often reasonably small compared to the radius of the Earth, one can neglect the variation of [g](https://en.wikipedia.org/wiki/Gravity "Gravity"). Under these circumstances, one can transport out of the integral the density and the gravity acceleration and the law is simplified into the formula Δ p ( z ) \= ρ g Δ z , {\\displaystyle \\Delta p(z)=\\rho g\\Delta z,} ![{\\displaystyle \\Delta p(z)=\\rho g\\Delta z,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9afa9a1da855d0dde3239d2e26ad69a22c9354c6) where Δ z {\\displaystyle \\Delta z} ![{\\displaystyle \\Delta z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf59b5e4dc02e6b2b89b426a27e02c84861c0caa) is the height z − z0 of the liquid column between the test volume and the zero reference point of the pressure. This formula is often called [Stevin's](https://en.wikipedia.org/wiki/Simon_Stevin "Simon Stevin") law.[\[1\]](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_note-1)[\[2\]](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_note-2) 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: ρ g ( x , y , z ) \= − ρ g k ^ {\\displaystyle \\rho \\mathbf {g} (x,y,z)=-\\rho g{\\hat {k}}} ![{\\displaystyle \\rho \\mathbf {g} (x,y,z)=-\\rho g{\\hat {k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/286c83b8ad7070465180a3bda0f08ac24ca1b7a7) is conservative, so one can write the body force density as: ρ g \= ∇ ( − ρ g z ) {\\displaystyle \\rho \\mathbf {g} =\\nabla (-\\rho gz)} ![{\\displaystyle \\rho \\mathbf {g} =\\nabla (-\\rho gz)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f7a68af5b40d5e2bb679d68fe7d38ad381d1fc) Then the body force density has a simple [scalar potential](https://en.wikipedia.org/wiki/Scalar_potential "Scalar potential"): ϕ ( z ) \= − ρ g z {\\displaystyle \\phi (z)=-\\rho gz} ![{\\displaystyle \\phi (z)=-\\rho gz}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad0108c644e333a75e7b0ae3c46a697b51d08b03) And the pressure difference follows another time the Stevin's law: Δ p \= − Δ ϕ \= ρ g Δ z {\\displaystyle \\Delta p=-\\Delta \\phi =\\rho g\\Delta z} ![{\\displaystyle \\Delta p=-\\Delta \\phi =\\rho g\\Delta z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/787b041c209cf18ccbb6b1aa4368c9aee99b2066) 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 ρliquid and ρ(z′)above. For example, the [absolute pressure](https://en.wikipedia.org/wiki/Pressure_measurement#Absolute,_gauge_and_differential_pressures_-_zero_reference "Pressure measurement") compared to vacuum is p \= ρ g Δ z \+ p 0 , {\\displaystyle p=\\rho g\\Delta z+p\_{\\mathrm {0} },} ![{\\displaystyle p=\\rho g\\Delta z+p\_{\\mathrm {0} },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65dc7e8253c8fa7fade48f906aef2b45e821c398) where Δ z {\\displaystyle \\Delta z} ![{\\displaystyle \\Delta z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf59b5e4dc02e6b2b89b426a27e02c84861c0caa) is the total height of the liquid column above the test area to the surface, and p0 is the [atmospheric pressure](https://en.wikipedia.org/wiki/Atmospheric_pressure "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](https://en.wikipedia.org/wiki/Pressure_prism "Pressure prism"). ## Applications \[[edit](https://en.wikipedia.org/w/index.php?title=Hydrostatic_pressure&action=edit&section=4 "Edit section: Applications")\] Hydrostatic pressure has been used in the preservation of foods in a process called [pascalization](https://en.wikipedia.org/wiki/Pascalization "Pascalization").[\[3\]](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_note-3) ### Medicine \[[edit](https://en.wikipedia.org/w/index.php?title=Hydrostatic_pressure&action=edit&section=5 "Edit section: Medicine")\] In medicine, hydrostatic pressure in [blood vessels](https://en.wikipedia.org/wiki/Blood_vessel "Blood vessel") is the pressure of the blood against the wall. It is the opposing force to [oncotic pressure](https://en.wikipedia.org/wiki/Oncotic_pressure "Oncotic pressure"). In capillaries, hydrostatic pressure (also known as capillary blood pressure) is higher than the opposing “colloid [osmotic pressure](https://en.wikipedia.org/wiki/Osmotic_pressure "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.[\[4\]](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_note-Openstax_Anatomy_&_Physiology_attribution-4) ### Atmospheric pressure \[[edit](https://en.wikipedia.org/w/index.php?title=Hydrostatic_pressure&action=edit&section=6 "Edit section: Atmospheric pressure")\] [Statistical mechanics](https://en.wikipedia.org/wiki/Statistical_mechanics "Statistical mechanics") shows that, for a pure [ideal gas](https://en.wikipedia.org/wiki/Ideal_gas "Ideal gas") of constant temperature T in the earth gravitational field, its pressure, p will vary with height, h, as p ( h ) \= p ( 0 ) e − M g h k T {\\displaystyle p(h)=p(0)e^{-{\\frac {Mgh}{kT}}}} ![{\\displaystyle p(h)=p(0)e^{-{\\frac {Mgh}{kT}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa359443bd71846a0f795d8af50a298f02fc900b) where - g is the [acceleration due to gravity](https://en.wikipedia.org/wiki/Standard_gravity "Standard gravity") - T is the [absolute temperature](https://en.wikipedia.org/wiki/Absolute_temperature "Absolute temperature") - k is [Boltzmann constant](https://en.wikipedia.org/wiki/Boltzmann_constant "Boltzmann constant") - M is the [molecular mass](https://en.wikipedia.org/wiki/Molecular_mass "Molecular mass") of the gas - p is the pressure - h is the height This is known as the [barometric formula](https://en.wikipedia.org/wiki/Barometric_formula "Barometric formula"), and may be derived from assuming the pressure is [hydrostatic](). If there are multiple types of molecules in the gas, the [partial pressure](https://en.wikipedia.org/wiki/Partial_pressure "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 \[[edit](https://en.wikipedia.org/w/index.php?title=Hydrostatic_pressure&action=edit&section=7 "Edit section: Buoyancy")\] Main article: [Buoyancy](https://en.wikipedia.org/wiki/Buoyancy "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 \= ρ g V {\\displaystyle F=\\rho gV} ![{\\displaystyle F=\\rho gV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b1b65f31d86a2c06763a4f93131781af9867fba) 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.[\[5\]](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_note-F-M-5) In the case of a [ship](https://en.wikipedia.org/wiki/Ship "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.\[[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")\] Discovery of the principle of buoyancy is attributed to [Archimedes](https://en.wikipedia.org/wiki/Archimedes "Archimedes"). ### Hydrostatic force on submerged surfaces \[[edit](https://en.wikipedia.org/w/index.php?title=Hydrostatic_pressure&action=edit&section=8 "Edit section: Hydrostatic force on submerged surfaces")\] The horizontal and vertical components of the hydrostatic force acting on a submerged surface are given by the following formula:[\[5\]](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_note-F-M-5) F h \= p c A F v \= ρ g V {\\displaystyle {\\begin{aligned}F\_{\\mathrm {h} }&=p\_{\\mathrm {c} }A\\\\F\_{\\mathrm {v} }&=\\rho gV\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}F\_{\\mathrm {h} }&=p\_{\\mathrm {c} }A\\\\F\_{\\mathrm {v} }&=\\rho gV\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8812be401acd9d7effb3ae8d7e6e4da2e9ee95a6) where - pc is the pressure at the centroid 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 ## See also \[[edit](https://en.wikipedia.org/w/index.php?title=Hydrostatic_pressure&action=edit&section=9 "Edit section: See also")\] - [Hydrostatics](https://en.wikipedia.org/wiki/Hydrostatics "Hydrostatics") - [Vertical pressure variation](https://en.wikipedia.org/wiki/Vertical_pressure_variation "Vertical pressure variation") ## References \[[edit](https://en.wikipedia.org/w/index.php?title=Hydrostatic_pressure&action=edit&section=10 "Edit section: References")\] 1. [^](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_ref-1) Bettini, Alessandro (2016). A Course in Classical Physics 2—Fluids and Thermodynamics. Springer. p. 8. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-3-319-30685-8](https://en.wikipedia.org/wiki/Special:BookSources/978-3-319-30685-8 "Special:BookSources/978-3-319-30685-8") . 2. [^](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_ref-2) Mauri, Roberto (8 April 2015). [Transport Phenomena in Multiphase Flow](https://books.google.com/books?id=S3L0BwAAQBAJ&pg=PA24). Springer. p. 24. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-3-319-15792-4](https://en.wikipedia.org/wiki/Special:BookSources/978-3-319-15792-4 "Special:BookSources/978-3-319-15792-4") . Retrieved 3 February 2017. 3. [^](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_ref-3) Brown, Amy Christian (2007). [Understanding Food: Principles and Preparation](https://books.google.com/books?id=edPzm5KSMmYC) (3 ed.). Cengage Learning. p. 546. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-495-10745-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-495-10745-3 "Special:BookSources/978-0-495-10745-3") . 4. [^](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_ref-Openstax_Anatomy_&_Physiology_attribution_4-0) [![](https://upload.wikimedia.org/wikipedia/commons/thumb/6/65/Creative_Commons_by_small.svg/120px-Creative_Commons_by_small.svg.png)](https://creativecommons.org/licenses/by/4.0/ "creativecommons:by/4.0/") This article incorporates [text](https://openstax.org/books/anatomy-and-physiology/pages/26-1-body-fluids-and-fluid-compartments) available under the [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/ "creativecommons:by/4.0/") license. Betts, J Gordon; Desaix, Peter; Johnson, Eddie; Johnson, Jody E; Korol, Oksana; Kruse, Dean; Poe, Brandon; Wise, James; Womble, Mark D; Young, Kelly A (September 16, 2023). Anatomy & Physiology. Houston: OpenStax CNX. 26.1 Body fluids and fluid compartments. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-947172-04-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-947172-04-3 "Special:BookSources/978-1-947172-04-3") . 5. ^ [*a](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_ref-F-M_5-0) [b](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_ref-F-M_5-1) Fox, Robert; McDonald, Alan; Pritchard, Philip (2012). Fluid Mechanics* (8 ed.). [John Wiley & Sons](https://en.wikipedia.org/wiki/John_Wiley_%26_Sons "John Wiley & Sons"). pp. 76–83\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-118-02641-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-118-02641-0 "Special:BookSources/978-1-118-02641-0") . ![](https://en.wikipedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=1) Retrieved from "<https://en.wikipedia.org/w/index.php?title=Hydrostatic_pressure&oldid=1341733806>" [Categories](https://en.wikipedia.org/wiki/Help:Category "Help:Category"): - [Hydrostatics](https://en.wikipedia.org/wiki/Category:Hydrostatics "Category:Hydrostatics") - [Pressure](https://en.wikipedia.org/wiki/Category:Pressure "Category:Pressure") Hidden categories: - [Articles with imported Creative Commons Attribution 4.0 text](https://en.wikipedia.org/wiki/Category:Articles_with_imported_Creative_Commons_Attribution_4.0_text "Category:Articles with imported Creative Commons Attribution 4.0 text") - [Articles with short description](https://en.wikipedia.org/wiki/Category:Articles_with_short_description "Category:Articles with short description") - [Short description is different from Wikidata](https://en.wikipedia.org/wiki/Category:Short_description_is_different_from_Wikidata "Category:Short description is different from Wikidata") - [All articles with unsourced statements](https://en.wikipedia.org/wiki/Category:All_articles_with_unsourced_statements "Category:All articles with unsourced statements") - [Articles with unsourced statements from July 2022](https://en.wikipedia.org/wiki/Category:Articles_with_unsourced_statements_from_July_2022 "Category:Articles with unsourced statements from July 2022") - This page was last edited on 4 March 2026, at 19:25 (UTC). - Text is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://en.wikipedia.org/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License "Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License"); additional terms may apply. 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Readable Markdown	From Wikipedia, the free encyclopedia Hydrostatic pressure is the [static pressure](https://en.wikipedia.org/wiki/Static_pressure "Static pressure") exerted at a point of interest by the [weight](https://en.wikipedia.org/wiki/Weight "Weight") of a [fluid](https://en.wikipedia.org/wiki/Fluid "Fluid") column above the point. Due to the fundamental nature of fluids, a fluid cannot remain at rest under the presence of a [shear stress](https://en.wikipedia.org/wiki/Shear_stress "Shear stress"). However, fluids can exert [pressure](https://en.wikipedia.org/wiki/Pressure "Pressure") [normal](https://en.wikipedia.org/wiki/Surface_normal "Surface 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](https://en.wikipedia.org/wiki/Fluid_pressure "Fluid pressure") on a fluid at rest is [isotropic](https://en.wikipedia.org/wiki/Isotropic "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](https://en.wikipedia.org/wiki/Blaise_Pascal "Blaise Pascal"), and is now called [Pascal's law](https://en.wikipedia.org/wiki/Pascal%27s_law "Pascal's law").\[[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")\] 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 V = 0 is applied to the [Navier–Stokes equations](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations "Navier–Stokes equations") for viscous fluids or [Euler equations (fluid dynamics)](https://en.wikipedia.org/wiki/Euler_equations_\(fluid_dynamics\) "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: Navier–Stokes momentum equation (convective form) ![{\\displaystyle \\rho {\\frac {\\mathrm {D} \\mathbf {u} }{\\mathrm {D} t}}=-\\nabla \[p-\\zeta (\\nabla \\cdot \\mathbf {u} )\]+\\nabla \\cdot \\left\\{\\mu \\left\[\\nabla \\mathbf {u} +(\\nabla \\mathbf {u} )^{\\mathrm {T} }-{\\tfrac {2}{3}}(\\nabla \\cdot \\mathbf {u} )\\mathbf {I} \\right\]\\right\\}+\\rho \\mathbf {g} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fd98d1bcacc93a84fa4fc981a05a200e680be04) By setting the [flow velocity](https://en.wikipedia.org/wiki/Flow_velocity "Flow velocity") :![{\\displaystyle \\mathbf {u} =\\mathbf {0} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/92a3de96f1315a7b1aa90a60f2322d7e5012b419), they become simply: ![{\\displaystyle \\mathbf {0} =-\\nabla p+\\rho \\mathbf {g} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fadede56aa656a73a1903ef4ef33c276184ecac1) or: ![{\\displaystyle \\nabla p=\\rho \\mathbf {g} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5d9f468c7d46b3471f2b1534701b7bbed1fc9e4) This is the general form of Stevin's law: the [pressure gradient](https://en.wikipedia.org/wiki/Pressure_gradient "Pressure gradient") equals the [body force](https://en.wikipedia.org/wiki/Body_force "Body force") [force density](https://en.wikipedia.org/wiki/Force_density "Force density") field. Let us now consider two particular cases of this law. In case of a [conservative](https://en.wikipedia.org/wiki/Conservative_force "Conservative force") body force with [scalar potential](https://en.wikipedia.org/wiki/Scalar_potential "Scalar potential") :![{\\displaystyle \\phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4): ![{\\displaystyle \\rho \\mathbf {g} =-\\nabla \\phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb954c118e31f5c60fef95bbfe11eaa52a62b9b9) the Stevin equation becomes: ![{\\displaystyle \\nabla p=-\\nabla \\phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5810d8e16907c3b14f06f080fa7406811594ee27) That can be integrated to give: ![{\\displaystyle \\Delta p=-\\Delta \\phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc17ceebfece38c57a6b1ccf34447dbe8b824f92) 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: ![{\\displaystyle \\mathbf {g} =-g(x,y,z){\\hat {k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a192db8a1f1df66661ab6991e2a9ef35311d56b8) the generalised Stevin's law above becomes: ![{\\displaystyle {\\frac {\\partial p}{\\partial z}}=-\\rho (x,y,z)g(x,y,z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/836715bf560634b1be927d83a788bd92964b1c2f) That can be integrated to give another (less-) generalised Stevin's law: ![{\\displaystyle p(x,y,z)-p\_{0}(x,y)=-\\int \_{0}^{z}\\rho (x,y,z')g(x,y,z')dz'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78bcc6804783378024effbb00548d25e28827d18) where: ### Simplification for liquids \[[edit](https://en.wikipedia.org/w/index.php?title=Hydrostatic_pressure&action=edit&section=3 "Edit section: Simplification for liquids")\] 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](https://en.wikipedia.org/wiki/Incompressible_flow "Incompressible flow"), 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 ![{\\displaystyle \\Delta z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf59b5e4dc02e6b2b89b426a27e02c84861c0caa) of the fluid column between z and z0 is often reasonably small compared to the radius of the Earth, one can neglect the variation of [g](https://en.wikipedia.org/wiki/Gravity "Gravity"). Under these circumstances, one can transport out of the integral the density and the gravity acceleration and the law is simplified into the formula ![{\\displaystyle \\Delta p(z)=\\rho g\\Delta z,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9afa9a1da855d0dde3239d2e26ad69a22c9354c6) where ![{\\displaystyle \\Delta z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf59b5e4dc02e6b2b89b426a27e02c84861c0caa) is the height z − z0 of the liquid column between the test volume and the zero reference point of the pressure. This formula is often called [Stevin's](https://en.wikipedia.org/wiki/Simon_Stevin "Simon Stevin") law.[\[1\]](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_note-1)[\[2\]](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_note-2) 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: ![{\\displaystyle \\rho \\mathbf {g} (x,y,z)=-\\rho g{\\hat {k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/286c83b8ad7070465180a3bda0f08ac24ca1b7a7) is conservative, so one can write the body force density as: ![{\\displaystyle \\rho \\mathbf {g} =\\nabla (-\\rho gz)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f7a68af5b40d5e2bb679d68fe7d38ad381d1fc) Then the body force density has a simple [scalar potential](https://en.wikipedia.org/wiki/Scalar_potential "Scalar potential"): ![{\\displaystyle \\phi (z)=-\\rho gz}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad0108c644e333a75e7b0ae3c46a697b51d08b03) And the pressure difference follows another time the Stevin's law: ![{\\displaystyle \\Delta p=-\\Delta \\phi =\\rho g\\Delta z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/787b041c209cf18ccbb6b1aa4368c9aee99b2066) 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 ρliquid and ρ(z′)above. For example, the [absolute pressure](https://en.wikipedia.org/wiki/Pressure_measurement#Absolute,_gauge_and_differential_pressures_-_zero_reference "Pressure measurement") compared to vacuum is ![{\\displaystyle p=\\rho g\\Delta z+p\_{\\mathrm {0} },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65dc7e8253c8fa7fade48f906aef2b45e821c398) where ![{\\displaystyle \\Delta z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf59b5e4dc02e6b2b89b426a27e02c84861c0caa) is the total height of the liquid column above the test area to the surface, and p0 is the [atmospheric pressure](https://en.wikipedia.org/wiki/Atmospheric_pressure "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](https://en.wikipedia.org/wiki/Pressure_prism "Pressure prism"). Hydrostatic pressure has been used in the preservation of foods in a process called [pascalization](https://en.wikipedia.org/wiki/Pascalization "Pascalization").[\[3\]](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_note-3) In medicine, hydrostatic pressure in [blood vessels](https://en.wikipedia.org/wiki/Blood_vessel "Blood vessel") is the pressure of the blood against the wall. It is the opposing force to [oncotic pressure](https://en.wikipedia.org/wiki/Oncotic_pressure "Oncotic pressure"). In capillaries, hydrostatic pressure (also known as capillary blood pressure) is higher than the opposing “colloid [osmotic pressure](https://en.wikipedia.org/wiki/Osmotic_pressure "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.[\[4\]](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_note-Openstax_Anatomy_&_Physiology_attribution-4) ### Atmospheric pressure \[[edit](https://en.wikipedia.org/w/index.php?title=Hydrostatic_pressure&action=edit&section=6 "Edit section: Atmospheric pressure")\] [Statistical mechanics](https://en.wikipedia.org/wiki/Statistical_mechanics "Statistical mechanics") shows that, for a pure [ideal gas](https://en.wikipedia.org/wiki/Ideal_gas "Ideal gas") of constant temperature T in the earth gravitational field, its pressure, p will vary with height, h, as ![{\\displaystyle p(h)=p(0)e^{-{\\frac {Mgh}{kT}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa359443bd71846a0f795d8af50a298f02fc900b) where - g is the [acceleration due to gravity](https://en.wikipedia.org/wiki/Standard_gravity "Standard gravity") - T is the [absolute temperature](https://en.wikipedia.org/wiki/Absolute_temperature "Absolute temperature") - k is [Boltzmann constant](https://en.wikipedia.org/wiki/Boltzmann_constant "Boltzmann constant") - M is the [molecular mass](https://en.wikipedia.org/wiki/Molecular_mass "Molecular mass") of the gas - p is the pressure - h is the height This is known as the [barometric formula](https://en.wikipedia.org/wiki/Barometric_formula "Barometric formula"), and may be derived from assuming the pressure is [hydrostatic](). If there are multiple types of molecules in the gas, the [partial pressure](https://en.wikipedia.org/wiki/Partial_pressure "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. 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, ![{\\displaystyle F=\\rho gV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b1b65f31d86a2c06763a4f93131781af9867fba) 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.[\[5\]](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_note-F-M-5) In the case of a [ship](https://en.wikipedia.org/wiki/Ship "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.\[[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")\] Discovery of the principle of buoyancy is attributed to [Archimedes](https://en.wikipedia.org/wiki/Archimedes "Archimedes"). ### Hydrostatic force on submerged surfaces \[[edit](https://en.wikipedia.org/w/index.php?title=Hydrostatic_pressure&action=edit&section=8 "Edit section: Hydrostatic force on submerged surfaces")\] The horizontal and vertical components of the hydrostatic force acting on a submerged surface are given by the following formula:[\[5\]](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_note-F-M-5) ![{\\displaystyle {\\begin{aligned}F\_{\\mathrm {h} }&=p\_{\\mathrm {c} }A\\\\F\_{\\mathrm {v} }&=\\rho gV\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8812be401acd9d7effb3ae8d7e6e4da2e9ee95a6) where - pc is the pressure at the centroid 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 - [Hydrostatics](https://en.wikipedia.org/wiki/Hydrostatics "Hydrostatics") - [Vertical pressure variation](https://en.wikipedia.org/wiki/Vertical_pressure_variation "Vertical pressure variation") 1. [^](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_ref-1) Bettini, Alessandro (2016). A Course in Classical Physics 2—Fluids and Thermodynamics. Springer. p. 8. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-3-319-30685-8](https://en.wikipedia.org/wiki/Special:BookSources/978-3-319-30685-8 "Special:BookSources/978-3-319-30685-8") . 2. [^](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_ref-2) Mauri, Roberto (8 April 2015). [Transport Phenomena in Multiphase Flow](https://books.google.com/books?id=S3L0BwAAQBAJ&pg=PA24). Springer. p. 24. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-3-319-15792-4](https://en.wikipedia.org/wiki/Special:BookSources/978-3-319-15792-4 "Special:BookSources/978-3-319-15792-4") . Retrieved 3 February 2017. 3. [^](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_ref-3) Brown, Amy Christian (2007). [Understanding Food: Principles and Preparation](https://books.google.com/books?id=edPzm5KSMmYC) (3 ed.). Cengage Learning. p. 546. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-495-10745-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-495-10745-3 "Special:BookSources/978-0-495-10745-3") . 4. [^](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_ref-Openstax_Anatomy_&_Physiology_attribution_4-0) [![](https://upload.wikimedia.org/wikipedia/commons/thumb/6/65/Creative_Commons_by_small.svg/120px-Creative_Commons_by_small.svg.png)](https://creativecommons.org/licenses/by/4.0/ "creativecommons:by/4.0/") This article incorporates [text](https://openstax.org/books/anatomy-and-physiology/pages/26-1-body-fluids-and-fluid-compartments) available under the [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/ "creativecommons:by/4.0/") license. Betts, J Gordon; Desaix, Peter; Johnson, Eddie; Johnson, Jody E; Korol, Oksana; Kruse, Dean; Poe, Brandon; Wise, James; Womble, Mark D; Young, Kelly A (September 16, 2023). Anatomy & Physiology. Houston: OpenStax CNX. 26.1 Body fluids and fluid compartments. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-947172-04-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-947172-04-3 "Special:BookSources/978-1-947172-04-3") . 5. ^ [*a](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_ref-F-M_5-0) [b](https://en.wikipedia.org/wiki/Hydrostatic_pressure#cite_ref-F-M_5-1) Fox, Robert; McDonald, Alan; Pritchard, Philip (2012). Fluid Mechanics* (8 ed.). [John Wiley & Sons](https://en.wikipedia.org/wiki/John_Wiley_%26_Sons "John Wiley & Sons"). pp. 76–83\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-118-02641-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-118-02641-0 "Special:BookSources/978-1-118-02641-0") .
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