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1. Shard Calculation

Query:

Response:

Calculated Shard: 62 (from laksa055)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa062.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://www.britannica.com/science/hydrostatics\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.britannica.com/science/hydrostatics\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.britannica.com\/science\/hydrostatics","crawl_time":1775910882,"first_indexed_time":1477631570,"http_code":200,"src_unparsed":"com,britannica!www,\/science\/hydrostatics s443","src_root_hash":"5455945239613777662","history_drop_reason":null,"meta_title":"Hydrostatics | Fluid Pressure, Buoyancy & Equilibrium | Britannica","meta_descriptions":["Hydrostatics, Branch of physics that deals with the characteristics of fluids at rest, particularly with the pressure in a fluid or exerted by a fluid (gas or liquid) on an immersed body. In applications, the principles of hydrostatics are used for problems relating to pressure in deep water"],"attrs_boilerpipe_text":"Top Questions\nWhat is fluid mechanics?\nWhat is a fluid in science?\nWhat are some important properties of fluids?\nHow is pressure created and measured in fluids?\nWhat is buoyancy, and how does it affect objects in water?\nHow is fluid mechanics used in real-life situations like airplanes and weather?\nfluid mechanics\n,\nscience\nconcerned with the\nresponse of fluids\nto forces exerted upon them. It is a branch of classical\nphysics\nwith applications of great importance in hydraulic and\naeronautical engineering\n,\nchemical engineering\n,\nmeteorology\n, and zoology.\nThe most familiar fluid is of course\nwater\n, and an encyclopaedia of the 19th century probably would have dealt with the subject under the separate headings of\nhydrostatics\n, the science of water at rest, and hydrodynamics, the science of water in\nmotion\n.\nArchimedes\nfounded hydrostatics in about 250\nbc\nwhen, according to\nlegend\n, he leapt out of his bath and ran naked through the streets of Syracuse crying “Eureka!”; it has undergone rather little development since. The foundations of hydrodynamics, on the other hand, were not laid until the 18th century when mathematicians such as\nLeonhard Euler\nand\nDaniel Bernoulli\nbegan to explore the consequences, for a virtually continuous medium like water, of the\ndynamic\nprinciples that Newton had enunciated for systems composed of discrete particles. Their work was continued in the 19th century by several mathematicians and physicists of the first rank, notably G.G. Stokes and William Thomson. By the end of the century explanations had been found for a host of intriguing phenomena having to do with the flow of water through tubes and orifices, the waves that ships moving through water leave behind them, raindrops on windowpanes, and the like. There was still no proper understanding, however, of problems as fundamental as that of water flowing past a fixed obstacle and exerting a\ndrag force\nupon it; the theory of potential flow, which worked so well in other\ncontexts\n, yielded results that at relatively high flow rates were grossly at variance with experiment. This problem was not properly understood until 1904, when the German physicist\nLudwig Prandtl\nintroduced the concept of the\nboundary layer\n(see below\nHydrodynamics: Boundary layers and separation\n). Prandtl’s career continued into the period in which the first manned aircraft were developed. Since that time, the flow of\nair\nhas been of as much interest to physicists and engineers as the flow of water, and hydrodynamics has, as a consequence, become fluid\ndynamics\n. The term fluid mechanics, as used here, embraces both fluid\ndynamics\nand the subject still generally referred to as hydrostatics.\nOne other representative of the 20th century who deserves mention here besides Prandtl is\nGeoffrey Taylor\nof England. Taylor remained a classical physicist while most of his contemporaries were turning their attention to the problems of atomic structure and\nquantum mechanics\n, and he made several unexpected and important discoveries in the\nfield\nof fluid mechanics. The richness of fluid mechanics is due in large part to a term in the basic equation of the motion of fluids which is nonlinear—\ni.e.,\none that involves the fluid\nvelocity\ntwice over. It is characteristic of systems described by nonlinear equations that under certain conditions they become unstable and begin behaving in ways that seem at first sight to be totally chaotic. In the case of fluids, chaotic behaviour is very common and is called turbulence. Mathematicians have now begun to recognize patterns in\nchaos\nthat can be analyzed fruitfully, and this development suggests that fluid mechanics will remain a field of active research well into the 21st century. (For a discussion of the concept of\nchaos\n, see\nphysical science, principles of\n.)\nFluid mechanics is a subject with almost endless ramifications, and the account that follows is necessarily incomplete. Some knowledge of the basic properties of fluids will be needed; a survey of the most relevant properties is given in the next section. For further details, see\nthermodynamics\nand\nliquid\n.\nBasic properties of\nfluids\nFluids are not strictly continuous media in the way that all the successors of Euler and Bernoulli have assumed, for they are composed of discrete molecules. The molecules, however, are so small and, except in gases at very low pressures, the number of molecules per millilitre is so enormous that they need not be viewed as individual\nentities\n. There are a few liquids, known as liquid crystals, in which the molecules are packed together in such a way as to make the properties of the medium locally anisotropic, but the vast majority of fluids (including air and water) are\nisotropic\n. In fluid mechanics, the state of an isotropic fluid may be completely described by defining its mean mass per unit volume, or\ndensity\n(ρ), its\ntemperature\n(\nT\n), and its velocity (\nv\n) at every point in space, and just what the connection is between these macroscopic properties and the positions and velocities of individual molecules is of no direct relevance.\nBritannica Quiz\nAll About Physics Quiz\nA word perhaps is needed about the difference between\ngases\nand\nliquids\n, though the difference is easier to perceive than to describe. In gases the molecules are sufficiently far apart to move almost independently of one another, and gases tend to expand to fill any volume available to them. In liquids the molecules are more or less in contact, and the short-range attractive forces between them make them cohere; the molecules are moving too fast to settle down into the ordered\narrays\nthat are characteristic of solids, but not so fast that they can fly apart. Thus, samples of liquid can exist as drops or as jets with free surfaces, or they can sit in beakers constrained only by gravity, in a way that samples of\ngas\ncannot. Such samples may evaporate in time, as molecules one by one pick up enough speed to escape across the free surface and are not replaced. The lifetime of liquid drops and jets, however, is normally long enough for evaporation to be ignored.\nThere are two sorts of\nstress\nthat may exist in any solid or fluid medium, and the difference between them may be illustrated by reference to a brick held between two hands. If the holder moves his hands toward each other, he exerts\npressure\non the brick; if he moves one hand toward his body and the other away from it, then he exerts what is called a\nshear stress\n. A solid substance such as a brick can withstand stresses of both types, but fluids, by definition, yield to shear stresses no matter how small these stresses may be. They do so at a rate determined by the fluid’s\nviscosity\n. This property, about which more will be said later, is a measure of the\nfriction\nthat arises when\nadjacent\nlayers of\nfluid\nslip over one another. It follows that the shear stresses are everywhere zero in a fluid at rest and in\nequilibrium\n, and from this it follows that the pressure (that is,\nforce\nper unit area) acting perpendicular to all planes in the fluid is the same irrespective of their orientation (\nPascal’s law\n). For an isotropic fluid in equilibrium there is only one value of the local pressure (\np\n) consistent with the stated values for ρ and\nT\n. These three quantities are linked together by what is called the\nequation of state\nfor the fluid.\nSmart, reliable knowledge for professionals, students, and curious minds everywhere.\nSUBSCRIBE\nFor gases at low pressures the equation of state is simple and well known. It is\nwhere\nR\nis the\nuniversal gas constant\n(8.3 joules per degree Celsius per mole) and\nM\nis the\nmolar mass\n, or an average molar mass if the gas is a mixture; for air, the\nappropriate\naverage is about 29 × 10\n−3\nkilogram per mole. For other fluids knowledge of the equation of state is often incomplete. Except under very extreme conditions, however, all one needs to know is how the density\nchanges\nwhen the pressure is changed by a small amount, and this is described by the\ncompressibility\nof the\nfluid—either the\nisothermal compressibility, β\nT\n, or the\nadiabatic compressibility, β\nS\n, according to circumstance. When an element of fluid is compressed, the work done on it tends to heat it up. If the heat has time to drain away to the surroundings and the temperature of the fluid remains essentially unchanged throughout, then β\nT\nis the relevant quantity. If virtually none of the heat escapes, as is more commonly the case in flow problems because the\nthermal conductivity\nof most fluids is poor, then the flow is said to be\nadiabatic, and β\nS\nis needed instead. (The\nS\nrefers to\nentropy\n, which remains constant in an\nadiabatic process\nprovided that it takes place slowly enough to be treated as “reversible” in the thermodynamic sense.) For gases that obey equation (\n118\n), it is evident that\np\nand ρ are proportional to one another in an isothermal process, and\nIn reversible adiabatic processes for such gases, however, the temperature rises on compression at a rate such that\nand\nwhere γ is about 1.4 for air and takes similar values for other common gases. For liquids the ratio between the isothermal and adiabatic compressibilities is much closer to\nunity\n. For liquids, however, both compressibilities are normally much less than\np\n−1\n, and the simplifying assumption that they are zero is often justified.\nThe factor γ is not only the ratio between two compressibilities; it is also the ratio between two principal\nspecific heats\n. The molar specific heat is the amount of heat required to raise the temperature of one mole through one degree. This is greater if the substance is allowed to expand as it is heated, and therefore to do work, than if its volume is fixed. The principal molar specific heats,\nC\nP\nand\nC\nV\n, refer to heating at constant pressure and constant volume, respectively, and\nFor air,\nC\nP\nis about 3.5\nR\n.\nSolids can be stretched without breaking, and liquids, though not gases, can withstand stretching, too. Thus, if the pressure is steadily reduced in a\nspecimen\nof very pure water, bubbles will ultimately appear, but they may not do so until the pressure is negative and well below -10\n7\nnewton per square metre; this is 100 times greater in magnitude than the (positive) pressure exerted by the Earth’s\natmosphere\n. Water owes its high ideal strength to the fact that rupture involves breaking links of attraction between molecules on either side of the plane on which rupture occurs; work must be done to break these links. However, its strength is drastically reduced by anything that provides a nucleus at which the process known as\ncavitation\n(formation of vapour- or gas-filled cavities) can begin, and a liquid containing suspended dust particles or\ndissolved\ngases is liable to cavitate quite easily.\nWork also must be done if a free liquid drop of spherical shape is to be drawn out into a long thin cylinder or deformed in any other way that increases its surface area. Here again work is needed to break intermolecular links. The surface of a liquid behaves, in fact, as if it were an elastic membrane under tension, except that the tension exerted by an elastic membrane increases when the membrane is stretched in a way that the tension exerted by a liquid surface does not.\nSurface tension\nis what causes liquids to rise up capillary tubes, what supports hanging liquid drops, what limits the formation of ripples on the surface of liquids, and so on.","attrs_markdown":"[![Encyclopedia Britannica](https:\/\/cdn.britannica.com\/mendel\/eb-logo\/MendelNewThistleLogo.png)](https:\/\/www.britannica.com\/)\n\n[![Encyclopedia Britannica](https:\/\/cdn.britannica.com\/mendel\/eb-logo\/MendelNewThistleLogo.png)](https:\/\/www.britannica.com\/)\n\n[SUBSCRIBE](https:\/\/premium.britannica.com\/premium-membership\/?utm_source=premium&utm_medium=global-nav&utm_campaign=blue-evergreen)\n\n[SUBSCRIBE](https:\/\/premium.britannica.com\/premium-membership\/?utm_source=premium&utm_medium=global-nav-mobile&utm_campaign=blue-evergreen)\n\nLogin\n\nhttps:\/\/premium.britannica.com\/premium-membership\/?utm\\_source=premium\\&utm\\_medium=nav-login-box\\&utm\\_campaign=evergreen\n\n[SUBSCRIBE](https:\/\/premium.britannica.com\/premium-membership\/?utm_source=premium&utm_medium=hamburger-menu&utm_campaign=blue)\n\n[Ask the Chatbot](https:\/\/www.britannica.com\/chatbot)\n\n[Games & Quizzes](https:\/\/www.britannica.com\/quiz\/browse) [History & Society](https:\/\/www.britannica.com\/History-Society) [Science & Tech](https:\/\/www.britannica.com\/Science-Tech) [Biographies](https:\/\/www.britannica.com\/Biographies) [Animals & Nature](https:\/\/www.britannica.com\/Animals-Nature) [Geography & Travel](https:\/\/www.britannica.com\/Geography-Travel) [Arts & Culture](https:\/\/www.britannica.com\/Arts-Culture) [ProCon](https:\/\/www.britannica.com\/procon) [Money](https:\/\/www.britannica.com\/money) [Videos](https:\/\/www.britannica.com\/videos)\n\n[hydrostatics](https:\/\/www.britannica.com\/science\/hydrostatics)\n\n[Introduction](https:\/\/www.britannica.com\/science\/hydrostatics)   [References & Edit History](https:\/\/www.britannica.com\/science\/hydrostatics\/additional-info)  [Related Topics](https:\/\/www.britannica.com\/facts\/hydrostatics)\n\nQuizzes\n\n[![Italian-born physicist Dr. Enrico Fermi draws a diagram at a blackboard with mathematical equations. circa 1950.](https:\/\/cdn.britannica.com\/01\/115001-131-7278E518\/Enrico-Fermi-Italian-problem-physics-1950.jpg?w=200&h=200&c=crop)](https:\/\/www.britannica.com\/quiz\/physics-and-natural-law)\n\n[Physics and Natural Law](https:\/\/www.britannica.com\/quiz\/physics-and-natural-law)\n\n[![Italian physicist Guglielmo Marconi at work in the wireless room of his yacht Electra, c. 1920.](https:\/\/cdn.britannica.com\/03\/221503-131-D7173869\/Italian-physicist-Guglielmo-Marconi-at-work-on-his-yacht-Electra.jpg?w=200&h=200&c=crop)](https:\/\/www.britannica.com\/quiz\/all-about-physics-quiz)\n\n[All About Physics Quiz](https:\/\/www.britannica.com\/quiz\/all-about-physics-quiz)\n\n![Britannica AI Icon](https:\/\/cdn.britannica.com\/mendel-resources\/3-179\/images\/chatbot\/star-ai.svg?v=3.179.9)\n\nContents\n\nAsk Anything\n\n[Science](https:\/\/www.britannica.com\/browse\/Science)    [Physics](https:\/\/www.britannica.com\/browse\/Physics)\n\nCITE\n\nShare\n\nFeedback\n\nExternal Websites\n\n# hydrostatics\nphysics\n\nHomework Help\n\nWritten and fact-checked by\n\n[Britannica Editors Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree....](https:\/\/www.britannica.com\/editor\/The-Editors-of-Encyclopaedia-Britannica\/4419)\n\nBritannica Editors\n\n[History](https:\/\/www.britannica.com\/science\/hydrostatics\/additional-info#history)\n\n![Britannica AI Icon](https:\/\/cdn.britannica.com\/mendel-resources\/3-179\/images\/chatbot\/star-ai.svg?v=3.179.9) Britannica AI\n\nAsk Anything\n\nTable of Contents\n\nTable of Contents\n\nAsk Anything\n\n**hydrostatics**, Branch of [physics](https:\/\/www.britannica.com\/science\/physics-science) that deals with the [characteristics](https:\/\/www.britannica.com\/dictionary\/characteristics) of [fluids](https:\/\/www.britannica.com\/science\/fluid-physics) at rest, particularly with the [pressure](https:\/\/www.britannica.com\/science\/pressure) in a [fluid](https:\/\/www.britannica.com\/science\/fluid-physics) or exerted by a fluid (gas or liquid) on an immersed body. In applications, the principles of hydrostatics are used for problems relating to pressure in deep water (pressure increases with depth) and high in the [atmosphere](https:\/\/www.britannica.com\/science\/atmosphere) (pressure lessens with altitude).\n\nThis article was most recently revised and updated by [Robert Curley](https:\/\/www.britannica.com\/editor\/Robert-Curley\/6737).\n\n![Britannica AI Icon](https:\/\/cdn.britannica.com\/mendel-resources\/3-179\/images\/chatbot\/star-ai.svg?v=3.179.9)Britannica AI\n\n*chevron\\_right*\n\nHydrostatics\n\n*close*\n\n[AI-generated answers](https:\/\/www.britannica.com\/about-britannica-ai) from Britannica articles. AI makes mistakes, so verify using Britannica articles.\n\n[fluid mechanics](https:\/\/www.britannica.com\/science\/fluid-mechanics)\n\n- [Introduction](https:\/\/www.britannica.com\/science\/fluid-mechanics)\n- [Basic properties of fluids](https:\/\/www.britannica.com\/science\/fluid-mechanics#ref77478)\n- [Hydrostatics](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Hydrostatics)\n  - [Differential manometers](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Hydrostatics#ref77480)\n  \n  - [Archimedes’ principle](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Hydrostatics#ref77481)\n  \n  - [Surface tension of liquids](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Hydrostatics#ref77482)\n- [Hydrodynamics](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Hydrodynamics)\n  - [Bernoulli’s law](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Hydrodynamics#ref77484)\n  \n  - [Waves on shallow water](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Waves-on-shallow-water)\n  \n  - [Compressible flow in gases](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Waves-on-shallow-water#ref77486)\n  \n  - [Viscosity](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Viscosity)\n    - [Stresses in laminar motion](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Viscosity#ref77488)\n    - [Bulk viscosity](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Viscosity#ref77489)\n    - [Measurement of shear viscosity](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Viscosity#ref77490)\n  \n  - [Navier-stokes equation](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Navier-stokes-equation)\n  \n  - [Potential flow](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Navier-stokes-equation#ref77492)\n  \n  - [Potential flow with circulation: vortex lines](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Navier-stokes-equation#ref77493)\n  \n  - [Waves on deep water](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Waves-on-deep-water)\n  \n  - [Boundary layers and separation](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Waves-on-deep-water#ref77495)\n  \n  - [Drag](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Drag)\n  \n  - [Lift](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Drag#ref77497)\n  \n  - [Turbulence](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Turbulence)\n  \n  - [Convection](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Turbulence#ref77499)\n\n[References & Edit History](https:\/\/www.britannica.com\/science\/fluid-mechanics\/additional-info)  [Related Topics](https:\/\/www.britannica.com\/facts\/fluid-mechanics)\n\n[Images & Videos](https:\/\/www.britannica.com\/science\/fluid-mechanics\/images-videos)\n\n[![differential manometer, Torricellian barometer, and siphon](https:\/\/cdn.britannica.com\/08\/2508-004-2696813B\/representations-manometer-barometer-Torricellian.jpg)](https:\/\/cdn.britannica.com\/08\/2508-004-2696813B\/representations-manometer-barometer-Torricellian.jpg)  [![Torricelli, Evangelista](https:\/\/cdn.britannica.com\/21\/195421-004-97DAD799\/Evangelista-Torricelli-physicist-Italian-mercury-barometer.jpg)](https:\/\/cdn.britannica.com\/21\/195421-050-FC98CA10\/Evangelista-Torricelli-physicist-Italian-mercury-barometer.jpg)  [![possible orientations of a square prism in liquid](https:\/\/cdn.britannica.com\/09\/2509-004-50CBDBFD\/orientations-prism-liquid-orientation-density.jpg)](https:\/\/cdn.britannica.com\/09\/2509-050-305BFADB\/orientations-prism-liquid-orientation-density.jpg)\n\n[![The science of surface tension explained](https:\/\/cdn.britannica.com\/76\/151076-138-927F881C\/Explanation-surface-tension.jpg?w=400&h=225&c=crop)](https:\/\/www.britannica.com\/video\/Explanation-surface-tension\/-159381)\n\n[![capillarity](https:\/\/cdn.britannica.com\/10\/2510-004-EF2E6BED\/Capillarity.jpg)](https:\/\/cdn.britannica.com\/10\/2510-004-EF2E6BED\/Capillarity.jpg)  [![formation of a liquid drop](https:\/\/cdn.britannica.com\/11\/2511-004-97E4DF70\/Stages-formation-drop.jpg)](https:\/\/cdn.britannica.com\/11\/2511-004-97E4DF70\/Stages-formation-drop.jpg)  [![venturi tube and pitot tube](https:\/\/cdn.britannica.com\/12\/2512-004-1BB4C5D1\/representation-venturi-tube-pitot.jpg)](https:\/\/cdn.britannica.com\/12\/2512-004-1BB4C5D1\/representation-venturi-tube-pitot.jpg)  [![surface of shallow water](https:\/\/cdn.britannica.com\/13\/2513-004-32814157\/Steps-surface-water.jpg)](https:\/\/cdn.britannica.com\/13\/2513-004-32814157\/Steps-surface-water.jpg)  [![interaction of two solitons](https:\/\/cdn.britannica.com\/14\/2514-004-4A861A32\/Interaction-solitons.jpg)](https:\/\/cdn.britannica.com\/14\/2514-004-4A861A32\/Interaction-solitons.jpg)  [![Mach's construction](https:\/\/cdn.britannica.com\/15\/2515-004-3EDFC935\/construction-Mach-U-speed-of-sound-VS.jpg)](https:\/\/cdn.britannica.com\/15\/2515-004-3EDFC935\/construction-Mach-U-speed-of-sound-VS.jpg)\n\nAt a Glance\n\n[![default image](https:\/\/cdn.britannica.com\/mendel-resources\/3-179\/images\/shared\/new-thistle.svg?v=3.179.9)](https:\/\/www.britannica.com\/summary\/fluid-mechanics)\n\n[fluid mechanics summary](https:\/\/www.britannica.com\/summary\/fluid-mechanics)\n\nQuizzes\n\n[![Italian-born physicist Dr. Enrico Fermi draws a diagram at a blackboard with mathematical equations. circa 1950.](https:\/\/cdn.britannica.com\/01\/115001-131-7278E518\/Enrico-Fermi-Italian-problem-physics-1950.jpg?w=200&h=200&c=crop)](https:\/\/www.britannica.com\/quiz\/physics-and-natural-law)\n\n[Physics and Natural Law](https:\/\/www.britannica.com\/quiz\/physics-and-natural-law)\n\n[![Italian physicist Guglielmo Marconi at work in the wireless room of his yacht Electra, c. 1920.](https:\/\/cdn.britannica.com\/03\/221503-131-D7173869\/Italian-physicist-Guglielmo-Marconi-at-work-on-his-yacht-Electra.jpg?w=200&h=200&c=crop)](https:\/\/www.britannica.com\/quiz\/all-about-physics-quiz)\n\n[All About Physics Quiz](https:\/\/www.britannica.com\/quiz\/all-about-physics-quiz)\n\nRelated Questions\n\n- [What led to Archimedes’ discovering his principle?](https:\/\/www.britannica.com\/question\/What-led-to-Archimedes-discovering-his-principle)\n- [What is Archimedes’ principle?](https:\/\/www.britannica.com\/question\/What-is-Archimedes-principle)\n- [What is Archimedes’ principle used for?](https:\/\/www.britannica.com\/question\/What-is-Archimedes-principle-used-for)\n- [What is the formula for buoyant force?](https:\/\/www.britannica.com\/question\/What-is-the-formula-for-buoyant-force)\n\n![Britannica AI Icon](https:\/\/cdn.britannica.com\/mendel-resources\/3-179\/images\/chatbot\/star-ai.svg?v=3.179.9)\n\nAsk Anything\n\nQuick Summary\n\n[Science](https:\/\/www.britannica.com\/browse\/Science)    [Physics](https:\/\/www.britannica.com\/browse\/Physics)\n\nCITE\n\nShare\n\nFeedback\n\nExternal Websites\n\n# fluid mechanics\nphysics\n\nHomework Help\n\nWritten by\n\n[Thomas E. Faber Lecturer in Physics, University of Cambridge, 1959–93; Fellow of Corpus Christi College, Cambridge. Author of *Fluid Dynamics for Physicists* and others.](https:\/\/www.britannica.com\/contributor\/Thomas-E-Faber\/3527)\n\nThomas E. Faber\n\nFact-checked by\n\n[Britannica Editors Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree....](https:\/\/www.britannica.com\/editor\/The-Editors-of-Encyclopaedia-Britannica\/4419)\n\nBritannica Editors\n\nLast updated\n\nMar. 26, 2026\n\n•[History](https:\/\/www.britannica.com\/science\/fluid-mechanics\/additional-info#history)\n\n![Britannica AI Icon](https:\/\/cdn.britannica.com\/mendel-resources\/3-179\/images\/chatbot\/star-ai.svg?v=3.179.9) Britannica AI\n\nAsk Anything\n\nQuick Summary\n\nTable of Contents\n\nTable of Contents\n\nQuick Summary\n\nAsk Anything\n\nTop Questions\n\n- What is fluid mechanics?\n- What is a fluid in science?\n- What are some important properties of fluids?\n- How is pressure created and measured in fluids?\n- What is buoyancy, and how does it affect objects in water?\n- How is fluid mechanics used in real-life situations like airplanes and weather?\n\nShow more\n\nShow less\n\n**fluid mechanics**, [science](https:\/\/www.britannica.com\/science\/science) concerned with the [response of fluids](https:\/\/www.britannica.com\/science\/fluid-flow) to forces exerted upon them. It is a branch of classical [physics](https:\/\/www.britannica.com\/science\/physics-science) with applications of great importance in hydraulic and [aeronautical engineering](https:\/\/www.britannica.com\/technology\/aerospace-engineering), [chemical engineering](https:\/\/www.britannica.com\/technology\/chemical-engineering), [meteorology](https:\/\/www.britannica.com\/science\/meteorology), and zoology.\n\nThe most familiar fluid is of course [water](https:\/\/www.britannica.com\/science\/water), and an encyclopaedia of the 19th century probably would have dealt with the subject under the separate headings of [hydrostatics](https:\/\/www.britannica.com\/science\/hydrostatics), the science of water at rest, and hydrodynamics, the science of water in [motion](https:\/\/www.britannica.com\/science\/motion-mechanics). [Archimedes](https:\/\/www.britannica.com\/biography\/Archimedes) founded hydrostatics in about 250 bc when, according to [legend](https:\/\/www.merriam-webster.com\/dictionary\/legend), he leapt out of his bath and ran naked through the streets of Syracuse crying “Eureka!”; it has undergone rather little development since. The foundations of hydrodynamics, on the other hand, were not laid until the 18th century when mathematicians such as [Leonhard Euler](https:\/\/www.britannica.com\/biography\/Leonhard-Euler) and [Daniel Bernoulli](https:\/\/www.britannica.com\/biography\/Daniel-Bernoulli) began to explore the consequences, for a virtually continuous medium like water, of the [dynamic](https:\/\/www.merriam-webster.com\/dictionary\/dynamic) principles that Newton had enunciated for systems composed of discrete particles. Their work was continued in the 19th century by several mathematicians and physicists of the first rank, notably G.G. Stokes and William Thomson. By the end of the century explanations had been found for a host of intriguing phenomena having to do with the flow of water through tubes and orifices, the waves that ships moving through water leave behind them, raindrops on windowpanes, and the like. There was still no proper understanding, however, of problems as fundamental as that of water flowing past a fixed obstacle and exerting a [drag force](https:\/\/www.britannica.com\/science\/drag) upon it; the theory of potential flow, which worked so well in other [contexts](https:\/\/www.merriam-webster.com\/dictionary\/contexts), yielded results that at relatively high flow rates were grossly at variance with experiment. This problem was not properly understood until 1904, when the German physicist [Ludwig Prandtl](https:\/\/www.britannica.com\/biography\/Ludwig-Prandtl) introduced the concept of the [boundary layer](https:\/\/www.britannica.com\/science\/boundary-layer) (see below [Hydrodynamics: Boundary layers and separation](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Waves-on-deep-water#ref77495)). Prandtl’s career continued into the period in which the first manned aircraft were developed. Since that time, the flow of [air](https:\/\/www.britannica.com\/science\/aerodynamics) has been of as much interest to physicists and engineers as the flow of water, and hydrodynamics has, as a consequence, become fluid [dynamics](https:\/\/www.britannica.com\/science\/dynamics-physics). The term fluid mechanics, as used here, embraces both fluid [dynamics](https:\/\/www.merriam-webster.com\/dictionary\/dynamics) and the subject still generally referred to as hydrostatics.\n\nOne other representative of the 20th century who deserves mention here besides Prandtl is [Geoffrey Taylor](https:\/\/www.britannica.com\/biography\/Geoffrey-Ingram-Taylor) of England. Taylor remained a classical physicist while most of his contemporaries were turning their attention to the problems of atomic structure and [quantum mechanics](https:\/\/www.britannica.com\/science\/quantum-mechanics-physics), and he made several unexpected and important discoveries in the [field](https:\/\/www.britannica.com\/science\/field-physics) of fluid mechanics. The richness of fluid mechanics is due in large part to a term in the basic equation of the motion of fluids which is nonlinear—*i.e.,* one that involves the fluid [velocity](https:\/\/www.britannica.com\/science\/velocity) twice over. It is characteristic of systems described by nonlinear equations that under certain conditions they become unstable and begin behaving in ways that seem at first sight to be totally chaotic. In the case of fluids, chaotic behaviour is very common and is called turbulence. Mathematicians have now begun to recognize patterns in [chaos](https:\/\/www.britannica.com\/science\/chaos-theory) that can be analyzed fruitfully, and this development suggests that fluid mechanics will remain a field of active research well into the 21st century. (For a discussion of the concept of [chaos](https:\/\/www.merriam-webster.com\/dictionary\/chaos), see [physical science, principles of](https:\/\/www.britannica.com\/science\/principles-of-physical-science).)\n\nFluid mechanics is a subject with almost endless ramifications, and the account that follows is necessarily incomplete. Some knowledge of the basic properties of fluids will be needed; a survey of the most relevant properties is given in the next section. For further details, see [thermodynamics](https:\/\/www.britannica.com\/science\/thermodynamics) and [liquid](https:\/\/www.britannica.com\/science\/liquid-state-of-matter).\n\n## Basic properties of [fluids](https:\/\/www.britannica.com\/science\/fluid-physics)\nFluids are not strictly continuous media in the way that all the successors of Euler and Bernoulli have assumed, for they are composed of discrete molecules. The molecules, however, are so small and, except in gases at very low pressures, the number of molecules per millilitre is so enormous that they need not be viewed as individual [entities](https:\/\/www.britannica.com\/dictionary\/entities). There are a few liquids, known as liquid crystals, in which the molecules are packed together in such a way as to make the properties of the medium locally anisotropic, but the vast majority of fluids (including air and water) are [isotropic](https:\/\/www.britannica.com\/science\/isotropy). In fluid mechanics, the state of an isotropic fluid may be completely described by defining its mean mass per unit volume, or [density](https:\/\/www.britannica.com\/science\/density) (ρ), its [temperature](https:\/\/www.britannica.com\/science\/temperature) (*T*), and its velocity (*v*) at every point in space, and just what the connection is between these macroscopic properties and the positions and velocities of individual molecules is of no direct relevance.\n\n[![Italian physicist Guglielmo Marconi at work in the wireless room of his yacht Electra, c. 1920.](https:\/\/cdn.britannica.com\/03\/221503-131-D7173869\/Italian-physicist-Guglielmo-Marconi-at-work-on-his-yacht-Electra.jpg) Britannica Quiz All About Physics Quiz](https:\/\/www.britannica.com\/quiz\/all-about-physics-quiz)\n\nA word perhaps is needed about the difference between [gases](https:\/\/www.britannica.com\/science\/gas-state-of-matter) and [liquids](https:\/\/www.britannica.com\/science\/liquid-state-of-matter), though the difference is easier to perceive than to describe. In gases the molecules are sufficiently far apart to move almost independently of one another, and gases tend to expand to fill any volume available to them. In liquids the molecules are more or less in contact, and the short-range attractive forces between them make them cohere; the molecules are moving too fast to settle down into the ordered [arrays](https:\/\/www.britannica.com\/dictionary\/arrays) that are characteristic of solids, but not so fast that they can fly apart. Thus, samples of liquid can exist as drops or as jets with free surfaces, or they can sit in beakers constrained only by gravity, in a way that samples of [gas](https:\/\/www.britannica.com\/science\/gas-state-of-matter) cannot. Such samples may evaporate in time, as molecules one by one pick up enough speed to escape across the free surface and are not replaced. The lifetime of liquid drops and jets, however, is normally long enough for evaporation to be ignored.\n\nThere are two sorts of [stress](https:\/\/www.britannica.com\/science\/stress-physics) that may exist in any solid or fluid medium, and the difference between them may be illustrated by reference to a brick held between two hands. If the holder moves his hands toward each other, he exerts [pressure](https:\/\/www.britannica.com\/science\/pressure) on the brick; if he moves one hand toward his body and the other away from it, then he exerts what is called a [shear stress](https:\/\/www.britannica.com\/science\/shear-stress). A solid substance such as a brick can withstand stresses of both types, but fluids, by definition, yield to shear stresses no matter how small these stresses may be. They do so at a rate determined by the fluid’s [viscosity](https:\/\/www.britannica.com\/science\/viscosity). This property, about which more will be said later, is a measure of the [friction](https:\/\/www.britannica.com\/science\/friction) that arises when [adjacent](https:\/\/www.merriam-webster.com\/dictionary\/adjacent) layers of [fluid](https:\/\/www.britannica.com\/science\/fluid-pressure) slip over one another. It follows that the shear stresses are everywhere zero in a fluid at rest and in [equilibrium](https:\/\/www.merriam-webster.com\/dictionary\/equilibrium), and from this it follows that the pressure (that is, [force](https:\/\/www.britannica.com\/science\/force-physics) per unit area) acting perpendicular to all planes in the fluid is the same irrespective of their orientation ([Pascal’s law](https:\/\/www.britannica.com\/science\/Pascals-principle)). For an isotropic fluid in equilibrium there is only one value of the local pressure (*p*) consistent with the stated values for ρ and *T*. These three quantities are linked together by what is called the [equation of state](https:\/\/www.britannica.com\/science\/equation-of-state) for the fluid.\n\nExplore Britannica Premium\\!\n\nSmart, reliable knowledge for professionals, students, and curious minds everywhere.\n\n[SUBSCRIBE](https:\/\/premium.britannica.com\/premium-membership\/?utm_source=premium&utm_medium=inline-cta&utm_campaign=smart-2026)\n\n![Penguin, ship, mountain, atlas](https:\/\/cdn.britannica.com\/marketing\/inline-left.webp)\n\n![shohei ohtani, plants, andy wharhol art](https:\/\/cdn.britannica.com\/marketing\/inline-right.webp)\n\n![Mobile](https:\/\/cdn.britannica.com\/marketing\/inline-mobile.webp?w=400)\n\nFor gases at low pressures the equation of state is simple and well known. It is![Equation.](https:\/\/cdn.britannica.com\/19\/15119-004-4B2AB8EC\/Equation.jpg)where *R* is the [universal gas constant](https:\/\/www.britannica.com\/science\/universal-gas-constant) (8.3 joules per degree Celsius per mole) and *M* is the [molar mass](https:\/\/www.britannica.com\/science\/molar-mass), or an average molar mass if the gas is a mixture; for air, the [appropriate](https:\/\/www.britannica.com\/dictionary\/appropriate) average is about 29 × 10−3 kilogram per mole. For other fluids knowledge of the equation of state is often incomplete. Except under very extreme conditions, however, all one needs to know is how the density [changes](https:\/\/www.britannica.com\/science\/isothermal-change) when the pressure is changed by a small amount, and this is described by the [compressibility](https:\/\/www.britannica.com\/science\/compressibility) of the fluid—either the isothermal compressibility, β*T*, or the adiabatic compressibility, β*S*, according to circumstance. When an element of fluid is compressed, the work done on it tends to heat it up. If the heat has time to drain away to the surroundings and the temperature of the fluid remains essentially unchanged throughout, then β*T* is the relevant quantity. If virtually none of the heat escapes, as is more commonly the case in flow problems because the [thermal conductivity](https:\/\/www.britannica.com\/science\/thermal-conductivity) of most fluids is poor, then the flow is said to be adiabatic, and β*S* is needed instead. (The *S* refers to [entropy](https:\/\/www.merriam-webster.com\/dictionary\/entropy), which remains constant in an [adiabatic process](https:\/\/www.britannica.com\/science\/adiabatic-process) provided that it takes place slowly enough to be treated as “reversible” in the thermodynamic sense.) For gases that obey equation ([118](https:\/\/www.britannica.com\/science\/hydrostatics#ref-15119)), it is evident that *p* and ρ are proportional to one another in an isothermal process, and![Equation.](https:\/\/cdn.britannica.com\/18\/15118-004-794918C3\/Equation.jpg)\n\nIn reversible adiabatic processes for such gases, however, the temperature rises on compression at a rate such that![Equations.](https:\/\/cdn.britannica.com\/17\/15117-004-EC759D42\/Equations.jpg)and![Equation.](https:\/\/cdn.britannica.com\/16\/15116-004-70DDB541\/Equation.jpg)where γ is about 1.4 for air and takes similar values for other common gases. For liquids the ratio between the isothermal and adiabatic compressibilities is much closer to [unity](https:\/\/www.britannica.com\/dictionary\/unity). For liquids, however, both compressibilities are normally much less than *p*−1, and the simplifying assumption that they are zero is often justified.\n\nThe factor γ is not only the ratio between two compressibilities; it is also the ratio between two principal [specific heats](https:\/\/www.britannica.com\/science\/specific-heat). The molar specific heat is the amount of heat required to raise the temperature of one mole through one degree. This is greater if the substance is allowed to expand as it is heated, and therefore to do work, than if its volume is fixed. The principal molar specific heats, *C**P* and *C**V*, refer to heating at constant pressure and constant volume, respectively, and![Equation.](https:\/\/cdn.britannica.com\/15\/15115-004-1FDA2C13\/Equation.jpg)\n\nFor air, *C**P* is about 3.5 *R*.\n\nSolids can be stretched without breaking, and liquids, though not gases, can withstand stretching, too. Thus, if the pressure is steadily reduced in a [specimen](https:\/\/www.britannica.com\/dictionary\/specimen) of very pure water, bubbles will ultimately appear, but they may not do so until the pressure is negative and well below -107 newton per square metre; this is 100 times greater in magnitude than the (positive) pressure exerted by the Earth’s [atmosphere](https:\/\/www.britannica.com\/science\/atmosphere). Water owes its high ideal strength to the fact that rupture involves breaking links of attraction between molecules on either side of the plane on which rupture occurs; work must be done to break these links. However, its strength is drastically reduced by anything that provides a nucleus at which the process known as [cavitation](https:\/\/www.britannica.com\/science\/cavitation) (formation of vapour- or gas-filled cavities) can begin, and a liquid containing suspended dust particles or [dissolved](https:\/\/www.britannica.com\/dictionary\/dissolved) gases is liable to cavitate quite easily.\n\nKey People:\n\n[Sir James Lighthill](https:\/\/www.britannica.com\/biography\/James-Lighthill)\n\n[Sir George Gabriel Stokes, 1st Baronet](https:\/\/www.britannica.com\/biography\/Sir-George-Gabriel-Stokes-1st-Baronet)\n\n[Sir Horace Lamb](https:\/\/www.britannica.com\/biography\/Horace-Lamb)\n\n[Henri-Émile Bazin](https:\/\/www.britannica.com\/biography\/Henri-Emile-Bazin)\n\n[Henri Pitot](https:\/\/www.britannica.com\/biography\/Henri-Pitot)\n\n*(Show more)*\n\nRelated Topics:\n\n[continuum mechanics](https:\/\/www.britannica.com\/science\/continuum-mechanics)\n\n[aerostatics](https:\/\/www.britannica.com\/science\/aerostatics)\n\n[d’Alembert’s paradox](https:\/\/www.britannica.com\/science\/dAlemberts-paradox)\n\n[hydrodynamics](https:\/\/www.britannica.com\/science\/hydrodynamics-physics)\n\n[hydrokinetics](https:\/\/www.britannica.com\/science\/hydrokinetics)\n\n*(Show more)*\n\nOn the Web:\n\n[Physics LibreTexts - Fluid Mechanics](https:\/\/phys.libretexts.org\/Bookshelves\/University_Physics\/Book%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_%28Martin_Neary_Rinaldo_and_Woodman%29\/15%3A_Fluid_Mechanics) (Mar. 26, 2026)\n\n*(Show more)*\n\n[See all related content](https:\/\/www.britannica.com\/facts\/fluid-mechanics)\n\nWork also must be done if a free liquid drop of spherical shape is to be drawn out into a long thin cylinder or deformed in any other way that increases its surface area. Here again work is needed to break intermolecular links. The surface of a liquid behaves, in fact, as if it were an elastic membrane under tension, except that the tension exerted by an elastic membrane increases when the membrane is stretched in a way that the tension exerted by a liquid surface does not. [Surface tension](https:\/\/www.britannica.com\/science\/surface-tension) is what causes liquids to rise up capillary tubes, what supports hanging liquid drops, what limits the formation of ripples on the surface of liquids, and so on.\n\n![Britannica AI Icon](https:\/\/cdn.britannica.com\/mendel-resources\/3-179\/images\/chatbot\/star-ai.svg?v=3.179.9)Britannica AI\n\n*chevron\\_right*\n\nFluid mechanics\n\n*close*\n\n[AI-generated answers](https:\/\/www.britannica.com\/about-britannica-ai) from Britannica articles. AI makes mistakes, so verify using Britannica articles.\n\nLoad Next Page\n\nFeedback\n\nThank you for your feedback\n\nOur editors will review what you’ve submitted and determine whether to revise the article.\n\n*verified*Cite\n\nWhile every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions.\n\nSelect Citation Style\n\nBritannica Editors. \"hydrostatics\". *Encyclopedia Britannica*, 29 Apr. 2009, https:\/\/www.britannica.com\/science\/hydrostatics. Accessed 11 April 2026.\n\nCopy Citation\n\nShare\n\nShare to social media\n\n[Facebook](https:\/\/www.facebook.com\/BRITANNICA\/) [X](https:\/\/x.com\/britannica)\n\nURL\n\n<https:\/\/www.britannica.com\/science\/hydrostatics>\n\nFeedback\n\nThank you for your feedback\n\nOur editors will review what you’ve submitted and determine whether to revise the article.\n\n*print* Print\n\nPlease select which sections you would like to print:\n\n*verified*Cite\n\nWhile every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions.\n\nSelect Citation Style\n\nFaber, Thomas E.. \"fluid mechanics\". *Encyclopedia Britannica*, 26 Mar. 2026, https:\/\/www.britannica.com\/science\/fluid-mechanics. Accessed 11 April 2026.\n\nCopy Citation\n\nShare\n\nShare to social media\n\n[Facebook](https:\/\/www.facebook.com\/BRITANNICA\/) [X](https:\/\/x.com\/britannica)\n\nURL\n\n<https:\/\/www.britannica.com\/science\/fluid-mechanics>\n\nExternal Websites\n\n- [University of Central Florida Pressbooks - University Physics Volume 1 - Fluid Dynamics](https:\/\/pressbooks.online.ucf.edu\/osuniversityphysics\/chapter\/14-5-fluid-dynamics\/)\n- [MIT OpenCourseWare - Introduction to basic principles of fluid mechanics (PDF)](https:\/\/ocw.mit.edu\/courses\/2-016-hydrodynamics-13-012-fall-2005\/b97992061b6ee90cd549e01a678b08e5_2005reading3.pdf)\n- [OpenStax - University Physics Volume 1 - Introduction](https:\/\/openstax.org\/books\/university-physics-volume-1\/pages\/14-introduction)\n- [Physics LibreTexts - Fluid Mechanics](https:\/\/phys.libretexts.org\/Bookshelves\/University_Physics\/Book%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_%28Martin_Neary_Rinaldo_and_Woodman%29\/15%3A_Fluid_Mechanics)\n- [Western University - Department of Physics and Astronomy - Fluid Mechanics](https:\/\/physics.uwo.ca\/~mhoude2\/courses\/PDF%20files\/physics1501\/Ch6-Fluid_Mechanics.pdf)\n- [Wolfram Research - Eric Weisstein's World of Physics - Fluid Mechanics](https:\/\/scienceworld.wolfram.com\/physics\/FluidMechanics.html)","attrs_readable_markdown":"Top Questions\n\n- What is fluid mechanics?\n- What is a fluid in science?\n- What are some important properties of fluids?\n- How is pressure created and measured in fluids?\n- What is buoyancy, and how does it affect objects in water?\n- How is fluid mechanics used in real-life situations like airplanes and weather?\n\n**fluid mechanics**, [science](https:\/\/www.britannica.com\/science\/science) concerned with the [response of fluids](https:\/\/www.britannica.com\/science\/fluid-flow) to forces exerted upon them. It is a branch of classical [physics](https:\/\/www.britannica.com\/science\/physics-science) with applications of great importance in hydraulic and [aeronautical engineering](https:\/\/www.britannica.com\/technology\/aerospace-engineering), [chemical engineering](https:\/\/www.britannica.com\/technology\/chemical-engineering), [meteorology](https:\/\/www.britannica.com\/science\/meteorology), and zoology.\n\nThe most familiar fluid is of course [water](https:\/\/www.britannica.com\/science\/water), and an encyclopaedia of the 19th century probably would have dealt with the subject under the separate headings of [hydrostatics](https:\/\/www.britannica.com\/science\/hydrostatics), the science of water at rest, and hydrodynamics, the science of water in [motion](https:\/\/www.britannica.com\/science\/motion-mechanics). [Archimedes](https:\/\/www.britannica.com\/biography\/Archimedes) founded hydrostatics in about 250 bc when, according to [legend](https:\/\/www.merriam-webster.com\/dictionary\/legend), he leapt out of his bath and ran naked through the streets of Syracuse crying “Eureka!”; it has undergone rather little development since. The foundations of hydrodynamics, on the other hand, were not laid until the 18th century when mathematicians such as [Leonhard Euler](https:\/\/www.britannica.com\/biography\/Leonhard-Euler) and [Daniel Bernoulli](https:\/\/www.britannica.com\/biography\/Daniel-Bernoulli) began to explore the consequences, for a virtually continuous medium like water, of the [dynamic](https:\/\/www.merriam-webster.com\/dictionary\/dynamic) principles that Newton had enunciated for systems composed of discrete particles. Their work was continued in the 19th century by several mathematicians and physicists of the first rank, notably G.G. Stokes and William Thomson. By the end of the century explanations had been found for a host of intriguing phenomena having to do with the flow of water through tubes and orifices, the waves that ships moving through water leave behind them, raindrops on windowpanes, and the like. There was still no proper understanding, however, of problems as fundamental as that of water flowing past a fixed obstacle and exerting a [drag force](https:\/\/www.britannica.com\/science\/drag) upon it; the theory of potential flow, which worked so well in other [contexts](https:\/\/www.merriam-webster.com\/dictionary\/contexts), yielded results that at relatively high flow rates were grossly at variance with experiment. This problem was not properly understood until 1904, when the German physicist [Ludwig Prandtl](https:\/\/www.britannica.com\/biography\/Ludwig-Prandtl) introduced the concept of the [boundary layer](https:\/\/www.britannica.com\/science\/boundary-layer) (see below [Hydrodynamics: Boundary layers and separation](https:\/\/www.britannica.com\/science\/fluid-mechanics\/Waves-on-deep-water#ref77495)). Prandtl’s career continued into the period in which the first manned aircraft were developed. Since that time, the flow of [air](https:\/\/www.britannica.com\/science\/aerodynamics) has been of as much interest to physicists and engineers as the flow of water, and hydrodynamics has, as a consequence, become fluid [dynamics](https:\/\/www.britannica.com\/science\/dynamics-physics). The term fluid mechanics, as used here, embraces both fluid [dynamics](https:\/\/www.merriam-webster.com\/dictionary\/dynamics) and the subject still generally referred to as hydrostatics.\n\nOne other representative of the 20th century who deserves mention here besides Prandtl is [Geoffrey Taylor](https:\/\/www.britannica.com\/biography\/Geoffrey-Ingram-Taylor) of England. Taylor remained a classical physicist while most of his contemporaries were turning their attention to the problems of atomic structure and [quantum mechanics](https:\/\/www.britannica.com\/science\/quantum-mechanics-physics), and he made several unexpected and important discoveries in the [field](https:\/\/www.britannica.com\/science\/field-physics) of fluid mechanics. The richness of fluid mechanics is due in large part to a term in the basic equation of the motion of fluids which is nonlinear—*i.e.,* one that involves the fluid [velocity](https:\/\/www.britannica.com\/science\/velocity) twice over. It is characteristic of systems described by nonlinear equations that under certain conditions they become unstable and begin behaving in ways that seem at first sight to be totally chaotic. In the case of fluids, chaotic behaviour is very common and is called turbulence. Mathematicians have now begun to recognize patterns in [chaos](https:\/\/www.britannica.com\/science\/chaos-theory) that can be analyzed fruitfully, and this development suggests that fluid mechanics will remain a field of active research well into the 21st century. (For a discussion of the concept of [chaos](https:\/\/www.merriam-webster.com\/dictionary\/chaos), see [physical science, principles of](https:\/\/www.britannica.com\/science\/principles-of-physical-science).)\n\nFluid mechanics is a subject with almost endless ramifications, and the account that follows is necessarily incomplete. Some knowledge of the basic properties of fluids will be needed; a survey of the most relevant properties is given in the next section. For further details, see [thermodynamics](https:\/\/www.britannica.com\/science\/thermodynamics) and [liquid](https:\/\/www.britannica.com\/science\/liquid-state-of-matter).\n\n## Basic properties of [fluids](https:\/\/www.britannica.com\/science\/fluid-physics)\nFluids are not strictly continuous media in the way that all the successors of Euler and Bernoulli have assumed, for they are composed of discrete molecules. The molecules, however, are so small and, except in gases at very low pressures, the number of molecules per millilitre is so enormous that they need not be viewed as individual [entities](https:\/\/www.britannica.com\/dictionary\/entities). There are a few liquids, known as liquid crystals, in which the molecules are packed together in such a way as to make the properties of the medium locally anisotropic, but the vast majority of fluids (including air and water) are [isotropic](https:\/\/www.britannica.com\/science\/isotropy). In fluid mechanics, the state of an isotropic fluid may be completely described by defining its mean mass per unit volume, or [density](https:\/\/www.britannica.com\/science\/density) (ρ), its [temperature](https:\/\/www.britannica.com\/science\/temperature) (*T*), and its velocity (*v*) at every point in space, and just what the connection is between these macroscopic properties and the positions and velocities of individual molecules is of no direct relevance.\n\n[![Italian physicist Guglielmo Marconi at work in the wireless room of his yacht Electra, c. 1920.](https:\/\/cdn.britannica.com\/03\/221503-131-D7173869\/Italian-physicist-Guglielmo-Marconi-at-work-on-his-yacht-Electra.jpg) Britannica Quiz All About Physics Quiz](https:\/\/www.britannica.com\/quiz\/all-about-physics-quiz)\n\nA word perhaps is needed about the difference between [gases](https:\/\/www.britannica.com\/science\/gas-state-of-matter) and [liquids](https:\/\/www.britannica.com\/science\/liquid-state-of-matter), though the difference is easier to perceive than to describe. In gases the molecules are sufficiently far apart to move almost independently of one another, and gases tend to expand to fill any volume available to them. In liquids the molecules are more or less in contact, and the short-range attractive forces between them make them cohere; the molecules are moving too fast to settle down into the ordered [arrays](https:\/\/www.britannica.com\/dictionary\/arrays) that are characteristic of solids, but not so fast that they can fly apart. Thus, samples of liquid can exist as drops or as jets with free surfaces, or they can sit in beakers constrained only by gravity, in a way that samples of [gas](https:\/\/www.britannica.com\/science\/gas-state-of-matter) cannot. Such samples may evaporate in time, as molecules one by one pick up enough speed to escape across the free surface and are not replaced. The lifetime of liquid drops and jets, however, is normally long enough for evaporation to be ignored.\n\nThere are two sorts of [stress](https:\/\/www.britannica.com\/science\/stress-physics) that may exist in any solid or fluid medium, and the difference between them may be illustrated by reference to a brick held between two hands. If the holder moves his hands toward each other, he exerts [pressure](https:\/\/www.britannica.com\/science\/pressure) on the brick; if he moves one hand toward his body and the other away from it, then he exerts what is called a [shear stress](https:\/\/www.britannica.com\/science\/shear-stress). A solid substance such as a brick can withstand stresses of both types, but fluids, by definition, yield to shear stresses no matter how small these stresses may be. They do so at a rate determined by the fluid’s [viscosity](https:\/\/www.britannica.com\/science\/viscosity). This property, about which more will be said later, is a measure of the [friction](https:\/\/www.britannica.com\/science\/friction) that arises when [adjacent](https:\/\/www.merriam-webster.com\/dictionary\/adjacent) layers of [fluid](https:\/\/www.britannica.com\/science\/fluid-pressure) slip over one another. It follows that the shear stresses are everywhere zero in a fluid at rest and in [equilibrium](https:\/\/www.merriam-webster.com\/dictionary\/equilibrium), and from this it follows that the pressure (that is, [force](https:\/\/www.britannica.com\/science\/force-physics) per unit area) acting perpendicular to all planes in the fluid is the same irrespective of their orientation ([Pascal’s law](https:\/\/www.britannica.com\/science\/Pascals-principle)). For an isotropic fluid in equilibrium there is only one value of the local pressure (*p*) consistent with the stated values for ρ and *T*. These three quantities are linked together by what is called the [equation of state](https:\/\/www.britannica.com\/science\/equation-of-state) for the fluid.\n\nSmart, reliable knowledge for professionals, students, and curious minds everywhere.\n\n[SUBSCRIBE](https:\/\/premium.britannica.com\/premium-membership\/?utm_source=premium&utm_medium=inline-cta&utm_campaign=smart-2026)\n\n![Penguin, ship, mountain, atlas](https:\/\/cdn.britannica.com\/marketing\/inline-left.webp)\n\n![shohei ohtani, plants, andy wharhol art](https:\/\/cdn.britannica.com\/marketing\/inline-right.webp)\n\n![Mobile](https:\/\/cdn.britannica.com\/marketing\/inline-mobile.webp?w=400)\n\nFor gases at low pressures the equation of state is simple and well known. It is![Equation.](https:\/\/cdn.britannica.com\/19\/15119-004-4B2AB8EC\/Equation.jpg)where *R* is the [universal gas constant](https:\/\/www.britannica.com\/science\/universal-gas-constant) (8.3 joules per degree Celsius per mole) and *M* is the [molar mass](https:\/\/www.britannica.com\/science\/molar-mass), or an average molar mass if the gas is a mixture; for air, the [appropriate](https:\/\/www.britannica.com\/dictionary\/appropriate) average is about 29 × 10−3 kilogram per mole. For other fluids knowledge of the equation of state is often incomplete. Except under very extreme conditions, however, all one needs to know is how the density [changes](https:\/\/www.britannica.com\/science\/isothermal-change) when the pressure is changed by a small amount, and this is described by the [compressibility](https:\/\/www.britannica.com\/science\/compressibility) of the fluid—either the isothermal compressibility, β*T*, or the adiabatic compressibility, β*S*, according to circumstance. When an element of fluid is compressed, the work done on it tends to heat it up. If the heat has time to drain away to the surroundings and the temperature of the fluid remains essentially unchanged throughout, then β*T* is the relevant quantity. If virtually none of the heat escapes, as is more commonly the case in flow problems because the [thermal conductivity](https:\/\/www.britannica.com\/science\/thermal-conductivity) of most fluids is poor, then the flow is said to be adiabatic, and β*S* is needed instead. (The *S* refers to [entropy](https:\/\/www.merriam-webster.com\/dictionary\/entropy), which remains constant in an [adiabatic process](https:\/\/www.britannica.com\/science\/adiabatic-process) provided that it takes place slowly enough to be treated as “reversible” in the thermodynamic sense.) For gases that obey equation ([118](https:\/\/www.britannica.com\/science\/hydrostatics#ref-15119)), it is evident that *p* and ρ are proportional to one another in an isothermal process, and![Equation.](https:\/\/cdn.britannica.com\/18\/15118-004-794918C3\/Equation.jpg)\n\nIn reversible adiabatic processes for such gases, however, the temperature rises on compression at a rate such that![Equations.](https:\/\/cdn.britannica.com\/17\/15117-004-EC759D42\/Equations.jpg)and![Equation.](https:\/\/cdn.britannica.com\/16\/15116-004-70DDB541\/Equation.jpg)where γ is about 1.4 for air and takes similar values for other common gases. For liquids the ratio between the isothermal and adiabatic compressibilities is much closer to [unity](https:\/\/www.britannica.com\/dictionary\/unity). For liquids, however, both compressibilities are normally much less than *p*−1, and the simplifying assumption that they are zero is often justified.\n\nThe factor γ is not only the ratio between two compressibilities; it is also the ratio between two principal [specific heats](https:\/\/www.britannica.com\/science\/specific-heat). The molar specific heat is the amount of heat required to raise the temperature of one mole through one degree. This is greater if the substance is allowed to expand as it is heated, and therefore to do work, than if its volume is fixed. The principal molar specific heats, *C**P* and *C**V*, refer to heating at constant pressure and constant volume, respectively, and![Equation.](https:\/\/cdn.britannica.com\/15\/15115-004-1FDA2C13\/Equation.jpg)\n\nFor air, *C**P* is about 3.5 *R*.\n\nSolids can be stretched without breaking, and liquids, though not gases, can withstand stretching, too. Thus, if the pressure is steadily reduced in a [specimen](https:\/\/www.britannica.com\/dictionary\/specimen) of very pure water, bubbles will ultimately appear, but they may not do so until the pressure is negative and well below -107 newton per square metre; this is 100 times greater in magnitude than the (positive) pressure exerted by the Earth’s [atmosphere](https:\/\/www.britannica.com\/science\/atmosphere). Water owes its high ideal strength to the fact that rupture involves breaking links of attraction between molecules on either side of the plane on which rupture occurs; work must be done to break these links. However, its strength is drastically reduced by anything that provides a nucleus at which the process known as [cavitation](https:\/\/www.britannica.com\/science\/cavitation) (formation of vapour- or gas-filled cavities) can begin, and a liquid containing suspended dust particles or [dissolved](https:\/\/www.britannica.com\/dictionary\/dissolved) gases is liable to cavitate quite easily.\n\nWork also must be done if a free liquid drop of spherical shape is to be drawn out into a long thin cylinder or deformed in any other way that increases its surface area. Here again work is needed to break intermolecular links. The surface of a liquid behaves, in fact, as if it were an elastic membrane under tension, except that the tension exerted by an elastic membrane increases when the membrane is stretched in a way that the tension exerted by a liquid surface does not. [Surface tension](https:\/\/www.britannica.com\/science\/surface-tension) is what causes liquids to rise up capillary tubes, what supports hanging liquid drops, what limits the formation of ripples on the surface of liquids, and so on.","meta_canonical":null}

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Boilerpipe Text	Top Questions What is fluid mechanics? What is a fluid in science? What are some important properties of fluids? How is pressure created and measured in fluids? What is buoyancy, and how does it affect objects in water? How is fluid mechanics used in real-life situations like airplanes and weather? fluid mechanics , science concerned with the response of fluids to forces exerted upon them. It is a branch of classical physics with applications of great importance in hydraulic and aeronautical engineering , chemical engineering , meteorology , and zoology. The most familiar fluid is of course water , and an encyclopaedia of the 19th century probably would have dealt with the subject under the separate headings of hydrostatics , the science of water at rest, and hydrodynamics, the science of water in motion . Archimedes founded hydrostatics in about 250 bc when, according to legend , he leapt out of his bath and ran naked through the streets of Syracuse crying “Eureka!”; it has undergone rather little development since. The foundations of hydrodynamics, on the other hand, were not laid until the 18th century when mathematicians such as Leonhard Euler and Daniel Bernoulli began to explore the consequences, for a virtually continuous medium like water, of the dynamic principles that Newton had enunciated for systems composed of discrete particles. Their work was continued in the 19th century by several mathematicians and physicists of the first rank, notably G.G. Stokes and William Thomson. By the end of the century explanations had been found for a host of intriguing phenomena having to do with the flow of water through tubes and orifices, the waves that ships moving through water leave behind them, raindrops on windowpanes, and the like. There was still no proper understanding, however, of problems as fundamental as that of water flowing past a fixed obstacle and exerting a drag force upon it; the theory of potential flow, which worked so well in other contexts , yielded results that at relatively high flow rates were grossly at variance with experiment. This problem was not properly understood until 1904, when the German physicist Ludwig Prandtl introduced the concept of the boundary layer (see below Hydrodynamics: Boundary layers and separation ). Prandtl’s career continued into the period in which the first manned aircraft were developed. Since that time, the flow of air has been of as much interest to physicists and engineers as the flow of water, and hydrodynamics has, as a consequence, become fluid dynamics . The term fluid mechanics, as used here, embraces both fluid dynamics and the subject still generally referred to as hydrostatics. One other representative of the 20th century who deserves mention here besides Prandtl is Geoffrey Taylor of England. Taylor remained a classical physicist while most of his contemporaries were turning their attention to the problems of atomic structure and quantum mechanics , and he made several unexpected and important discoveries in the field of fluid mechanics. The richness of fluid mechanics is due in large part to a term in the basic equation of the motion of fluids which is nonlinear— i.e., one that involves the fluid velocity twice over. It is characteristic of systems described by nonlinear equations that under certain conditions they become unstable and begin behaving in ways that seem at first sight to be totally chaotic. In the case of fluids, chaotic behaviour is very common and is called turbulence. Mathematicians have now begun to recognize patterns in chaos that can be analyzed fruitfully, and this development suggests that fluid mechanics will remain a field of active research well into the 21st century. (For a discussion of the concept of chaos , see physical science, principles of .) Fluid mechanics is a subject with almost endless ramifications, and the account that follows is necessarily incomplete. Some knowledge of the basic properties of fluids will be needed; a survey of the most relevant properties is given in the next section. For further details, see thermodynamics and liquid . Basic properties of fluids Fluids are not strictly continuous media in the way that all the successors of Euler and Bernoulli have assumed, for they are composed of discrete molecules. The molecules, however, are so small and, except in gases at very low pressures, the number of molecules per millilitre is so enormous that they need not be viewed as individual entities . There are a few liquids, known as liquid crystals, in which the molecules are packed together in such a way as to make the properties of the medium locally anisotropic, but the vast majority of fluids (including air and water) are isotropic . In fluid mechanics, the state of an isotropic fluid may be completely described by defining its mean mass per unit volume, or density (ρ), its temperature ( T ), and its velocity ( v ) at every point in space, and just what the connection is between these macroscopic properties and the positions and velocities of individual molecules is of no direct relevance. Britannica Quiz All About Physics Quiz A word perhaps is needed about the difference between gases and liquids , though the difference is easier to perceive than to describe. In gases the molecules are sufficiently far apart to move almost independently of one another, and gases tend to expand to fill any volume available to them. In liquids the molecules are more or less in contact, and the short-range attractive forces between them make them cohere; the molecules are moving too fast to settle down into the ordered arrays that are characteristic of solids, but not so fast that they can fly apart. Thus, samples of liquid can exist as drops or as jets with free surfaces, or they can sit in beakers constrained only by gravity, in a way that samples of gas cannot. Such samples may evaporate in time, as molecules one by one pick up enough speed to escape across the free surface and are not replaced. The lifetime of liquid drops and jets, however, is normally long enough for evaporation to be ignored. There are two sorts of stress that may exist in any solid or fluid medium, and the difference between them may be illustrated by reference to a brick held between two hands. If the holder moves his hands toward each other, he exerts pressure on the brick; if he moves one hand toward his body and the other away from it, then he exerts what is called a shear stress . A solid substance such as a brick can withstand stresses of both types, but fluids, by definition, yield to shear stresses no matter how small these stresses may be. They do so at a rate determined by the fluid’s viscosity . This property, about which more will be said later, is a measure of the friction that arises when adjacent layers of fluid slip over one another. It follows that the shear stresses are everywhere zero in a fluid at rest and in equilibrium , and from this it follows that the pressure (that is, force per unit area) acting perpendicular to all planes in the fluid is the same irrespective of their orientation ( Pascal’s law ). For an isotropic fluid in equilibrium there is only one value of the local pressure ( p ) consistent with the stated values for ρ and T . These three quantities are linked together by what is called the equation of state for the fluid. Smart, reliable knowledge for professionals, students, and curious minds everywhere. SUBSCRIBE For gases at low pressures the equation of state is simple and well known. It is where R is the universal gas constant (8.3 joules per degree Celsius per mole) and M is the molar mass , or an average molar mass if the gas is a mixture; for air, the appropriate average is about 29 × 10 −3 kilogram per mole. For other fluids knowledge of the equation of state is often incomplete. Except under very extreme conditions, however, all one needs to know is how the density changes when the pressure is changed by a small amount, and this is described by the compressibility of the fluid—either the isothermal compressibility, β T , or the adiabatic compressibility, β S , according to circumstance. When an element of fluid is compressed, the work done on it tends to heat it up. If the heat has time to drain away to the surroundings and the temperature of the fluid remains essentially unchanged throughout, then β T is the relevant quantity. If virtually none of the heat escapes, as is more commonly the case in flow problems because the thermal conductivity of most fluids is poor, then the flow is said to be adiabatic, and β S is needed instead. (The S refers to entropy , which remains constant in an adiabatic process provided that it takes place slowly enough to be treated as “reversible” in the thermodynamic sense.) For gases that obey equation ( 118 ), it is evident that p and ρ are proportional to one another in an isothermal process, and In reversible adiabatic processes for such gases, however, the temperature rises on compression at a rate such that and where γ is about 1.4 for air and takes similar values for other common gases. For liquids the ratio between the isothermal and adiabatic compressibilities is much closer to unity . For liquids, however, both compressibilities are normally much less than p −1 , and the simplifying assumption that they are zero is often justified. The factor γ is not only the ratio between two compressibilities; it is also the ratio between two principal specific heats . The molar specific heat is the amount of heat required to raise the temperature of one mole through one degree. This is greater if the substance is allowed to expand as it is heated, and therefore to do work, than if its volume is fixed. The principal molar specific heats, C P and C V , refer to heating at constant pressure and constant volume, respectively, and For air, C P is about 3.5 R . Solids can be stretched without breaking, and liquids, though not gases, can withstand stretching, too. Thus, if the pressure is steadily reduced in a specimen of very pure water, bubbles will ultimately appear, but they may not do so until the pressure is negative and well below -10 7 newton per square metre; this is 100 times greater in magnitude than the (positive) pressure exerted by the Earth’s atmosphere . Water owes its high ideal strength to the fact that rupture involves breaking links of attraction between molecules on either side of the plane on which rupture occurs; work must be done to break these links. However, its strength is drastically reduced by anything that provides a nucleus at which the process known as cavitation (formation of vapour- or gas-filled cavities) can begin, and a liquid containing suspended dust particles or dissolved gases is liable to cavitate quite easily. Work also must be done if a free liquid drop of spherical shape is to be drawn out into a long thin cylinder or deformed in any other way that increases its surface area. Here again work is needed to break intermolecular links. The surface of a liquid behaves, in fact, as if it were an elastic membrane under tension, except that the tension exerted by an elastic membrane increases when the membrane is stretched in a way that the tension exerted by a liquid surface does not. Surface tension is what causes liquids to rise up capillary tubes, what supports hanging liquid drops, what limits the formation of ripples on the surface of liquids, and so on.
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In applications, the principles of hydrostatics are used for problems relating to pressure in deep water (pressure increases with depth) and high in the [atmosphere](https://www.britannica.com/science/atmosphere) (pressure lessens with altitude). This article was most recently revised and updated by [Robert Curley](https://www.britannica.com/editor/Robert-Curley/6737). ![Britannica AI Icon](https://cdn.britannica.com/mendel-resources/3-179/images/chatbot/star-ai.svg?v=3.179.9)Britannica AI chevron\_right Hydrostatics close [AI-generated answers](https://www.britannica.com/about-britannica-ai) from Britannica articles. AI makes mistakes, so verify using Britannica articles. [fluid mechanics](https://www.britannica.com/science/fluid-mechanics) - [Introduction](https://www.britannica.com/science/fluid-mechanics) - [Basic properties of fluids](https://www.britannica.com/science/fluid-mechanics#ref77478) - [Hydrostatics](https://www.britannica.com/science/fluid-mechanics/Hydrostatics) - [Differential manometers](https://www.britannica.com/science/fluid-mechanics/Hydrostatics#ref77480) - [Archimedes’ principle](https://www.britannica.com/science/fluid-mechanics/Hydrostatics#ref77481) - [Surface tension of liquids](https://www.britannica.com/science/fluid-mechanics/Hydrostatics#ref77482) - [Hydrodynamics](https://www.britannica.com/science/fluid-mechanics/Hydrodynamics) - [Bernoulli’s law](https://www.britannica.com/science/fluid-mechanics/Hydrodynamics#ref77484) - [Waves on shallow water](https://www.britannica.com/science/fluid-mechanics/Waves-on-shallow-water) - [Compressible flow in gases](https://www.britannica.com/science/fluid-mechanics/Waves-on-shallow-water#ref77486) - [Viscosity](https://www.britannica.com/science/fluid-mechanics/Viscosity) - [Stresses in laminar motion](https://www.britannica.com/science/fluid-mechanics/Viscosity#ref77488) - [Bulk viscosity](https://www.britannica.com/science/fluid-mechanics/Viscosity#ref77489) - [Measurement of shear viscosity](https://www.britannica.com/science/fluid-mechanics/Viscosity#ref77490) - [Navier-stokes equation](https://www.britannica.com/science/fluid-mechanics/Navier-stokes-equation) - [Potential flow](https://www.britannica.com/science/fluid-mechanics/Navier-stokes-equation#ref77492) - [Potential flow with circulation: vortex lines](https://www.britannica.com/science/fluid-mechanics/Navier-stokes-equation#ref77493) - [Waves on deep water](https://www.britannica.com/science/fluid-mechanics/Waves-on-deep-water) - [Boundary layers and separation](https://www.britannica.com/science/fluid-mechanics/Waves-on-deep-water#ref77495) - [Drag](https://www.britannica.com/science/fluid-mechanics/Drag) - [Lift](https://www.britannica.com/science/fluid-mechanics/Drag#ref77497) - [Turbulence](https://www.britannica.com/science/fluid-mechanics/Turbulence) - [Convection](https://www.britannica.com/science/fluid-mechanics/Turbulence#ref77499) [References & Edit History](https://www.britannica.com/science/fluid-mechanics/additional-info) [Related Topics](https://www.britannica.com/facts/fluid-mechanics) [Images & Videos](https://www.britannica.com/science/fluid-mechanics/images-videos) [![differential manometer, Torricellian barometer, and siphon](https://cdn.britannica.com/08/2508-004-2696813B/representations-manometer-barometer-Torricellian.jpg)](https://cdn.britannica.com/08/2508-004-2696813B/representations-manometer-barometer-Torricellian.jpg) [![Torricelli, Evangelista](https://cdn.britannica.com/21/195421-004-97DAD799/Evangelista-Torricelli-physicist-Italian-mercury-barometer.jpg)](https://cdn.britannica.com/21/195421-050-FC98CA10/Evangelista-Torricelli-physicist-Italian-mercury-barometer.jpg) [![possible orientations of a square prism in liquid](https://cdn.britannica.com/09/2509-004-50CBDBFD/orientations-prism-liquid-orientation-density.jpg)](https://cdn.britannica.com/09/2509-050-305BFADB/orientations-prism-liquid-orientation-density.jpg) [![The science of surface tension explained](https://cdn.britannica.com/76/151076-138-927F881C/Explanation-surface-tension.jpg?w=400&h=225&c=crop)](https://www.britannica.com/video/Explanation-surface-tension/-159381) [![capillarity](https://cdn.britannica.com/10/2510-004-EF2E6BED/Capillarity.jpg)](https://cdn.britannica.com/10/2510-004-EF2E6BED/Capillarity.jpg) [![formation of a liquid drop](https://cdn.britannica.com/11/2511-004-97E4DF70/Stages-formation-drop.jpg)](https://cdn.britannica.com/11/2511-004-97E4DF70/Stages-formation-drop.jpg) [![venturi tube and pitot tube](https://cdn.britannica.com/12/2512-004-1BB4C5D1/representation-venturi-tube-pitot.jpg)](https://cdn.britannica.com/12/2512-004-1BB4C5D1/representation-venturi-tube-pitot.jpg) [![surface of shallow water](https://cdn.britannica.com/13/2513-004-32814157/Steps-surface-water.jpg)](https://cdn.britannica.com/13/2513-004-32814157/Steps-surface-water.jpg) [![interaction of two solitons](https://cdn.britannica.com/14/2514-004-4A861A32/Interaction-solitons.jpg)](https://cdn.britannica.com/14/2514-004-4A861A32/Interaction-solitons.jpg) [![Mach's construction](https://cdn.britannica.com/15/2515-004-3EDFC935/construction-Mach-U-speed-of-sound-VS.jpg)](https://cdn.britannica.com/15/2515-004-3EDFC935/construction-Mach-U-speed-of-sound-VS.jpg) At a Glance [![default image](https://cdn.britannica.com/mendel-resources/3-179/images/shared/new-thistle.svg?v=3.179.9)](https://www.britannica.com/summary/fluid-mechanics) [fluid mechanics summary](https://www.britannica.com/summary/fluid-mechanics) Quizzes [![Italian-born physicist Dr. Enrico Fermi draws a diagram at a blackboard with mathematical equations. circa 1950.](https://cdn.britannica.com/01/115001-131-7278E518/Enrico-Fermi-Italian-problem-physics-1950.jpg?w=200&h=200&c=crop)](https://www.britannica.com/quiz/physics-and-natural-law) [Physics and Natural Law](https://www.britannica.com/quiz/physics-and-natural-law) [![Italian physicist Guglielmo Marconi at work in the wireless room of his yacht Electra, c. 1920.](https://cdn.britannica.com/03/221503-131-D7173869/Italian-physicist-Guglielmo-Marconi-at-work-on-his-yacht-Electra.jpg?w=200&h=200&c=crop)](https://www.britannica.com/quiz/all-about-physics-quiz) [All About Physics Quiz](https://www.britannica.com/quiz/all-about-physics-quiz) Related Questions - [What led to Archimedes’ discovering his principle?](https://www.britannica.com/question/What-led-to-Archimedes-discovering-his-principle) - [What is Archimedes’ principle?](https://www.britannica.com/question/What-is-Archimedes-principle) - [What is Archimedes’ principle used for?](https://www.britannica.com/question/What-is-Archimedes-principle-used-for) - [What is the formula for buoyant force?](https://www.britannica.com/question/What-is-the-formula-for-buoyant-force) ![Britannica AI Icon](https://cdn.britannica.com/mendel-resources/3-179/images/chatbot/star-ai.svg?v=3.179.9) Ask Anything Quick Summary [Science](https://www.britannica.com/browse/Science) [Physics](https://www.britannica.com/browse/Physics) CITE Share Feedback External Websites # fluid mechanics physics Homework Help Written by [Thomas E. Faber Lecturer in Physics, University of Cambridge, 1959–93; Fellow of Corpus Christi College, Cambridge. Author of Fluid Dynamics for Physicists and others.](https://www.britannica.com/contributor/Thomas-E-Faber/3527) Thomas E. Faber Fact-checked by [Britannica Editors Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree....](https://www.britannica.com/editor/The-Editors-of-Encyclopaedia-Britannica/4419) Britannica Editors Last updated Mar. 26, 2026 •[History](https://www.britannica.com/science/fluid-mechanics/additional-info#history) ![Britannica AI Icon](https://cdn.britannica.com/mendel-resources/3-179/images/chatbot/star-ai.svg?v=3.179.9) Britannica AI Ask Anything Quick Summary Table of Contents Table of Contents Quick Summary Ask Anything Top Questions - What is fluid mechanics? - What is a fluid in science? - What are some important properties of fluids? - How is pressure created and measured in fluids? - What is buoyancy, and how does it affect objects in water? - How is fluid mechanics used in real-life situations like airplanes and weather? Show more Show less fluid mechanics, [science](https://www.britannica.com/science/science) concerned with the [response of fluids](https://www.britannica.com/science/fluid-flow) to forces exerted upon them. It is a branch of classical [physics](https://www.britannica.com/science/physics-science) with applications of great importance in hydraulic and [aeronautical engineering](https://www.britannica.com/technology/aerospace-engineering), [chemical engineering](https://www.britannica.com/technology/chemical-engineering), [meteorology](https://www.britannica.com/science/meteorology), and zoology. The most familiar fluid is of course [water](https://www.britannica.com/science/water), and an encyclopaedia of the 19th century probably would have dealt with the subject under the separate headings of [hydrostatics](https://www.britannica.com/science/hydrostatics), the science of water at rest, and hydrodynamics, the science of water in [motion](https://www.britannica.com/science/motion-mechanics). [Archimedes](https://www.britannica.com/biography/Archimedes) founded hydrostatics in about 250 bc when, according to [legend](https://www.merriam-webster.com/dictionary/legend), he leapt out of his bath and ran naked through the streets of Syracuse crying “Eureka!”; it has undergone rather little development since. The foundations of hydrodynamics, on the other hand, were not laid until the 18th century when mathematicians such as [Leonhard Euler](https://www.britannica.com/biography/Leonhard-Euler) and [Daniel Bernoulli](https://www.britannica.com/biography/Daniel-Bernoulli) began to explore the consequences, for a virtually continuous medium like water, of the [dynamic](https://www.merriam-webster.com/dictionary/dynamic) principles that Newton had enunciated for systems composed of discrete particles. Their work was continued in the 19th century by several mathematicians and physicists of the first rank, notably G.G. Stokes and William Thomson. By the end of the century explanations had been found for a host of intriguing phenomena having to do with the flow of water through tubes and orifices, the waves that ships moving through water leave behind them, raindrops on windowpanes, and the like. There was still no proper understanding, however, of problems as fundamental as that of water flowing past a fixed obstacle and exerting a [drag force](https://www.britannica.com/science/drag) upon it; the theory of potential flow, which worked so well in other [contexts](https://www.merriam-webster.com/dictionary/contexts), yielded results that at relatively high flow rates were grossly at variance with experiment. This problem was not properly understood until 1904, when the German physicist [Ludwig Prandtl](https://www.britannica.com/biography/Ludwig-Prandtl) introduced the concept of the [boundary layer](https://www.britannica.com/science/boundary-layer) (see below [Hydrodynamics: Boundary layers and separation](https://www.britannica.com/science/fluid-mechanics/Waves-on-deep-water#ref77495)). Prandtl’s career continued into the period in which the first manned aircraft were developed. Since that time, the flow of [air](https://www.britannica.com/science/aerodynamics) has been of as much interest to physicists and engineers as the flow of water, and hydrodynamics has, as a consequence, become fluid [dynamics](https://www.britannica.com/science/dynamics-physics). The term fluid mechanics, as used here, embraces both fluid [dynamics](https://www.merriam-webster.com/dictionary/dynamics) and the subject still generally referred to as hydrostatics. One other representative of the 20th century who deserves mention here besides Prandtl is [Geoffrey Taylor](https://www.britannica.com/biography/Geoffrey-Ingram-Taylor) of England. Taylor remained a classical physicist while most of his contemporaries were turning their attention to the problems of atomic structure and [quantum mechanics](https://www.britannica.com/science/quantum-mechanics-physics), and he made several unexpected and important discoveries in the [field](https://www.britannica.com/science/field-physics) of fluid mechanics. The richness of fluid mechanics is due in large part to a term in the basic equation of the motion of fluids which is nonlinear—i.e., one that involves the fluid [velocity](https://www.britannica.com/science/velocity) twice over. It is characteristic of systems described by nonlinear equations that under certain conditions they become unstable and begin behaving in ways that seem at first sight to be totally chaotic. In the case of fluids, chaotic behaviour is very common and is called turbulence. Mathematicians have now begun to recognize patterns in [chaos](https://www.britannica.com/science/chaos-theory) that can be analyzed fruitfully, and this development suggests that fluid mechanics will remain a field of active research well into the 21st century. (For a discussion of the concept of [chaos](https://www.merriam-webster.com/dictionary/chaos), see [physical science, principles of](https://www.britannica.com/science/principles-of-physical-science).) Fluid mechanics is a subject with almost endless ramifications, and the account that follows is necessarily incomplete. Some knowledge of the basic properties of fluids will be needed; a survey of the most relevant properties is given in the next section. For further details, see [thermodynamics](https://www.britannica.com/science/thermodynamics) and [liquid](https://www.britannica.com/science/liquid-state-of-matter). ## Basic properties of [fluids](https://www.britannica.com/science/fluid-physics) Fluids are not strictly continuous media in the way that all the successors of Euler and Bernoulli have assumed, for they are composed of discrete molecules. The molecules, however, are so small and, except in gases at very low pressures, the number of molecules per millilitre is so enormous that they need not be viewed as individual [entities](https://www.britannica.com/dictionary/entities). There are a few liquids, known as liquid crystals, in which the molecules are packed together in such a way as to make the properties of the medium locally anisotropic, but the vast majority of fluids (including air and water) are [isotropic](https://www.britannica.com/science/isotropy). In fluid mechanics, the state of an isotropic fluid may be completely described by defining its mean mass per unit volume, or [density](https://www.britannica.com/science/density) (ρ), its [temperature](https://www.britannica.com/science/temperature) (T), and its velocity (v) at every point in space, and just what the connection is between these macroscopic properties and the positions and velocities of individual molecules is of no direct relevance. [![Italian physicist Guglielmo Marconi at work in the wireless room of his yacht Electra, c. 1920.](https://cdn.britannica.com/03/221503-131-D7173869/Italian-physicist-Guglielmo-Marconi-at-work-on-his-yacht-Electra.jpg) Britannica Quiz All About Physics Quiz](https://www.britannica.com/quiz/all-about-physics-quiz) A word perhaps is needed about the difference between [gases](https://www.britannica.com/science/gas-state-of-matter) and [liquids](https://www.britannica.com/science/liquid-state-of-matter), though the difference is easier to perceive than to describe. In gases the molecules are sufficiently far apart to move almost independently of one another, and gases tend to expand to fill any volume available to them. In liquids the molecules are more or less in contact, and the short-range attractive forces between them make them cohere; the molecules are moving too fast to settle down into the ordered [arrays](https://www.britannica.com/dictionary/arrays) that are characteristic of solids, but not so fast that they can fly apart. Thus, samples of liquid can exist as drops or as jets with free surfaces, or they can sit in beakers constrained only by gravity, in a way that samples of [gas](https://www.britannica.com/science/gas-state-of-matter) cannot. Such samples may evaporate in time, as molecules one by one pick up enough speed to escape across the free surface and are not replaced. The lifetime of liquid drops and jets, however, is normally long enough for evaporation to be ignored. There are two sorts of [stress](https://www.britannica.com/science/stress-physics) that may exist in any solid or fluid medium, and the difference between them may be illustrated by reference to a brick held between two hands. If the holder moves his hands toward each other, he exerts [pressure](https://www.britannica.com/science/pressure) on the brick; if he moves one hand toward his body and the other away from it, then he exerts what is called a [shear stress](https://www.britannica.com/science/shear-stress). A solid substance such as a brick can withstand stresses of both types, but fluids, by definition, yield to shear stresses no matter how small these stresses may be. They do so at a rate determined by the fluid’s [viscosity](https://www.britannica.com/science/viscosity). This property, about which more will be said later, is a measure of the [friction](https://www.britannica.com/science/friction) that arises when [adjacent](https://www.merriam-webster.com/dictionary/adjacent) layers of [fluid](https://www.britannica.com/science/fluid-pressure) slip over one another. It follows that the shear stresses are everywhere zero in a fluid at rest and in [equilibrium](https://www.merriam-webster.com/dictionary/equilibrium), and from this it follows that the pressure (that is, [force](https://www.britannica.com/science/force-physics) per unit area) acting perpendicular to all planes in the fluid is the same irrespective of their orientation ([Pascal’s law](https://www.britannica.com/science/Pascals-principle)). For an isotropic fluid in equilibrium there is only one value of the local pressure (p) consistent with the stated values for ρ and T. These three quantities are linked together by what is called the [equation of state](https://www.britannica.com/science/equation-of-state) for the fluid. Explore Britannica Premium\! Smart, reliable knowledge for professionals, students, and curious minds everywhere. [SUBSCRIBE](https://premium.britannica.com/premium-membership/?utm_source=premium&utm_medium=inline-cta&utm_campaign=smart-2026) ![Penguin, ship, mountain, atlas](https://cdn.britannica.com/marketing/inline-left.webp) ![shohei ohtani, plants, andy wharhol art](https://cdn.britannica.com/marketing/inline-right.webp) ![Mobile](https://cdn.britannica.com/marketing/inline-mobile.webp?w=400) For gases at low pressures the equation of state is simple and well known. It is![Equation.](https://cdn.britannica.com/19/15119-004-4B2AB8EC/Equation.jpg)where R is the [universal gas constant](https://www.britannica.com/science/universal-gas-constant) (8.3 joules per degree Celsius per mole) and M is the [molar mass](https://www.britannica.com/science/molar-mass), or an average molar mass if the gas is a mixture; for air, the [appropriate](https://www.britannica.com/dictionary/appropriate) average is about 29 × 10−3 kilogram per mole. For other fluids knowledge of the equation of state is often incomplete. Except under very extreme conditions, however, all one needs to know is how the density [changes](https://www.britannica.com/science/isothermal-change) when the pressure is changed by a small amount, and this is described by the [compressibility](https://www.britannica.com/science/compressibility) of the fluid—either the isothermal compressibility, βT, or the adiabatic compressibility, βS, according to circumstance. When an element of fluid is compressed, the work done on it tends to heat it up. If the heat has time to drain away to the surroundings and the temperature of the fluid remains essentially unchanged throughout, then βT is the relevant quantity. If virtually none of the heat escapes, as is more commonly the case in flow problems because the [thermal conductivity](https://www.britannica.com/science/thermal-conductivity) of most fluids is poor, then the flow is said to be adiabatic, and βS is needed instead. (The S refers to [entropy](https://www.merriam-webster.com/dictionary/entropy), which remains constant in an [adiabatic process](https://www.britannica.com/science/adiabatic-process) provided that it takes place slowly enough to be treated as “reversible” in the thermodynamic sense.) For gases that obey equation ([118](https://www.britannica.com/science/hydrostatics#ref-15119)), it is evident that p and ρ are proportional to one another in an isothermal process, and![Equation.](https://cdn.britannica.com/18/15118-004-794918C3/Equation.jpg) In reversible adiabatic processes for such gases, however, the temperature rises on compression at a rate such that![Equations.](https://cdn.britannica.com/17/15117-004-EC759D42/Equations.jpg)and![Equation.](https://cdn.britannica.com/16/15116-004-70DDB541/Equation.jpg)where γ is about 1.4 for air and takes similar values for other common gases. For liquids the ratio between the isothermal and adiabatic compressibilities is much closer to [unity](https://www.britannica.com/dictionary/unity). For liquids, however, both compressibilities are normally much less than p−1, and the simplifying assumption that they are zero is often justified. The factor γ is not only the ratio between two compressibilities; it is also the ratio between two principal [specific heats](https://www.britannica.com/science/specific-heat). The molar specific heat is the amount of heat required to raise the temperature of one mole through one degree. This is greater if the substance is allowed to expand as it is heated, and therefore to do work, than if its volume is fixed. The principal molar specific heats, CP and CV, refer to heating at constant pressure and constant volume, respectively, and![Equation.](https://cdn.britannica.com/15/15115-004-1FDA2C13/Equation.jpg) For air, CP is about 3.5 R. Solids can be stretched without breaking, and liquids, though not gases, can withstand stretching, too. Thus, if the pressure is steadily reduced in a [specimen](https://www.britannica.com/dictionary/specimen) of very pure water, bubbles will ultimately appear, but they may not do so until the pressure is negative and well below -107 newton per square metre; this is 100 times greater in magnitude than the (positive) pressure exerted by the Earth’s [atmosphere](https://www.britannica.com/science/atmosphere). Water owes its high ideal strength to the fact that rupture involves breaking links of attraction between molecules on either side of the plane on which rupture occurs; work must be done to break these links. However, its strength is drastically reduced by anything that provides a nucleus at which the process known as [cavitation](https://www.britannica.com/science/cavitation) (formation of vapour- or gas-filled cavities) can begin, and a liquid containing suspended dust particles or [dissolved](https://www.britannica.com/dictionary/dissolved) gases is liable to cavitate quite easily. Key People: [Sir James Lighthill](https://www.britannica.com/biography/James-Lighthill) [Sir George Gabriel Stokes, 1st Baronet](https://www.britannica.com/biography/Sir-George-Gabriel-Stokes-1st-Baronet) [Sir Horace Lamb](https://www.britannica.com/biography/Horace-Lamb) [Henri-Émile Bazin](https://www.britannica.com/biography/Henri-Emile-Bazin) [Henri Pitot](https://www.britannica.com/biography/Henri-Pitot) (Show more) Related Topics: [continuum mechanics](https://www.britannica.com/science/continuum-mechanics) [aerostatics](https://www.britannica.com/science/aerostatics) [d’Alembert’s paradox](https://www.britannica.com/science/dAlemberts-paradox) [hydrodynamics](https://www.britannica.com/science/hydrodynamics-physics) [hydrokinetics](https://www.britannica.com/science/hydrokinetics) (Show more) On the Web: [Physics LibreTexts - Fluid Mechanics](https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_%28Martin_Neary_Rinaldo_and_Woodman%29/15%3A_Fluid_Mechanics) (Mar. 26, 2026) (Show more) [See all related content](https://www.britannica.com/facts/fluid-mechanics) Work also must be done if a free liquid drop of spherical shape is to be drawn out into a long thin cylinder or deformed in any other way that increases its surface area. Here again work is needed to break intermolecular links. The surface of a liquid behaves, in fact, as if it were an elastic membrane under tension, except that the tension exerted by an elastic membrane increases when the membrane is stretched in a way that the tension exerted by a liquid surface does not. [Surface tension](https://www.britannica.com/science/surface-tension) is what causes liquids to rise up capillary tubes, what supports hanging liquid drops, what limits the formation of ripples on the surface of liquids, and so on. ![Britannica AI Icon](https://cdn.britannica.com/mendel-resources/3-179/images/chatbot/star-ai.svg?v=3.179.9)Britannica AI chevron\_right Fluid mechanics close [AI-generated answers](https://www.britannica.com/about-britannica-ai) from Britannica articles. AI makes mistakes, so verify using Britannica articles. Load Next Page Feedback Thank you for your feedback Our editors will review what you’ve submitted and determine whether to revise the article. verifiedCite While every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions. Select Citation Style Britannica Editors. "hydrostatics". Encyclopedia Britannica, 29 Apr. 2009, https://www.britannica.com/science/hydrostatics. Accessed 11 April 2026. Copy Citation Share Share to social media [Facebook](https://www.facebook.com/BRITANNICA/) [X](https://x.com/britannica) URL <https://www.britannica.com/science/hydrostatics> Feedback Thank you for your feedback Our editors will review what you’ve submitted and determine whether to revise the article. print Print Please select which sections you would like to print: verifiedCite While every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions. Select Citation Style Faber, Thomas E.. "fluid mechanics". Encyclopedia Britannica, 26 Mar. 2026, https://www.britannica.com/science/fluid-mechanics. Accessed 11 April 2026. Copy Citation Share Share to social media [Facebook](https://www.facebook.com/BRITANNICA/) [X](https://x.com/britannica) URL <https://www.britannica.com/science/fluid-mechanics> External Websites - [University of Central Florida Pressbooks - University Physics Volume 1 - Fluid Dynamics](https://pressbooks.online.ucf.edu/osuniversityphysics/chapter/14-5-fluid-dynamics/) - [MIT OpenCourseWare - Introduction to basic principles of fluid mechanics (PDF)](https://ocw.mit.edu/courses/2-016-hydrodynamics-13-012-fall-2005/b97992061b6ee90cd549e01a678b08e5_2005reading3.pdf) - [OpenStax - University Physics Volume 1 - Introduction](https://openstax.org/books/university-physics-volume-1/pages/14-introduction) - [Physics LibreTexts - Fluid Mechanics](https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_%28Martin_Neary_Rinaldo_and_Woodman%29/15%3A_Fluid_Mechanics) - [Western University - Department of Physics and Astronomy - Fluid Mechanics](https://physics.uwo.ca/~mhoude2/courses/PDF%20files/physics1501/Ch6-Fluid_Mechanics.pdf) - [Wolfram Research - Eric Weisstein's World of Physics - Fluid Mechanics](https://scienceworld.wolfram.com/physics/FluidMechanics.html)
Readable Markdown	Top Questions - What is fluid mechanics? - What is a fluid in science? - What are some important properties of fluids? - How is pressure created and measured in fluids? - What is buoyancy, and how does it affect objects in water? - How is fluid mechanics used in real-life situations like airplanes and weather? fluid mechanics, [science](https://www.britannica.com/science/science) concerned with the [response of fluids](https://www.britannica.com/science/fluid-flow) to forces exerted upon them. It is a branch of classical [physics](https://www.britannica.com/science/physics-science) with applications of great importance in hydraulic and [aeronautical engineering](https://www.britannica.com/technology/aerospace-engineering), [chemical engineering](https://www.britannica.com/technology/chemical-engineering), [meteorology](https://www.britannica.com/science/meteorology), and zoology. The most familiar fluid is of course [water](https://www.britannica.com/science/water), and an encyclopaedia of the 19th century probably would have dealt with the subject under the separate headings of [hydrostatics](https://www.britannica.com/science/hydrostatics), the science of water at rest, and hydrodynamics, the science of water in [motion](https://www.britannica.com/science/motion-mechanics). [Archimedes](https://www.britannica.com/biography/Archimedes) founded hydrostatics in about 250 bc when, according to [legend](https://www.merriam-webster.com/dictionary/legend), he leapt out of his bath and ran naked through the streets of Syracuse crying “Eureka!”; it has undergone rather little development since. The foundations of hydrodynamics, on the other hand, were not laid until the 18th century when mathematicians such as [Leonhard Euler](https://www.britannica.com/biography/Leonhard-Euler) and [Daniel Bernoulli](https://www.britannica.com/biography/Daniel-Bernoulli) began to explore the consequences, for a virtually continuous medium like water, of the [dynamic](https://www.merriam-webster.com/dictionary/dynamic) principles that Newton had enunciated for systems composed of discrete particles. Their work was continued in the 19th century by several mathematicians and physicists of the first rank, notably G.G. Stokes and William Thomson. By the end of the century explanations had been found for a host of intriguing phenomena having to do with the flow of water through tubes and orifices, the waves that ships moving through water leave behind them, raindrops on windowpanes, and the like. There was still no proper understanding, however, of problems as fundamental as that of water flowing past a fixed obstacle and exerting a [drag force](https://www.britannica.com/science/drag) upon it; the theory of potential flow, which worked so well in other [contexts](https://www.merriam-webster.com/dictionary/contexts), yielded results that at relatively high flow rates were grossly at variance with experiment. This problem was not properly understood until 1904, when the German physicist [Ludwig Prandtl](https://www.britannica.com/biography/Ludwig-Prandtl) introduced the concept of the [boundary layer](https://www.britannica.com/science/boundary-layer) (see below [Hydrodynamics: Boundary layers and separation](https://www.britannica.com/science/fluid-mechanics/Waves-on-deep-water#ref77495)). Prandtl’s career continued into the period in which the first manned aircraft were developed. Since that time, the flow of [air](https://www.britannica.com/science/aerodynamics) has been of as much interest to physicists and engineers as the flow of water, and hydrodynamics has, as a consequence, become fluid [dynamics](https://www.britannica.com/science/dynamics-physics). The term fluid mechanics, as used here, embraces both fluid [dynamics](https://www.merriam-webster.com/dictionary/dynamics) and the subject still generally referred to as hydrostatics. One other representative of the 20th century who deserves mention here besides Prandtl is [Geoffrey Taylor](https://www.britannica.com/biography/Geoffrey-Ingram-Taylor) of England. Taylor remained a classical physicist while most of his contemporaries were turning their attention to the problems of atomic structure and [quantum mechanics](https://www.britannica.com/science/quantum-mechanics-physics), and he made several unexpected and important discoveries in the [field](https://www.britannica.com/science/field-physics) of fluid mechanics. The richness of fluid mechanics is due in large part to a term in the basic equation of the motion of fluids which is nonlinear—i.e., one that involves the fluid [velocity](https://www.britannica.com/science/velocity) twice over. It is characteristic of systems described by nonlinear equations that under certain conditions they become unstable and begin behaving in ways that seem at first sight to be totally chaotic. In the case of fluids, chaotic behaviour is very common and is called turbulence. Mathematicians have now begun to recognize patterns in [chaos](https://www.britannica.com/science/chaos-theory) that can be analyzed fruitfully, and this development suggests that fluid mechanics will remain a field of active research well into the 21st century. (For a discussion of the concept of [chaos](https://www.merriam-webster.com/dictionary/chaos), see [physical science, principles of](https://www.britannica.com/science/principles-of-physical-science).) Fluid mechanics is a subject with almost endless ramifications, and the account that follows is necessarily incomplete. Some knowledge of the basic properties of fluids will be needed; a survey of the most relevant properties is given in the next section. For further details, see [thermodynamics](https://www.britannica.com/science/thermodynamics) and [liquid](https://www.britannica.com/science/liquid-state-of-matter). ## Basic properties of [fluids](https://www.britannica.com/science/fluid-physics) Fluids are not strictly continuous media in the way that all the successors of Euler and Bernoulli have assumed, for they are composed of discrete molecules. The molecules, however, are so small and, except in gases at very low pressures, the number of molecules per millilitre is so enormous that they need not be viewed as individual [entities](https://www.britannica.com/dictionary/entities). There are a few liquids, known as liquid crystals, in which the molecules are packed together in such a way as to make the properties of the medium locally anisotropic, but the vast majority of fluids (including air and water) are [isotropic](https://www.britannica.com/science/isotropy). In fluid mechanics, the state of an isotropic fluid may be completely described by defining its mean mass per unit volume, or [density](https://www.britannica.com/science/density) (ρ), its [temperature](https://www.britannica.com/science/temperature) (T), and its velocity (v) at every point in space, and just what the connection is between these macroscopic properties and the positions and velocities of individual molecules is of no direct relevance. [![Italian physicist Guglielmo Marconi at work in the wireless room of his yacht Electra, c. 1920.](https://cdn.britannica.com/03/221503-131-D7173869/Italian-physicist-Guglielmo-Marconi-at-work-on-his-yacht-Electra.jpg) Britannica Quiz All About Physics Quiz](https://www.britannica.com/quiz/all-about-physics-quiz) A word perhaps is needed about the difference between [gases](https://www.britannica.com/science/gas-state-of-matter) and [liquids](https://www.britannica.com/science/liquid-state-of-matter), though the difference is easier to perceive than to describe. In gases the molecules are sufficiently far apart to move almost independently of one another, and gases tend to expand to fill any volume available to them. In liquids the molecules are more or less in contact, and the short-range attractive forces between them make them cohere; the molecules are moving too fast to settle down into the ordered [arrays](https://www.britannica.com/dictionary/arrays) that are characteristic of solids, but not so fast that they can fly apart. Thus, samples of liquid can exist as drops or as jets with free surfaces, or they can sit in beakers constrained only by gravity, in a way that samples of [gas](https://www.britannica.com/science/gas-state-of-matter) cannot. Such samples may evaporate in time, as molecules one by one pick up enough speed to escape across the free surface and are not replaced. The lifetime of liquid drops and jets, however, is normally long enough for evaporation to be ignored. There are two sorts of [stress](https://www.britannica.com/science/stress-physics) that may exist in any solid or fluid medium, and the difference between them may be illustrated by reference to a brick held between two hands. If the holder moves his hands toward each other, he exerts [pressure](https://www.britannica.com/science/pressure) on the brick; if he moves one hand toward his body and the other away from it, then he exerts what is called a [shear stress](https://www.britannica.com/science/shear-stress). A solid substance such as a brick can withstand stresses of both types, but fluids, by definition, yield to shear stresses no matter how small these stresses may be. They do so at a rate determined by the fluid’s [viscosity](https://www.britannica.com/science/viscosity). This property, about which more will be said later, is a measure of the [friction](https://www.britannica.com/science/friction) that arises when [adjacent](https://www.merriam-webster.com/dictionary/adjacent) layers of [fluid](https://www.britannica.com/science/fluid-pressure) slip over one another. It follows that the shear stresses are everywhere zero in a fluid at rest and in [equilibrium](https://www.merriam-webster.com/dictionary/equilibrium), and from this it follows that the pressure (that is, [force](https://www.britannica.com/science/force-physics) per unit area) acting perpendicular to all planes in the fluid is the same irrespective of their orientation ([Pascal’s law](https://www.britannica.com/science/Pascals-principle)). For an isotropic fluid in equilibrium there is only one value of the local pressure (p) consistent with the stated values for ρ and T. These three quantities are linked together by what is called the [equation of state](https://www.britannica.com/science/equation-of-state) for the fluid. Smart, reliable knowledge for professionals, students, and curious minds everywhere. [SUBSCRIBE](https://premium.britannica.com/premium-membership/?utm_source=premium&utm_medium=inline-cta&utm_campaign=smart-2026) ![Penguin, ship, mountain, atlas](https://cdn.britannica.com/marketing/inline-left.webp) ![shohei ohtani, plants, andy wharhol art](https://cdn.britannica.com/marketing/inline-right.webp) ![Mobile](https://cdn.britannica.com/marketing/inline-mobile.webp?w=400) For gases at low pressures the equation of state is simple and well known. It is![Equation.](https://cdn.britannica.com/19/15119-004-4B2AB8EC/Equation.jpg)where R is the [universal gas constant](https://www.britannica.com/science/universal-gas-constant) (8.3 joules per degree Celsius per mole) and M is the [molar mass](https://www.britannica.com/science/molar-mass), or an average molar mass if the gas is a mixture; for air, the [appropriate](https://www.britannica.com/dictionary/appropriate) average is about 29 × 10−3 kilogram per mole. For other fluids knowledge of the equation of state is often incomplete. Except under very extreme conditions, however, all one needs to know is how the density [changes](https://www.britannica.com/science/isothermal-change) when the pressure is changed by a small amount, and this is described by the [compressibility](https://www.britannica.com/science/compressibility) of the fluid—either the isothermal compressibility, βT, or the adiabatic compressibility, βS, according to circumstance. When an element of fluid is compressed, the work done on it tends to heat it up. If the heat has time to drain away to the surroundings and the temperature of the fluid remains essentially unchanged throughout, then βT is the relevant quantity. If virtually none of the heat escapes, as is more commonly the case in flow problems because the [thermal conductivity](https://www.britannica.com/science/thermal-conductivity) of most fluids is poor, then the flow is said to be adiabatic, and βS is needed instead. (The S refers to [entropy](https://www.merriam-webster.com/dictionary/entropy), which remains constant in an [adiabatic process](https://www.britannica.com/science/adiabatic-process) provided that it takes place slowly enough to be treated as “reversible” in the thermodynamic sense.) For gases that obey equation ([118](https://www.britannica.com/science/hydrostatics#ref-15119)), it is evident that p and ρ are proportional to one another in an isothermal process, and![Equation.](https://cdn.britannica.com/18/15118-004-794918C3/Equation.jpg) In reversible adiabatic processes for such gases, however, the temperature rises on compression at a rate such that![Equations.](https://cdn.britannica.com/17/15117-004-EC759D42/Equations.jpg)and![Equation.](https://cdn.britannica.com/16/15116-004-70DDB541/Equation.jpg)where γ is about 1.4 for air and takes similar values for other common gases. For liquids the ratio between the isothermal and adiabatic compressibilities is much closer to [unity](https://www.britannica.com/dictionary/unity). For liquids, however, both compressibilities are normally much less than p−1, and the simplifying assumption that they are zero is often justified. The factor γ is not only the ratio between two compressibilities; it is also the ratio between two principal [specific heats](https://www.britannica.com/science/specific-heat). The molar specific heat is the amount of heat required to raise the temperature of one mole through one degree. This is greater if the substance is allowed to expand as it is heated, and therefore to do work, than if its volume is fixed. The principal molar specific heats, CP and CV, refer to heating at constant pressure and constant volume, respectively, and![Equation.](https://cdn.britannica.com/15/15115-004-1FDA2C13/Equation.jpg) For air, CP is about 3.5 R. Solids can be stretched without breaking, and liquids, though not gases, can withstand stretching, too. Thus, if the pressure is steadily reduced in a [specimen](https://www.britannica.com/dictionary/specimen) of very pure water, bubbles will ultimately appear, but they may not do so until the pressure is negative and well below -107 newton per square metre; this is 100 times greater in magnitude than the (positive) pressure exerted by the Earth’s [atmosphere](https://www.britannica.com/science/atmosphere). Water owes its high ideal strength to the fact that rupture involves breaking links of attraction between molecules on either side of the plane on which rupture occurs; work must be done to break these links. However, its strength is drastically reduced by anything that provides a nucleus at which the process known as [cavitation](https://www.britannica.com/science/cavitation) (formation of vapour- or gas-filled cavities) can begin, and a liquid containing suspended dust particles or [dissolved](https://www.britannica.com/dictionary/dissolved) gases is liable to cavitate quite easily. Work also must be done if a free liquid drop of spherical shape is to be drawn out into a long thin cylinder or deformed in any other way that increases its surface area. Here again work is needed to break intermolecular links. The surface of a liquid behaves, in fact, as if it were an elastic membrane under tension, except that the tension exerted by an elastic membrane increases when the membrane is stretched in a way that the tension exerted by a liquid surface does not. [Surface tension](https://www.britannica.com/science/surface-tension) is what causes liquids to rise up capillary tubes, what supports hanging liquid drops, what limits the formation of ripples on the surface of liquids, and so on.
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