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| {{Infobox Physical quantity
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| | name = Viscosity
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| | image = [[File:Viscosity.gif]]
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| | caption = A simulation of substances with different viscosities. The substance above has lower viscosity than the substance below
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| | unit = [[Pascal (unit)|Pa]]·[[Second|s]] = [[kilogram|kg]]/([[second|s]]·[[metre|m]])
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| | symbols = [[Eta (letter)|η]], [[Mu (letter)|μ]]
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| | derivations = μ = [[Shear modulus|G]]·[[time|t]]
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| }}
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| {{pp-move-indef}}
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| {{Continuum mechanics|cTopic=[[Fluid mechanics]]}}
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| The '''viscosity''' of a [[fluid]] is a measure of its [[drag (physics)|resistance]] to gradual deformation by [[shear stress]] or [[tensile stress]]. For liquids, it corresponds to the informal notion of "thickness". For example, [[honey]] has a higher viscosity than [[water]].<ref>
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| {{cite book
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| | author = Symon, Keith
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| | title = Mechanics
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| | edition = Third
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| | publisher = Addison-Wesley
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| | year = 1971
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| | isbn = 0-201-07392-7
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| }}
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| </ref>
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| Viscosity is due to the [[friction]] between neighboring particles in a fluid that are moving at different [[velocity|velocities]]. When the fluid is forced through a tube, the fluid generally moves faster near the axis and very slowly near the walls; therefore, some [[stress (physics)|stress]] (such as a [[pressure]] difference between the two ends of the tube) is needed to overcome the friction between layers and keep the fluid moving. For the same velocity pattern, the stress required is proportional to the fluid's viscosity. A liquid's viscosity depends on the size and shape of its particles and the attractions between the particles.{{citation needed|date=April 2013}}
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| A fluid that has no resistance to shear stress is known as an '''ideal fluid''' or '''inviscid fluid'''. Zero viscosity is observed only at [[cryogenics|very low temperatures]], in [[superfluid]]s. Otherwise all fluids have positive viscosity. If the viscosity is very high, for instance in [[pitch (resin)|pitch]], the fluid will appear to be a solid in the short term. A liquid whose viscosity is less than that of water is sometimes known as a '''mobile''' liquid, while a substance with a viscosity substantially greater than water is called a '''viscous''' liquid.
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| ==Etymology==
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| The word "viscosity" is derived from the [[Latin]] "{{lang|la|viscum}}", meaning "anything sticky, birdlime made from mistletoe, [[mistletoe]]". A viscous glue called [[birdlime]] was made from mistletoe berries and was used for lime-twigs to catch birds.<ref>{{cite web|url=http://www.etymonline.com/index.php?term=viscous |title=The Online Etymology Dictionary |publisher=Etymonline.com |accessdate=2010-09-14}}</ref>
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| ==Definition==
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| === Dynamic (shear) viscosity ===
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| [[File:Laminar shear.svg|thumb|right|320px|Laminar shear of fluid between two plates. Friction between the fluid and the moving boundaries causes the fluid to shear. The force required for this action is a measure of the fluid's viscosity.]]
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| [[File:Laminar shear flow.svg|thumb|right|320px|In a general parallel flow (such as could occur in a straight pipe), the shear stress is proportional to the gradient of the velocity]]
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| The '''dynamic (shear) viscosity''' of a fluid expresses its resistance to shearing flows, where adjacent layers move parallel to each other with different speeds. It can be defined through the idealized situation known as a [[Couette flow]], where a layer of fluid is trapped between two horizontal plates, one fixed and one moving horizontally at constant speed <math>u</math>. (The plates are assumed to be very large, so that one need not consider what happens near their edges.)
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| If the speed of the top plate is small enough, the fluid particles will move [[parallel (geometry)|parallel]] to it, and their speed will vary [[linear function|linearly]] from zero at the bottom to <math>u</math> at the top. Each layer of fluid will move faster than the one just below it, and friction between them will give rise to a [[force (physics)|force]] resisting their relative motion. In particular, the fluid will apply on the top plate a force in the direction opposite to its motion, and an equal but opposite one to the bottom plate. An external force is therefore required in order to keep the top plate moving at constant speed.
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| The magnitude <math>F</math> of this force is found to be proportional to the speed <math>u</math> and the area <math>A</math> of each plate, and inversely proportional to their separation <math>y</math>. That is,
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| :<math> F=\mu A \frac{u}{y}</math>
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| The proportionality factor ''μ'' in this formula is the viscosity (specifically, the '''dynamic viscosity''') of the fluid.
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| The ratio <math>u/y</math> is called the ''rate of shear deformation'' or ''[[shear velocity]]'', and is the [[derivative]] of the fluid speed in the direction [[perpendicular]] to the plates. [[Isaac Newton]] expressed the viscous forces by the [[differential equation]]
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| :<math>\tau=\mu \frac{\partial u}{\partial y}</math>
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| where <math>\tau = F/A</math> and <math>{\partial u}/{\partial y}</math> is the local shear velocity. This formula assumes that the flow is moving along parallel lines and the <math>y</math> axis, perpendicular to the flow, points in the direction of maximum shear velocity. This equation can be used where the velocity does not vary linearly with <math>y</math>, such as in fluid flowing through a pipe. | |
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| Use of the [[mu (letter)|Greek letter mu]] (''μ'') for the dynamic stress viscosity is common among mechanical and chemical engineers, as well as physicists.<ref>Streeter, Victor Lyle; Wylie, E. Benjamin and Bedford, Keith W. (1998) ''Fluid Mechanics'', McGraw-Hill, ISBN 0-07-062537-9</ref><ref>Holman, J. P. (2002) ''Heat Transfer'', McGraw-Hill, ISBN 0-07-122621-4</ref><ref>Incropera, Frank P. and DeWitt, David P. (2007) ''Fundamentals of Heat and Mass Transfer'', Wiley, ISBN 0-471-45728-0</ref> However, the [[eta (letter)|Greek letter eta]] (''η'') is also used by chemists, physicists, and the [[IUPAC]].<ref>{{cite book|chapter=dynamic viscosity, η|doi=10.1351/goldbook|title=IUPAC Compendium of Chemical Terminology|year=1997|publisher=Blackwell Scientific Publications|place= Oxford |editor1-last=Nič|editor1-first=Miloslav|editor2-last=Jirát|editor2-first=Jiří|editor3-last=Košata|editor3-first=Bedřich|editor4-last=Jenkins|editor4-first=Aubrey|isbn=0-9678550-9-8}}</ref>
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| === Kinematic viscosity ===
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| The '''kinematic viscosity''' is the ratio of the dynamic viscosity ''μ'' to the [[density]] of the fluid ''ρ''. It is usually denoted by the [[nu (letter)|Greek letter nu]] (''ν'').
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| : <math>\nu = \frac{\mu}{\rho}</math>
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| It is a convenient concept when analyzing the [[Reynolds number]], that expresses the ratio of the [[inertia]]l forces to the viscous forces:
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| : <math>Re = \frac{\rho u L}{\mu} = \frac{uL}{\nu} \;,</math>
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| where <math>L</math> is a typical length scale in the system.
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| === Bulk viscosity ===
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| When a [[compressible fluid]] is compressed or expanded evenly, without shear, it may still exhibit
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| a form of internal friction that resists its flow. These forces are related to the rate of compression or expansion by a factor ''σ'', called the '''[[volume viscosity]]''', '''bulk viscosity''' or '''second viscosity'''.
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| The bulk viscosity is important only when the fluid is being rapidly compressed or expanded, such as in [[sound]] and [[shock wave]]s. Bulk viscosity explains the loss of energy in those waves, as described by [[Stokes' law (sound attenuation)|Stokes' law of sound attenuation]].
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| ===Viscosity tensor===
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| {{Main|Viscous stress tensor}}
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| In general, the [[stress (mechanics)|stresses]] within a flow can be attributed partly to the [[deformation (mechanics)|deformation]] of the material from some rest state ([[Elasticity (physics)|elastic]] stress), and partly to the [[strain rate|rate of change of the deformation]] over time (viscous stress). In a fluid, by definition, the elastic stress includes only the [[hydrostatic pressure]].
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| In very general terms, the fluid's viscosity is the relation between the strain rate and the viscous stress. In the [[Newtonian fluid]] model, the relationship is by definition a linear map, described by a viscosity tensor that, multiplied by the [[strain rate tensor]] (which is the [[gradient]] of the flow's velocity), gives the viscous stress tensor.
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| The viscosity tensor has nine independent [[degrees of freedom]] in general. For [[isotropic]] Newtonian fluids, these can be reduced to two independent parameters. The most usual decomposition yields the stress viscosity ''μ'' and the bulk viscosity ''σ''.
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| ==Newtonian and non-Newtonian fluids==
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| [[File:Viscous regimes chart.png|thumb|right|320px|Viscosity, the slope of each line, varies among materials]]
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| Newton's law of viscosity is a [[constitutive equation]] (like [[Hooke's law]], [[Fick's law]], [[Ohm's law]]): it is not a fundamental law of nature but an approximation that holds in some materials and fails in others.
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| A fluid that behaves according to Newton's law, with a viscosity ''μ'' that is independent of the stress, is said to be [[Newtonian fluid|Newtonian]]. [[Gas]]es, [[water]] and many common liquids can be considered Newtonian in ordinary conditions and contexts. There are many [[non-Newtonian fluid]]s that significantly deviate from that law in some way or other. For example:
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| *[[Shear thickening]] liquids, whose viscosity increases with the rate of shear stress.
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| *[[Shear thinning]] liquids, whose viscosity decreases with the rate of shear stress.
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| *[[Thixotropic]] liquids, that become less viscous over time when shaken, agitated, or otherwise stressed.
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| *[[Rheopectic]] liquids, that become more viscous over time when shaken, agitated, or otherwise stressed.
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| *[[Bingham plastic]]s that behave as a solid at low stresses but flows as a viscous fluid at high stresses.
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| Shear thinning liquids are very commonly, but misleadingly, described as thixotropic.
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| Even for a Newtonian fluid, the viscosity usually depends on its composition and temperature. For gases and other [[compressible fluid]]s, it depends on temperature and varies very slowly with pressure.
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| The viscosity of some fluids may depend on other factors. A [[magnetorheological fluid]], for example, becomes thicker when subjected to a [[magnetic field]], possibly to the point of behaving like a solid.
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| ==Viscosity in solids==
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| The viscous forces that arise during fluid flow must not be confused with the [[Elasticity (physics)|elastic]] forces that arise in a solid in response to shear, compression or extension stresses. While in the latter the stress is proportional to the ''amount'' of shear deformation, in a fluid it is proportional to the ''rate'' of deformation over time. (For this reason, [[James Clerk Maxwell|Maxwell]] used the term '''fugitive elasticity''' for fluid viscosity.)
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| However, many liquids (including water) will briefly react like elastic solids when subjected to sudden stress. Conversely, many "solids" (even [[granite]]) will flow like liquids, albeit very slowly, even under arbitrarily small stress.<ref>{{cite journal
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| |last = Kumagai
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| |first = Naoichi
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| |coauthors = Sadao Sasajima, Hidebumi Ito
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| |title = Long-term Creep of Rocks: Results with Large Specimens Obtained in about 20 Years and Those with Small Specimens in about 3 Years
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| |journal = Journal of the Society of Materials Science (Japan)
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| |volume = 27
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| |issue = 293
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| |pages = 157–161
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| |publisher = Japan Energy Society
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| |url = http://translate.google.com/translate?hl=en&sl=ja&u=http://ci.nii.ac.jp/naid/110002299397/&sa=X&oi=translate&resnum=4&ct=result&prev=/search%3Fq%3DIto%2BHidebumi%26hl%3Den
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| |date = 15 February 1978
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| |accessdate = 2008-06-16}}</ref> Such materials are therefore best described as possessing both elasticity (reaction to deformation) and viscosity (reaction to rate of deformation); that is, being [[viscoelasticity|viscoelastic]].
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| Indeed, some authors have claimed that [[amorphous solid]]s, such as [[glass]] and many [[polymers]], are actually liquids with a very high viscosity (e.g.~greater than 10<sup>12</sup> Pa·s).
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| <!--<ref>[http://web.umr.edu/~brow/PDF_viscosity.pdf The Properties of Glass ], page 6, retrieved on August 1, 2007</ref>--><ref name=r1>{{cite web|url=http://hypertextbook.com/physics/matter/viscosity/|work=The Physics Hypertextbook| last=Elert|first=Glenn|title=Viscosity}}</ref> However, other authors dispute this hypothesis, claiming instead that there is some threshold for the stress, below which most solids will not flow at all,<ref>{{cite web|last = Gibbs
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| |first = Philip
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| |title = Is Glass a Liquid or a Solid?
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| |url = http://math.ucr.edu/home/baez/physics/General/Glass/glass.html
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| |accessdate = 2007-07-31}}</ref> and that alleged instances of glass flow in window panes of old buildings are due to the crude manufacturing process of older eras rather than to the viscosity of glass.<ref>{{cite journal|doi=10.1021/ed066p994|url=http://dwb.unl.edu/Teacher/NSF/C01/C01Links/www.ualberta.ca/~bderksen/windowpane.html|title=Antique windowpanes and the flow of supercooled liquids|year=1989|last1=Plumb|first1=Robert C.|journal=Journal of Chemical Education|volume=66|issue=12|pages=994}}</ref>
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| Viscoelastic solids may exhibit both stress viscosity and bulk viscosity. The '''[[extensional viscosity]]''' is a [[linear combination]] of the shear and bulk viscosities that describes the reaction of a solid elastic material to elongation. It is widely used for characterizing polymers.
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| In [[geology]], earth materials that exhibit viscous deformation at least three times greater than their elastic deformation are sometimes called [[rheid]]s.<ref>{{cite journal|doi=10.1016/0022-3093(88)90086-5|title=Viscoelasticity in silica gel|year=1988|last1=Scherer|first1=George W.|last2=Pardenek|first2=Sandra A.|last3=Swiatek|first3=Rose M.|journal=Journal of Non-Crystalline Solids|volume=107|pages=14}}</ref>
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| ==Viscosity measurement==
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| {{Main|Viscometer}}
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| Viscosity is measured with various types of [[viscometer]]s and [[rheometer]]s. A rheometer is used for those fluids that cannot be defined by a single value of viscosity and therefore require more parameters to be set and measured than is the case for a viscometer. Close temperature control of the fluid is essential to acquire accurate measurements, particularly in materials like lubricants, whose viscosity can double with a change of only 5 °C.
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| For some fluids, viscosity is a constant over a wide range of shear rates ([[Newtonian fluids]]). The fluids without a constant viscosity ([[non-Newtonian fluid]]s) cannot be described by a single number. Non-Newtonian fluids exhibit a variety of different correlations between shear stress and shear rate.
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| One of the most common instruments for measuring kinematic viscosity is the glass capillary viscometer.
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| In [[coating]] industries, viscosity may be measured with a cup in which the [[efflux time]] is measured. There are several sorts of cup- e.g. [[Zahn cup]], [[Ford viscosity cup]]- with usage of each type varying mainly according to the industry. The efflux time can also be converted to kinematic viscosities (centistokes, cSt) through the conversion equations.<ref>[http://www.byk.com/fileadmin/BYK/downloads/support-downloads/instruments/theory/physical-properties/en/Intro_Viscosity.pdf Viscosity]. BYK-Gardner GmbH</ref>
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| Also used in coatings, a Stormer viscometer uses load-based rotation in order to determine viscosity. The viscosity is reported in Krebs units (KU), which are unique to Stormer viscometers.
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| Vibrating viscometers can also be used to measure viscosity. These models such as the ''Dynatrol'' use vibration rather than rotation to measure viscosity.
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| ''Extensional viscosity'' can be measured with various [[rheometer]]s that apply [[extensional stress]].
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| [[Volume viscosity]] can be measured with an [[acoustic rheometer]].
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| [[Apparent viscosity]] is a calculation derived from tests performed on [[drilling fluid]] used in oil or gas well development. These calculations and tests help engineers develop and maintain the properties of the drilling fluid to the specifications required.
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| ==Units==
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| ===Dynamic viscosity===
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| The [[SI]] [[physical unit]] of dynamic viscosity is the [[pascal (unit)|pascal]]-[[second]] (Pa·s), (equivalent to (N·s)/m<sup>2</sup>, or kg/(m·s)). If a [[fluid]] with a viscosity of one Pa·s is placed between two plates, and one plate is pushed sideways with a [[shear stress]] of one [[pascal (unit)|pascal]], it moves a distance equal to the thickness of the layer between the plates in one [[second]]. Water at 20 °C has a viscosity of 0.001002 Pa·s, while a typical motor oil could have a viscosity of about 0.250 Pa·s.<ref>{{cite book|author=Serway, Raymond A. | title=Physics for Scientists & Engineers|edition=4th| publisher=Saunders College Publishing| year=1996|isbn=0-03-005932-1}}</ref>
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| The [[cgs]] [[physical unit]] for dynamic viscosity is the ''[[poise]]''<ref>{{cite web|url=http://www.iupac.org/goldbook/P04705.pdf#search=%22poise%20iupac%22 |title=IUPAC definition of the Poise |accessdate=2010-09-14}}</ref> (P), named after [[Jean Léonard Marie Poiseuille]]. It is more commonly expressed, particularly in [[ASTM]] standards, as ''centipoise'' (cP). Water at 20 °C has a viscosity of 1.0020 cP.
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| :1 P = 0.1 Pa·s,
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| :1 cP = 1 mPa·s = 0.001 Pa·s = 0.001 N.s/m^2.
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| ===Kinematic viscosity===
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| The SI unit of kinematic viscosity is m<sup>2</sup>/s.
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| The cgs physical unit for kinematic viscosity is the ''stokes'' (St), named after [[George Gabriel Stokes]]. It is sometimes expressed in terms of ''centistokes'' (cSt). In U.S. usage, ''stoke'' is sometimes used as the singular form.
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| :1 St = 1 cm<sup>2</sup>·s<sup>−1</sup> = 10<sup>−4</sup> m<sup>2</sup>·s<sup>−1</sup>.
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| :1 cSt = 1 mm<sup>2</sup>·s<sup>−1</sup> = 10<sup>−6</sup>m<sup>2</sup>·s<sup>−1</sup>.
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| Water at 20 °C has a kinematic viscosity of about 1 cSt.
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| The kinematic viscosity is sometimes referred to as '''diffusivity of momentum''', because it is analogous to [[thermal diffusivity|diffusivity of heat]] and [[diffusion coefficient|diffusivity of mass]]. It is therefore used in [[dimensionless number]]s which compare the ratio of the diffusivities.
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| ===Fluidity===
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| The [[Multiplicative inverse|reciprocal]] of viscosity is ''fluidity'', usually symbolized by ''φ'' = 1 / ''μ'' or ''F'' = 1 / ''μ'', depending on the convention used, measured in ''reciprocal poise'' ([[centimetre|cm]]·[[second|s]]·[[gram|g]]<sup>−1</sup>), sometimes called the ''rhe''. ''Fluidity'' is seldom used in [[engineering]] practice.
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| The concept of fluidity can be used to determine the viscosity of an [[ideal solution]]. For two components <math>a</math> and <math>b</math>, the fluidity when ''a'' and ''b'' are mixed is
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| :<math>F \approx \chi_a F_a + \chi_b F_b,</math>,
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| which is only slightly simpler than the equivalent equation in terms of viscosity:
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| :<math>\mu \approx \frac{1}{\chi_a /\mu_a + \chi_b/\mu_b},</math>
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| where ''χ<sub>a</sub>'' and ''χ<sub>b</sub>'' is the mole fraction of component ''a'' and ''b'' respectively, and ''μ<sub>a</sub>'' and ''μ<sub>b</sub>'' are the components' pure viscosities.
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| ===Non-standard units===
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| The [[Reyn]] is a British unit of dynamic viscosity.
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| [[Viscosity index]] is a measure for the change of [[kinematic viscosity]] with temperature. It is used to characterise lubricating oil in the automotive industry.
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| At one time the petroleum industry relied on measuring [[kinematic viscosity]] by means of the Saybolt viscometer, and expressing kinematic viscosity in units of ''[[Saybolt Universal Second]]s'' (SUS).<ref>ASTM D 2161 (2005) "Standard Practice for Conversion of Kinematic Viscosity to Saybolt Universal Viscosity or to Saybolt Furol Viscosity", p. 1</ref> Other abbreviations such as SSU (''Saybolt Seconds Universal'') or SUV (''Saybolt Universal Viscosity'') are sometimes used. Kinematic viscosity in centistoke can be converted from SUS according to the arithmetic and the reference table provided in [[ASTM]] D 2161.<ref>{{cite web|url=http://www.uniteasy.com/en/unitguide/Viscosity.htm |title=Quantities and Units of Viscosity |publisher=Uniteasy.com |accessdate=2010-09-14}}</ref>
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| ==Molecular origins==
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| [[File:University of Queensland Pitch drop experiment-white bg.jpg|thumb|right|[[pitch drop experiment|Pitch]] has a viscosity approximately 230 billion (2.3{{e|11}}) times that of water.<ref>{{cite web|url=http://www.physics.uq.edu.au/physics_museum/pitchdrop.shtml|title=The pitch drop experiment|first1=R.|last1=Edgeworth|first2=B.J.|last2=Dalton|first3=T.|last3=Parnell|publisher=[[University of Queensland]]|accessdate=2009-03-31}}. A copy of: ''European Journal of Physics (1984) pp. 198–200.''</ref>]]
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| The viscosity of a system is determined by how molecules constituting the system interact. There are no simple but correct expressions for the viscosity of a fluid. The simplest exact expressions are the [[Green–Kubo relations]] for the linear shear viscosity or the [[Transient Time Correlation Function]] expressions derived by Evans and Morriss in 1985.<ref>{{cite journal | title = Transient-time-correlation functions and the rheology of fluids | journal = Physical Review A | date = October 15, 1988 | first = Denis J. | last = Evans | coauthors = Gary P. Morriss | volume = 38 | issue = 8 | pages = 4142–4148 | doi = 10.1103/PhysRevA.38.4142 | url = http://link.aps.org/doi/10.1103/PhysRevA.38.4142 | accessdate = 2012-10-24|bibcode = 1988PhRvA..38.4142E | pmid = 9900865 }}</ref> Although these expressions are each exact, in order to calculate the viscosity of a dense fluid using these relations currently requires the use of [[molecular dynamics]] computer simulations.
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| ===Gases===
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| Viscosity in gases arises principally from the molecular diffusion that transports momentum between layers of flow. The kinetic theory of gases allows accurate prediction of the behavior of gaseous viscosity.
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| Within the regime where the theory is applicable:
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| *Viscosity is independent of pressure and
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| *Viscosity increases as temperature increases.<ref name=physicsinfo>{{cite web|author=Elert, Glenn |url=http://physics.info/viscosity/ |title=The Physics Hypertextbook-Viscosity |publisher=Physics.info |accessdate=2010-09-14}}</ref>
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| [[James Clerk Maxwell]] published a famous paper in 1866 using the kinetic theory of gases to study gaseous viscosity.<ref name=Maxwell1866>{{Cite journal
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| |author = Maxwell, J. C.
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| |year = 1866
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| |title = On the viscosity or internal friction of air and other gases
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| |journal = Philosophical Transactions of the Royal Society of London
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| |volume = 156
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| |pages = 249–268
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| |doi = 10.1098/rstl.1866.0013
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| }}</ref> To understand why the viscosity is independent of pressure, consider two adjacent boundary layers (A and B) moving with respect to each other. The internal friction (the viscosity) of the gas is determined by the probability a particle of layer A enters layer B with a corresponding transfer of momentum. Maxwell's calculations show that the viscosity coefficient is proportional to the density, the mean free path, and the mean velocity of the atoms. On the other hand, the ''mean free path'' is inversely proportional to the density. So an increase in density due to an increase in pressure doesn't result in any change in viscosity.
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| ====Relation to mean free path of diffusing particles====
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| In relation to diffusion, the kinematic viscosity provides a better understanding of the behavior of mass transport of a dilute species. Viscosity is related to shear stress and the rate of shear in a fluid, which illustrates its dependence on the mean free path, ''λ'', of the diffusing particles.
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| From [[fluid mechanics]], for a [[Newtonian fluid]], the [[shear stress]], ''τ'', on a unit area moving parallel to itself, is found to be proportional to the rate of change of velocity with distance perpendicular to the unit area:
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| :<math>\tau = \mu \frac{\mathrm{d}u_x}{\mathrm{d}y}</math>
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| for a unit area parallel to the x-z plane, moving along the x axis.
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| We will derive this formula and show how ''μ'' is related to ''λ''.
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| Interpreting shear stress as the time rate of change of [[momentum]], ''p'', per unit area ''A'' (rate of momentum flux) of an arbitrary control surface gives
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| :<math>\tau = \frac{\dot{p}}{A} = \frac{\dot{m} \langle u_x \rangle}{A}.</math>
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| where <math>\langle u_x \rangle</math> is the average velocity, along the x axis, of fluid molecules hitting the unit area, with respect to the unit area.
| |
| | |
| Further manipulation will show<ref>{{Cite book|title=Lectures on geophysical fluid dynamics|first=R.L.|last=Salmon|publisher=Oxford University Press|year=1998|isbn=0-19-510808-6}}, pp. 23–26.</ref>
| |
| | |
| :<math>\dot{m} = \rho \bar{u} A </math>
| |
| :<math>\langle u_x \rangle = \frac12\, \lambda\frac{\mathrm{d}u_x}{\mathrm{d}y}</math>, assuming that molecules hitting the unit area come from all distances between 0 and ''λ'' (equally distributed), and that their average velocities change linearly with distance (always true for small enough ''λ''). From this follows:
| |
| :<math>\tau = \underbrace{\frac12\, \rho \bar{u} \lambda}_{\mu} \cdot \frac{\mathrm{d}u_x}{\mathrm{d}y} \; \; \Rightarrow \; \; \nu = \frac{\mu}{\rho} = \tfrac12\, \bar{u} \lambda,</math>
| |
| | |
| where
| |
| :<math>\dot{m}</math> is the rate of fluid mass hitting the surface,
| |
| :''ρ'' is the density of the fluid,
| |
| :''ū'' is the average molecular speed (<math>\bar{u} = \sqrt{\langle u^2 \rangle}</math>),
| |
| :''μ'' is the dynamic viscosity.
| |
| | |
| ====Effect of temperature on the viscosity of a gas====
| |
| ''Sutherland's formula'' can be used to derive the dynamic viscosity of an [[ideal gas]] as a function of the temperature:<ref>Smits, Alexander J. and Dussauge, Jean-Paul (2006) [http://books.google.com/books?id=oRx6U4T8zcIC&pg=PA46 Turbulent shear layers in supersonic flow], Birkhäuser, ISBN 0-387-26140-0 p. 46</ref>
| |
| | |
| :<math> {\mu} = {\mu}_0 \frac {T_0+C} {T + C} \left (\frac {T} {T_0} \right )^{3/2}.</math>
| |
| | |
| This in turn is equal to
| |
| | |
| :<math>\lambda\,\frac{T^{3/2}}{T+C}\,,</math> where <math>\lambda = \frac{\mu_0(T_0+C)}{T_0^{3/2}}\,</math> is a constant for the gas.
| |
| | |
| in Sutherland's formula:
| |
| *''μ'' = dynamic viscosity in (Pa·s) at input temperature ''T'',
| |
| *''μ<sub>0</sub>'' = reference viscosity in (Pa·s) at reference temperature ''T<sub>0</sub>'',
| |
| *''T'' = input temperature in kelvins,
| |
| *''T<sub>0</sub>'' = reference temperature in kelvins,
| |
| *''C'' = Sutherland's constant for the gaseous material in question.
| |
| | |
| Valid for temperatures between 0 < ''T'' < 555 K with an error due to pressure less than 10% below 3.45 MPa.
| |
| | |
| According to Sutherland's formula, if the absolute temperature is less than C, the relative change in viscosity for a small change in temperature is greater than the relative change in the absolute temperature, but it is smaller when T is above C. The kinematic viscosity though always increases faster than the temperature (that is, d log(ν)/d log(T) is greater than 1).
| |
| | |
| Sutherland's constant, reference values and λ values for some gases:
| |
| {| class="wikitable sortable"
| |
| |- bgcolor="#efefef"
| |
| ! Gas
| |
| ! ''C''
| |
| [K]
| |
| ! ''T<sub>0</sub>''
| |
| [K]
| |
| ! ''μ<sub>0</sub>''
| |
| [μPa s]
| |
| ! ''λ''
| |
| [μPa s K<sup>-1/2</sup>]
| |
| <!--
| |
| |-
| |
| |
| |
| | -
| |
| | K
| |
| | μPa s
| |
| -->
| |
| |-
| |
| | [[air]]
| |
| | 120
| |
| | 291.15
| |
| | 18.27
| |
| | 1.512041288
| |
| |-
| |
| | [[nitrogen]]
| |
| | 111
| |
| | 300.55
| |
| | 17.81
| |
| | 1.406732195
| |
| |-
| |
| | [[oxygen]]
| |
| | 127
| |
| | 292.25
| |
| | 20.18
| |
| | 1.693411300
| |
| |-
| |
| | [[carbon dioxide]]
| |
| | 240
| |
| | 293.15
| |
| | 14.8
| |
| | 1.572085931
| |
| |-
| |
| | [[carbon monoxide]]
| |
| | 118
| |
| | 288.15
| |
| | 17.2
| |
| | 1.428193225
| |
| |-
| |
| | [[hydrogen]]
| |
| | 72
| |
| | 293.85
| |
| | 8.76
| |
| | 0.636236562
| |
| |-
| |
| | [[ammonia]]
| |
| | 370
| |
| | 293.15
| |
| | 9.82
| |
| | 1.297443379
| |
| |-
| |
| | [[sulfur dioxide]]
| |
| | 416
| |
| | 293.65
| |
| | 12.54
| |
| | 1.768466086
| |
| |-
| |
| | [[helium]]
| |
| | 79.4<ref>{{cite journal |title=Numerical analysis of flow characteristics of an atmospheric plasma torch|author=Kim, Youn J.; Kim, You-Jae and Han, J.-G. |journal=12th International Congress on Plasma Physics, 25–29 October 2004, Nice (France)|arxiv=physics/0410237.pdf |year=1970}}</ref>
| |
| | 273
| |
| | 19<ref name=hyper>[http://hyperphysics.phy-astr.gsu.edu/Hbase/tables/viscosity.html Viscosity of liquids and gases]. hyperphysics.phy-astr.gsu.edu</ref>
| |
| | 1.484381490
| |
| |}
| |
| | |
| ====Viscosity of a dilute gas====
| |
| The [[Chapman-Enskog theory|Chapman-Enskog equation]]<ref>{{cite book|author= Hirshfelder, J.O.; Curtis, C.F. and Bird, R.B. |title=Molecular theory of gases and liquids|edition=First|publisher= Wiley|year=1964|isbn=0-471-40065-3}}</ref> may be used to estimate viscosity for a dilute gas. This equation is based on a semi-theoretical assumption by Chapman and Enskog. The equation requires three empirically determined parameters: the collision diameter (''σ''), the maximum energy of attraction divided by the [[Boltzmann constant]] (''є''/''к'') and the collision integral (''ω''(''T<sup>*</sup>'')).
| |
| | |
| :<math> {\mu}_0 \times 10^6 = {2.6693}\frac {(MT)^{1/2}} {\sigma^{2}\omega(T^*)},</math>
| |
| | |
| with
| |
| *''T<sup>*</sup>'' = ''κT/ε'' — reduced temperature (dimensionless),
| |
| *''μ<sub>0</sub>'' = viscosity for dilute gas (μPa.s),
| |
| *''M'' = molecular mass (g/mol),
| |
| *''T'' = temperature (K),
| |
| *''σ'' = the collision diameter (Å),
| |
| *''ε'' / ''κ'' = the maximum energy of attraction divided by the Boltzmann constant (K),
| |
| *''ω<sub>μ</sub>'' = the collision integral.
| |
| | |
| ===Liquids===
| |
| [[File:Viscosity video science museum.ogv|thumb|right|Video showing three liquids with different Viscosities]]
| |
| In liquids, the additional forces between molecules become important. This leads to an additional contribution to the shear stress though the exact mechanics of this are still controversial.{{Citation needed|date=February 2007}} Thus, in liquids:
| |
| | |
| *Viscosity is independent of pressure (except at very high pressure); and
| |
| *Viscosity tends to fall as temperature increases (for example, water viscosity goes from 1.79 cP to 0.28 cP in the temperature range from 0 °C to 100 °C); see [[temperature dependence of liquid viscosity]] for more details.
| |
| | |
| The dynamic viscosities of liquids are typically several orders of magnitude higher than dynamic viscosities of gases.
| |
| | |
| ====Viscosity of blends of liquids====
| |
| The viscosity of the blend of two or more liquids can be estimated using the Refutas equation.<ref>{{cite book|author=Maples, Robert E. |title=Petroleum Refinery Process Economics|edition=2nd|publisher=Pennwell Books|year=2000|isbn=0-87814-779-9}}</ref><!--- dead link ref>Baird, C.T. (1989), ''Guide to Petroleum Product Blending'', HPI Consultants, Inc. [http://www.hpiconsultants.com/blending/index.htm HPI website], ISBN 0685547841.</ref ---> The calculation is carried out in three steps.
| |
| | |
| The first step is to calculate the Viscosity Blending Number (VBN) (also called the Viscosity Blending Index) of each component of the blend:
| |
| | |
| :(1) {{pad|3em}} <math>\mbox{VBN} = 14.534 \times \ln\left[ \ln(\nu + 0.8) \right] + 10.975\,</math>
| |
| | |
| where ''ν'' is the kinematic viscosity in centistokes (cSt). It is important that the kinematic viscosity of each component of the blend be obtained at the same temperature.
| |
| | |
| The next step is to calculate the VBN of the blend, using this equation:
| |
| | |
| :(2) {{pad|3em}} <math>\mbox{VBN}_\text{Blend} = \left[ x_A \times \mbox{VBN}_A \right] + \left[x_B \times \mbox{VBN}_B\right] + \cdots + \left[x_N \times \mbox{VBN}_N\right]\,</math>
| |
| | |
| where ''x<sub>X</sub>'' is the [[mass fraction (chemistry)|mass fraction]] of each component of the blend.
| |
| | |
| Once the viscosity blending number of a blend has been calculated using equation (2), the final step is to determine the kinematic viscosity of the blend by solving equation (1) for ''ν'':
| |
| | |
| :(3) {{pad|3em}} <math>\nu = \exp\left( \exp\left( \frac{\text{VBN}_\text{Blend} - 10.975}{14.534} \right) \right) - 0.8,</math>
| |
| | |
| where ''VBN<sub>Blend</sub>'' is the viscosity blending number of the blend.
| |
| | |
| ==Viscosity of selected substances==
| |
| | |
| ===Air===
| |
| [[File:Air dry dynamic visocity on pressure temperature.svg|thumb|right|Pressure dependence of the dynamic viscosity of dry air at the temperatures of 300, 400 and 500 K]]
| |
| The viscosity of air depends mostly on the temperature.
| |
| At 15 °C, the viscosity of air is 1.81{{e|−5}} kg/(m·s), 18.1 μPa.s or 1.81{{e|−5}} Pa.s. The kinematic viscosity at 15 °C is 1.48{{e|-5}} m<sup>2</sup>/s or 14.8 cSt. At 25 °C, the viscosity is 18.6 μPa.s and the kinematic viscosity 15.7 cSt. One can get the viscosity of air as a function of temperature from the [http://www.lmnoeng.com/Flow/GasViscosity.htm Gas Viscosity Calculator]
| |
| | |
| ===Water===
| |
| [[File:Dynamic Viscosity of Water.png|thumb|Dynamic viscosity of water]]
| |
| The [[dynamics (mechanics)|dynamic]] viscosity of [[water]] is 8.90 × 10<sup>−4</sup> [[Pascal (unit)|Pa]]·[[second|s]] or 8.90 × 10<sup>−3</sup> dyn·s/cm<sup>2</sup> or 0.890 cP at about 25 °C.<br>
| |
| Water has a viscosity of 0.0091 poise at 25 °C, or 1 centipoise at 20 °C.<br>
| |
| As a function of temperature ''T'' (K): (Pa·s) = ''A'' × 10<sup>''B''/(''T''−''C'')</sup><br>
| |
| where ''A''=2.414 × 10<sup>−5</sup> Pa·s ; ''B'' = 247.8 K ; and ''C'' = 140 K.{{Citation needed|date=January 2012}}
| |
| | |
| Viscosity of liquid water at different temperatures up to the normal boiling point is listed below.
| |
| | |
| {| class="wikitable sortable"
| |
| |- style="background:#efefef;"
| |
| !Temperature
| |
| [°C]
| |
| !Viscosity
| |
| [mPa·s]
| |
| |-
| |
| |10
| |
| |1.308
| |
| |-
| |
| |20
| |
| |1.002
| |
| |-
| |
| |30
| |
| |0.7978
| |
| |-
| |
| |40
| |
| |0.6531
| |
| |-
| |
| |50
| |
| |0.5471
| |
| |-
| |
| |60
| |
| |0.4658
| |
| |-
| |
| |70
| |
| |0.4044
| |
| |-
| |
| |80
| |
| |0.3550
| |
| |-
| |
| |90
| |
| |0.3150
| |
| |-
| |
| |100
| |
| |0.2822
| |
| |}
| |
| | |
| ===Other substances===
| |
| [[File:Drop 0.jpg|thumb|right|Example of the viscosity of milk and water. Liquids with higher viscosities make smaller splashes when poured at the same velocity.]]
| |
| | |
| [[File:Runny hunny.jpg|thumb|[[Honey]] being drizzled.]]
| |
| [[File:PeanutButter.jpg|thumb|[[Peanut butter]] is a [[semi-solid]] and can therefore hold peaks.]]
| |
| | |
| Some dynamic viscosities of Newtonian fluids are listed below:
| |
| | |
| {| class="wikitable sortable"
| |
| |+ Viscosity of selected gases at 100 kPa, [μPa·s]
| |
| |- style="background:#efefef;"
| |
| ![[Gas]]
| |
| ! at 0 °[[Celsius|C]] (273 K)
| |
| ! at 27 °C (300 K)<ref name="CRC83">{{RubberBible86th}}</ref>
| |
| |-
| |
| |[[Earth's atmosphere|air]]
| |
| |17.4
| |
| |18.6
| |
| |-
| |
| |[[hydrogen]]
| |
| |8.4
| |
| |9.0
| |
| |-
| |
| |[[helium]]
| |
| |
| |
| |20.0
| |
| |-
| |
| |[[argon]]
| |
| |
| |
| |22.9
| |
| |-
| |
| |[[xenon]]
| |
| |21.2
| |
| |23.2
| |
| |-
| |
| |[[carbon dioxide]]
| |
| |
| |
| |15.0
| |
| |-
| |
| |[[methane]]
| |
| |
| |
| |11.2
| |
| |-
| |
| |[[ethane]]
| |
| |
| |
| |9.5
| |
| |}
| |
| | |
| {| class="wikitable sortable"
| |
| |- style="background:#efefef;"
| |
| |+Viscosity of [[fluid]]s with variable compositions
| |
| !Fluid
| |
| !Viscosity
| |
| [Pa·s]
| |
| !Viscosity
| |
| [cP]
| |
| |-
| |
| |[[blood]] (37 °C)<ref name=r1/>
| |
| |(3–4){{e|-3}}
| |
| |3–4
| |
| |-
| |
| |[[honey]]
| |
| |2–10
| |
| |2,000–10,000
| |
| |-
| |
| |[[molasses]]
| |
| |5–10
| |
| |5,000–10,000
| |
| |-
| |
| |molten [[glass]]
| |
| |10–1,000
| |
| |10,000–1,000,000
| |
| |-
| |
| |[[chocolate syrup]]
| |
| |10–25
| |
| |10,000–25,000
| |
| |-
| |
| |molten [[chocolate]]<sup>*</sup>
| |
| |45–130<ref>{{cite web |url=http://www.brookfieldengineering.com/education/applications/laboratory-chocolate-processing.asp |title=Chocolate Processing |accessdate=2007-12-03 |work=[[Brookfield Engineering]] website}}</ref>
| |
| |45,000–130,000
| |
| |-
| |
| |[[ketchup]]<sup>*</sup>
| |
| |50–100
| |
| |50,000–100,000
| |
| |-
| |
| |[[lard]]
| |
| |≈ 100
| |
| |≈ 100,000
| |
| |-
| |
| |[[peanut butter]]<sup>*</sup>
| |
| |≈ 250
| |
| |≈ 250,000
| |
| |-
| |
| |[[shortening]]<sup>*</sup>
| |
| |≈ 250
| |
| |≈ 250,000
| |
| |}
| |
| {| class="wikitable sortable"
| |
| |+Viscosity of [[liquid]]s <br>(at 25 °[[Celsius|C]] unless otherwise specified)
| |
| |- style="background:#efefef;"
| |
| !Liquid :
| |
| !Viscosity
| |
| [Pa·s]
| |
| !Viscosity
| |
| [cP=mPa·s]
| |
| |-
| |
| |[[acetone]]<ref name="CRC83"/>
| |
| |3.06{{e|-4}}
| |
| |0.306
| |
| |-
| |
| |[[benzene]]<ref name="CRC83"/>
| |
| |6.04{{e|-4}}
| |
| |0.604
| |
| |-
| |
| |[[castor bean|castor oil]]<ref name="CRC83"/>
| |
| |0.985
| |
| |985
| |
| |-
| |
| |[[corn syrup]]<ref name="CRC83"/>
| |
| |1.3806
| |
| |1380.6
| |
| |-
| |
| |[[ethanol]]<ref name="CRC83"/>
| |
| |1.074{{e|-3}}
| |
| |1.074
| |
| |-
| |
| |[[ethylene glycol]]
| |
| |1.61{{e|-2}}
| |
| |16.1
| |
| |-
| |
| |[[glycerol]] (at 20 °[[Celsius|C]])<ref name=hyper/>
| |
| |1.2
| |
| |1200
| |
| |-
| |
| |[[Fuel oil|HFO-380]]
| |
| |2.022
| |
| |2022
| |
| |-
| |
| |[[mercury (element)|mercury]]<ref name="CRC83"/>
| |
| |1.526{{e|-3}}
| |
| |1.526
| |
| |-
| |
| |[[methanol]]<ref name="CRC83"/>
| |
| |5.44{{e|-4}}
| |
| |0.544
| |
| |-
| |
| |[[motor oil]] SAE 10 (20 °C)<ref name=physicsinfo/>
| |
| |0.065
| |
| |65
| |
| |-
| |
| |[[motor oil]] SAE 40 (20 °C)<ref name=physicsinfo/>
| |
| |0.319
| |
| |319
| |
| |-
| |
| |[[nitrobenzene]]<ref name="CRC83"/>
| |
| |1.863{{e|-3}}
| |
| |1.863
| |
| |-
| |
| |[[liquid nitrogen]] @ 77K
| |
| |1.58{{e|-4}}
| |
| |0.158
| |
| |-
| |
| |[[Propan-1-ol|propanal]]<ref name="CRC83"/>
| |
| |1.945{{e|-3}}
| |
| |1.945
| |
| |-
| |
| |[[olive oil]]
| |
| |.081
| |
| |81
| |
| |-
| |
| |[[pitch (resin)|pitch]]
| |
| |2.3{{e|8}}
| |
| |2.3{{e|11}}
| |
| |-
| |
| |[[sulfuric acid]]<ref name="CRC83"/>
| |
| |2.42{{e|-2}}
| |
| |24.2
| |
| |-
| |
| |[[water]]
| |
| |8.94{{e|-4}}
| |
| |0.894
| |
| |}
| |
| {| class="wikitable sortable"
| |
| |- style="background:#efefef;"
| |
| |+Viscosity of [[solid]]s
| |
| !Solid
| |
| !Viscosity
| |
| [Pa·s]
| |
| !Temperature
| |
| [K]
| |
| |-
| |
| |[[asthenosphere]]<ref name=asthenosphere>{{cite journal|last=Fjeldskaar|first=W.|title=Viscosity and thickness of the asthenosphere detected from the Fennoscandian uplift|journal=Earth and Planetary Science Letters|year=1994|volume=126|issue=4|pages=399–410|doi=10.1016/0012-821X(94)90120-1|bibcode = 1994E&PSL.126..399F }}</ref>
| |
| |7.0{{e|19}}
| |
| |900 °C
| |
| |-
| |
| |[[upper mantle]]<ref name=asthenosphere/>
| |
| |(0.7-1.0){{e|21}}
| |
| |1300-3000 °C
| |
| |-
| |
| |[[lower mantle]]
| |
| |(1.0-2.0){{e|21}}
| |
| |3000-4000 °C
| |
| |}
| |
| | |
| <nowiki>*</nowiki> These materials are highly [[non-Newtonian fluid|non-Newtonian]].
| |
| | |
| <small>Note: Higher viscosity means thicker substance</small>
| |
| | |
| ==Viscosity of slurry==
| |
| [[File:Slurry Viscosity Plot.png|300px|thumb|Plot of slurry relative viscosity ''μ<sub>r</sub>'' as calculated by empirical correlations from Einstein,<ref name=Einstein /> Guth and Simha,<ref name=GuthAndSimha /> Thomas,<ref name=Thomas /> and Kitano ''et al.''.<ref name=KitanoEtAl />]]
| |
| The term [[slurry]] describes mixtures of a liquid and solid particles that retain some fluidity. The viscosity of slurry can be described as relative to the viscosity of the liquid phase:
| |
| | |
| :<math>\mu_s = \mu_r \cdot \mu_l,</math>
| |
| | |
| where ''μ<sub>s</sub>'' and ''μ<sub>l</sub>'' are respectively the dynamic viscosity of the slurry and liquid (Pa·s), and ''μ<sub>r</sub>'' is the relative viscosity (dimensionless).
| |
| | |
| Depending on the size and concentration of the solid particles, several models exist that describe the relative viscosity as a function of [[Slurry#Volumetric fraction from mass fraction|volume fraction]] ''ɸ'' of solid particles.
| |
| | |
| In the case of extremely low concentrations of fine particles, Einstein's equation<ref name=Einstein>{{cite journal|last=Einstein |first=A.|year=1906|journal=Annalen der Physik|volume=19|pages=289|bibcode = 1906AnP...324..289E |doi = 10.1002/andp.19063240204|title=Eine neue Bestimmung der Moleküldimensionen|issue=2 }}</ref> may be used:
| |
| | |
| :<math>\mu_r = 1 + 2.5 \cdot \phi</math>
| |
| | |
| In the case of higher concentrations, a modified equation was proposed by Guth and Simha,<ref name=GuthAndSimha>{{cite journal|author=Guth, E., Simha, R.|year=1936|journal=Kolloid Z.|volume=74|pages=266|doi=10.1007/BF01428643|title=Untersuchungen über die Viskosität von Suspensionen und Lösungen. 3. Über die Viskosität von Kugelsuspensionen|issue=3}}</ref> which takes into account interaction between the solid particles:
| |
| | |
| :<math>\mu_r = 1 + 2.5 \cdot \phi + 14.1 \cdot \phi^2</math>
| |
| | |
| Further modification of this equation was proposed by Thomas<ref name=Thomas>{{cite journal|author=Thomas, D. G.|year=1965|journal=J. Colloid Sci.|volume=20|pages=267|doi=10.1016/0095-8522(65)90016-4|title=Transport characteristics of suspension: VIII. A note on the viscosity of Newtonian suspensions of uniform spherical particles|issue=3}}</ref> from the fitting of empirical data:
| |
| | |
| :<math>\mu_r = 1 + 2.5 \cdot \phi + 10.05 \cdot \phi^2 + A \cdot e^{B \cdot \phi},</math>
| |
| | |
| where ''A = 0.00273'' and ''B = 16.6''.
| |
| | |
| In the case of very high concentrations, another empirical equation was proposed by Kitano ''et al.'':<ref name=KitanoEtAl>{{cite journal|author=Kitano, T., Kataoka, T., and Shirota, T.|year=1981|journal=Rheologica Acta|volume=20|pages=207|doi=10.1007/BF01513064|title=An empirical equation of the relative viscosity of polymer melts filled with various inorganic fillers|issue=2}}</ref>
| |
| | |
| :<math>\mu_r = (1 - \frac{\phi}{A})^{-2},</math>
| |
| | |
| where ''A = 0.68'' for smooth spherical particles.
| |
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| ==Viscosity of amorphous materials==
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| [[File:Glassviscosityexamples.png|300px|thumb|Common [[glass]] viscosity curves.<ref>{{cite web|author=Fluegel, Alexander |url=http://www.glassproperties.com/viscosity/ |title=Viscosity calculation of glasses |publisher=Glassproperties.com |accessdate=2010-09-14}}</ref>]]
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| Viscous flow in [[Amorphous solid|amorphous materials]] (e.g. in [[glass]]es and melts)<ref>{{cite journal|author=Doremus, R.H.|year=2002|title=Viscosity of silica|journal=J. Appl. Phys.|volume=92|issue=12 |pages=7619–7629|doi = 10.1063/1.1515132
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| |bibcode = 2002JAP....92.7619D }}</ref><ref>{{cite journal|author=Ojovan, M.I. and Lee, W.E. |year=2004 |title=Viscosity of network liquids within Doremus approach |journal=J. Appl. Phys.|volume=95|issue=7|pages=3803–3810|doi = 10.1063/1.1647260|month=|bibcode = 2004JAP....95.3803O }}</ref><ref>{{cite journal|author=Ojovan, M.I.; Travis, K.P. and Hand, R.J. |year=2000|title=Thermodynamic parameters of bonds in glassy materials from viscosity-temperature relationships|journal=J. Phys.: Condensed matter|volume=19|issue=41 |page=415107|doi=10.1088/0953-8984/19/41/415107|bibcode = 2007JPCM...19O5107O }}</ref> is a thermally activated process:
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| :<math>\mu = A \cdot e^{Q/RT},</math>
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| where ''Q'' is activation energy, ''T'' is temperature, ''R'' is the molar gas constant and ''A'' is approximately a constant.
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| The viscous flow in amorphous materials is characterized by a deviation from the [[Arrhenius equation|Arrhenius-type]] behavior: ''Q'' changes from a high value ''Q<sub>H</sub>'' at low temperatures (in the glassy state) to a low value ''Q<sub>L</sub>'' at high temperatures (in the liquid state). Depending on this change, amorphous materials are classified as either
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| *strong when: ''Q<sub>H</sub>'' − ''Q<sub>L</sub>'' < ''Q<sub>L</sub>'' or
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| *fragile when: ''Q<sub>H</sub>'' − ''Q<sub>L</sub>'' ≥ ''Q<sub>L</sub>''.
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| The fragility of amorphous materials is numerically characterized by the Doremus’ fragility ratio:
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| :<math>R_D = \frac{Q_H}{Q_L}</math>
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| and strong material have ''R<sub>D</sub>'' < 2 whereas fragile materials have ''R<sub>D</sub>'' ≥ 2.
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| [[Image:B2O3 viscosoty.jpg|thumb|Common log of viscosity vs temperature for B<sub>2</sub>O<sub>3</sub>, showing two regimes]]
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| The viscosity of amorphous materials is quite exactly described by a two-exponential equation:
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| :<math>\mu = A_1 \cdot T \cdot \left[1 + A_2 \cdot e^{B/RT}] \cdot [1 + C \cdot e^{D/RT} \right],</math>
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| with constants ''A<sub>1</sub>'', ''A<sub>2</sub>'', ''B'', ''C'' and ''D'' related to thermodynamic parameters of joining bonds of an amorphous material.
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| Not very far from the [[glass transition temperature]], ''T<sub>g</sub>'', this equation can be approximated by a [[Vogel-Fulcher-Tammann equation|Vogel-Fulcher-Tammann]] (VFT) equation.
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| If the temperature is significantly lower than the glass transition temperature, ''T'' {{unicode|≪}} ''T<sub>g</sub>'', then the two-exponential equation simplifies to an Arrhenius type equation:
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| :<math>\mu = A_LT \cdot e^{Q_H/RT}</math>
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| with:
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| :<math>Q_H = H_d + H_m,\,</math>
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| where ''H<sub>d</sub>'' is the [[enthalpy of formation]] of broken bonds (termed [http://www.wikidoc.org/index.php/Configuron configuron] s) and ''H<sub>m</sub>'' is the [[enthalpy]] of their motion. When the temperature is less than the glass transition temperature, ''T'' < ''T<sub>g</sub>'', the activation energy of viscosity is high because the amorphous materials are in the glassy state and most of their joining bonds are intact.
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| If the temperature is highly above the glass transition temperature, ''T'' {{unicode|≫}} ''T<sub>g</sub>'', the two-exponential equation also simplifies to an Arrhenius type equation:
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| :<math>\mu = A_HT\cdot e^{Q_L/RT},</math>
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| with:
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| :<math>Q_L = H_m.\,</math>
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| When the temperature is higher than the glass transition temperature, ''T'' > ''T<sub>g</sub>'', the activation energy of viscosity is low because amorphous materials are melted and have most of their joining bonds broken, which facilitates flow.
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| ==Eddy viscosity==
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| In the study of [[turbulence]] in [[fluid]]s, a common practical strategy for calculation is to ignore the small-scale ''vortices'' (or ''eddies'') in the motion and to calculate a large-scale motion with an ''eddy viscosity'' that characterizes the transport and dissipation of [[energy]] in the smaller-scale flow (see ''[[large eddy simulation]]''). Values of eddy viscosity used in modeling [[ocean]] circulation may be from 5×10<sup>4</sup> to 10<sup>6</sup> Pa·s depending upon the resolution of the numerical grid.
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| ==See also==
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| {{colbegin|3}}
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| *[[Deborah number]]
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| *[[Dilatant]]
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| *[[Herschel–Bulkley fluid]]
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| *[[Hyperviscosity syndrome]]
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| *[[Intrinsic viscosity]]
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| *[[Inviscid flow]]
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| *[[Morton number]]
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| *[[Relative viscosity]]
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| *[[Reyn]]
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| *[[Reynolds number]]
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| *[[Trouton's ratio]]
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| *[[Two-dimensional point vortex gas]]
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| *[[Viscoelasticity]]
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| *[[Viscosity index]]
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| *[[Joback method]] (estimation of the liquid viscosity from molecular structure)
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| *[[Microviscosity]]
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| *[[Rheology]]
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| *[[Superfluid helium-4]]
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| *[[Stokes flow]]
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| {{colend}}
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| ==References==
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| {{reflist|30em}}
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| ==Further reading==
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| * Hatschek, Emil (1928). ''The Viscosity of Liquids''. New York: [[Van Nostrand]]. {{oclc|53438464}}.
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| *{{cite book
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| | author = Massey, B. S.
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| | coauthors = Ward-Smith, A. J.
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| | title = Mechanics of Fluids
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| | edition = Ninth
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| | publisher= Spon Press
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| | location = London; New York
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| | year = 2011 |url=http://science.fire.ustc.edu.cn/download/job.php?job=download&id=141&did=0
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| | oclc = 690084654
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| | isbn = 978-0-415-60259-4}}
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| ==External links==
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| {{Wiktionary}}
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| *[http://webbook.nist.gov/chemistry/fluid/ Fluid properties] High accuracy calculation of viscosity and other physical properties of frequent used pure liquids and gases.
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| *[http://www.enggcyclopedia.com/calculators/physical-properties/gas-viscosity/ Gas viscosity calculator as function of temperature]
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| *[http://www.enggcyclopedia.com/calculators/physical-properties/air-viscosity-calculator/ Air viscosity calculator as function of temperature and pressure]
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| *[http://www.engineersedge.com/fluid_flow/fluid_data.htm Fluid Characteristics Chart] A table of viscosities and vapor pressures for various fluids
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| *[http://web.ics.purdue.edu/~alexeenk/GDT/index.html Gas Dynamics Toolbox] Calculate coefficient of viscosity for mixtures of gases
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| *[http://glassproperties.com/viscosity/ViscosityMeasurement.htm Glass Viscosity Measurement] Viscosity measurement, viscosity units and fixpoints, glass viscosity calculation
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| *[http://www.diracdelta.co.uk/science/source/k/i/kinematic%20viscosity/source.html Kinematic Viscosity] conversion between kinematic and dynamic viscosity.
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| *[http://www.thermexcel.com/english/tables/eau_atm.htm Physical Characteristics of Water] A table of water viscosity as a function of temperature
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| *[http://www.iop.org/EJ/abstract/0953-8984/12/46/305 Vogel–Tammann–Fulcher Equation Parameters]
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| *[http://ddbonline.ddbst.de/VogelCalculation/VogelCalculationCGI.exe Calculation of temperature-dependent dynamic viscosities for some common components]
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| *[http://www.epa.gov/EPA-AIR/2005/July/Day-13/a11534d.htm "Test Procedures for Testing Highway and Nonroad Engines and Omnibus Technical Amendments"]. [[United States Environmental Protection Agency]]
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| *[http://www.astro.uu.se/~bf/course/numhd_course/2_5_2Artificial_viscosity.html Artificial viscosity]
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| {{Physics-footer}}
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| [[Category:Concepts in physics]]
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| [[Category:Glass engineering and science]]
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| [[Category:Viscosity| ]]
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