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| symbols = {{mvar|[[Eta (letter)|η]]}}, {{mvar|[[Mu (letter)|μ]]}}
| derivations = {{math|1=''μ'' = ''[[Shear modulus|G]]''·''[[time|t]]''}}
| dimension = <math>\mathsf{M} \mathsf{L}^{-1} \mathsf{T}^{-1}</math>
}}
{{continuum mechanics|cTopic=fluid}}
Viscosity quantifies the internal [[friction|frictional force]] between adjacent layers of fluid that are in relative motion.<ref name="Britanica"/> For instance, when a viscous fluid is forced through a tube, it flows more quickly near the tube's
In general, viscosity depends on a fluid's state, such as its temperature, pressure, and rate of deformation. However, the dependence on some of these properties is negligible in certain cases. For example, the viscosity of a [[Newtonian fluid]] does not vary significantly with the rate of deformation.
Zero viscosity (no resistance to [[shear stress]]) is observed only at [[cryogenics|very low temperatures]] in [[Superfluidity|superfluids]]; otherwise, the [[second law of thermodynamics]] requires all fluids to have positive viscosity.{{sfn|Balescu|1975|pp=428–429}}{{sfn|Landau|Lifshitz|1987|p=}} A fluid that has zero viscosity (non-viscous) is called ''ideal'' or ''inviscid''.
For [[non-Newtonian fluid]]'s viscosity, there are [[pseudoplastic]], [[plastic flow|plastic]], and [[dilatant]] flows that are time-independent, and there are [[thixotropic]] and [[rheopectic]] flows that are time-dependent.
{{toclimit|limit=3}}
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[[File:Laminar shear flow.svg|thumb|In a general parallel flow, the shear stress is proportional to the gradient of the velocity.]]
In [[materials science]] and [[engineering]],
Viscosity is the material property which relates the viscous stresses in a material to the rate of change of a deformation (the strain rate). Although it applies to general flows, it is easy to visualize and define in a simple shearing flow, such as a planar [[Couette flow]].
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</math> [[pressure]] multiplied by [[time]] <math>=</math> energy per unit volume multiplied by time.
The aforementioned 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 [[
:<math>\tau=\mu \frac{\partial u}{\partial y},</math>
where <math>\tau = F / A</math>, and <math>\partial u / \partial y</math> is the local shear velocity. This expression is referred to as [[Newton's law of viscosity]]. In shearing flows with planar symmetry, it is what ''defines'' <math>\mu</math>. It is a special case of the general definition of viscosity (see below), which can be expressed in coordinate-free form.
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</math>
where <math>\mu_{ijk\ell}</math> is a viscosity tensor that maps the [[velocity gradient]] tensor <math>\partial v_k / \partial r_\ell</math> onto the viscous stress tensor <math>\tau_{ij}</math>.{{sfn|Bird| Stewart| Lightfoot| 2007| p= 18|ps=: This source uses an alternative sign convention, which has been reversed here.}} Since the indices in this expression can vary from 1 to 3, there are 81 "viscosity coefficients" <math>\mu_{ijkl}</math> in total. However, assuming that the viscosity rank-
:<math>
\mu_{ijk\ell} = \alpha \delta_{ij}\delta_{k\ell} + \beta \delta_{ik}\delta_{j\ell} + \gamma \delta_{i\ell}\delta_{jk},
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and furthermore, it is assumed that no viscous forces may arise when the fluid is undergoing simple rigid-body rotation, thus <math>\beta = \gamma</math>, leaving only two independent parameters.{{sfn|Landau|Lifshitz|1987|pp=44–45}} The most usual decomposition is in terms of the standard (scalar) viscosity <math>\mu</math> and the [[bulk viscosity]] <math>\kappa</math> such that <math>\alpha = \kappa - \tfrac{2}{3}\mu</math> and <math>\beta = \gamma = \mu</math>. In vector notation this appears as:
:<math>
\boldsymbol{\tau} = \mu \left[\nabla \mathbf{v} + (\nabla \mathbf{v})^{\
</math>
where <math>\mathbf{\delta}</math> is the unit tensor
The bulk viscosity (also called volume viscosity) expresses a type of internal friction that resists the shearless compression or expansion of a fluid. Knowledge of <math>\kappa</math> is frequently not necessary in fluid dynamics problems. For example, an incompressible fluid satisfies <math>\nabla \cdot \mathbf{v} = 0</math> and so the term containing <math>\kappa</math> drops out. Moreover, <math>\kappa</math> is often assumed to be negligible for gases since it is <math>0</math> in a [[monatomic]] [[ideal gas]].{{sfn|Bird| Stewart| Lightfoot| 2007| p=19}} One situation in which <math>\kappa</math> can be important is the calculation of energy loss in [[sound]] and [[shock wave]]s, described by [[Stokes' law (sound attenuation)|Stokes' law of sound attenuation]], since these phenomena involve rapid expansions and compressions.
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==In solids==
The viscous forces that arise during fluid flow are distinct from the [[Elasticity (physics)|elastic]] forces that occur 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
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.{{sfn|Kumagai|Sasajima |Ito|1978|pp=157–161}} Such materials are best described as [[viscoelasticity|viscoelastic]]—that is, possessing both elasticity (reaction to deformation) and viscosity (reaction to rate of deformation).
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==Measurement==
{{Main|Viscometer}}
Viscosity is measured with various types of [[viscometer]]s and [[rheometer]]s. Close temperature control of the fluid is essential to obtain accurate measurements, particularly in materials like lubricants, whose viscosity can double with a change of only 5 °C. A rheometer is used for 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
For some fluids, the viscosity is 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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Kinematic viscosity has units of square feet per second (ft<sup>2</sup>/s) in both the BG and EE systems.
Nonstandard units include the [[reyn]] (lbf·s/in<sup>2</sup>), a British unit of dynamic viscosity.<ref>{{cite web |title=What is the unit called a reyn? |url=https://www.sizes.com/units/reyn.htm |website=sizes.com |access-date=23 December 2023 |language=en}}</ref> In the automotive industry the [[viscosity index]] is used to describe the change of viscosity with temperature.
The [[Multiplicative inverse|reciprocal]] of viscosity is ''fluidity'', usually symbolized by <math>\phi = 1 / \mu</math> or <math>F = 1 / \mu</math>, depending on the convention used, measured in ''reciprocal poise'' (P<sup>−1</sup>, or [[centimetre|cm]]·[[second|s]]·[[gram|g]]<sup>−1</sup>), sometimes called the ''rhe''. Fluidity is seldom used in [[engineering]] practice.{{Citation needed|date=January 2022}}
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\mu = \mu_0 \left(\frac{T}{T_0}\right)^{\!\!3/2}\ \frac{T_0 + S}{T + S},
</math>
where <math>\mu_0</math> is the viscosity at temperature <math>T_0</math>. This expression is usually named Sutherland's formula.{{sfn|Sutherland|1893|pp=507–531}} If <math>\mu</math> is known from experiments at <math>T = T_0</math> and at least one other temperature, then <math>S</math> can be calculated. Expressions for <math>\mu</math> obtained in this way are qualitatively accurate for a number of simple gases. Slightly more sophisticated models, such as the [[Lennard-Jones potential]], or the more flexible [[Mie potential]], may provide better agreement with experiments, but only at the cost of a more opaque dependence on temperature. A further advantage of these more complex interaction potentials is that they can be used to develop accurate models for a wide variety of properties using the same potential parameters. In situations where little experimental data is available, this makes it possible to obtain model parameters from fitting to properties such as pure-fluid [[Vapor–liquid equilibrium|vapour-liquid equilibria]], before using the parameters thus obtained to predict the viscosities of interest with reasonable accuracy.
In some systems, the assumption of [[spherical symmetry]] must be abandoned, as is the case for vapors with highly [[polar molecules]] like [[Properties of water|H<sub>2</sub>O]]. In these cases, the Chapman–Enskog analysis is significantly more complicated.{{sfn|Bird| Stewart| Lightfoot| 2007|pp=25–27}}{{sfn|Chapman|Cowling|1970|pp= 235–237}}
==== Bulk viscosity ====
{{See also|Bulk viscosity}}
In the kinetic-molecular picture, a non-zero bulk viscosity arises in gases whenever there are non-negligible relaxational timescales governing the exchange of energy between the translational energy of molecules and their internal energy, e.g. [[Rotational energy|rotational]] and [[vibration]]al. As such, the bulk viscosity is <math>0</math> for a monatomic ideal gas, in which the internal energy of molecules is negligible, but is nonzero for a gas like [[carbon dioxide]], whose molecules possess both rotational and vibrational energy.{{sfn|Chapman|Cowling|1970|pp= 197, 214–216}}{{sfn|Cramer|2012|p=066102-2}}
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Errors as large as 30% can be encountered using equation ({{EquationNote|1}}), compared with fitting equation ({{EquationNote|2}}) to experimental data.{{sfn|Bird| Stewart| Lightfoot| 2007| pp=29–31}} More fundamentally, the physical assumptions underlying equation ({{EquationNote|1}}) have been criticized.{{sfn|Hildebrand|1977|p=}} It has also been argued that the exponential dependence in equation ({{EquationNote|1}}) does not necessarily describe experimental observations more accurately than simpler, non-exponential expressions.{{sfn|Hildebrand|1977|p=37}}{{sfn|Egelstaff|1992|p=264}}
In light of these shortcomings, the development of a less ad hoc model is a matter of practical interest. Foregoing simplicity in favor of precision, it is possible to write rigorous expressions for viscosity starting from the fundamental equations of motion for molecules. A classic example of this approach is Irving–Kirkwood theory.{{sfn|Irving|Kirkwood|1949|pp=817–829}} On the other hand, such expressions are given as averages over multiparticle [[Correlation function (statistical mechanics)|correlation functions]] and are therefore difficult to apply in practice.▼
▲Foregoing simplicity in favor of precision, it is possible to write rigorous expressions for viscosity starting from the fundamental equations of motion for molecules. A classic example of this approach is Irving–Kirkwood theory.{{sfn|Irving|Kirkwood|1949|pp=817–829}} On the other hand, such expressions
In general, empirically derived expressions (based on existing viscosity measurements) appear to be the only consistently reliable means of calculating viscosity in liquids.{{sfn|Reid|Sherwood|1958|pp=206–209}}
Local atomic structure changes observed in undercooled liquids on cooling below the equilibrium melting temperature either in terms of radial distribution function ''g''(''r'')<ref>{{Cite journal |last=Louzguine-Luzgin |first=D. V. |date=2022-10-18 |title=Structural Changes in Metallic Glass-Forming Liquids on Cooling and Subsequent Vitrification in Relationship with Their Properties |journal=Materials |language=en |volume=15 |issue=20 |pages=7285 |doi=10.3390/ma15207285 |doi-access=free |issn=1996-1944 |pmc=9610435 |pmid=36295350|bibcode=2022Mate...15.7285L }}</ref> or structure factor ''S''(''Q'')<ref>{{Cite journal |last=Kelton |first=K F |date=2017-01-18 |title=Kinetic and structural fragility—a correlation between structures and dynamics in metallic liquids and glasses |url=https://iopscience.iop.org/article/10.1088/0953-8984/29/2/023002 |journal=Journal of Physics: Condensed Matter |volume=29 |issue=2 |pages=023002 |doi=10.1088/0953-8984/29/2/023002 |pmid=27841996 |bibcode=2017JPCM...29b3002K |issn=0953-8984|url-access=subscription }}</ref> are found to be directly responsible for the liquid fragility: deviation of the temperature dependence of viscosity of the undercooled liquid from the Arrhenius equation (2) through modification of the activation energy for viscous flow. At the same time equilibrium liquids follow the Arrhenius equation.
===Mixtures and blends===
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The same molecular-kinetic picture of a single component gas can also be applied to a gaseous mixture. For instance, in the [[Chapman–Enskog theory|Chapman–Enskog]] approach the viscosity <math>\mu_{\text{mix}}</math> of a binary mixture of gases can be written in terms of the individual component viscosities <math>\mu_{1,2}</math>, their respective volume fractions, and the intermolecular interactions.{{sfn|Chapman|Cowling|1970|p=}}
As for the single-component gas, the dependence of <math>\mu_{\text{mix}}</math> on the parameters of the intermolecular interactions enters through various collisional integrals which may not be expressible in [[Closed-form expression#Symbolic integration|closed form]]. To obtain usable expressions for <math>\mu_{\text{mix}}</math> which reasonably match experimental data, the collisional integrals may be computed numerically or from correlations.<ref name=":0" /> In some cases, the collision integrals are regarded as fitting parameters, and are fitted directly to experimental data.<ref>{{Cite journal |last1=Lemmon |first1=E. W. |last2=Jacobsen |first2=R. T. |date=2004-01-01 |title=Viscosity and Thermal Conductivity Equations for Nitrogen, Oxygen, Argon, and Air |url=https://doi.org/10.1023/B:IJOT.0000022327.04529.f3 |journal=International Journal of Thermophysics |language=en |volume=25 |issue=1 |pages=21–69 |doi=10.1023/B:IJOT.0000022327.04529.f3 |bibcode=2004IJT....25...21L |s2cid=119677367 |issn=1572-9567|url-access=subscription }}</ref> This is a common approach in the development of [[reference equations]] for gas-phase viscosities. An example of such a procedure is the Sutherland approach for the single-component gas, discussed above.
For gas mixtures consisting of simple molecules, [[Revised Enskog theory|Revised Enskog Theory]] has been shown to accurately represent both the density- and temperature dependence of the viscosity over a wide range of conditions.<ref>{{Cite journal |last1=López de Haro |first1=M. |last2=Cohen |first2=E. G. D. |last3=Kincaid |first3=J. M. |date=1983-03-01 |title=The Enskog theory for multicomponent mixtures. I. Linear transport theory |url=https://doi.org/10.1063/1.444985 |journal=The Journal of Chemical Physics |volume=78 |issue=5 |pages=2746–2759 |doi=10.1063/1.444985 |bibcode=1983JChPh..78.2746L |issn=0021-9606|url-access=subscription }}</ref><ref name=":0" />
====Blends of liquids====
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where <math>\alpha</math> is an empirical parameter, and <math>x_{1,2}</math> and <math>\mu_{1,2}</math> are the respective [[mole fractions]] and viscosities of the component liquids.{{sfn|Zhmud|2014|p=22}}
Since blending is an important process in the lubricating and oil industries, a variety of empirical and
===Solutions and suspensions===
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:<math>\mu = AT \exp\left(\frac{B}{RT}\right) \left[ 1 + C \exp\left(\frac{D}{RT}\right) \right],</math>
where <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math> are all constants, provides a good fit to experimental data over the entire range of temperatures, while at the same time reducing to the correct Arrhenius form in the low and high temperature limits. This expression, also known as Duouglas-Doremus-Ojovan model,<ref>P. Hrma, P. Ferkl, P., A.A.Kruger. Arrhenian to non-Arrhenian crossover in glass melt viscosity. J. Non-Cryst. Solids, 619, 122556 (2023). https://doi.org/10.1016/j.jnoncrysol.2023.122556</ref> can be motivated from various theoretical models of amorphous materials at the atomic level.{{sfn|Ojovan|Travis|Hand|2007|p=415107}}
A two-exponential equation for the viscosity can be derived within the Dyre shoving model of supercooled liquids, where the Arrhenius energy barrier is identified with the high-frequency [[shear modulus]] times a characteristic shoving volume.{{sfn|Dyre|Olsen|Christensen|1996|p=2171}}{{sfn | Hecksher | Dyre | 2015 | p=}} Upon specifying the temperature dependence of the shear modulus via thermal expansion and via the repulsive part of the intermolecular potential, another two-exponential equation is retrieved:{{sfn|Krausser|Samwer|Zaccone|2015|p=13762}}
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For the simplest fluids, such as dilute monatomic gases and their mixtures, ''[[ab initio]]'' [[Quantum mechanics|quantum mechanical]] computations can accurately predict viscosity in terms of fundamental atomic constants, i.e., without reference to existing viscosity measurements.{{sfn | Sharipov | Benites | 2020| p=}} For the special case of dilute helium, [[Measurement uncertainty|uncertainties]] in the ''ab initio'' calculated viscosity are two order of magnitudes smaller than uncertainties in experimental values.{{sfn | Rowland | Al Ghafri | May | 2020 | p=}}
For slightly more complex fluids and mixtures at moderate densities (i.e. [[Critical point (thermodynamics)|sub-critical densities]]) [[Revised Enskog theory|Revised Enskog Theory]] can be used to predict viscosities with some accuracy.<ref name=":0">{{Cite journal |last1=Jervell |first1=Vegard G. |last2=Wilhelmsen |first2=Øivind |date=2023-06-08 |title=Revised Enskog theory for Mie fluids: Prediction of diffusion coefficients, thermal diffusion coefficients, viscosities, and thermal conductivities |url=https://doi.org/10.1063/5.0149865 |journal=The Journal of Chemical Physics |volume=158 |issue=22 |doi=10.1063/5.0149865 |pmid=37290070 |bibcode=2023JChPh.158v4101J |s2cid=259119498 |issn=0021-9606|url-access=subscription }}</ref> Revised Enskog Theory is predictive in the sense that predictions for viscosity can be obtained using parameters fitted to other, pure-fluid [[List of thermodynamic properties|thermodynamic properties]] or [[Transport phenomena|transport properties]], thus requiring no ''a priori'' experimental viscosity measurements.
For most fluids, high-accuracy, first-principles computations are not feasible. Rather, theoretical or empirical expressions must be fit to existing viscosity measurements. If such an expression is fit to high-fidelity data over a large range of temperatures and pressures, then it is called a "reference correlation" for that fluid. Reference correlations have been published for many pure fluids; a few examples are [[water]], [[carbon dioxide]], [[ammonia]], [[benzene]], and [[xenon]].{{sfn | Huber | Perkins | Laesecke | Friend | 2009 | p=}}{{sfn | Laesecke | Muzny | 2017 | p=}}{{sfn | Monogenidou | Assael | Huber | 2018 | p=}}{{sfn | Avgeri | Assael | Huber | Perkins | 2014 | p=}}{{sfn | Velliadou | Tasidou | Antoniadis | Assael | 2021 | p=}} Many of these cover temperature and pressure ranges that encompass gas, liquid, and [[Supercritical fluid|supercritical]] phases.
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|+Viscosity of water <br>at various temperatures{{sfn|Rumble|2018|p=}}
|- style="background:#efefef;"
!Temperature
!Viscosity
|-
| align="center" | 10
| 1.
|-
| align="center" | 20
| 1.
|-
| align="center" | 30
| 0.
|-
| align="center" | 50
| 0.
|-
| align="center" | 70
| 0.
|-
| align="center" | 90
| 0.
|}
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|-
|[[Mercury (element)|Mercury]]
| 1.526
| 25
|-
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|[[Mantle (geology)]]
||≈ 10<sup>19</sup> to 10<sup>24</sup>
|{{sfn|Walzer|Hendel|Baumgardner
|}
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|edition = 3rd
|year = 1970
|isbn =
}}
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| pages = 86–96
| doi = 10.1007/s003970000120
| bibcode = 2001AcRhe..40...86C
| s2cid = 94555820
}}
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*{{cite journal|last1=Dyre|first1=J.C.|last2=Olsen|first2=N. B.|last3=Christensen|first3=T.|date=1996|title=Local elastic expansion model for viscous-flow activation energies of glass-forming molecular liquids|journal=Physical Review B|volume=53|issue=5|pages=2171–2174|doi=10.1103/PhysRevB.53.2171|pmid=9983702|bibcode=1996PhRvB..53.2171D|doi-access=free|s2cid=39833708 }}
*{{cite journal|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|access-date=2009-03-31|journal=European Journal of Physics|date=1984|volume=5|issue=4|pages=198–200|doi=10.1088/0143-0807/5/4/003|bibcode=1984EJPh....5..198E|s2cid=250769509 |archive-date=2013-03-28|archive-url=https://web.archive.org/web/20130328064508/http://www.physics.uq.edu.au/physics_museum/pitchdrop.shtml|url-status=live|url-access=subscription}}
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| edition = 3rd
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*{{cite book |last1=Hannan |first1=Henry |title=Technician's Formulation Handbook for Industrial and Household Cleaning Products |date=2007 |publisher=Kyral LLC |location=Waukesha, Wisconsin |isbn=978-0-
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*{{cite journal|title=The viscosity of five gaseous hydrocarbons|journal=The Journal of Chemical Physics|date=1977|last1=Kestin|first1=J.|last2= Khalifa|first2=H.E.|last3=Wakeham|first3= W.A.|volume= 66|issue=3|page=1132|doi=10.1063/1.434048|bibcode=1977JChPh..66.1132K}}
*{{cite journal|last1=Koocheki|first1=Arash|last2=Ghandi|first2=Amir|last3=Razavi|first3=Seyed M. A.|last4=Mortazavi|first4=Seyed Ali|last5=Vasiljevic|first5=Todor|display-authors=1|title=The rheological properties of ketchup as a function of different hydrocolloids and temperature|year=2009|journal=International Journal of Food Science & Technology|volume=44|issue=3|pages=596–602|doi = 10.1111/j.1365-2621.2008.01868.x}}
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* {{cite journal | last1=Laesecke | first1=Arno | last2=Muzny | first2=Chris D. | title=Reference Correlation for the Viscosity of Carbon Dioxide | journal=Journal of Physical and Chemical Reference Data | publisher=AIP Publishing | volume=46 | issue=1 | year=2017 | issn=0047-2689 | doi=10.1063/1.4977429 | page=013107| pmid=28736460 | pmc=5514612 | bibcode=2017JPCRD..46a3107L }}
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* {{cite book|title=IUPAC. Compendium of Chemical Terminology (the "Gold Book")|edition= 2nd |first1= A. D. |last1=McNaught |first2=A. |last2=Wilkinson|publisher= Blackwell Scientific |location= Oxford |date=1997|others= S. J. Chalk|isbn= 0-9678550-9-8|doi=10.1351/goldbook|chapter=poise}}
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* {{cite journal | last1=Rowland | first1=Darren | last2=Al Ghafri | first2=Saif Z. S. | last3=May | first3=Eric F. | title=Wide-Ranging Reference Correlations for Dilute Gas Transport Properties Based on Ab Initio Calculations and Viscosity Ratio Measurements | journal=Journal of Physical and Chemical Reference Data | publisher=AIP Publishing | volume=49 | issue=1 | date=2020-03-01 | issn=0047-2689 | doi=10.1063/1.5125100 | page=013101| bibcode=2020JPCRD..49a3101X | s2cid=213794612 | url=https://research-repository.uwa.edu.au/en/publications/2a380e21-0c3d-459a-aa07-cd95d75e08bf }}
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