Content deleted Content added
→In solids: full name when first presented |
→top: restore specific facet of over-broad ambiguous "dynamic" (Strunk's maxim is "omit *needless* words.") |
||
| (34 intermediate revisions by 28 users not shown) | |||
Line 10:
| 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}}
Line 32 ⟶ 36:
[[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]].
Line 50 ⟶ 54:
</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.
Line 80 ⟶ 84:
</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},
Line 86 ⟶ 90:
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.
Line 139 ⟶ 143:
==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.
Line 168 ⟶ 172:
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}}
Line 186 ⟶ 190:
In general, however, the viscosity of a system depends in detail on how the molecules constituting the system interact, and there are no simple but correct formulas for it. 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 1988.{{sfn|Evans | Morriss | 1988 | p=}} Although these expressions are each exact, calculating the viscosity of a dense fluid using these relations currently requires the use of [[molecular dynamics]] computer simulations. Somewhat more progress can be made for a dilute gas, as elementary assumptions about how gas molecules move and interact lead to a basic understanding of the molecular origins of viscosity. More sophisticated treatments can be constructed by systematically coarse-graining the [[equations of motion]] of the gas molecules. An example of such a treatment is [[Chapman–Enskog theory]], which derives expressions for the viscosity of a dilute gas from the [[Boltzmann equation]].{{sfn|Chapman|Cowling|1970|p=}}
===Pure gases===
Line 243 ⟶ 246:
\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}}
Line 279 ⟶ 282:
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===
Line 292 ⟶ 295:
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====
Line 306 ⟶ 309:
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===
Line 383 ⟶ 386:
:<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}}
Line 400 ⟶ 403:
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.
Line 429 ⟶ 432:
|+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.
|}
Line 656 ⟶ 659:
|[[Mantle (geology)]]
||≈ 10<sup>19</sup> to 10<sup>24</sup>
|{{sfn|Walzer|Hendel|Baumgardner
|}
Line 775 ⟶ 778:
|edition = 3rd
|year = 1970
|isbn =
}}
Line 792 ⟶ 795:
| pages = 86–96
| doi = 10.1007/s003970000120
| bibcode = 2001AcRhe..40...86C
| s2cid = 94555820
}}
Line 816 ⟶ 820:
*{{cite journal|last=Doremus|first=R.H.|date=2002|title=Viscosity of silica|journal=J. Appl. Phys.|volume=92|issue=12 |pages=7619–7629|doi = 10.1063/1.1515132 |bibcode = 2002JAP....92.7619D }}
*{{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}}
*{{cite book
| last1 = Egelstaff
Line 827 ⟶ 831:
| isbn = 978-0-19-851012-3
}}
* {{cite book | last1 = Evans | first1 = Denis J. | last2 = Morriss | first2 = Gary P. | author-link1 = Denis Evans | url = http://www.jstor.org/stable/j.ctt24h99q | title = Statistical Mechanics of Nonequilibrium Liquids | publisher = ANU Press | year = 2007 | jstor = j.ctt24h99q | isbn =
*{{cite journal | title = Transient-time-correlation functions and the rheology of fluids | journal = Physical Review A | date = October 15, 1988 | first1 = Denis J. | last1 = Evans |first2=Gary P. |last2=Morriss | volume = 38 | issue = 8 | pages = 4142–4148 | doi = 10.1103/PhysRevA.38.4142 |bibcode = 1988PhRvA..38.4142E | pmid = 9900865 }}
*{{cite book
Line 835 ⟶ 839:
| publisher = Woodhead
| year = 2009
| isbn = 978-
| edition = 3rd
}}
Line 841 ⟶ 845:
*{{cite web|last=Fluegel|first=Alexander|url=http://www.glassproperties.com/viscosity/|title=Viscosity calculation of glasses|publisher=Glassproperties.com|access-date=2010-09-14|date=2007|archive-date=2010-11-27|archive-url=https://web.archive.org/web/20101127050312/http://www.glassproperties.com/viscosity/|url-status=live}}
*{{cite web |title=Is glass liquid or solid? |last=Gibbs |first=Philip |work=math.ucr.edu |date=January 1997 |access-date=19 September 2019 |url=http://math.ucr.edu/home/baez/physics/General/Glass/glass.html |archive-date=29 March 2007 |archive-url=https://web.archive.org/web/20070329154027/http://math.ucr.edu/home/baez/physics/General/Glass/glass.html |url-status=live }}
*{{cite encyclopedia |last=Gyllenbok |first=Jan |title=Encyclopaedia of Historical Metrology, Weights, and Measures: Volume 1 |author-link=Jan Gyllenbok |encyclopedia=Encyclopaedia of Historical Metrology, Weights, and Measures|volume= 1 |year=2018 |publisher=Birkhäuser |isbn=
*{{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-
*{{Cite journal|last1=Hecksher|first1=Tina|last2=Dyre|first2=Jeppe C.|date=2015-01-01|title=A review of experiments testing the shoving model|url=https://www.sciencedirect.com/science/article/pii/S0022309314004529|journal=Journal of Non-Crystalline Solids|series=7th IDMRCS: Relaxation in Complex Systems|language=en|volume=407|pages=14–22|doi=10.1016/j.jnoncrysol.2014.08.056|bibcode=2015JNCS..407...14H|issn=0022-3093|access-date=2021-10-17|archive-date=2022-02-15|archive-url=https://web.archive.org/web/20220215003710/https://www.sciencedirect.com/science/article/abs/pii/S0022309314004529|url-status=live|url-access=subscription}}
*{{cite book
| last1 = Hildebrand
Line 872 ⟶ 876:
}}
*{{cite journal|last1=Kestin|first1=J.|last2=Ro|first2=S. T.|last3=Wakeham|first3=W. A.|title=Viscosity of the Noble Gases in the Temperature Range 25–700 °C|journal=The Journal of Chemical Physics|volume=56|issue=8|year=1972|pages=4119–4124|issn=0021-9606|doi=10.1063/1.1677824|bibcode=1972JChPh..56.4119K|doi-access=free}}
*{{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}}
Line 899 ⟶ 903:
* {{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 }}
* {{cite book|last1=Landau|first1=L. D.|last2=Lifshitz|first2=E.M.|author-link1=Lev Landau|author-link2=Evgeny Lifshitz|title=Fluid Mechanics|url=https://books.google.com/books?id=eVKbCgAAQBAJ|edition=2nd|year=1987|publisher=Elsevier|isbn=978-0-08-057073-0|access-date=2019-09-18|archive-date=2020-03-21|archive-url=https://web.archive.org/web/20200321213113/https://books.google.com/books?id=eVKbCgAAQBAJ|url-status=live}}
* {{cite journal | last1=Maginn | first1=Edward J. | last2=Messerly | first2=Richard A. | last3=Carlson | first3=Daniel J. | last4=Roe | first4=Daniel R. | last5=Elliott | first5=J. Richard | title=Best Practices for Computing Transport Properties 1. Self-Diffusivity and Viscosity from Equilibrium Molecular Dynamics
* {{cite journal | last1=Monogenidou | first1=S. A. | last2=Assael | first2=M. J. | last3=Huber | first3=M. L. | title=Reference Correlation for the Viscosity of Ammonia from the Triple Point to 725 K and up to 50 MPa | journal=Journal of Physical and Chemical Reference Data | publisher=AIP Publishing | volume=47 | issue=2 | year=2018 | issn=0047-2689 | doi=10.1063/1.5036724 | page=023102| pmid=31092958 | pmc=6512859 | bibcode=2018JPCRD..47b3102M }}
* {{cite book|first=Marcel|last=Lesieur|title=Turbulence in Fluids: Stochastic and Numerical Modelling|url=https://books.google.com/books?id=QILpCAAAQBAJ&pg=PR2|date=2012|publisher=Springer|isbn=978-94-009-0533-7|access-date=2018-11-30|archive-date=2020-03-14|archive-url=https://web.archive.org/web/20200314133321/https://books.google.com/books?id=QILpCAAAQBAJ&pg=PR2|url-status=live}}
Line 905 ⟶ 909:
* {{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}}
* {{cite book | last=Millat | first=Jorgen | title=Transport Properties of Fluids : Their Correlation, Prediction and Estimation | publisher=Cambridge University Press | publication-place=Cambridge | year=1996 | isbn=978-0-521-02290-3 | oclc=668204060}}
* {{cite journal|last1=Mueller|first1=S.|last2=Llewellin|first2=E. W.|last3=Mader|first3=H. M.|title=The rheology of suspensions of solid particles|journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=466|issue=2116|year=2009|pages=1201–1228|issn=1364-5021|doi=10.1098/rspa.2009.0445|doi-access=
* {{cite book|chapter=dynamic viscosity, ''η''|doi=10.1351/goldbook|title=IUPAC Compendium of Chemical Terminology|date=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|display-editors=1|isbn=978-0-9678550-9-7}}
* {{cite journal|last1=Ojovan|first1=M.I.|last2=Lee|first2=W.E.|date=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|bibcode = 2004JAP....95.3803O }}
* {{cite journal|last1=Ojovan|first1=M.I.|last2=Travis|first2=K. P.|last3=Hand|first3=R.J.|date=2007|title=Thermodynamic parameters of bonds in glassy materials from viscosity-temperature relationships|journal=J. Phys.: Condens. Matter|volume=19|issue=41|page=415107|doi=10.1088/0953-8984/19/41/415107|pmid=28192319|bibcode=2007JPCM...19O5107O|s2cid=24724512 |url=http://eprints.whiterose.ac.uk/4058/1/ojovanm1.pdf|access-date=2019-09-27|archive-date=2018-07-25|archive-url=https://web.archive.org/web/20180725094232/http://eprints.whiterose.ac.uk/4058/1/ojovanm1.pdf|url-status=live}}
* {{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|date=1989|last1=Plumb|first1=Robert C.|journal=Journal of Chemical Education|volume=66|issue=12|page=994|bibcode=1989JChEd..66..994P|access-date=2013-12-25|archive-date=2005-08-26|archive-url=https://web.archive.org/web/20050826033925/http://dwb.unl.edu/Teacher/NSF/C01/C01Links/www.ualberta.ca/~bderksen/windowpane.html|url-status=dead|url-access=subscription}}
* {{cite book | last1 = Rapaport | first1 = D.C. | url = https://www.cambridge.org/us/academic/subjects/physics/mathematical-methods/art-molecular-dynamics-simulation-2nd-edition?format=HB&isbn=
* {{cite book|last1=Reid|first1= Robert C.|last2= Sherwood|first2= Thomas K. |date=1958|title= The Properties of Gases and Liquids|publisher= McGraw-Hill }}
* {{Citation|last1 = Reif|first1 = F.|title = Fundamentals of Statistical and Thermal Physics|publisher = McGraw-Hill|year = 1965}}. An advanced treatment.
Line 916 ⟶ 920:
* {{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 }}
* {{cite journal|last1=Różańska|first1=S.|first2=J.|last2=Różański|first3=M.|last3=Ochowiak|first4=P. T.|last4=Mitkowski|title=Extensional viscosity measurements of concentrated emulsions with the use of the opposed nozzles device|journal=Brazilian Journal of Chemical Engineering|volume=31|issue=1|date=2014|pages=47–55|doi=10.1590/S0104-66322014000100006|issn=0104-6632|url=http://www.scielo.br/pdf/bjce/v31n1/06.pdf|doi-access=free|access-date=2019-09-19|archive-date=2020-05-08|archive-url=https://web.archive.org/web/20200508203430/https://www.scielo.br/pdf/bjce/v31n1/06.pdf|url-status=live}}
*{{cite book|editor-first=John R. |editor-last=Rumble|title=CRC Handbook of Chemistry and Physics|edition=99th|year=2018|publisher=CRC Press|location=Boca Raton, FL|isbn=978-
* {{cite journal | last1=Sagdeev | first1=Damir | last2=Gabitov | first2=Il'giz | last3=Isyanov | first3=Chingiz | last4=Khairutdinov | first4=Vener | last5=Farakhov | first5=Mansur | last6=Zaripov | first6=Zufar | last7=Abdulagatov | first7=Ilmutdin | title=Densities and Viscosities of Oleic Acid at Atmospheric Pressure | journal=Journal of the American Oil Chemists' Society | publisher=Wiley | volume=96 | issue=6 | date=2019-04-22 | issn=0003-021X | doi=10.1002/aocs.12217 | pages=647–662| s2cid=150156106 }}
*{{cite journal|doi=10.1016/0022-3093(88)90086-5|title=Viscoelasticity in silica gel|date=1988|last1=Scherer|first1=George W.|last2=Pardenek|first2=Sandra A.|last3=Swiatek|first3=Rose M.|journal=Journal of Non-Crystalline Solids|volume=107|issue=1|page=14|bibcode = 1988JNCS..107...14S }}
Line 927 ⟶ 931:
* {{cite journal | last1=Trachenko | first1=K. | last2=Brazhkin | first2=V. V. | title=Minimal quantum viscosity from fundamental physical constants | journal=Science Advances | publisher=American Association for the Advancement of Science (AAAS) | volume=6 | issue=17 | date=2020-04-22 | pages=eaba3747 | issn=2375-2548 | doi=10.1126/sciadv.aba3747| pmid=32426470 | pmc=7182420 | arxiv=1912.06711 | bibcode=2020SciA....6.3747T }}
* {{cite journal | last1=Trachenko | first1=Kostya | last2=Brazhkin | first2=Vadim V. | title=The quantum mechanics of viscosity | journal=Physics Today | publisher=AIP Publishing | volume=74 | issue=12 | date=2021-12-01 | issn=0031-9228 | doi=10.1063/pt.3.4908 | pages=66–67 | bibcode=2021PhT....74l..66T | s2cid=244831744 | url=https://ccmmp.ph.qmul.ac.uk/~kostya/The%20Purcell%20question.pdf | access-date=2022-01-10 | archive-date=2022-01-10 | archive-url=https://web.archive.org/web/20220110000400/https://ccmmp.ph.qmul.ac.uk/~kostya/The%20Purcell%20question.pdf | url-status=live }}
*{{cite journal|last1=Trouton|first1=Fred. T.|title=On the Coefficient of Viscous Traction and Its Relation to that of Viscosity|journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=77|issue=519|year=1906|pages=426–440|issn=1364-5021|doi=10.1098/rspa.1906.0038|bibcode=1906RSPSA..77..426T|doi-access=free}}
* {{cite journal | last1=Velliadou | first1=Danai | last2=Tasidou | first2=Katerina A. | last3=Antoniadis | first3=Konstantinos D. | last4=Assael | first4=Marc J. | last5=Perkins | first5=Richard A. | last6=Huber | first6=Marcia L. | title=Reference Correlation for the Viscosity of Xenon from the Triple Point to 750 K and up to 86 MPa | journal=International Journal of Thermophysics | publisher=Springer Science and Business Media LLC | volume=42 | issue=5 | date=2021-03-25 | page=74 | issn=0195-928X | doi=10.1007/s10765-021-02818-9| pmid=34393314 | pmc=8356199 | bibcode=2021IJT....42...74V }}
*{{cite book|last1=Viswanath|first1=D.S.|last2=Natarajan|first2=G.|title=Data Book on the Viscosity of Liquids|publisher=Hemisphere Publishing Corporation|year=1989|isbn=0-89116-778-1}}
*{{cite book|last1=Viswanath|first1=Dabir S.|last2=Ghosh|first2=Tushar K.|last3=Prasad|first3=Dasika H.L.|last4=Dutt|first4=Nidamarty V.K.|last5=Rani|first5=Kalipatnapu Y.|display-authors=1|title=Viscosity of Liquids: Theory, Estimation, Experiment, and Data|publisher=Springer|year=2007|isbn=978-1-4020-5481-5}}
*{{citation|title=Mantle Viscosity and the Thickness of the Convective Downwellings|first1=Uwe|last1=Walzer|first2=Roland|last2=Hendel|first3=John|last3=Baumgardner |
*{{cite journal
|last1=Xie
| |||