Viscosity
40492
225953020
2008-07-16T04:42:08Z
128.250.80.15
/* Viscosity measurement */
{{otheruses}}
{{Continuum mechanics}}
'''Viscosity''' is a measure of the [[Drag (physics)|resistance]] of a [[fluid]] which is being deformed by either [[shear stress]] or [[extensional stress]]. In general terms it is the resistance of a liquid to flow, or its "thickness". Viscosity describes a fluid's internal resistance to flow and may be thought of as a measure of fluid [[friction]]. Thus, [[water]] is "thin", having a lower viscosity, while [[vegetable oil]] is "thick" having a higher viscosity. All real fluids (except [[superfluid]]s) have some resistance to [[Stress (physics)|stress]], but a fluid which has no resistance to shear stress is known as an '''ideal fluid''' or '''inviscid fluid'''. For example a high viscosity magma will create a tall volcano, because it cannot spread fast enough, low viscosity lava will create a shield volcano, which is large and wide.<ref>{{cite book
| author = Symon, Keith
| title = Mechanics
| edition= Third Edition
| publisher = Addison-Wesley
| year = 1971
| id = ISBN 0-201-07392-7}}</ref> The study of viscosity is known as [[rheology]].
==Etymology==
The word "viscosity" derives from the [[Latin]] word "{{lang|la|viscum}}" for [[mistletoe]]. A viscous glue was made from mistletoe berries and used for lime-twigs to catch birds.<ref> [http://www.etymonline.com/index.php?term=viscous The Online Etymology Dictionary]</ref>
==Viscosity coefficients==
When looking at a value for viscosity, the number that one most often sees is the coefficient of viscosity. There are several different viscosity coefficients depending on the nature of applied stress and nature of the fluid. They are introduced in the main books on [[hydrodynamics]]<ref> Happel, J. and Brenner , H. "Low Reynolds number hydrodynamics", ''Prentice-Hall'', (1965)</ref><ref> Landau, L.D. and Lifshitz, E.M. "Fluid mechanics", ''Pergamon Press'',(1959)</ref> and [[rheology]].<ref> Barnes, H.A. "A Handbook of Elementary Rheology", Institute of Non-Newtonian Fluid mechanics, UK (2000)</ref>
*'''Dynamic viscosity''' determines the dynamics of an [[incompressible]] [[Newtonian fluid]];
*'''Kinematic viscosity''' is the ''dynamic viscosity'' divided by the density for a Newtonian fluid;
*'''[[Volume viscosity]]''' determines the dynamics of a compressible [[Newtonian fluid]];
*'''[[Bulk viscosity]]''' is the same as ''volume viscosity''
*'''Shear viscosity''' is the viscosity coefficient when the applied stress is a [[shear stress]] (valid for non-Newtonian fluids);
*'''[[Extensional viscosity]]''' is the viscosity coefficient when the applied stress is an [[extensional stress]] (valid for non-Newtonian fluids).
''Shear viscosity'' and ''dynamic viscosity'' are much better known than the others. That is why they are often referred to as simply ''viscosity''.
Simply put, this quantity is the ratio between the pressure exerted on the surface of a fluid, in the lateral or horizontal direction, to the change in velocity of the fluid as you move down in the fluid (this is what is referred to as a velocity [[gradient]]). For example, at room temperature, water has a nominal viscosity of 1.0 × 10<sup>-3</sup> Pa∙s and motor oil has a nominal apparent viscosity of 250 × 10<sup>-3</sup> Pa∙s.<ref>{{cite book|author=Raymond A. Serway| title=Physics for Scientists & Engineers|edition=4th Edition| publisher=Saunders College Publishing| year=1996|id=ISBN 0-03-005932-1}}</ref>
:''Extensional viscosity'' is widely used for characterizing polymers.
:''Volume viscosity'' is essential for [[Acoustics]] in fluids, see [[Stokes' law (sound attenuation)]] <ref> Dukhin, A.S. and Goetz, P.J. "Ultrasound for characterizing colloids", Elsevier, (2002)</ref>
==Newton's theory==
[[Image:Laminar shear.png|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. This type of flow is known as a [[Couette flow]].]]
[[Image:Laminar shear flow.PNG|thumb|right|320px|Laminar shear, the non-constant gradient, is a result of the geometry the fluid is flowing through (e.g. a pipe).]]
In general, in any flow, layers move at different [[velocity|velocities]] and the fluid's viscosity arises from the shear stress between the layers that ultimately opposes any applied force.
[[Isaac Newton]] postulated that, for straight, [[Parallel (geometry)|parallel]] and uniform flow, the shear stress, τ, between layers is proportional to the [[velocity]] [[gradient]], ∂''u''/∂''y'', in the direction [[perpendicular]] to the layers.
:<math>\tau=\eta \frac{\partial u}{\partial y}</math>.
Here, the constant η is known as the ''coefficient of viscosity'', the ''viscosity'', the ''dynamic viscosity'', or the ''Newtonian viscosity''. Many [[fluid]]s, such as [[water]] and most [[gas]]es, satisfy Newton's criterion and are known as [[Newtonian fluid]]s. [[Non-Newtonian fluid]]s exhibit a more complicated relationship between shear stress and velocity gradient than simple linearity.
The relationship between the shear stress and the velocity gradient can also be obtained by considering two plates closely spaced apart at a distance ''y'', and separated by a [[heterogeneous|homogeneous]] substance. Assuming that the plates are very large, with a large area ''A'', such that edge effects may be ignored, and that the lower plate is fixed, let a force ''F'' be applied to the upper plate. If this force causes the substance between the plates to undergo shear flow (as opposed to just [[deformation|shearing]] [[elasticity (solid mechanics)|elastically]] until the shear stress in the substance balances the applied force), the substance is called a fluid. The applied force is proportional to the area and velocity of the plate and inversely proportional to the distance between the plates. Combining these three relations results in the equation ''F = η(Au/y)'', where η is the proportionality factor called the ''absolute viscosity'' (with units Pa·s = kg/(m·s) or slugs/(ft·s)). The absolute viscosity is also known as the ''dynamic viscosity'', and is often shortened to simply ''viscosity''. The equation can be expressed in terms of shear stress; ''τ = F/A = η(u/y)''. The rate of shear deformation is <math>u/y</math> and can be also written as a shear velocity, ''du/dy''. Hence, through this method, the relation between the shear stress and the velocity gradient can be obtained.
[[James Clerk Maxwell]] called viscosity ''fugitive elasticity'' because of the analogy that elastic deformation opposes shear stress in [[solid]]s, while in viscous [[fluid]]s, shear stress is opposed by ''rate'' of deformation.
==Viscosity measurement==
Dynamic viscosity is measured with various types of [[rheometer]]. Close temperature control of the fluid is essential to accurate measurements, particularly in materials like lubricants, whose viscosity can double with a change of only 5 °C. For some fluids, it is a constant over a wide range of shear rates. These are [[Newtonian fluids]].
The fluids without a constant viscosity are called [[Non-Newtonian fluid]]s. Their viscosity cannot be described by a single number. Non-Newtonian fluids exhibit a variety of different correlations between shear stress and shear rate.
One of the most common instruments for measuring kinematic viscosity is the glass capillary viscometer.
In paint industries, viscosity is commonly measured with a [[Zahn cup]], in which the [[efflux time]] is determined and given to customers. The efflux time can also be converted to kinematic viscosities (cSt) through the conversion equations.
Also used in paint, 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.
Vibrating viscometers can also be used to measure viscosity. These models such as the ''Dynatrol'' use vibration rather than rotation to measure viscosity.
''Extensional viscosity'' can be measured with various [[rheometer]]s that apply [[extensional stress]]
[[Volume viscosity]] can be measured with [[acoustic rheometer]].
===Units of measure===
====Viscosity (dynamic/absolute viscosity)====
Dynamic viscosity and absolute viscosity are synonymous. The [[IUPAC]] symbol for dynamic viscosity is the Greek letter eta (<math>{\eta}</math>) <ref>[http://goldbook.iupac.org/D01877.html IUPAC Gold Book, Definition of (dynamic) viscocity]</ref>, but it is also commonly referred to using the Greek symbol mu (<math>{\mu}</math>). The [[SI]] [[physical unit]] of dynamic viscosity is the [[pascal (unit)|pascal]]-[[second]] (Pa·s), which is identical to [[kilogram|kg]]·m<sup>−1</sup>·s<sup>−1</sup>. 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]].
The name [[poiseuille]] (Pl) was proposed for this unit (after [[Jean Louis Marie Poiseuille]] who formulated [[Poiseuille's law]] of viscous flow), but not accepted internationally. Care must be taken in not confusing the poiseuille with the [[poise]] named after the same person.
The [[cgs]] [[physical unit]] for dynamic viscosity is the ''poise''<ref>[http://www.iupac.org/goldbook/P04705.pdf#search=%22poise%20iupac%22 IUPAC definition of the Poise]</ref> (P), named after [[Jean Louis 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.
:1 P = 1 g·cm<sup>−1</sup>·s<sup>−1</sup>
The relation between poise and pascal-seconds is:
:10 P = 1 kg·m<sup>−1</sup>·s<sup>−1</sup> = 1 Pa·s
:1 cP = 0.001 Pa·s = 1 mPa·s
====Kinematic viscosity====
In many situations, we are concerned with the ratio of the viscous force to the [[inertia]]l force, the latter characterised by the [[fluid]] [[density]] ρ. This ratio is characterised by the ''kinematic viscosity'' (<math>\nu </math>), defined as follows:
:<math>\nu = \frac {\mu} {\rho}</math>,
or
:<math>\nu = \frac{\eta}{\rho}</math>.
where <math>\mu \ {\rm or} \ \eta</math> is the (dynamic or absolute) viscosity (in centipoise cP),
and <math>\rho</math> is the density (in grams/cm^3),
and <math>\nu </math> is the kinematic viscosity (in centistokes cSt ).
Kinematic viscosity (Greek symbol: <math>{\nu}</math>) has SI units Pa.s/(kg/m<sup>3</sup>) = m<sup>2</sup>·s<sup>−1</sup>. 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 or ctsk). In U.S. usage, ''stoke'' is sometimes used as the singular form.
:1 stokes = 100 centistokes = 1 cm<sup>2</sup>·s<sup>−1</sup> = 0.0001 m<sup>2</SUP>·s<sup>−1</sup>.
:1 centistokes = 1 mm<sup>2</sup>·s<sup>-1</sup> = 10<sup>-6</sup>m<sup>2</sup>·s<sup>−1</sup>
====Saybolt Universal Viscosity ====
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
Seconds (SUS). <ref> ASTM D 2161, Page one,(2005)</ref> Kinematic viscosity in centistoke can be converted from SUS according to the arithmetic and the reference tabel provided in [[ASTM]] D 2161. It can also be converted in computerized method, or vice versa.<ref>[http://www.uniteasy.com/en/unitguide/Viscosity.htm Quantities and Units of Viscosity]</ref>
====Relation to Mean Free Path of Diffusing Particles====
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, <math> \lambda </math>, of the diffusing particles.
From [[fluid mechanics]], [[shear stress]], <math> \tau </math>, is the rate of change of velocity with distance perpendicular to the direction of movement.
:<math>\tau = \mu \frac{du}{dx}</math>.
Interpreting shear stress as the time rate of change of [[momentum]],p, per unit area (rate of momentum flux) of an arbitrary control surface gives
:<math>\tau = \frac{\dot{p}}{A} = \frac{\dot{m} u}{A}</math>.
Further manipulation will show
:<math>\frac{\dot{p}}{u} = \dot{m} = \rho \bar{u} A \; \; \Rightarrow \; \; \tau = \underbrace{2 \rho \bar{u} \lambda}_{\mu} \cdot \frac{du}{dx} \; \; \Rightarrow \; \; \nu = \frac{\mu}{\rho} = 2 \bar{u} \lambda</math>
where
:<math>\dot{m}</math> is the rate of change of mass
:<math>\rho</math> is the density of the fluid
:<math>\bar{u}</math> is the average molecular speed
:<math>\mu</math> is the dynamic viscosity.
====Dynamic versus kinematic viscosity====
Conversion between kinematic and dynamic viscosity is given by <math>\nu \rho = \mu</math>.
For example,
:if <math>\nu = </math>0.0001 m<sup>2</sup>·s<sup>-1</sup> and <math>\rho = </math>1000 kg m<sup>-3</sup> then <math>\mu = \nu \rho = </math>0.1 kg·m<sup>−1</sup>·s<sup>−1</sup> = 0.1 Pa·s
:if <math>\nu = </math>1 St (= 1 cm<sup>2</sup>·s<sup>−1</sup>) and <math>\rho = </math>1 g cm<sup>-3</sup> then <math>\mu = \nu \rho = </math>1 g·cm<sup>−1</sup>·s<sup>−1</sup> = 1 P
A plot of the kinematic viscosity of air as a function of absolute temperature is available on the Internet.<ref>
[http://users.wpi.edu/~ierardi/PDF/air_nu_plot.PDF James Ierardi's Fire Protection Engineering Site]</ref>
====Example: viscosity of water====
Because of its density of <math>\rho</math> = 1 g/cm<sup>3</sup> (varies slightly with temperature), and its dynamic viscosity is near 1 mPa·s (see [[#Viscosity of water]] section), the viscosity values of water are, to rough precision, all powers of ten:
Dynamic viscosity:
:<math>{\mu}</math> = 1 mPa·s = 10<sup>-3</sup> Pa·s = 1 cP = 10<sup>-2</sup> poise
Kinematic viscosity:
:<math>{\nu}</math> = 1 cSt = 10<sup>-2</sup> stokes = 1 mm²/s
==Molecular origins==
[[Image:University of Queensland Pitch drop experiment-6-2.jpg|thumb|right|250px|[[pitch drop experiment|Pitch]] has a viscosity approximately 100 billion times that of water.]]
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. Although these expressions are each exact in order to calculate the viscosity of a dense fluid, using these relations requires the use of [[molecular dynamics]] computer simulations.
===Gases===
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.
Within the regime where the theory is applicable:
*Viscosity is independent of pressure and
*Viscosity increases as temperature increases.
[[James Clerk Maxwell]] published a famous paper in 1866 using the kinetic theory of gases to study gaseous viscosity. (Reference: J.C. Maxwell, "On the viscosity or internal friction of air and other gases", Philosophical Transactions of the Royal Society of London, vol. 156 (1866), pp. 249-268.)
====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:
:<math> {\eta} = {\eta}_0 \frac {T_0+C} {T + C} \left (\frac {T} {T_0} \right )^{3/2} </math>
where:
*<math>{\eta}</math> = viscosity in (Pa·s) at input temperature <math>T</math>
*<math>{\eta}_0</math> = reference viscosity in (Pa·s) at reference temperature <math>T_0</math>
*<math>T</math> = input temperature in kelvin
*<math>T_0</math> = reference temperature in kelvin
*<math>C</math> = Sutherland's constant for the gaseous material in question
Valid for temperatures between 0 < <math>T</math> < 555 K with an error due to pressure less than 10% below 3.45 MPa
Sutherland's constant and reference temperature for some gases
{| class="wikitable"
|- bgcolor="#efefef"
! Gas
! <math>C</math>
[K]
! <math>T_0</math>
[K]
! <math>{\eta}_0</math>
[10<sup>-6</sup> Pa s]
<!--
|-
|
| -
| K
| 10<sup>-6</sup> Pa s
-->
|-
| [[air]]
| 120
| 291.15
| 18.27
|-
| [[nitrogen]]
| 111
| 300.55
| 17.81
|-
| [[oxygen]]
| 127
| 292.25
| 20.18
|-
| [[carbon dioxide]]
| 240
| 293.15
| 14.8
|-
| [[carbon monoxide]]
| 118
| 288.15
| 17.2
|-
| [[hydrogen]]
| 72
| 293.85
| 8.76
|-
| [[ammonia]]
| 370
| 293.15
| 9.82
|-
| [[sulfur dioxide]]
| 416
| 293.65
| 12.54
|-
| [[helium]]
| 79.4 <ref>[http://arxiv.org/pdf/physics/0410237.pdf data constants for sutherland's formula]</ref>
| 273
| 19 <ref>[http://hyperphysics.phy-astr.gsu.edu/Hbase/tables/viscosity.html Viscosity of liquids and gases]</ref>
|}
(also see: <ref>http://www.epa.gov/EPA-AIR/2005/July/Day-13/a11534d.htm</ref>)
====Viscosity of a dilute gas====
The [[Chapman-Enskog equation]]<ref>{{cite book|author= J.O. Hirshfelder, C.F. Curtis and R.B. Bird|title=Molecular theory of gases and liquids|edition=First Edition|publisher= Wiley|year=1964|id=ISBN 0-471-40065-3}}</ref> may be used to estimate viscosity for a dilute gas. This equation is based on semi-theorethical assumption by Chapman and Enskoq. 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*)).
:<math> {\eta}_0 \times 10^7 = {266.93}\frac {(MT)^{1/2}} {\sigma^{2}\omega(T^*)}</math>
*T*=κT/ε Reduced temperature (dimensionless)
*<math> {\eta}_0 </math> = viscosity for dilute gas (uP)
*<math> M </math> = molecular mass (g/mol)
*<math> T </math> = temperature (K)
*<math> {\sigma}</math> = the collision diameter (Å)
*<math>{\epsilon}/{\kappa} </math> = the maximum energy of attraction divided by the Boltzmann constant (K)
*<math> {\omega}_{\eta } </math> = the collision integral
===Liquids===
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.{{Fact|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=Robert E. Maples|title=Petroleum Refinery Process Economics|edition=2nd Edition|publisher=Pennwell Books|date=2000|id=ISBN 0-87814-779-9}}</ref><ref>C.T. Baird (1989), ''Guide to Petroleum Product Blending'', HPI Consultants, Inc. [http://www.hpiconsultants.com/blending/index.htm HPI website]</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) <math>\mbox{VBN} = 14.534 \times ln[ln(v + 0.8)] + 10.975\,</math>
where ''v'' 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) <math>\mbox{VBN}_\mbox{Blend} = [x_A \times \mbox{VBN}_A] + [x_B \times \mbox{VBN}_B] + ... + [x_N \times \mbox{VBN}_N]\,</math>
where <math>x_X</math> 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 ''v'':
:(3) <math>v = e^{e^{\frac{VBN_{Blend} - 10.975}{14.534}}} - 0.8</math>
where <math>VBN_{Blend}</math> is the viscosity blending number of the blend.
==Viscosity of selected substances==
The viscosity of air and water are by far the two most important materials for aviation aerodynamics and shipping fluid dynamics. Temperature plays the main role in determining viscosity.
===Viscosity of air===
The viscosity of air depends mostly on the temperature.
At 15.0 °C, the viscosity of air is 1.78 × 10<sup>−5</sup> kg/(m·s) or 1.78 × 10<sup>−4</sup> P. One can get the viscosity of air as a function of temperature from the [http://www.lmnoeng.com/Flow/GasViscosity.htm Gas Viscosity Calculator]
===Viscosity of water===
The viscosity of water is 8.90 × 10<sup>−4</sup> Pa·s or 8.90 × 10<sup>−3</sup> dyn·s/cm<sup>2</sup> or 0.890 cP at about 25 °C.<br>
As a function of temperature ''T'' (K):
''μ''(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.
Viscosity of water at different temperatures is listed below.
{| class="wikitable"
|- bgcolor="#efefef"
!Temperature
[ºC]
!viscosity
[Pa·s]
|-
|10
|1.308 × 10<sup>−3</sup>
|-
|20
|1.003 × 10<sup>−3</sup>
|-
|30
|7.978 × 10<sup>−4</sup>
|-
|40
|6.531 × 10<sup>−4</sup>
|-
|50
|5.471 × 10<sup>−4</sup>
|-
|60
|4.668 × 10<sup>−4</sup>
|-
|70
|4.044 × 10<sup>−4</sup>
|-
|80
|3.550 × 10<sup>−4</sup>
|-
|90
|3.150 × 10<sup>−4</sup>
|-
|100
|2.822 × 10<sup>−4</sup>
|}
===Viscosity of various materials===
[[Image:Drop 0.jpg|thumb|right|200px|Example of the viscosity of milk and water. Liquids with higher viscosities will not make such a splash when poured at the same velocity.]]
[[Image:Runny hunny.jpg|thumb|[[Honey]] being drizzled.]]
[[Image:PeanutButter.jpg|thumb|[[Peanut butter]] is a [[semi-solid]] and so can hold peaks.]]
Some dynamic viscosities of Newtonian fluids are listed below:
[[Gas]]es (at 0 °[[celsius|C]]):
{| class="wikitable"
|- bgcolor="#efefef"
!
!viscosity
[Pa·s]
|-
|[[hydrogen]]
|8.4 × 10<sup>−6</sup>
|-
|[[Earth's atmosphere|air]]
|17.4 × 10<sup>−6</sup>
|-
|[[xenon]]
|2.12 × 10<sup>−5</sup>
|}
[[Liquid]]s (at 25 °[[celsius|C]]):
{| class="wikitable"
|- bgcolor="#efefef"
!
!viscosity
[Pa·s]
!viscosity
[cP]
|-
|[[liquid nitrogen]] @ 77K
|1.58 × 10<sup>−4</sup>
|0.158
|-
|[[acetone]]*
|3.06 × 10<sup>−4</sup>
|0.306
|-
|[[methanol]]*
|5.44 × 10<sup>−4</sup>
|0.544
|-
|[[benzene]]*
|6.04 × 10<sup>−4</sup>
|0.604
|-
|[[water]]
|8.94 × 10<sup>−4</sup>
|0.894
|-
|[[ethanol]]*
|1.074 × 10<sup>−3</sup>
|1.074
|-
|[[mercury (element)|mercury]]*
|1.526 × 10<sup>−3</sup>
|1.526
|-
|[[nitrobenzene]]*
|1.863 × 10<sup>−3</sup>
|1.863
|-
|[[Propan-1-ol|propanol]]*
|1.945 × 10<sup>−3</sup>
|1.945
|-
|[[Ethylene glycol]]
|1.61 × 10<sup>−2</sup>
|16.1
|-
|[[sulfuric acid]]*
|2.42 × 10<sup>−2</sup>
|24.2
|-
|[[olive oil]]
|.081
|81
|-
|[[glycerol]]*
|.934
|934
|-
|[[castor bean|castor oil]]*
|.985
|985
|-
|[[corn syrup]]*
|1.3806
|1380.6
|-
|[[Fuel oil|HFO-380]]
|2.022
|2022
|-
|[[pitch (resin)|pitch]]
|2.3 × 10<sup>8</sup>
|2.3 × 10<sup>11</sup>
|}
<nowiki>*</nowiki> Data from CRC Handbook of Chemistry and Physics, 73<sup>rd</sup> edition, 1992-1993.
[[Fluid]]s with variable compositions, such as [[honey]], can have a wide range of viscosities.
A more complete table can be found at [http://xtronics.com/reference/viscosity.htm Transwiki], including the following:
{| class="wikitable"
|- bgcolor="#efefef"
!
!viscosity
[cP]
|-
|[[honey]]
|2,000–10,000
|-
|[[molasses]]
|5,000–10,000
|-
|molten [[glass]]
|10,000–1,000,000
|-
|[[chocolate syrup]]
|10,000–25,000
|-
|molten [[chocolate]]<sup>*</sup>
| 45,000–130,000 <ref>{{cite web |url=http://www.brookfieldengineering.com/education/applications/laboratory-chocolate-processing.asp |title=Chocolate Processing |accessdate=2007-12-03 |format= |work=[[Brookfield Engineering]] website}}</ref>
|-
|[[ketchup]]<sup>*</sup>
|50,000–100,000
|-
|[[peanut butter]]
|~250,000
|-
|[[shortening]]<sup>*</sup>
|~250,000
|}
<nowiki>*</nowiki> These materials are highly [[non-Newtonian fluid|non-Newtonian]].
== Viscosity of solids ==
On the basis that all solids, such as [[granite]]<ref>{{cite journal
| last = Kumagai
| first = Naoichi
| coauthors = Sadao Sasajima, Hidebumi Ito
| title = Long-term Creep of Rocks: Results with Large Specimens Obtained in about 20 Years and Those with Small Specimens in about 3 Years
| journal = Journal of the Society of Materials Science (Japan)
| volume = 27
| issue = 293
| pages = 157–161
| publisher = Japan Energy Society
| 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
| date = 15 February 1978
| accessdate = 06-16-2008}}</ref> flow to a small extent in response to [[shear stress]] some researchers<ref>{{cite web|url=http://hypertextbook.com/physics/matter/viscosity/|work=The Physics Hypertextbook| last=Elert|first=Glenn|title=Viscosity}}</ref><!--<ref>[http://web.umr.edu/~brow/PDF_viscosity.pdf The Properties of Glass ], page 6, retrieved on August 1, 2007</ref>--> have contended that substances known as [[amorphous solid]]s, such as [[glass]] and many [[polymers]], may be considered to have viscosity. This has led some to the view that [[solid]]s are simply [[liquid]]s with a very high viscosity, typically greater than 10<sup>12</sup> Pa·s. This position is often adopted by supporters of the widely held misconception that [[Glass#Behavior_of_antique_glass|glass flow]] can be observed in old buildings. This distortion is more likely the result of the glass making process rather than the viscosity of glass.<ref>"Antique windowpanes and the flow of supercooled liquids", by Robert C. Plumb, (Worcester Polytech. Inst., Worcester, MA, 01609, USA), J. Chem. Educ. (1989), 66 (12), 994-6</ref>
However, others argue that [[solid]]s are, in general, elastic for small stresses while [[fluid]]s are not.<ref>{{cite web|
|last = Gibbs
|first = Philip
|title = Is Glass a Liquid or a Solid?
|url = http://math.ucr.edu/home/baez/physics/General/Glass/glass.html
|accessdate = 2007-07-31}}</ref> Even if [[solid]]s flow at higher stresses, they are characterized by their low-stress behavior. Viscosity may be an appropriate characteristic for [[solid]]s in a [[plasticity (physics)|plastic]] regime. The situation becomes somewhat confused as the term ''viscosity'' is sometimes used for solid materials, for example [[Maxwell material]]s, to describe the relationship between stress and the rate of change of strain, rather than rate of shear.
These distinctions may be largely resolved by considering the constitutive equations of the material in question, which take into account both its viscous and elastic behaviors. Materials for which both their viscosity and their elasticity are important in a particular range of deformation and deformation rate are called [[viscoelasticity|''viscoelastic'']]. In [[geology]], earth materials that exhibit viscous deformation at least three times greater than their elastic deformation are sometimes called [[rheid]]s.
<!-- SEE ALSO:
GEORGE W. SCHERER, SANDRA A. PARDENEK and ROSE M. SWIATEK; “Viscoelasticity in silica gel”; Journal of Non-Crystalline Solids; Elsevier Science, Amsterdam; 02 December 1988; 107 (1): pp. 14–22.
-->
== Viscosity of amorphous materials ==
[[Image:Glassviscosityexamples.png|300px|thumb|Common [[glass]] viscosity curves.<ref>[http://www.glassproperties.com/viscosity/ Viscosity calculation of glasses]</ref>]]
Viscous flow in [[Amorphous solid|amorphous materials]] (e.g. in [[glass]]es and melts)<ref>{{cite journal|author=R.H.Doremus|year=2002|month= |
title=Viscosity of silica|journal=J. Appl. Phys.|volume=92|issue=12 |pages=7619–7629|issn=0021-8979 | doi = 10.1063/1.1515132 <!--Retrieved from CrossRef by DOI bot-->
}}</ref><ref>{{cite journal|author=M.I. Ojovan and W.E. Lee|year=2004 |title=Viscosity of network liquids within Doremus approach |journal=J. Appl. Phys.|volume=95|issue=7|pages=3803–3810 |issn=0021-8979 | doi = 10.1063/1.1647260 <!--Retrieved from CrossRef by DOI bot--> |unused_data=|month}}</ref><ref>{{cite journal|author=M.I. Ojovan, K.P. Travis and R.J. Hand|year=2000|moth= |title=Thermodynamic parameters of bonds in glassy materials from viscosity-temperature relationships|journal=J. Phys.: Condensed matter|volume=19|issue=41 |pages=415107|issn=0953-8984|doi=10.1088/0953-8984/19/41/415107}}</ref> is a thermally activated process:
<math>\eta = A \cdot e^{Q/RT}</math>
where <math>Q</math> is activation energy, <math>T</math> is temperature, <math>R</math> is the molar gas constant and <math>A</math> is approximately a constant.
The viscous flow in amorphous materials is characterized by a deviation from the [[Arrhenius equation|Arrhenius-type]] behavior: <math>Q</math> changes from a high value <math>Q_H</math> at low temperatures (in the glassy state) to a low value <math>Q_L</math> at high temperatures (in the liquid state). Depending on this change, amorphous materials are classified as either
*strong when: <math>Q_H - Q_L < Q_L</math> or
*fragile when: <math>Q_H - Q_L \ge Q_L</math>
The fragility of amorphous materials is numerically characterized by the Doremus’ fragility ratio:
<math>R_D = Q_H/Q_L</math>
and strong material have <math>R_D < 2\;</math> whereas fragile materials have <math>R_D \ge 2</math>
The viscosity of amorphous materials is quite exactly described by a two-exponential equation:
<math>\eta = A_1 \cdot T \cdot [1 + A_2 \cdot e^{B/RT}] \cdot [1 + C \cdot e^{D/RT}]</math>
with constants <math>A_1 , A_2 , B, C</math> and <math>D</math> related to thermodynamic parameters of joining bonds of an amorphous material.
Not very far from the [[glass transition temperature]], <math>T_g</math>, this equation can be approximated by a [[Vogel-Tammann-Fulcher equation|Vogel-Tammann-Fulcher]] (VTF) equation or a [[Kohlrausch-Williams-Watts function|Kohlrausch-type]] stretched-exponential law.
If the temperature is significantly lower than the glass transition temperature, <math>T < T_g</math>, then the two-exponential equation simplifies to an Arrhenius type equation:
<math>\eta = A_LT \cdot e^{Q_H/RT}</math>
with:
<math>Q_H = H_d + H_m</math>
where <math>H_d</math> is the [[enthalpy of formation]] of broken bonds (termed [[configuron]]s) and <math>H_m</math> is the [[enthalpy]] of their motion. When the temperature is less than the glass transition temperature, <math>T < T_g</math>, the activation energy of viscosity is high because the amorphous materials are in the glassy state and most of their joining bonds are intact.
If the temperature is highly above the glass transition temperature, <math>T > T_g</math>, the two-exponential equation also simplifies to an Arrhenius type equation:
<math>\eta = A_HT\cdot e^{Q_L/RT}</math>
with:
<math>Q_L = H_m</math>
When the temperature is higher than the glass transition temperature, <math>T > T_g</math>, the activation energy of viscosity is low because amorphous materials are melt and have most of their joining bonds broken which facilitates flow.
==Volume (bulk) viscosity==
The negative-one-third of the [[Trace (linear algebra)|trace]] of the [[Stress (physics)|stress]] [[tensor]] is often identified with the thermodynamic [[pressure]],
<blockquote>
<math>-{1\over3}T_a^a = p</math>,
</blockquote>
which only depends upon the equilibrium state potentials like temperature and density ([[equation of state]]). In general, the trace of the stress tensor is the sum of thermodynamic pressure contribution plus another contribution which is proportional to the divergence of the velocity field. This constant of proportionality is called the [[volume viscosity]].
==Eddy viscosity==
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 5x10<sup>4</sup> to 10<sup>6</sup> Pa·s depending upon the resolution of the numerical grid.
==Fluidity==
The [[reciprocal]] of viscosity is ''fluidity'', usually symbolized by <math>\phi = 1/\eta</math> or <math>F=1/\eta</math>, 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.
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 <math>a</math> and <math>b</math> are mixed is
:<math>F \approx \chi_a F_a + \chi_b F_b</math>
which is only slightly simpler than the equivalent equation in terms of viscosity:
:<math>\eta \approx \frac{1}{\chi_a /\eta_a + \chi_b/\eta_b}</math>
where <math>\chi_a</math> and <math>\chi_b</math> is the mole fraction of component <math>a</math> and <math>b</math> respectively, and <math>\eta_a</math> and <math>\eta_b</math> are the components pure viscosities.
== The linear viscous stress tensor ==
(See ''[[Hooke's law]]'' and ''[[strain tensor]]'' for an analogous development for linearly elastic materials.)
Viscous forces in a fluid are a function of the rate at which the fluid velocity is changing over distance. The velocity at any point <math>\mathbf{r}</math> is specified by the velocity field <math>\mathbf{v}(\mathbf{r})</math>. The velocity at a small distance <math>d\mathbf{r}</math> from point <math>\mathbf{r}</math> may be written as a [[Taylor series]]:
:<math>\mathbf{v}(\mathbf{r}+d\mathbf{r}) = \mathbf{v}(\mathbf{r})+\frac{d\mathbf{v}}{d\mathbf{r}}d\mathbf{r}+\ldots</math>
where <math>\frac{d\mathbf{v}}{d\mathbf{r}}</math> is shorthand for the dyadic product of the del operator and the velocity:
:
<math>\frac{d\mathbf{v}}{d\mathbf{r}} = \begin{bmatrix}
\frac{\partial v_x}{\partial x} & \frac{\partial v_x}{\partial y} & \frac{\partial v_x}{\partial z}\\
\frac{\partial v_y}{\partial x} & \frac{\partial v_y}{\partial y} & \frac{\partial v_y}{\partial z}\\
\frac{\partial v_z}{\partial x} & \frac{\partial v_z}{\partial y}&\frac{\partial v_z}{\partial z}
\end{bmatrix}
</math>
This is just the [[Jacobian matrix|Jacobian]] of the velocity field. Viscous forces are the result of relative motion between elements of the fluid, and so are expressible as a function of the velocity field. In other words, the forces at <math>\mathbf{r}</math> are a function of <math>\mathbf{v}(\mathbf{r})</math> and all derivatives of <math>\mathbf{v}(\mathbf{r})</math> at that point. In the case of linear viscosity, the viscous force will be a function of the Jacobian [[tensor]] alone. For almost all practical situations, the linear approximation is sufficient.
If we represent ''x'', ''y'', and ''z'' by indices 1, 2, and 3 respectively, the ''i,j'' component of the Jacobian may be written as <math>\partial_i v_j</math> where <math>\partial_i</math> is shorthand for <math>\partial /\partial x_i</math>. Note that when the first and higher derivative terms are zero, the velocity of all fluid elements is parallel, and there are no viscous forces.
Any matrix may be written as the sum of an [[antisymmetric matrix]] and a [[symmetric matrix]], and this decomposition is independent of coordinate system, and so has physical significance. The velocity field may be approximated as:
:<math>v_i(\mathbf{r}+d\mathbf{r}) = v_i(\mathbf{r})+\frac{1}{2}\left(\partial_i v_j-\partial_j v_i\right)dr_i + \frac{1}{2}\left(\partial_i v_j+\partial_j v_i\right)dr_i</math>
where [[Einstein notation]] is now being used in which repeated indices in a product are implicitly summed. The second term from the right is the asymmetric part of the first derivative term, and it represents a rigid rotation of the fluid about <math>\mathbf{r}</math> with angular velocity <math>\omega</math> where:
:<math>\omega=\frac12 \mathbf{\nabla}\times \mathbf{v}=\frac{1}{2}\begin{bmatrix}
\partial_2 v_3-\partial_3 v_2\\
\partial_3 v_1-\partial_1 v_3\\
\partial_1 v_2-\partial_2 v_1
\end{bmatrix}
</math>
For such a rigid rotation, there is no change in the relative positions of the fluid elements, and so there is no viscous force associated with this term. The remaining symmetric term is responsible for the viscous forces in the fluid. Assuming the fluid is [[isotropic]] (i.e. its properties are the same in all directions), then the most general way that the symmetric term (the rate-of-strain tensor) can be broken down in a coordinate-independent (and therefore physically real) way is as the sum of a constant tensor (the rate-of-expansion tensor) and a traceless symmetric tensor (the rate-of-shear tensor):
:<math>
\frac{1}{2}\left(\partial_i v_j+\partial_j v_i\right)
=
\underbrace{\frac{1}{3}\partial_k v_k \delta_{ij}}_{\text{rate-of-expansion tensor}}
+
\underbrace{\left(\frac{1}{2}\left(\partial_i v_j+\partial_j v_i\right)-\frac{1}{3}\partial_k v_k \delta_{ij}\right)}_{\text{rate-of-shear tensor}}
</math>
where <math>\delta_{ij}</math> is the [[Kronecker delta|unit tensor]]. The most general linear relationship between the stress tensor <math>\mathbf{\sigma}</math> and the rate-of-strain tensor is then a linear combination of these two tensors:<ref>{{cite book| author= L.D. Landau and E.M. Lifshitz (translated from Russian by J.B. Sykes and W.H. Reid)|title=Fluid Mechanics|edition=2nd Edition| publisher=Butterworth Heinemann|year=1997|id=ISBN 0-7506-2767-0}}</ref>
:<math>\sigma_{visc;ij} = \zeta\partial_k v_k \delta_{ij}+
\eta\left(\partial_i v_j+\partial_j v_i-\frac{2}{3}\partial_k v_k \delta_{ij}\right)
</math>
where <math>\zeta</math> is the coefficient of bulk viscosity (or "second viscosity") and <math>\eta</math> is the coefficient of (shear) viscosity.
The forces in the fluid are due to the velocities of the individual molecules. The velocity of a molecule may be thought of as the sum of the fluid velocity and the thermal velocity. The viscous stress tensor described above gives the force due to the fluid velocity only. The force on an area element in the fluid due to the thermal velocities of the molecules is just the hydrostatic [[pressure]]. This pressure term (<math>-p\delta_{ij}</math>) must be added to the viscous stress tensor to obtain the total stress tensor for the fluid.
:<math>\sigma_{ij} = -p\delta_{ij}+\sigma_{visc;ij}\,</math>
The infinitesimal force <math>dF_i</math> on an infinitesimal area <math>dA_i</math> is then given by the usual relationship:
:<math>dF_i=\sigma_{ij}dA_j\,</math>
==See also==
*[[Deborah number]]
*[[Dilatant]]
*[[Hyperviscosity syndrome]]
*[[Inviscid flow]]
*[[Reynold's number]]
*[[Rheology]]
*[[Thixotropy]]
*[[Viscometer]]
*[[Viscometry]]
*[[Viscoelasticity]]
*[[Viscosity index]]
*[[Joback method]] (Estimation of the liquid viscosity from molecular structure)
==References==
{{reflist}}
==Additional reading==
{{wiktionary}}
* {{cite book
| author = Massey, B. S.
| title = Mechanics of Fluids
| edition = Fifth Edition
| publisher= Van Nostrand Reinhold (UK)
| year = 1983
| id = ISBN 0-442-30552-4 }}
== External links ==
*[http://www.widman.biz/Seleccion/Viscosidad/SAE-ISO/sae-iso.html SAE-ISO-AGMA comparison chart]
*[http://www.widman.biz/Seleccion/Viscosidad/SAE_J300/SAE_J300_English/sae_j300_english.html SAE J300 Motor Oil Viscosity Chart]
*[http://www.widman.biz/Seleccion/Viscosidad/SAE_J306/SAE_J306_English/sae_j306_english.html SAE J306 Automotive Gear Oil Viscosity Chart]
* [http://web.ics.purdue.edu/~alexeenk/GDT/index.html Gas Dynamics Toolbox] Calculate coefficient of viscosity for mixtures of gases
* [http://www.thermexcel.com/english/tables/eau_atm.htm Physical Characteristics of Water] A table of water viscosity as a function of temperature
* [http://glassproperties.com/viscosity/ViscosityMeasurement.htm Glass Viscosity Measurement] Viscosity measurement, viscosity units and fixpoints, glass viscosity calculation
* [http://www.diracdelta.co.uk/science/source/k/i/kinematic%20viscosity/source.html diracdelta.co.uk] conversion between kinematic and dynamic viscosity.
* [http://www.iop.org/EJ/abstract/0953-8984/12/46/305 Vogel-Tammann-Fulcher Equation Parameters]
* [http://www.dispersion.com/ Dispersion Technology]
* [http://www.widman.biz/Corvair/html/oils.html The effects of viscosity in a car engine]
{{Physics-footer}}
[[Category:Fundamental physics concepts]]
[[Category:Glass engineering and science]]
[[Category:Viscosity|Viscosity]]
[[Category:Petroleum engineering]]
[[Category:Oilfield terminology]]
[[ar:لزوجة]]
[[bg:Вискозитет]]
[[bn:সান্দ্রতা]]
[[bs:Viskoznost]]
[[ca:Viscositat]]
[[cs:Viskozita]]
[[da:Viskositet]]
[[de:Viskosität]]
[[el:Ιξώδες]]
[[eo:Viskozeco]]
[[es:Viscosidad]]
[[et:Viskoossus]]
[[eu:Biskositate zinematiko]]
[[fa:گرانروی]]
[[fi:Viskositeetti]]
[[fr:Viscosité]]
[[he:צמיגות]]
[[hr:Viskoznost]]
[[hu:Viszkozitás]]
[[id:Viskositas]]
[[is:Seigja]]
[[it:Viscosità]]
[[ja:粘度]]
[[lb:Viskositéit]]
[[lt:Klampumas]]
[[lv:Viskozitāte]]
[[ms:Kelikatan]]
[[nl:Viscositeit]]
[[nn:Viskositet]]
[[no:Viskositet]]
[[pl:Lepkość]]
[[pt:Viscosidade]]
[[ro:Viscozitate]]
[[ru:Вязкость]]
[[sk:Viskozita]]
[[sl:Viskoznost]]
[[sv:Viskositet]]
[[ta:பிசுக்குமை]]
[[tr:Viskozite]]
[[uk:В'язкість]]
[[vi:Độ nhớt]]
[[zh:粘性]]