Viscosity – Ethernet fiber optic modem – 2-channel optic video multiplexer
Etymology
The word “viscosity” derives from the Latin word “viscum” for mistletoe. A viscous glue was made from mistletoe berries and used for lime-twigs to catch birds.
Viscosity coefficients
Viscosity coefficients can be defined in two ways:
Dynamic viscosity, also absolute viscosity, the more usual one;
Kinematic viscosity is the dynamic viscosity divided by the density.
Viscosity is a tensorial quantity that can be decomposed in different ways into two independent components. The most usual decomposition yields the following viscosity coefficients:
Shear viscosity, the most important one, often referred to as simply viscosity, describing the reaction to applied shear stress; simply put, it 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).
Volume viscosity or bulk viscosity, describes the reaction to compression, essential for acoustics in fluids, see Stokes’ law (sound attenuation).
Alternatively,
Extensional viscosity, a linear combination of shear and bulk viscosity, describes the reaction to elongation, widely used for characterizing polymers.
For example, at room temperature, water has a dynamic shear viscosity of about 1.0 103 Pa and motor oil of about 250 103 Pa.
Newton’s theory
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.
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 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 and uniform flow, the shear stress, , between layers is proportional to the velocity gradient, /, in the direction perpendicular to the layers.
Here, the constant is known as the coefficient of viscosity, the viscosity, the dynamic viscosity, or the Newtonian viscosity.
This is a constitutive equation (like Hooke’s law, Fick’s law, Ohm’s law). This means: it is not a fundamental law of nature, but a reasonable first approximation that holds in some materials and fails in others. Many fluids, such as water and most gases, satisfy Newton’s criterion and are known as Newtonian fluids. Non-Newtonian fluids 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 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 shearing 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 = (Au/y), where is the proportionality factor called the dynamic viscosity (also called absolute viscosity, or simply viscosity). The equation can be expressed in terms of shear stress; = F/A = (u / y). The rate of shear deformation is u / y 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 solids, while in viscous fluids, shear stress is opposed by rate of deformation.
Viscosity measurement
Main article: Viscometer
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 fluids. 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 (centistokes, cSt) through the conversion equations.
A Ford viscosity cup measures the rate of flow of a liquid. This, under ideal conditions, is proportional to the kinematic viscosity.
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 rheometers that apply extensional stress.
Volume viscosity can be measured with acoustic rheometer.
Units
Dynamic viscosity
The usual symbol for dynamic viscosity used by mechanical and chemical engineers as well as fluid dynamicists is the Greek letter mu (). The symbol is also used by chemists, physicists, and the IUPAC.
The SI physical unit of dynamic viscosity is the pascalmb-second (Pas), which is identical to Nm2s. If a fluid with a viscosity of one Pas is placed between two plates, and one plate is pushed sideways with a shear stress of one pascal, it moves a distance equal to the thickness of the layer between the plates in one second.
The cgs physical unit for dynamic viscosity is the poise (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 or 0.001002 kilogram/meter second.
1 P = 1 gcm1s1.
The relation to the SI unit is
1 P = 0.1 Pas,
1 cP = 1 mPas = 0.001 Pas.
Kinematic viscosity
In many situations, we are concerned with the ratio of the viscous force to the inertial force, the latter characterised by the fluid density . This ratio is characterised by the kinematic viscosity (Greek letter nu, ), defined as follows:
The SI unit of is m2/s.
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 St = 1 cm2s1 = 104 m2s1.
1 cSt = 1 mm2s1 = 106m2s1.
Water at 20 C has a kinematic viscosity of about 1 cSt.
The kinematic viscosity is sometimes referred to as diffusivity of momentum, because it has the same unit as and is comparable to diffusivity of heat and diffusivity of mass. It is therefore used in dimensionless numbers which compare the ratio of the diffusivities.
Non-standard units
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). Other abbreviations such as SSU (Saybolt Seconds Universal) or SUV (Saybolt Universal Viscosity) are sometimes used. Kinematic viscosity in centistoke can be converted from SUS according to the arithmetic and the reference table provided in ASTM D 2161.
Molecular origins
Pitch has a viscosity approximately 230 billion (2.3 1011) 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 Greenubo 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. To understand why the viscosity is independent of pressure consider two adjacent boundary layers (A and B) moving with respect to each other. The internal friction (the viscosity) of the gas is determined by the probability a particle of layer A enters layer B with a corresponding transfer of momentum. Maxwell’s calculations showed him that the viscosity coefficient is proportional to both the density, the mean free path and the mean velocity of the atoms. On the other hand, the mean free path is inversely proportional to the density. So an increase of pressure doesn’t result in any change of the viscosity.
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, , of the diffusing particles.
From fluid mechanics, shear stress, , on a unit area moving parallel to itself, is found to be proportional to the rate of change of velocity with distance perpendicular to the unit area:
for a unit area parallel to the x-z plane, moving along the x axis. We will derive this formula and show how is related to .
Interpreting shear stress as the time rate of change of momentum, p, per unit area A (rate of momentum flux) of an arbitrary control surface gives
where is the average velocity along x of fluid molecules hitting the unit area, with respect to the unit area.
Further manipulation will show
, assuming that molecules hitting the unit area come from all distances between 0 and (equally distributed), and that their average velocities change linearly with distance (always true for small enough ). From this follows:
where
is the rate of fluid mass hitting the surface,
is the density of the fluid,
is the average molecular speed (),
is the dynamic viscosity.
Viscosity of a dilute gas
The Chapman-Enskog equation may be used to estimate viscosity for a dilute gas. This equation is based on a semi-theoretical assumption by Chapman and Enskog. The equation requires three empirically determined parameters: the collision diameter (), the maximum energy of attraction divided by the Boltzmann constant (/) and the collision integral ((T*)).
with
T* = T/ reduced temperature (dimensionless),
0 = viscosity for dilute gas (Pa.s),
M = molecular mass (g/mol),
T = temperature (K),
= the collision diameter (),
/ = the maximum energy of attraction divided by the Boltzmann constant (K),
= the collision integral.
Liquids
Video showing three liquids with different Viscosities
In liquids, the additional forces between molecules become important. This leads to an additional contribution to the shear stress though the exact mechanics of this are still controversial.[citation needed] 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. 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)
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)
where xX is the 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)
where VBNBlend 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 1.78 105 kg/(ms), 17.8 mPa.s or 1.78 104 P. One can get the viscosity of air as a function of temperature from the Gas Viscosity Calculator
Viscosity of water
Dynamic Viscosity of Water
The dynamic viscosity of water is 8.90 104 Pas or 8.90 103 dyns/cm2 or 0.890 cP at about 25 C.
Water has a viscosity of 0.0091 poise at 25 C, or 1 centipoise at 20 C.
As a function of temperature T (K): (Pas) = A 10B/(T)
where A=2.414 105 Pas ; B = 247.8 K ; and C = 140 K[citation needed].
Viscosity of liquid water at different temperatures up to the normal boiling point is listed below.
Temperature
[C]
Viscosity
[mPas]
10
1.308
20
1.002
30
0.7978
40
0.6531
50
0.5471
60
0.4668
70
0.4044
80
0.3550
90
0.3150
100
0.2822
Viscosity of various materials
Example of the viscosity of milk and water. Liquids with higher viscosities will not make such a splash when poured at the same velocity.
Honey being drizzled.
Peanut butter is a semi-solid and can therefore hold peaks.
Some dynamic viscosities of Newtonian fluids are listed below:
Gases (at 0 C):
viscosity
[Pas]
hydrogen
8.4
air
17.4
xenon
21.2
Liquids (at 25 C):
viscosity
[Pas]
viscosity
[cP=mPa.s]
acetone
3.06e-4
0.306
benzene
6.04e-4
0.604
blood (37 C)
3e-3 to 4e-3
34
castor oil
0.985
985
corn syrup
1.3806
1380.6
ethanol
1.074e-3
1.074
ethylene glycol
1.61e-2
16.1
glycerol
1.49 (at 20 C)
1490
HFO-380
2.022
2022
mercury
1.526e-3
1.526
methanol
5.44e-4
0.544
nitrobenzene
1.863e-3
1.863
liquid nitrogen @ 77K
1.58e-4
0.158
propanol
1.945e-3
1.945
olive oil
.081
81
pitch
2.3e8
2.3e11
sulfuric acid
2.42e-2
24.2
water
8.94e-4
0.894
Fluids with variable compositions
viscosity
[Pas]
viscosity
[cP]
honey
210
2,00010,000
molasses
510
5,00010,000
molten glass
101,000
10,0001,000,000
chocolate syrup
1025
10,00025,000
molten chocolate*
45130
45,000130,000
ketchup*
50100
50,000100,000
peanut butter*
~250
~250,000
shortening*
~250
250,000
* These materials are highly non-Newtonian.
Viscosity of solids
On the basis that all solids such as granite flow to a small extent in response to small shear stress, some researchers have contended that substances known as amorphous solids, such as glass and many polymers, may be considered to have viscosity. This has led some to the view that solids are simply liquids with a very high viscosity, typically greater than 1012 Pas. This position is often adopted by supporters of the widely held misconception that 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.
However, others argue that solids are, in general, elastic for small stresses while fluids are not. Even if solids flow at higher stresses, they are characterized by their low-stress behavior. This distinction can become muddled if measurements are continued over long time periods, such as the Pitch drop experiment. Viscosity may be an appropriate characteristic for solids in a plastic regime. The situation becomes somewhat confused as the term viscosity is sometimes used for solid materials, for example Maxwell materials, 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 viscoelastic. In geology, earth materials that exhibit viscous deformation at least three times greater than their elastic deformation are sometimes called rheids.
Viscosity of amorphous materials
Common glass viscosity curves.
Viscous flow in amorphous materials (e.g. in glasses and melts) is a thermally activated process:
where Q is activation energy, T is temperature, R is the molar gas constant and A is approximately a constant.
The viscous flow in amorphous materials is characterized by a deviation from the Arrhenius-type behavior: Q changes from a high value QH at low temperatures (in the glassy state) to a low value QL at high temperatures (in the liquid state). Depending on this change, amorphous materials are classified as either
strong when: QH QL < QL or
fragile when: QH QL QL.
The fragility of amorphous materials is numerically characterized by the Doremus fragility ratio:
and strong material have RD < 2 whereas fragile materials have RD 2.
The viscosity of amorphous materials is quite exactly described by a two-exponential equation:
with constants A1, A2, B, C and D related to thermodynamic parameters of joining bonds of an amorphous material.
Not very far from the glass transition temperature, Tg, this equation can be approximated by a Vogel-Fulcher-Tammann (VFT) equation.
If the temperature is significantly lower than the glass transition temperature, T < Tg, then the two-exponential equation simplifies to an Arrhenius type equation:
with:
where Hd is the enthalpy of formation of broken bonds (termed configuron s) and Hm is the enthalpy of their motion. When the temperature is less than the glass transition temperature, T < Tg, 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, T > Tg, the two-exponential equation also simplifies to an Arrhenius type equation:
with:
When the temperature is higher than the glass transition temperature, T > Tg, 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 of the stress tensor is often identified with the thermodynamic pressure,
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 fluids, a common practical strategy for calculation is to ignore the small-scale vortices (or eddies) in the motion and to calculate a large-scale motion with an eddy viscosity that characterizes the transport and dissipation of energy in the smaller-scale flow (see large eddy simulation). Values of eddy viscosity used in modeling ocean circulation may be from 5×104 to 106 Pas depending upon the resolution of the numerical grid.
Fluidity
The reciprocal of viscosity is fluidity, usually symbolized by = 1 / or F = 1 / , depending on the convention used, measured in reciprocal poise (cmsg1), 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 a and b, the fluidity when a and b are mixed is
which is only slightly simpler than the equivalent equation in terms of viscosity:
where a and b is the mole fraction of component a and b respectively, and a and b are the components pure viscosities.
The linear viscous stress tensor
For more details on an analogous development for linearly elastic materials, see Hooke’s law and strain tensor.
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 r is specified by the velocity field v(r). The velocity at a small distance dr from point r may be written as a Taylor series:
where dv / dr is shorthand for the dyadic product of the del operator and the velocity:
This is just the 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 r are a function of v(r) and all derivatives of v(r) 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 vj where is shorthand for /i. 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:
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 r with angular velocity where:
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):
where ij is the unit tensor. The most general linear relationship between the stress tensor and the rate-of-strain tensor is then a linear combination of these two tensors:
where is the coefficient of bulk viscosity (or “second viscosity”) and 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 ( ij) must be added to the viscous stress tensor to obtain the total stress tensor for the fluid.
The infinitesimal force dFi on an infinitesimal area dAi is then given by the usual relationship:
See also
Deborah number
Dilatant
Hyperviscosity syndrome
Inviscid flow
Reyn
Reynolds number
Rheology
Thixotropy
Viscoelasticity
Viscosity index
Joback method (Estimation of the liquid viscosity from molecular structure)
References
^ Symon, Keith (1971). Mechanics (Third ed.). Addison-Wesley. ISBN 0-201-07392-7.
^ The Online Etymology Dictionary
^ Raymond A. Serway (1996). Physics for Scientists & Engineers (4th ed.). Saunders College Publishing. ISBN 0-03-005932-1.
^ ASHRAE handbook, 1989 edition
^ Streeter & Wylie Fluid Mechanics, McGraw-Hill, 1981
^ Holman Heat Transfer, McGraw-Hill, 2002
^ Incropera & DeWitt, Fundamentals of Heat and Mass Transfer, Wiley, 1996
^ IUPAC Gold Book, Definition of (dynamic) viscosity
^ IUPAC definition of the Poise
^ ASTM D 2161, Page one,(2005)
^ Quantities and Units of Viscosity
^ Edgeworth,, R.; Dalton, B.J.; Parnell, T.. “The pitch drop experiment”. University of Queensland. http://www.physics.uq.edu.au/physics_museum/pitchdrop.shtml. Retrieved 2009-03-31. . A copy of: European Journal of Physics (1984) pp. 198200.
^ http://physics.info/viscosity/ The Physics Hypertextbook-Viscosity
^ Maxwell, J. C. (1866), “On the viscosity or internal friction of air and other gases”, Philosophical Transactions of the Royal Society of London 156: 249268, doi:10.1098/rstl.1866.0013
^ Salmon, R.L. (1998), Lectures on geophysical fluid dynamics, Oxford University Press, ISBN 0195108086 , pp. 2326.
^ J.O. Hirshfelder, C.F. Curtis and R.B. Bird (1964). Molecular theory of gases and liquids (First ed.). Wiley. ISBN 0-471-40065-3.
^ Robert E. Maples (2000). Petroleum Refinery Process Economics (2nd ed.). Pennwell Books. ISBN 0-87814-779-9.
^ C.T. Baird (1989), Guide to Petroleum Product Blending, HPI Consultants, Inc. HPI website
^ a b c d e f g h i j CRC Handbook of Chemistry and Physics, 73rd edition, 19921993
^ Viscosity. The Physics Hypertextbook. by Glenn Elert
^ viscosity table at hyperphysics.phy-astr.gsu.edu, contains glycerin(=glycerol) viscosity
^ “Chocolate Processing”. Brookfield Engineering website. http://www.brookfieldengineering.com/education/applications/laboratory-chocolate-processing.asp. Retrieved 2007-12-03.
^ Kumagai, Naoichi; Sadao Sasajima, Hidebumi Ito (15 February 1978). “Long-term Creep of Rocks: Results with Large Specimens Obtained in about 20 Years and Those with Small Specimens in about 3 Years”. Journal of the Society of Materials Science (Japan) (Japan Energy Society) 27 (293): 157161. 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. Retrieved 2008-06-16.
^ Elert, Glenn. “Viscosity”. The Physics Hypertextbook. http://hypertextbook.com/physics/matter/viscosity/.
^ “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), 9946
^ Gibbs, Philip. “Is Glass a Liquid or a Solid?”. http://math.ucr.edu/home/baez/physics/General/Glass/glass.html. Retrieved 2007-07-31.
^ Viscosity calculation of glasses
^ R.H.Doremus (2002). “Viscosity of silica”. J. Appl. Phys. 92 (12): 76197629. doi:10.1063/1.1515132.
^ M.I. Ojovan and W.E. Lee (2004). “Viscosity of network liquids within Doremus approach”. J. Appl. Phys. 95 (7): 38033810. doi:10.1063/1.1647260.
^ M.I. Ojovan, K.P. Travis and R.J. Hand (2000). “Thermodynamic parameters of bonds in glassy materials from viscosity-temperature relationships”. J. Phys.: Condensed matter 19 (41): 415107. doi:10.1088/0953-8984/19/41/415107.
^ L.D. Landau and E.M. Lifshitz (translated from Russian by J.B. Sykes and W.H. Reid) (1997). Fluid Mechanics (2nd ed.). Butterworth Heinemann. ISBN 0-7506-2767-0.
Additional reading
Look up viscosity in Wiktionary, the free dictionary.
Massey, B. S. (1983). Mechanics of Fluids (Fifth ed.). Van Nostrand Reinhold (UK). ISBN 0-442-30552-4.
External links
Fluid properties High accuracy calculation of viscosity and other physical properties of frequent used pure liquids and gases.
Fluid Characteristics Chart A table of viscosities and vapor pressures for various fluids
Gas Dynamics Toolbox Calculate coefficient of viscosity for mixtures of gases
Glass Viscosity Measurement Viscosity measurement, viscosity units and fixpoints, glass viscosity calculation
Kinematic Viscosity conversion between kinematic and dynamic viscosity.
Physical Characteristics of Water A table of water viscosity as a function of temperature
Vogelammannulcher Equation Parameters
Calculation of temperature-dependent dynamic viscosities for some common components
v d e
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