Viscosity

Viscosity (dynamic)
A simulation of liquids with different viscosities. The liquid on the right has higher viscosity than the liquid on the left.
Common symbols
η, μ
Derivations from
other quantities
μ = G·t

The viscosity of a fluid is the measure of its resistance to gradual deformation by shear stress or tensile stress.[1] For liquids, it corresponds to the informal concept of "thickness": for example, honey has a higher viscosity than water.[2]

Viscosity is the property of a fluid which opposes the relative motion between two surfaces of the fluid that are moving at different velocities. In simple terms, viscosity means friction between the molecules of fluid. When the fluid is forced through a tube, the particles which compose the fluid generally move more quickly near the tube's axis and more slowly near its walls; therefore some stress (such as a pressure difference between the two ends of the tube) is needed to overcome the friction between particle layers to keep the fluid moving. For a given velocity pattern, the stress required is proportional to the fluid's viscosity.

A fluid that has no resistance to shear stress is known as an ideal or inviscid fluid. Zero viscosity is observed only at very low temperatures in superfluids. Otherwise, all fluids have positive viscosity and are technically said to be viscous or viscid. A fluid with a relatively high viscosity, such as pitch, may appear to be a solid.

Etymology

The word "viscosity" is derived from the Latin "viscum", meaning mistletoe and also a viscous glue made from mistletoe berries.[3]

Definition

Dynamic (shear) viscosity

Laminar shear of fluid between two plates and cups. Friction between the fluid and the moving boundaries/plates causes the fluid to shear and cut. The force required for this action is a measure of the fluid's viscosity.
In a general parallel flow (such as could occur in a straight pipe), the shear stress is proportional to the gradient of the velocity.

The dynamic viscosity of a fluid expresses its resistance to shearing flows, where adjacent layers move parallel to each other with different speeds. It can be defined through the idealized situation known as a Couette flow, where a layer of fluid is trapped between two horizontal plates, one fixed and one moving horizontally at constant speed ${\displaystyle u}$. This fluid has to be homogeneous in the layer and at different shear stresses. (The plates are assumed to be very large so that one need not consider what happens near their edges.)

If the speed of the top plate is low enough, the fluid particles will move parallel to it, and their speed will vary linearly from zero at the bottom to u at the top. Each layer of fluid will move faster than the one just below it, and friction between them will give rise to a force resisting their relative motion. In particular, the fluid will apply on the top plate a force in the direction opposite to its motion, and an equal but opposite one to the bottom plate. An external force is therefore required in order to keep the top plate moving at constant speed.

The magnitude F of this force is found to be proportional to the speed u and the area A of each plate, and inversely proportional to their separation y:

${\displaystyle F=\mu A{\frac {u}{y}}.}$

The proportionality factor μ in this formula is the viscosity (specifically, the dynamic viscosity) of the fluid, with units of ${\displaystyle {\text{Pa}}\cdot {\text{s}}}$ (pascal-second).

The ratio u/y is called the rate of shear deformation or shear velocity, and is the derivative of the fluid speed in the direction perpendicular to the plates (see illustrations to the right). Isaac Newton expressed the viscous forces by the differential equation

${\displaystyle \tau =\mu {\frac {\partial u}{\partial y}},}$

where τ = F/A, and u/y is the local shear velocity. This formula assumes that the flow is moving along parallel lines to x-axis. Furthermore, it assumes that the y-axis, perpendicular to the flow, points in the direction of maximum shear velocity. This equation can be used where the velocity does not vary linearly with y, such as in fluid flowing through a pipe. This equation is called the defining equation for shear viscosity. The viscosity is not a material constant, but a material property that depends on physical properties like temperature. The variation of the dynamic viscosity μ at a given temperature ${\displaystyle T}$ can be evaluated on the basis of the dynamic viscosity at room temperature (${\displaystyle T_{0}=296.15K}$) by the equation: [4]

${\displaystyle \mu =\mu _{0}{\frac {T_{0}+120}{T+120}}({\frac {T}{T_{0}}})^{\frac {3}{2}}}$

The functional relationship between viscosity and other physical properties is described by mathematical viscosity models called constitutive equations which are usually more complex than the defining equation for viscosity. There exist many viscosity models, and based on type of development-reasoning, some viscosity models are selected and presented in the article Viscosity models for mixtures.

Use of the Greek letter mu (μ) for the dynamic viscosity is common among mechanical and chemical engineers, as well as physicists.[5][6][7] However, the Greek letter eta (η) is also used by chemists, physicists, and the IUPAC.[8]

Kinematic viscosity

The kinematic viscosity (also called "momentum diffusivity") is the ratio of the dynamic viscosity μ to the density of the fluid ρ. It is usually denoted by the Greek letter nu (ν) and has units ${\displaystyle \mathrm {m^{2}/s} }$.

${\displaystyle \nu ={\frac {\mu }{\rho }}}$

A convenient concept when analyzing the Reynolds number, which expresses the ratio of the inertial forces to the viscous forces, is:

${\displaystyle \mathrm {Re} ={\frac {\rho VD}{\mu }}={\frac {VD}{\nu }},}$

where D is a diameter in the system, and V is the velocity of the fluid with respect to the object (m/s).

Bulk viscosity

When a compressible fluid is compressed or expanded evenly, without shear, it may still exhibit a form of internal friction that resists its flow. These forces are related to the rate of compression or expansion by a factor called the volume viscosity, bulk viscosity or second viscosity.

The bulk viscosity is important only when the fluid is being rapidly compressed or expanded, such as in sound and shock waves. Bulk viscosity explains the loss of energy in those waves, as described by Stokes' law of sound attenuation.

Viscosity tensor

In general, the stresses within a flow can be attributed partly to the deformation of the material from some rest state (elastic stress), and partly to the rate of change of the deformation over time (viscous stress). In a fluid, by definition, the elastic stress includes only the hydrostatic pressure.

In very general terms, the fluid's viscosity is the relation between the strain rate and the viscous stress. In the Newtonian fluid model, the relationship is by definition a linear map, described by a viscosity tensor that, multiplied by the strain rate tensor (which is the gradient of the flow's velocity), gives the viscous stress tensor.

The viscosity tensor has nine independent degrees of freedom in general. For isotropic Newtonian fluids, these can be reduced to two independent parameters. The most usual decomposition yields the dynamic viscosity μ and the bulk viscosity σ.

Newtonian and non-Newtonian fluids

Viscosity, the slope of each line, varies among materials.

Newton's law of viscosity is a constitutive equation (like Hooke's law, Fick's law, and Ohm's law): it is not a fundamental law of nature but an approximation that holds in some materials and fails in others.

A fluid that behaves according to Newton's law, with a viscosity μ that is independent of the stress, is said to be Newtonian. Gases, water, and many common liquids can be considered Newtonian in ordinary conditions and contexts. There are many non-Newtonian fluids that significantly deviate from that law in some way or other. For example:

• Shear-thickening liquids, whose viscosity increases with the rate of shear strain.
• Shear-thinning liquids, whose viscosity decreases with the rate of shear strain.
• Thixotropic liquids, that become less viscous over time when shaken, agitated, or otherwise stressed.
• Rheopectic (dilatant) liquids, that become more viscous over time when shaken, agitated, or otherwise stressed.
• Bingham plastics that behave as a solid at low stresses but flow as a viscous fluid at high stresses.

Shear-thinning liquids are very commonly, but misleadingly, described as thixotropic.

Even for a Newtonian fluid, the viscosity usually depends on its composition and temperature. For gases and other compressible fluids, it depends on temperature and varies very slowly with pressure.

The viscosity of some fluids may depend on other factors. A magnetorheological fluid, for example, becomes thicker when subjected to a magnetic field, possibly to the point of behaving like a solid.

In solids

The viscous forces that arise during fluid flow must not be confused with the elastic forces that arise in a solid in response to shear, compression or extension stresses. While in the latter the stress is proportional to the amount of shear deformation, in a fluid it is proportional to the rate of deformation over time. (For this reason, Maxwell used the term fugitive elasticity for fluid viscosity.)

However, many liquids (including water) will briefly react like elastic solids when subjected to sudden stress. Conversely, many "solids" (even granite) will flow like liquids, albeit very slowly, even under arbitrarily small stress.[9] Such materials are therefore best described as possessing both elasticity (reaction to deformation) and viscosity (reaction to rate of deformation); that is, being viscoelastic.

Indeed, some authors have claimed that amorphous solids, such as glass and many polymers, are actually liquids with a very high viscosity (greater than 1012 Pa·s). [10] However, other authors dispute this hypothesis, claiming instead that there is some threshold for the stress, below which most solids will not flow at all,[11] and that alleged instances of glass flow in window panes of old buildings are due to the crude manufacturing process of older eras rather than to the viscosity of glass.[12]

Viscoelastic solids may exhibit both shear viscosity and bulk viscosity. The extensional viscosity is a linear combination of the shear and bulk viscosities that describes the reaction of a solid elastic material to elongation. It is widely used for characterizing polymers.

In geology, earth materials that exhibit viscous deformation at least three orders of magnitude greater than their elastic deformation are sometimes called rheids.[13]

Measurement

Viscosity is measured with various types of viscometers and rheometers. A rheometer is used for those fluids that cannot be defined by a single value of viscosity and therefore require more parameters to be set and measured than is the case for a viscometer. Close temperature control of the fluid is essential to acquire accurate measurements, particularly in materials like lubricants, whose viscosity can double with a change of only 5 °C.

For some fluids, the viscosity is constant over a wide range of shear rates (Newtonian fluids). The fluids without a constant viscosity (non-Newtonian fluids) 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 coating industries, viscosity may be measured with a cup in which the efflux time is measured. There are several sorts of cup – such as the Zahn cup and the Ford viscosity cup – with the usage of each type varying mainly according to the industry. The efflux time can also be converted to kinematic viscosities (centistokes, cSt) through the conversion equations.[14]

Also used in coatings, a Stormer viscometer uses load-based rotation in order to determine viscosity. The viscosity is reported in Krebs units (KU), which are unique to Stormer viscometers.

Vibrating viscometers can also be used to measure viscosity. Resonant, or vibrational viscometers work by creating shear waves within the liquid. In this method, the sensor is submerged in the fluid and is made to resonate at a specific frequency. As the surface of the sensor shears through the liquid, energy is lost due to its viscosity. This dissipated energy is then measured and converted into a viscosity reading. A higher viscosity causes a greater loss of energy.[citation needed]

Extensional viscosity can be measured with various rheometers that apply extensional stress.

Volume viscosity can be measured with an acoustic rheometer.

Apparent viscosity is a calculation derived from tests performed on drilling fluid used in oil or gas well development. These calculations and tests help engineers develop and maintain the properties of the drilling fluid to the specifications required.

Units

Dynamic viscosity, μ

Both the physical unit of dynamic viscosity in SI units, the poiseuille (Pl), and cgs units, the poise (P), are named after Jean Léonard Marie Poiseuille. The poiseuille, which is rarely used, is equivalent to the pascal second (Pa·s), or (N·s)/m2, or kg/(m·s). If a fluid is placed between two plates with distance one meter, and one plate is pushed sideways with a shear stress of one pascal, and it moves at x meters per second, then it has viscosity of 1/x pascal seconds. For example, water at 20 °C has a viscosity of 1.002 mPa·s, while a typical motor oil could have a viscosity of about 250 mPa·s.[15] The units used in practice are either Pa·s and its submultiples or the cgs poise referred to below, and its submultiples.

The cgs physical unit for dynamic viscosity, the poise[16] (P), is also named after Jean Poiseuille. It is more commonly expressed, particularly in ASTM standards, as centipoise (cP) since the latter is equal to the SI multiple millipascal seconds (mPa·s). For example, water at 20 °C has a viscosity of 1.002 mPa·s = 1.002 cP.

1 Pl = 1 Pa·s
1 P = 1 dPa·s = 0.1 Pa·s = 0.1 kg·m−1·s−1
1 cP = 1 mPa·s = 0.001 Pa·s = 0.001 N·s·m−2 = 0.001 kg·m−1·s−1.

Kinematic viscosity, ν

The SI unit of kinematic viscosity is m2/s.

The cgs physical unit for kinematic viscosity is the stokes (St), named after Sir George Gabriel Stokes.[17] It is sometimes expressed in terms of centistokes (cSt). In U.S. usage, stoke is sometimes used as the singular form.

1 St = 1 cm2·s−1 = 10−4 m2·s−1.
1 cSt = 1 mm2·s−1 = 10−6 m2·s−1.

Water at 20 °C has a kinematic viscosity of about 10−6 m2·s−1 or 1 cSt.

The kinematic viscosity is sometimes referred to as diffusivity of momentum, because it is analogous to diffusivity of heat and diffusivity of mass. It is therefore used in dimensionless numbers which compare the ratio of the diffusivities.

Fluidity

The reciprocal of viscosity is fluidity, usually symbolized by φ = 1/μ or F = 1/μ, depending on the convention used, measured in reciprocal poise (P−1, or cm·s·g−1), 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

${\displaystyle F\approx \chi _{\mathrm {A} }F_{\mathrm {A} }+\chi _{\mathrm {B} }F_{\mathrm {B} },}$

which is only slightly simpler than the equivalent equation in terms of viscosity:

${\displaystyle \mu \approx {\frac {1}{{\dfrac {\chi _{\mathrm {A} }}{\mu _{\mathrm {A} }}}+{\dfrac {\chi _{\mathrm {B} }}{\mu _{\mathrm {B} }}}}},}$

where χA and χB are the mole fractions of components A and B respectively, and μA and μB are the components' pure viscosities.

Non-standard units

The reyn is a British unit of dynamic viscosity.

Viscosity index is a measure for the change of viscosity with temperature. It is used in the automotive industry to characterise lubricating oil.

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).[18] Other abbreviations such as SSU (Saybolt seconds universal) or SUV (Saybolt universal viscosity) are sometimes used. Kinematic viscosity in centistokes can be converted from SUS according to the arithmetic and the reference table provided in ASTM D 2161.[19]

Molecular origins

Pitch has a viscosity approximately 230 billion (2.3×1011) times that of water.[20]

In general, the viscosity of a system depends in detail on how the 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.[21] 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. On the other hand, much more progress can be made for a dilute gas. Even 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.[22]

Gases

Viscosity in gases arises principally from the molecular diffusion that transports momentum between layers of flow. An elementary calculation for a dilute gas at temperature ${\displaystyle T}$ and density ${\displaystyle \rho }$ gives

${\displaystyle \mu =\alpha \rho \lambda {\sqrt {\frac {2k_{B}T}{\pi m}}},}$

where ${\displaystyle k_{B}}$ is the Boltzmann constant, ${\displaystyle m}$ the molecular mass, and ${\displaystyle \alpha }$ a numerical constant on the order of ${\displaystyle 1}$. The quantity ${\displaystyle \lambda }$, the mean free path, measures the average distance a molecule travels between collisions. Even without a priori knowledge of ${\displaystyle \alpha }$, this expression has interesting implications. In particular, since ${\displaystyle \lambda }$ is typically inversely proportional to density and increases with temperature, ${\displaystyle \mu }$ itself should increase with temperature and be independent of density. In fact, both of these predictions persist in more sophisticated treatments, and accurately describe experimental observations. Note that this behavior runs counter to common intuition regarding liquids, for which viscosity typically decreases with temperature.[23][24]

For rigid elastic spheres of diameter ${\displaystyle \sigma }$, ${\displaystyle \lambda }$ can be computed, giving

${\displaystyle \mu ={\frac {\alpha }{\pi ^{3/2}}}{\frac {\sqrt {k_{B}mT}}{\sigma ^{2}}}.}$

In this case ${\displaystyle \lambda }$ is independent of temperature, so ${\displaystyle \mu \propto T^{1/2}}$. For more complicated molecular models, however, ${\displaystyle \lambda }$ depends on temperature in a non-trivial way, and simple kinetic arguments as used here are inadequate. More fundamentally, the notion of a mean free path becomes imprecise for particles that interact over a finite range, which limits the usefulness of the concept for describing real-world gases.[26]

Chapman–Enskog theory

A technique developed by Sydney Chapman and David Enskog in the early 1900s allows a more refined calculation of ${\displaystyle \mu }$.[22] It is based on the Boltzmann equation, which provides a systematic statistical description of a dilute gas in terms of intermolecular interactions.[27] As such, their technique allows accurate calculation of ${\displaystyle \mu }$ for more realistic molecular models, such as those incorporating intermolecular attraction rather than just hard-core repulsion.

It turns out that a more realistic modeling of interactions is essential for accurate prediction of the temperature dependence of ${\displaystyle \mu }$, which experiments show increases more rapidly than the ${\displaystyle T^{1/2}}$ trend predicted for rigid elastic spheres.[23] Indeed, the Chapman–Enskog analysis shows that the predicted temperature dependence can be tuned by varying the parameters in various molecular models. A simple example is the Sutherland model,[28] which describes rigid elastic spheres with weak mutual attraction. In such a case, the attractive force can be treated perturbatively, which leads to a particularly simple expression for ${\displaystyle \mu }$:

${\displaystyle \mu ={\frac {5}{16\sigma ^{2}}}\left({\frac {k_{B}mT}{\pi }}\right)^{1/2}\left(1+{\frac {S}{T}}\right)^{-1},}$

where ${\displaystyle S}$ is independent of temperature, being determined only by the parameters of the intermolecular attraction. To connect with experiment, it is convenient to rewrite as

${\displaystyle \mu =\mu _{0}\left({\frac {T}{T_{0}}}\right)^{3/2}{\frac {T_{0}+S}{T+S}},}$

where ${\displaystyle \mu _{0}}$ is the viscosity at temperature ${\displaystyle T_{0}}$. If ${\displaystyle \mu }$ is known from experiments at ${\displaystyle T=T_{0}}$ and at least one other temperature, then ${\displaystyle S}$ can be calculated. It turns out that expressions for ${\displaystyle \mu }$ obtained in this way are accurate for a number of gases over a sizable range of temperatures. On the other hand, Chapman and Cowling[22] argue that this success does not imply that molecules actually interact according to the Sutherland model. Rather, they interpret the prediction for ${\displaystyle \mu }$ as a simple interpolation which is valid for some gases over fixed ranges of temperature, but otherwise does not provide a picture of intermolecular interactions which is fundamentally correct and general. Slightly more sophisticated models, such as the Lennard–Jones potential, may provide a better picture, but only at the cost of a more opaque dependence on temperature. In some systems the assumption of spherical symmetry must be abandoned as well, as is the case for vapors with highly polar molecules like H2O.[29][30]

Liquids

Video showing three liquids with different viscosities
Experiment showing the behaviour of a viscous fluid with blue dye for visibility

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.[31][32] 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:

${\displaystyle \mathrm {VBN} =14.534\times \ln {\big (}\ln(\nu +0.8){\big )}+10.975\,}$   (1)

where ν is the kinematic viscosity in centistokes (cSt). It is important that the kinematic viscosity of each component of the blend be obtained at the same temperature.

The next step is to calculate the VBN of the blend, using this equation:

${\displaystyle \mathrm {VBN_{Blend}} =\left(x_{\mathrm {A} }\times \mathrm {VBN_{A}} \right)+\left(x_{\mathrm {B} }\times \mathrm {VBN_{B}} \right)+\cdots +\left(x_{\mathrm {N} }\times \mathrm {VBN_{N}} \right)\,}$   (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 ν:

${\displaystyle \nu =\exp \left(\exp \left({\frac {\mathrm {VBN_{Blend}} -10.975}{14.534}}\right)\right)-0.8,}$   (3)

where VBNBlend is the viscosity blending number of the blend.

alternatively use the more accurate Lederer-Roegiers equation [1]

${\displaystyle \ln \eta _{1,2}={\frac {x_{1}\ln \eta _{1}}{x_{1}+x_{2}\beta }}+{\frac {\beta x_{2}\ln \eta _{2}}{x_{1}+x_{2}\beta }}}$

• ${\displaystyle \beta }$ is based on the difference in intermolecular cohesion energies between the liquids
• ${\displaystyle \eta }$=dynamic viscosity
• x[i]=mole_fraction[i]

Selected substances

Air

Pressure dependence of the dynamic viscosity of dry air at the temperatures of 300, 400 and 500 kelvins

The viscosity of air depends mostly on the temperature. At 15 °C, the viscosity of air is 1.81×10−5 kg/(m·s), 18.1 μPa·s or 1.81×10−5 Pa·s. The kinematic viscosity at 15 °C is 1.48×10−5 m2/s or 14.8 cSt. At 25 °C, the viscosity is 18.6 μPa·s and the kinematic viscosity 15.7 cSt.

Water

Dynamic viscosity of water

The dynamic viscosity of water is 8.90×10−4 Pa·s or 8.90×10−3 dyn·s/cm2 or 0.890 cP at about 25 °C.

As a function of temperature T (in kelvins): μ = A × 10B/(TC), where A = 2.414×10−5 Pa·s, B = 247.8 K, and C = 140 K.[citation needed]

The dynamic viscosity of liquid water at different temperatures up to the normal boiling point is listed below.

Dynamic viscosity of water at various temperatures
Temperature (°C) Viscosity (mPa·s)
10 1.308
20 1.002
30 0.7978
40 0.6531
50 0.5471
60 0.4658
70 0.4044
80 0.3550
90 0.3150
100 0.2822

Other substances

Example of the viscosity of milk and water. Liquids with higher viscosities make smaller splashes 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:

Viscosity of selected gases at 100 kPa (μPa·s)
Gas at 0 °C (273 K) at 27 °C (300 K)[33]
air 17.4 18.6
hydrogen 8.4 9.0
helium 20.0
argon 22.9
xenon 21.2 23.2
carbon dioxide 15.0
methane 11.2
ethane 9.5
Viscosity of fluids with variable compositions
Fluid Viscosity (Pa·s) Viscosity (cP)
blood (37 °C)[10] 3×10−34×10−3 3–4
honey 2–10[34] 2000–10000
molasses 5–10 5000–10000
molten glass 10–1000 100001000000
chocolate syrup 10–25 1000025000
molten chocolate[a] 45–130[35] 45000130000
ketchup[a] 50–100 50000100000
lard ≈ 100 100000
peanut butter[a] ≈ 250 250000
shortening[a] ≈ 250 250000
Viscosity of liquids
(at 25 °C unless otherwise specified)
Liquid Viscosity (Pa·s) Viscosity (cP)
acetone[33] 3.06×10−4 0.306
benzene[33] 6.04×10−4 0.604
castor oil[33] 0.985 985
corn syrup[33] 1.3806 1380.6
ethanol[33] 1.074×10−3 1.074
ethylene glycol 1.61×10−2 16.1
glycerol (at 20 °C)[36] 1.2 1200
HFO-380 2.022 2022
mercury[33] 1.526×10−3 1.526
methanol[33] 5.44×10−4 0.544
motor oil SAE 10 (20 °C)[37] 0.065 65
motor oil SAE 40 (20 °C)[37] 0.319 319
nitrobenzene[33] 1.863×10−3 1.863
liquid nitrogen (−196 °C) 1.58×10−4 0.158
propanol[33] 1.945×10−3 1.945
olive oil 0.081 81
pitch 2.3×108 2.30×1011
sulfuric acid[33] 2.42×10−2 24.2
water 8.94×10−4 0.894
Viscosity of solids
Solid Viscosity (Pa·s) Temperature (°C)
granite[9] 3×1019 - 6×1019 25
asthenosphere[38] 7.0×1019 900
upper mantle[38] 7×10201×1021 1300–3000
lower mantle 1×10212×1021 3000–4000
1. ^ a b c d These materials are highly non-Newtonian.

Slurry

Plot of slurry relative viscosity μr as calculated by empirical correlations from Einstein,[39] Guth and Simha,[40] Thomas,[41] and Kitano et al.[42]

The term slurry describes mixtures of a liquid and solid particles that retain some fluidity. The viscosity of slurry can be described as relative to the viscosity of the liquid phase:

${\displaystyle \mu _{\mathrm {s} }=\mu _{\mathrm {r} }\mu _{\mathrm {l} },}$

where μs and μl are respectively the dynamic viscosity of the slurry and liquid (Pa·s), and μr is the relative viscosity (dimensionless).

Depending on the size and concentration of the solid particles, several models exist that describe the relative viscosity as a function of volume fraction φ of solid particles.

In the case of extremely low concentrations of fine particles, Einstein's equation[39] may be used:

${\displaystyle \mu _{\mathrm {r} }=1+2.5\varphi }$

In the case of higher concentrations, a modified equation was proposed by Guth and Simha,[40] which takes into account interaction between the solid particles:

${\displaystyle \mu _{\mathrm {r} }=1+2.5\varphi +14.1\varphi ^{2}}$

Further modification of this equation was proposed by Thomas[41] from the fitting of empirical data:

${\displaystyle \mu _{\mathrm {r} }=1+2.5\varphi +10.05\varphi ^{2}+Ae^{B\varphi },}$

where A = 0.00273 and B = 16.6.

In the case of high shear stress (above 1 kPa), another empirical equation was proposed by Kitano et al. for polymer melts:[42]

${\displaystyle \mu _{\mathrm {r} }=\left(1-{\frac {\varphi }{A}}\right)^{-2},}$

where A = 0.68 for smooth spherical particles.

Nanofluids

Alumina-ethylene glycol, particle sizes 8 and 43 nm at 10 Celsius
Magnatite-water, particle sizes 25nm at 20 Celsius

Nanofluid is a novel class of fluid, which is developed by dispersing nano-sized particles in base fluid.[43]

Einstein model

Einstein derived the applicable first theoretical formula for the estimation of viscosity values of composites or mixtures in 1906. This model developed while assuming linear viscous fluid including suspensions of rigid and spherical particles. Einstein’s model is valid for very low volume fraction ${\displaystyle \varnothing }$.[39]

${\displaystyle \mu _{nf}=\mu _{bf}(1+2.5\varnothing )}$

Brinkman model

Brinkman modified Einstein’s model for used with average particle volume fraction up to 4%[44]

${\displaystyle \mu _{nf}=\mu _{bf}{\Biggl (}{\frac {1}{\ (1-\varnothing )^{2.5}}}{\Biggr )}=\mu _{bf}{\Big (}1+2.5\varnothing +4.375\varnothing ^{2}+O(\varnothing ^{3}){\Big )}}$

Batchelor model

Batchelor reformed Einstein's theoretical model by presenting Brownian motion effect.[45]

${\displaystyle \mu _{nf}=\mu _{bf}(1+2.5\varnothing +6.5{\varnothing }^{2})}$

Wang et al. model

Wang et al. found a model to predict viscosity of nanofluid as follows.[46]

${\displaystyle \mu _{nf}=\mu _{bf}(1+7.3\varnothing +123{\varnothing }^{2})}$

Masoumi et al. model

Masoumi et al. suggested a new viscosity correlation by considering Brownian motion of nanoparticle in nanofluid.[47]

${\displaystyle \mu _{nf}=\mu _{bf}{\Biggl (}1+{\frac {\rho _{p}V_{B}{d_{p}}^{2}}{72\delta C}}{\Biggr )}}$

${\displaystyle V_{B}={\sqrt {\frac {18K_{B}T}{\pi \rho _{p}{d_{p}}^{3}}}}}$

${\displaystyle \delta ={\sqrt[{3}]{\frac {\pi {d_{p}}^{3}}{6\varnothing }}}}$

${\displaystyle C=\{{(-1.133d_{p}-2.771)\varnothing +(0.09d_{p}-0.393)}\}\times 10^{-6}}$

Udawattha et al. model

Udawattha et al. modified the Masoumi et al. model. The developed model valid for suspension containing micro-size particles.[48]

${\displaystyle \mu _{nf}=\mu _{bf}{\Biggl (}1+2.5\varnothing _{e}+{\frac {\rho _{p}V_{B}{d_{p}}^{2}}{72\delta [T\times 10^{-10}\times \varnothing ^{-0.002T-0.284}]}}{\Biggr )}}$

${\displaystyle \varnothing _{e}=\varnothing {{\Biggl (}1+{\frac {h}{r}}{\Biggr )}}^{3}}$

where

${\displaystyle \mu }$ is the viscosity of the sample, in [Pa·s]
${\displaystyle nf}$ is nanofluid
${\displaystyle bf}$ is basefluid
${\displaystyle p}$ is particle
${\displaystyle \varnothing }$ is volume fraction
${\displaystyle \rho }$ is density of the sample, in [kg·m−3]
${\displaystyle \delta }$ is distance between two particles
${\displaystyle V_{B}}$ is Brownian motion of particle
${\displaystyle K_{B}}$ is the Boltzmann constant
${\displaystyle T}$ is Temperature of the sample, in [K]
${\displaystyle d_{p}}$ is diameter of a particle
${\displaystyle h}$ is nanolayer thickness (1 nm)
${\displaystyle r}$ is radius of a particle

Amorphous materials

Common glass viscosity curves[49]

Viscous flow in amorphous materials (e.g. in glasses and melts)[50][51][52] is a thermally activated process:

${\displaystyle \mu =Ae^{\frac {Q}{RT}},}$

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: QHQL < QL or
• fragile when: QHQLQL.

The fragility of amorphous materials is numerically characterized by Doremus' fragility ratio:

${\displaystyle R_{\mathrm {D} }={\frac {Q_{\mathrm {H} }}{Q_{\mathrm {L} }}}}$

and strong materials have RD < 2 whereas fragile materials have RD ≥ 2.

Common logarithm of viscosity against temperature for B2O3, showing two regimes

The viscosity of amorphous materials is quite exactly described by a two-exponential equation:

${\displaystyle \mu =A_{1}T\left(1+A_{2}e^{\frac {B}{RT}}\right)\left(1+Ce^{\frac {D}{RT}}\right),}$

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, TTg, then the two-exponential equation simplifies to an Arrhenius-type equation:

${\displaystyle \mu =A_{\mathrm {L} }Te^{\frac {Q_{\mathrm {H} }}{RT}}}$

with

${\displaystyle Q_{\mathrm {H} }=H_{\mathrm {d} }+H_{\mathrm {m} },}$

where Hd is the enthalpy of formation of broken bonds (termed configurons[53]) 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 much higher than the glass transition temperature, TTg, the two-exponential equation also simplifies to an Arrhenius-type equation:

${\displaystyle \mu =A_{\mathrm {H} }Te^{\frac {Q_{\mathrm {L} }}{RT}},}$

with

${\displaystyle Q_{\mathrm {L} }=H_{\mathrm {m} }.}$

When the temperature is higher than the glass transition temperature, T > Tg, the activation energy of viscosity is low because amorphous materials are melted and have most of their joining bonds broken, which facilitates flow.

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 1×106 Pa·s depending upon the resolution of the numerical grid.

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