The viscosity of a fluid is a 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 higher viscosity than water.[2]
Contents 1 Etymology 2 Definition 2.1 Dynamic (shear) viscosity
2.2 Kinematic viscosity
2.3 Bulk viscosity
2.4
3 Newtonian and non-Newtonian fluids 4 In solids 5 Measurement 6 Units 6.1 Dynamic viscosity, μ 6.2 Kinematic viscosity, ν 6.3 Fluidity 6.4 Non-standard units 7 Molecular origins 7.1 Gases 7.1.1 Relation to mean free path of diffusing particles
7.1.2 Effect of temperature on the viscosity of a gas
7.1.3
7.2 Liquids 7.2.1
8 Selected substances 8.1 Air 8.2 Water 8.3 Other substances 9 Slurry 10 Nanofluids 11 Amorphous materials 12 Eddy viscosity 13 See also 14 References 15 Further reading 16 External links Etymology[edit]
The word "viscosity" is derived from the
Laminar shear of fluid between two plates and cups.
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 u 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: F = μ A u 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 P a ⋅ s displaystyle Pacdot 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 τ = μ ∂ u ∂ y , 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 functional relationship between viscosity and
other physical properties is described by a mathematical viscosity
model called a constitutive equation which is usually more complex
than the defining equation for viscosity. There exists many viscosity
models, and based on type of development-reasoning are some viscosity
models selected and presented in the article
m 2 / s displaystyle mathrm m^ 2 /s . ν = μ ρ displaystyle nu = frac mu rho It is a convenient concept when analyzing the Reynolds number, which expresses the ratio of the inertial forces to the viscous forces: R e = ρ u L μ = u L ν , displaystyle mathrm Re = frac rho uL mu = frac uL nu , where L is a typical length scale in the system, and u is the velocity
of the fluid with respect to the object (m/s).
Bulk viscosity[edit]
Main article: Volume 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, the slope of each line, varies among materials Newton's law of viscosity is a constitutive equation (like Hooke's
law, Fick's law, 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-
Shear-thickening liquids, whose viscosity increases with the rate of
shear strain.
Shear-thinning liquids, whose viscosity decreases with the rate of
shear strain.
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[edit]
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.[8] 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). [9] 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,[10] 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.[11]
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.[12]
Measurement[edit]
Main article: Viscometer
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[edit] The SI unit of kinematic viscosity 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). 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.
F ≈ χ A F A + χ B F B , displaystyle Fapprox 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: μ ≈ 1 χ A μ A + χ B μ B , 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[edit]
The reyn is a British unit of dynamic viscosity.
Pitch has a viscosity approximately 230 billion (7011229999999999999♠2.3×1011) times that of water.[18] 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
τ = μ d u x d y displaystyle tau =mu frac mathrm d u_ x mathrm d y for a unit area parallel to the xz-plane, moving along the x axis. We will derive this formula and show how μ is related to λ. 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 τ = p ˙ A = m ˙ ⟨ u x ⟩ A . displaystyle tau = frac dot p A = frac dot m leftlangle u_ x rightrangle A . where ⟨ux⟩ is the average velocity, along the x-axis, of fluid molecules hitting the unit area, with respect to the unit area and ṁ is the rate of fluid mass hitting the surface. By making simplified assumption that the velocity of the molecules depends linearly on the distance they are coming from, the mean velocity depends linearly on the mean distance: ⟨ u x ⟩ = λ d u x d y displaystyle leftlangle u_ x rightrangle =lambda frac mathrm d u_ x mathrm d y . Further manipulation will show,[22] m ˙ = ρ u ¯ A τ = ρ u ¯ λ ⏟ μ ⋅ d u x d y ⇒ ν = μ ρ = u ¯ λ , displaystyle begin aligned dot m &=rho bar u A\tau &=underbrace rho bar u lambda _ mu cdot frac mathrm d u_ x mathrm d y quad Rightarrow quad nu = frac mu rho = bar u lambda ,end aligned where ρ is the density of the fluid, ū is the root mean square molecular speed: ū = √⟨u2⟩, λ is the mean free path, μ is the dynamic viscosity. Note, that the mean free path itself typically depends (inversely) on the density. Effect of temperature on the viscosity of a gas[edit] Sutherland's formula can be used to derive the dynamic viscosity of an ideal gas as a function of the temperature:[23] μ = μ 0 T 0 + C T + C ( T T 0 ) 3 2 . displaystyle mu =mu _ 0 frac T_ 0 +C T+C left( frac T T_ 0 right)^ frac 3 2 . This, in turn, is equal to μ = λ T 3 2 T + C , displaystyle mu =lambda frac T^ frac 3 2 T+C , where λ = μ 0 ( T 0 + C ) T 0 3 2 displaystyle lambda = frac mu _ 0 left(T_ 0 +Cright) T_ 0 ^ frac 3 2 is a constant for the gas. in Sutherland's formula: μ = dynamic viscosity (Pa·s or μPa·s) at input temperature T, μ0 = reference viscosity (in the same units as μ) at reference temperature T0, T = input temperature (K), T0 = reference temperature (K), C = Sutherland's constant for the gaseous material in question. Valid for temperatures between 0 < T < 555 K with an error due to pressure less than 10% below 3.45 MPa. According to Sutherland's formula, if the absolute temperature is less than C, the relative change in viscosity for a small change in temperature is greater than the relative change in the absolute temperature, but it is smaller when T is above C. The kinematic viscosity though always increases faster than the temperature (that is, d log(ν)/d log(T) is greater than 1). Sutherland's constant, reference values and λ values for some gases: Gas C (K) T0 (K) μ0 (μPa·s) λ (μPa·s·K−1⁄2) air 120 291.15 18.27 7000151204128800000♠1.512041288 nitrogen 111 300.55 17.81 7000140673219500000♠1.406732195 oxygen 127 292.25 20.18 7000169341130000000♠1.693411300 carbon dioxide 240 293.15 14.8 7000157208593100000♠1.572085931 carbon monoxide 118 288.15 17.2 7000142819322500000♠1.428193225 hydrogen 72 293.85 8.76 6999636236562000000♠0.636236562 ammonia 370 293.15 9.82 7000129744337900000♠1.297443379 sulfur dioxide 416 293.65 12.54 7000176846608600000♠1.768466086 helium 79.4[24] 273 19[25] 7000148438149000000♠1.484381490
μ 0 × 10 6 = 2.6693 M T σ 2 ω ( T ∗ ) , displaystyle mu _ 0 times 10^ 6 = 2.6693 frac sqrt MT sigma ^ 2 omega (T^ * ) , with T* = κT/ε is reduced temperature (dimensionless), μ0 is viscosity for dilute gas (μPa·s), M is molecular mass (g/mol), T is temperature (K), σ is the collision diameter (Å), ε/к is the maximum energy of attraction divided by the Boltzmann constant (K), ωμ is the collision integral. Liquids[edit] Play media Video showing three liquids with different viscosities Play media 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:
The dynamic viscosities of liquids are typically several orders of
magnitude higher than dynamic viscosities of gases.
V B N = 14.534 × ln ( ln ( ν + 0.8 ) ) + 10.975 displaystyle mathrm VBN =14.534times 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: V B N B l e n d = ( x A × V B N A ) + ( x B × V B N B ) + ⋯ + ( x N × V B N N ) 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: ν = exp ( exp ( V B N B l e n d − 10.975 14.534 ) ) − 0.8 , 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] ln η 1 , 2 = x 1 ln η 1 x 1 + x 2 α + α x 2 ln η 2 x 1 + x 2 α displaystyle ln eta _ 1,2 = frac x_ 1 ln eta _ 1 x_ 1 +x_ 2 alpha + frac alpha x_ 2 ln eta _ 2 x_ 1 +x_ 2 alpha α displaystyle alpha is based on the difference in intermolecular cohesion energies between the liquids η displaystyle eta =dynamic viscosity x[i]=mole_fraction[i] Selected substances[edit] Air[edit]
The viscosity of air depends mostly on the temperature. At 15 °C, the viscosity of air is 6995181000000000000♠1.81×10−5 kg/(m·s), 18.1 μPa·s or 6995181000000000000♠1.81×10−5 Pa·s. The kinematic viscosity at 15 °C is 6995148000000000000♠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[edit] Dynamic viscosity of water The dynamic viscosity of water is 6996890000000000000♠8.90×10−4 Pa·s or 6997890000000000000♠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/(T − C), where A = 6995241400000000000♠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)
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[edit] Example of the viscosity of milk and water. Liquids with higher viscosities make smaller splashes when poured at the same velocity.
Some dynamic viscosities of
Gas at 0 °C (273 K) at 27 °C (300 K)[29] 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
Fluid
blood (37 °C)[9] 6997300000000000000♠3×10−3 – 6997400000000000000♠4×10−3 3–4 honey 2–10[30] 2000–7004100000000000000♠10000 molasses 5–10 5000–7004100000000000000♠10000 molten glass 10–1000 7004100000000000000♠10000–7006100000000000000♠1000000 chocolate syrup 10–25 7004100000000000000♠10000–7004250000000000000♠25000 molten chocolate[a] 45–130[31] 7004450000000000000♠45000–7005130000000000000♠130000 ketchup[a] 50–100 7004500000000000000♠50000–7005100000000000000♠100000 lard ≈ 100 ≈ 7005100000000000000♠100000 peanut butter[a] ≈ 250 ≈ 7005250000000000000♠250000 shortening[a] ≈ 250 ≈ 7005250000000000000♠250000
Liquid
acetone[29] 6996306000000000000♠3.06×10−4 0.306 benzene[29] 6996604000000000000♠6.04×10−4 0.604 castor oil[29] 0.985 985 corn syrup[29] 1.3806 1380.6 ethanol[29] 6997107400000000000♠1.074×10−3 7000107400000000000♠1.074 ethylene glycol 6998161000000000000♠1.61×10−2 7001161000000000000♠16.1 glycerol (at 20 °C)[25] 7000120000000000000♠1.2 7003120000000000000♠1200 HFO-380 7000202199999999999♠2.022 7003202200000000000♠2022 mercury[29] 6997152600000000000♠1.526×10−3 7000152600000000000♠1.526 methanol[29] 6996544000000000000♠5.44×10−4 6999544000000000000♠0.544 motor oil SAE 10 (20 °C)[20] 6998650000000000000♠0.065 7001650000000000000♠65 motor oil SAE 40 (20 °C)[20] 6999319000000000000♠0.319 7002319000000000000♠319 nitrobenzene[29] 6997186300000000000♠1.863×10−3 7000186300000000000♠1.863 liquid nitrogen (−196 °C) 6996158000000000000♠1.58×10−4 6999158000000000000♠0.158 propanol[29] 6997194500000000000♠1.945×10−3 7000194500000000000♠1.945 olive oil 6998810000000000000♠0.081 7001810000000000000♠81 pitch 7008229999999999999♠2.3×108 7011229999999999999♠2.30×1011 sulfuric acid[29] 6998242000000000000♠2.42×10−2 7001242000000000000♠24.2 water 6996894000000000000♠8.94×10−4 6999894000000000000♠0.894
Solid
granite[8] 7019300000000000000♠3×1019 - 7019600000000000000♠6×1019 25 asthenosphere[32] 7019700000000000000♠7.0×1019 900 upper mantle[32] 7020700000000000000♠7×1020 – 7021100000000000000♠1×1021 1300–3000 lower mantle 7021100000000000000♠1×1021 – 7021200000000000000♠2×1021 3000–4000 ^ a b c d These materials are highly non-Newtonian. Slurry[edit] Plot of slurry relative viscosity μr as calculated by empirical correlations from Einstein,[33] Guth and Simha,[34] Thomas,[35] and Kitano et al..[36] 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: μ s = μ r μ l , 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[33] may be used: μ r = 1 + 2.5 φ displaystyle mu _ mathrm r =1+2.5varphi In the case of higher concentrations, a modified equation was proposed by Guth and Simha,[34] which takes into account interaction between the solid particles: μ r = 1 + 2.5 φ + 14.1 φ 2 displaystyle mu _ mathrm r =1+2.5varphi +14.1varphi ^ 2 Further modification of this equation was proposed by Thomas[35] from the fitting of empirical data: μ r = 1 + 2.5 φ + 10.05 φ 2 + A e B φ , displaystyle mu _ mathrm r =1+2.5varphi +10.05varphi ^ 2 +Ae^ Bvarphi , where A = 6997272999999999999♠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:[36] μ r = ( 1 − φ A ) − 2 , displaystyle mu _ mathrm r =left(1- frac varphi A right)^ -2 , where A = 0.68 for smooth spherical particles. Nanofluids[edit] Alumina-Ethylene Glycol, Particle sizes 8 and 43 nm at 10 Celsius Magnatite-Water,Particle sizes 25nm at 20 Celsius
μ n f = μ b f ( 1 + 2.5 ∅ ) displaystyle mu _ nf =mu _ bf (1+2.5varnothing ) Brinkman model Brinkman modified Einstein’s model for used with average particle volume fraction up to 4%[38] μ n f = μ b f ( 1 ( 1 − ∅ ) 2.5 ) = μ b f ( 1 + 2.5 ∅ + 4.375 ∅ 2 + O ( ∅ 3 ) ) displaystyle mu _ nf =mu _ bf Biggl ( frac 1 (1-varnothing )^ 2.5 Biggr ) =mu _ bf Big ( 1+2.5varnothing +4.375varnothing ^ 2 +O(varnothing ^ 3 ) Big ) Batchelor model Batchelor reformed Einstein's theoretical model by presenting Brownian motion effect.[39] μ n f = μ b f ( 1 + 2.5 ∅ + 6.5 ∅ 2 ) displaystyle mu _ nf =mu _ bf (1+2.5varnothing +6.5 varnothing ^ 2 ) Wang et al. model Wang et al. found a model to predict viscosity of nanofluid as follows.[40] μ n f = μ b f ( 1 + 7.3 ∅ + 123 ∅ 2 ) displaystyle mu _ nf =mu _ bf (1+7.3varnothing +123 varnothing ^ 2 ) Masoumi et al. model Masoumi et al. suggested a new viscosity correlation by considering Brownian motion of nanoparticle in nanofluid.[41] μ n f = μ b f ( 1 + ρ p V B d p 2 72 δ C ) displaystyle mu _ nf =mu _ bf Biggl ( 1+ frac rho _ p V_ B d_ p ^ 2 72delta C Biggr ) V B = 18 K B T π ρ p d p 3 displaystyle V_ B = sqrt frac 18K_ B T pi rho _ p d_ p ^ 3 δ = π d p 3 6 ∅ 3 displaystyle delta = sqrt[ 3 ] frac pi d_ p ^ 3 6varnothing C = ( − 1.133 d p − 2.771 ) ∅ + ( 0.09 d p − 0.393 ) × 10 − 6 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.[42] μ n f = μ b f ( 1 + 2.5 ∅ e + ρ p V B d p 2 72 δ [ T × 10 − 10 × ∅ − 0.002 T − 0.284 ] ) displaystyle mu _ nf =mu _ bf Biggl ( 1+2.5varnothing _ e + frac rho _ p V_ B d_ p ^ 2 72delta [Ttimes 10^ -10 times varnothing ^ -0.002T-0.284 ] Biggr ) ∅ e = ∅ ( 1 + h r ) 3 displaystyle varnothing _ e =varnothing Biggl ( 1+ frac h r Biggr ) ^ 3 where μ displaystyle mu is the viscosity of the sample, in [Pa·s] n f displaystyle nf is nanofluid b f displaystyle bf is basefluid p displaystyle p is particle ∅ displaystyle varnothing is volume fraction ρ displaystyle rho is density of the sample, in [kg·m−3] δ displaystyle delta is distance between two particles V B displaystyle V_ B is Brownian motion of particle K B displaystyle K_ B is the Boltzmann constant T displaystyle T is Temperature of the sample, in [K] d p displaystyle d_ p is diameter of a particle h displaystyle h is nanolayer thickness (1 nm) r displaystyle r is radius of a particle Amorphous materials[edit] Common glass viscosity curves.[43] Viscous flow in amorphous materials (e.g. in glasses and melts)[44][45][46] is a thermally activated process: μ = A e Q R T , 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: QH − QL < QL or fragile when: QH − QL ≥ QL. The fragility of amorphous materials is numerically characterized by Doremus' fragility ratio: R D = Q H Q L displaystyle R_ mathrm D = frac Q_ mathrm H Q_ mathrm L and strong materials have RD < 2 whereas fragile materials have RD ≥ 2.
The viscosity of amorphous materials is quite exactly described by a two-exponential equation: μ = A 1 T ( 1 + A 2 e B R T ) ( 1 + C e D R T ) , displaystyle mu =A_ 1 Tleft(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, T ≪ Tg, then the two-exponential equation simplifies to an Arrhenius-type equation: μ = A L T e Q H R T displaystyle mu =A_ mathrm L Te^ frac Q_ mathrm H RT with Q H = H d + H m , displaystyle Q_ mathrm H =H_ mathrm d +H_ mathrm m , where Hd is the enthalpy of formation of broken bonds (termed configurons[47]) 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, T ≫ Tg, the two-exponential equation also simplifies to an Arrhenius-type equation: μ = A H T e Q L R T , displaystyle mu =A_ mathrm H Te^ frac Q_ mathrm L RT , with Q L = H m . 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[edit] 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 7004500000000000000♠5×104 to 7006100000000000000♠1×106 Pa·s depending upon the resolution of the numerical grid. See also[edit] Dashpot
Deborah number
Dilatant
Herschel–Bulkley fluid
Hyperviscosity syndrome
Intrinsic viscosity
Inviscid flow
References[edit] ^ "viscosity". Merriam-Webster Dictionary.
^ Symon, Keith (1971).
Further reading[edit] Hatschek, Emil (1928). The
External links[edit] Look up viscosity in Wiktionary, the free dictionary.
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