Viscosity is a measure of the
resistanceIn fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...
of a
fluidIn physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
which is being deformed by either
shearShear as a noun may refer to:*Bias , in clothing design, fabric may be cut on the shear*Cosmic shear, an effect of distortion of image of distant galaxies due to deflection of light by matter, as predicted by general relativity *Shear , a form of fault in rocks*Simple shear, a special case of...
or tensile stress. In everyday terms (and for fluids only), viscosity is "thickness" or "internal friction". Thus,
waterWater is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often coexists on Earth with its solid state, ice, and gaseous state . Water also exists in a...
is "thin", having a lower viscosity, while
honeyHoney is a sweet food made by bees using nectar from flowers. The variety produced by honey bees is the one most commonly referred to and is the type of honey collected by beekeepers and consumed by humans...
is "thick", having a higher viscosity. Put simply, the less viscous the fluid is, the greater its ease of movement (fluidity).
Viscosity describes a fluid's internal resistance to flow and may be thought of as a measure of fluid
frictionFriction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...
. For example, highviscosity felsic magma will create a tall, steep
stratovolcanoA stratovolcano, also known as a composite volcano, is a tall, conical volcano built up by many layers of hardened lava, tephra, pumice, and volcanic ash. Unlike shield volcanoes, stratovolcanoes are characterized by a steep profile and periodic, explosive eruptions...
, because it cannot flow far before it cools, while lowviscosity mafic lava will create a wide, shallowsloped
shield volcanoA shield volcano is a type of volcano usually built almost entirely of fluid lava flows. They are named for their large size and low profile, resembling a warrior's shield. This is caused by the highly fluid lava they erupt, which travels farther than lava erupted from more explosive volcanoes...
.
All real fluids (except
superfluidSuperfluidity is a state of matter in which the matter behaves like a fluid without viscosity and with extremely high thermal conductivity. The substance, which appears to be a normal liquid, will flow without friction past any surface, which allows it to continue to circulate over obstructions and...
s) have some resistance to
stressIn continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...
and therefore are viscous, but a fluid which has no resistance to shear stress is known as an ideal fluid or inviscid fluid.
The study of flowing matter is known as
rheologyRheology is the study of the flow of matter, primarily in the liquid state, but also as 'soft solids' or solids under conditions in which they respond with plastic flow rather than deforming elastically in response to an applied force....
, which includes viscosity and related concepts.
Etymology
The word "viscosity" is derived from the
LatinLatin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient ProtoIndoEuropean language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
"", meaning white
mistletoeMistletoe is the common name for obligate hemiparasitic plants in several families in the order Santalales. The plants in question grow attached to and within the branches of a tree or shrub.Mistletoe in the genus Viscum:...
. A viscous glue called
birdlimeBirdlime is an adhesive substance used in trapping birds. It is spread on a branch or twig, upon which a bird may land and be caught. Its use is illegal in many jurisdictions....
was made from mistletoe berries and was used for limetwigs to catch birds.
Overview
In general, in any flow, layers move at different
velocitiesIn physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
and the fluid's viscosity arises from the shear stress between the layers that ultimately opposes any applied force.
The relationship between the shear stress and the velocity gradient can be obtained by considering two plates closely spaced 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 with a velocity gradient u/y (as opposed to just
shearingDeformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body...
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 gradient in the fluid:
,
where μ is the proportionality factor called dynamic viscosity.
This equation can be expressed in terms of shear stress
. Thus as expressed in
differentialA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
form by
Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
for straight,
parallelParallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...
and uniform flow, the shear stress between layers is proportional to the
velocityIn physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
gradientIn vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
in the direction
perpendicularIn geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...
to the layers:
Hence, through this method, the relation between the shear stress and the velocity gradient can be obtained.
Note that the rate of shear deformation is
which can be also written as a
shear velocityShear velocity, also called friction velocity, is a form by which a shear stress may be rewritten in units of velocity. It is useful as a method in fluid mechanics to compare true velocities, such as the velocity of a flow in a stream, to a velocity that relates shear between layers of flow.Shear...
,
.
James Clerk MaxwellJames Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
called viscosity fugitive elasticity because of the analogy that elastic deformation opposes shear stress in
solidSolid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...
s, while in viscous
fluidIn physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
s, shear stress is opposed by rate of deformation.
Types of viscosity
Newton's law of viscosity, given above, is a
constitutive equationIn physics, a constitutive equation is a relation between two physical quantities that is specific to a material or substance, and approximates the response of that material to external forces...
(like
Hooke's lawIn mechanics, and physics, Hooke's law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load applied to it. Many materials obey this law as long as the load does not exceed the material's elastic limit. Materials for which Hooke's law...
, Fick's law,
Ohm's lawOhm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...
). It is not a fundamental law of nature but an approximation that holds in some materials and fails in others.
NonNewtonian fluidA nonNewtonian fluid is a fluid whose flow properties differ in any way from those of Newtonian fluids. Most commonly the viscosity of nonNewtonian fluids is not independent of shear rate or shear rate history...
s exhibit a more complicated relationship between shear stress and velocity gradient than simple linearity. Thus there exist a number of forms of viscosity:
 Newtonian
A Newtonian fluid is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.Definition:...
: fluids, such as waterWater is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often coexists on Earth with its solid state, ice, and gaseous state . Water also exists in a...
and most gasGas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...
es which have a constant viscosity.
 Shear thickening: viscosity increases with the rate of shear.
 Shear thinning
Shear thinning is an effect where a fluid's viscosity—the measure of a fluid's resistance to flow—decreases with an increasing rate of shear stress. Another name for a shear thinning fluid is a pseudoplastic. This property is found in certain complex solutions, such as lava, ketchup, whipped cream,...
: viscosity decreases with the rate of shear. Shear thinning liquids are very commonly, but misleadingly, described as thixotropic.
 Thixotropic: materials which become less viscous over time when shaken, agitated, or otherwise stressed.
 Rheopectic: materials which become more viscous over time when shaken, agitated, or otherwise stressed.
 A Bingham plastic
A Bingham plastic is a viscoplastic material that behaves as a rigid body at low stresses but flows as a viscous fluid at high stress. It is named after Eugene C. Bingham who proposed its mathematical form....
is a material that behaves as a solid at low stresses but flows as a viscous fluid at high stresses.
 A magnetorheological fluid
A magnetorheological fluid is a type of smart fluid in a carrier fluid, usually a type of oil. When subjected to a magnetic field, the fluid greatly increases its apparent viscosity, to the point of becoming a viscoelastic solid. Importantly, the yield stress of the fluid when in its active state...
is a type of "smart fluid" which, when subjected to a magnetic field, greatly increases its apparent viscosity, to the point of becoming a viscoelastic solid.
Viscosity coefficients
Viscosity coefficients can be defined in two ways:
 Dynamic viscosity, also absolute viscosity, the more usual one (typical units Pa·s, Poise, P);
 Kinematic viscosity is the dynamic viscosity divided by the density (typical units cm^{2}/s, Stokes, St).
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
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
).
 Volume viscosity (also called bulk viscosity or second viscosity) becomes important only for such effects where fluid compressibility is essential. Examples would include shock wave
A shock wave is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium or in some cases in the absence of a material medium, through a field such as the electromagnetic field...
s and soundSound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.Propagation of...
propagation. It appears in the Stokes' law (sound attenuation) that describes propagation of sound in Newtonian liquid.
Alternatively,
 Extensional viscosity
Extensional viscosity is a viscosity coefficient when applied stress is extensional stress.Extensional viscosity can be measured using rheometers that apply extensional stress...
, 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 and motor oil of about .
Viscosity measurement
Viscosity is measured with various types of
viscometerA viscometer is an instrument used to measure the viscosity of a fluid. For liquids with viscosities which vary with flow conditions, an instrument called a rheometer is used...
s and
rheometerA rheometer is a laboratory device used to measure the way in which a liquid, suspension or slurry flows in response to applied forces. It is used for those fluids which cannot be defined by a single value of viscosity and therefore require more parameters to be set and measured than is the case...
s. A rheometer is used for those fluids which 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 accurate measurements, particularly in materials like lubricants, whose viscosity can double with a change of only 5 °C.
For some fluids, viscosity is a constant over a wide range of shear rates (Newtonian fluids). The fluids without a constant viscosity (
nonNewtonian fluidA nonNewtonian fluid is a fluid whose flow properties differ in any way from those of Newtonian fluids. Most commonly the viscosity of nonNewtonian fluids is not independent of shear rate or shear rate history...
s) cannot be described by a single number. NonNewtonian 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 cupA Zahn cup is a viscosity measurement device widely used in the paint industry. It is commonly a stainless steel cup with a tiny hole drilled in the center of the bottom of the cup. There is also a long handle attached to the sides. There are five cup specifications, labeled Zahn cup #x, where x is...
, 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.
Also used in paint, a Stormer viscometer uses loadbased rotation in order to determine viscosity. The viscosity is reported in Krebs units (KU), which are unique to Stormer viscometers.
A
Ford viscosity cupThe Ford viscosity cup is a simple gravity device that permits the timed flow of a known volume of liquid passing through an orifice located at the bottom. Under ideal conditions, this rate of flow would be proportional to the kinematic viscosity that is dependent upon the specific gravity of the...
measures the rate of flow of a liquid. This, under ideal conditions, is proportional to the kinematic viscosity.
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
rheometerA rheometer is a laboratory device used to measure the way in which a liquid, suspension or slurry flows in response to applied forces. It is used for those fluids which cannot be defined by a single value of viscosity and therefore require more parameters to be set and measured than is the case...
s that apply extensional stress.
Volume viscosityVolume viscosity becomes important only for such effects where fluid compressibility is essential. Examples would include shock waves and sound propagation...
can be measured with an
acoustic rheometerAn acoustic rheometer employs piezoelectric crystal that can easily launch a successive wave of extensions and contractions into the fluid. It applies an oscillating extensional stress to the system...
.
Apparent viscosityApparent viscosity is a rheological property calculated from rheometer readings performed by a Mud engineer on drilling fluid. It is normally abbreviated as AV. It is expressed in cP . Centipoise is the amount of force required to move one layer of fluid in relation to another...
is a calculation derived from tests performed on
drilling fluidIn geotechnical engineering, drilling fluid is a fluid used to aid the drilling of boreholes into the earth. Often used while drilling oil and natural gas wells and on exploration drilling rigs, drilling fluids are also used for much simpler boreholes, such as water wells. Liquid 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.
Dynamic viscosity
The usual symbol for dynamic viscosity used by mechanical and chemical engineers — as well as fluid dynamicists — is the Greek letter mu (μ). The symbol η is also used by chemists, physicists, and the IUPAC.
The
SISi, si, or SI may refer to : Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
physical unit of dynamic viscosity is the
pascalThe pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre...

secondThe second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
(Pa·s), (equivalent to N·s/m
^{2}, or kg/(m·s)). If a
fluidIn physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
with a viscosity of one Pa·s is placed between two plates, and one plate is pushed sideways with a
shear stressA shear stress, denoted \tau\, , is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section...
of one
pascalThe pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre...
, it moves a distance equal to the thickness of the layer between the plates in one
secondThe second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
. Water at 20 °C has a viscosity of 0.001002 Pa·s.
The cgs physical unit for dynamic viscosity is the
poiseThe poise is the unit of dynamic viscosity in the centimetre gram second system of units. It is named after Jean Louis Marie Poiseuille ....
(P), named after
Jean Louis Marie PoiseuilleJean Louis Marie Poiseuille was a French physician and physiologist.Poiseuille was born in Paris, France.From 1815 to 1816 he studied at the École Polytechnique in Paris. He was trained in physics and mathematics. In 1828 he earned his D.Sc...
. 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 = 0.1 Pa·s,
 1 cP = 1 mPa·s = 0.001 Pa·s.
Kinematic viscosity
In many situations, we are concerned with the ratio of the
inertiaInertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...
l force to the viscous force (i.e. the
Reynolds number,
), the former characterized by the
fluidIn physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
densityThe mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
ρ. This ratio is characterized by the kinematic viscosity (Greek letter nu, ν), defined as follows:
The SI unit of ν is m
^{2}/s. The SI unit of ρ is kg/m
^{3}.
The cgs physical unit for kinematic viscosity is the stokes (St), named after
George Gabriel StokesSir George Gabriel Stokes, 1st Baronet FRS , was an Irish mathematician and physicist, who at Cambridge made important contributions to fluid dynamics , optics, and mathematical physics...
. It is sometimes expressed in terms of centiStokes (cSt). In U.S. usage, stoke is sometimes used as the singular form.
 1 St = 1 cm^{2}·s^{−1} = 10^{−4} m^{2}·s^{−1}.
 1 cSt = 1 mm^{2}·s^{−1} = 10^{−6}m^{2}·s^{−1}.
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 is analogous to
diffusivity of heatIn heat transfer analysis, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure. It has the SI unit of m²/s...
and diffusivity of mass. It is therefore used in dimensionless numbers which compare the ratio of the diffusivities.
Fluidity
The
reciprocalIn mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
of viscosity is fluidity, usually symbolized by φ = 1 / μ or F = 1 / μ, depending on the convention used, measured in reciprocal poise (
cmA centimetre is a unit of length in the metric system, equal to one hundredth of a metre, which is the SI base unit of length. Centi is the SI prefix for a factor of . Hence a centimetre can be written as or — meaning or respectively...
·
sThe second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
·
gThe gram is a metric system unit of mass....
^{−1}), sometimes called the rhe. Fluidity is seldom used in
engineeringEngineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
practice.
The concept of fluidity can be used to determine the viscosity of an
ideal solutionIn chemistry, an ideal solution or ideal mixture is a solution with thermodynamic properties analogous to those of a mixture of ideal gases. The enthalpy of solution is zero as is the volume change on mixing; the closer to zero the enthalpy of solution is, the more "ideal" the behavior of the...
. For two components
and
, the fluidity when a and b are mixed is
which is only slightly simpler than the equivalent equation in terms of viscosity:
where χ
_{a} and χ
_{b} is the mole fraction of component a and b respectively, and μ
_{a} and μ
_{b} are the components pure viscosities.
Nonstandard units
The
ReynIn fluid dynamics, the Reyn is a British unit of dynamic viscosity.Named in honour of Osbourne Reynolds, for whom Reynolds number is also named One Reyn is:Relation between Reyn and centipoise is approximately:...
is a British unit of dynamic viscosity.
Viscosity indexViscosity index is an arbitrary measure for the change of viscosity with temperature. It is used to characterize lubricating oil in the automotive industry....
is a measure for the change of kinematic viscosity with temperature. It is used to characterise lubricating oil in the automotive industry.
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 SecondSaybolt Universal second is a measure of kinematic viscosity used in classical mechanics. It is the time that 60 cm3 of oil takes to flow through a calibrated tube at a controlled temperature...
s (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
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 dynamicsMolecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
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 MaxwellJames Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
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, λ, of the diffusing particles.
From
fluid mechanicsFluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...
, for a
Newtonian fluidA Newtonian fluid is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.Definition:...
, the
shear stressA shear stress, denoted \tau\, , is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section...
, τ, 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 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
momentumIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
, 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 λ (equally distributed), and that their average velocities change linearly with distance (always true for small enough λ). From this follows:
where
is the rate of fluid mass hitting the surface,
 ρ is the density of the fluid,
 ū is the average molecular speed (),
 μ is the dynamic viscosity.
Effect of temperature on the viscosity of a gas
Sutherland's formula can be used to derive the dynamic viscosity of an
ideal gasAn ideal gas is a theoretical gas composed of a set of randomlymoving, noninteracting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
as a function of the temperature:
This in turn is equal to
where
is a constant.
in Sutherland's formula:
 μ = dynamic viscosity in (Pa·s) at input temperature T,
 μ_{0} = reference viscosity in (Pa·s) at reference temperature T_{0},
 T = input temperature in kelvins,
 T_{0} = reference temperature in kelvins,
 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.
Sutherland's constant and reference temperature for some gases
Gas 
C
[K] 
T_{0}
[K] 
μ_{0}
[μPa s] 
air 
120 
291.15 
18.27 
nitrogenNitrogen is a chemical element that has the symbol N, atomic number of 7 and atomic mass 14.00674 u. Elemental nitrogen is a colorless, odorless, tasteless, and mostly inert diatomic gas at standard conditions, constituting 78.08% by volume of Earth's atmosphere...

111 
300.55 
17.81 
oxygenOxygen is the element with atomic number 8 and represented by the symbol O. Its name derives from the Greek roots ὀξύς and γενής , because at the time of naming, it was mistakenly thought that all acids required oxygen in their composition...

127 
292.25 
20.18 
carbon dioxide Carbon dioxide is a naturally occurring chemical compound composed of two oxygen atoms covalently bonded to a single carbon atom...

240 
293.15 
14.8 
carbon monoxideCarbon monoxide , also called carbonous oxide, is a colorless, odorless, and tasteless gas that is slightly lighter than air. It is highly toxic to humans and animals in higher quantities, although it is also produced in normal animal metabolism in low quantities, and is thought to have some normal...

118 
288.15 
17.2 
hydrogenHydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...

72 
293.85 
8.76 
ammoniaAmmonia is a compound of nitrogen and hydrogen with the formula . It is a colourless gas with a characteristic pungent odour. Ammonia contributes significantly to the nutritional needs of terrestrial organisms by serving as a precursor to food and fertilizers. Ammonia, either directly or...

370 
293.15 
9.82 
sulfur dioxideSulfur dioxide is the chemical compound with the formula . It is released by volcanoes and in various industrial processes. Since coal and petroleum often contain sulfur compounds, their combustion generates sulfur dioxide unless the sulfur compounds are removed before burning the fuel...

416 
293.65 
12.54 
heliumHelium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, nontoxic, inert, monatomic gas that heads the noble gas group in the periodic table...

79.4 
273 
19 
See also
http://www.epa.gov/EPAAIR/2005/July/Day13/a11534d.htm.
Viscosity of a dilute gas
The
ChapmanEnskog equationChapmanEnskog theory presents accurate formulas for a multicomponent gas mixture under thermal and chemical equilibrium. In elastic gases the deviation from the Maxwell–Boltzmann distribution in the equilibrium is small and it can be treated as a perturbation. This method was aimed to obtain...
may be used to estimate viscosity for a dilute gas. This equation is based on a semitheoretical assumption by Chapman and Enskog. 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
^{*})).
with
 T^{*} = κT/ε — reduced temperature (dimensionless),
 μ_{0} = viscosity for dilute gas (μPa.s),
 M = molecular mass (g/mol),
 T = temperature (K),
 σ = the collision diameter (Å),
 ε / κ = the maximum energy of attraction divided by the Boltzmann constant (K),
 ω_{μ} = 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. 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
The temperature dependence of liquid viscosity is the phenomenon by which liquid viscosity tends to decrease as its temperature increases. This can be observed, for example, by watching how cooking oil appears to move more fluidly upon a frying pan after being heated by a stove...
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:
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:
where x
_{X} 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:
where VBN
_{Blend} 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 kg/(m·s), 17.8 μPa.s or 1.78 Pa.s.. One can get the viscosity of air as a function of temperature from the
Gas Viscosity Calculator
Viscosity of water
The
dynamicIn the field of physics, the study of the causes of motion and changes in motion is dynamics. In other words the study of forces and why objects are in motion. Dynamics includes the study of the effect of torques on motion...
viscosity of
waterWater is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often coexists on Earth with its solid state, ice, and gaseous state . Water also exists in a...
is 8.90 × 10
^{−4} PaThe pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre...
·
sThe second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
or 8.90 × 10
^{−3} dyn·s/cm
^{2} 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): (Pa·s) = A × 10
^{B/(T−C)}
where A=2.414 × 10
^{−5} Pa·s ; B = 247.8 K ; and C = 140 K.
Viscosity of liquid water at different temperatures up to the normal boiling point is listed below.
Temperature
[°C] 
Viscosity
[mPa·s] 
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
Some dynamic viscosities of Newtonian fluids are listed below:
Viscosity of selected gases at 100 kPa, [μPa·s]
Gas Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...

at 0 °CCelsius is a scale and unit of measurement for temperature. It is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death... (273 K) 
at 27 °C (300 K) 
airThe atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...

17.4 
18.6 
hydrogenHydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...

8.4 
9.0 
heliumHelium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, nontoxic, inert, monatomic gas that heads the noble gas group in the periodic table...


20.0 
argonArgon is a chemical element represented by the symbol Ar. Argon has atomic number 18 and is the third element in group 18 of the periodic table . Argon is the third most common gas in the Earth's atmosphere, at 0.93%, making it more common than carbon dioxide...


22.9 
xenonXenon is a chemical element with the symbol Xe and atomic number 54. The element name is pronounced or . A colorless, heavy, odorless noble gas, xenon occurs in the Earth's atmosphere in trace amounts...

21.2 
23.2 
carbon dioxide Carbon dioxide is a naturally occurring chemical compound composed of two oxygen atoms covalently bonded to a single carbon atom...


15.0 
methane Methane is a chemical compound with the chemical formula . It is the simplest alkane, the principal component of natural gas, and probably the most abundant organic compound on earth. The relative abundance of methane makes it an attractive fuel...


11.2 
ethaneEthane is a chemical compound with chemical formula C2H6. It is the only twocarbon alkane that is an aliphatic hydrocarbon. At standard temperature and pressure, ethane is a colorless, odorless gas....


9.5 
Viscosity of fluidIn physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
s with variable compositions
Fluid 
Viscosity
[Pa·s] 
Viscosity
[cP] 

honey Honey is a sweet food made by bees using nectar from flowers. The variety produced by honey bees is the one most commonly referred to and is the type of honey collected by beekeepers and consumed by humans...

2–10 
2,000–10,000 
molassesMolasses is a viscous byproduct of the processing of sugar cane, grapes or sugar beets into sugar. The word molasses comes from the Portuguese word melaço, which ultimately comes from mel, the Latin word for "honey". The quality of molasses depends on the maturity of the sugar cane or sugar beet,...

5–10 
5,000–10,000 
molten glassGlass is an amorphous solid material. Glasses are typically brittle and optically transparent.The most familiar type of glass, used for centuries in windows and drinking vessels, is sodalime glass, composed of about 75% silica plus Na2O, CaO, and several minor additives...

10–1,000 
10,000–1,000,000 
chocolate syrup Chocolate syrup is a chocolate flavored condiment. It is often used as a topping for various desserts, such as ice cream or mixed with milk to make chocolate milk.Basic ingredients:...

10–25 
10,000–25,000 
molten chocolateChocolate is a raw or processed food produced from the seed of the tropical Theobroma cacao tree. Cacao has been cultivated for at least three millennia in Mexico, Central and South America. Its earliest documented use is around 1100 BC... ^{*} 
45–130 
45,000–130,000 
ketchupKetchup is a sweetandtangy condiment typically made from tomatoes, vinegar, sugar or highfructose corn syrup and an assortment of... ^{*} 
50–100 
50,000–100,000 
lard Lard is pig fat in both its rendered and unrendered forms. Lard was commonly used in many cuisines as a cooking fat or shortening, or as a spread similar to butter. Its use in contemporary cuisine has diminished because of health concerns posed by its saturatedfat content and its often negative...

≈ 100 
≈ 100,000 
peanut butter Peanut butter is a food paste made primarily from ground dry roasted peanuts, popular in North America, Netherlands, United Kingdom, and parts of Asia, particularly the Philippines and Indonesia. It is mainly used as a sandwich spread, sometimes in combination as in the peanut butter and jelly... ^{*} 
≈ 250 
≈ 250,000 
shortening Shortening is any fat that is solid at room temperature and used to make crumbly pastry. The reason it is called shortening is because it prevents crosslinkage between gluten molecules. Cross linking is what causes doughs to be sticky. Seeing as cake is not meant to be sticky, shortening is used... ^{*} 
≈ 250 
≈ 250,000 
Viscosity of liquidLiquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...
s at 25 °CCelsius is a scale and unit of measurement for temperature. It is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death...
Liquid : 
Viscosity
[Pa·s] 
Viscosity
[cP=mPa·s] 
acetone Acetone is the organic compound with the formula 2CO, a colorless, mobile, flammable liquid, the simplest example of the ketones.Acetone is miscible with water and serves as an important solvent in its own right, typically as the solvent of choice for cleaning purposes in the laboratory...

3.06 
0.306 
benzene Benzene is an organic chemical compound. It is composed of 6 carbon atoms in a ring, with 1 hydrogen atom attached to each carbon atom, with the molecular formula C6H6....

6.04 
0.604 
bloodBlood is a specialized bodily fluid in animals that delivers necessary substances such as nutrients and oxygen to the cells and transports metabolic waste products away from those same cells.... (37 °C) 
(3–4) 
3–4 
castor oil 
0.985 
985 
corn syrupCorn syrup is a food syrup, which is made from the starch of maize and contains varying amounts of maltose and higher oligosaccharides, depending on the grade. Corn syrup is used in foods to soften texture, add volume, prevent crystallization of sugar, and enhance flavor...

1.3806 
1380.6 
ethanolEthanol, also called ethyl alcohol, pure alcohol, grain alcohol, or drinking alcohol, is a volatile, flammable, colorless liquid. It is a psychoactive drug and one of the oldest recreational drugs. Best known as the type of alcohol found in alcoholic beverages, it is also used in thermometers, as a...

1.074 
1.074 
ethylene glycolEthylene glycol is an organic compound widely used as an automotive antifreeze and a precursor to polymers. In its pure form, it is an odorless, colorless, syrupy, sweettasting liquid...

1.61 
16.1 
glycerol Glycerol is a simple polyol compound. It is a colorless, odorless, viscous liquid that is widely used in pharmaceutical formulations. Glycerol has three hydroxyl groups that are responsible for its solubility in water and its hygroscopic nature. The glycerol backbone is central to all lipids...

1.2 (at 20 °CCelsius is a scale and unit of measurement for temperature. It is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death... ) 
1200 
HFO380Fuel oil is a fraction obtained from petroleum distillation, either as a distillate or a residue. Broadly speaking, fuel oil is any liquid petroleum product that is burned in a furnace or boiler for the generation of heat or used in an engine for the generation of power, except oils having a flash...

2.022 
2022 
mercuryMercury is a chemical element with the symbol Hg and atomic number 80. It is also known as quicksilver or hydrargyrum...

1.526 
1.526 
methanol Methanol, also known as methyl alcohol, wood alcohol, wood naphtha or wood spirits, is a chemical with the formula CH3OH . It is the simplest alcohol, and is a light, volatile, colorless, flammable liquid with a distinctive odor very similar to, but slightly sweeter than, ethanol...

5.44 
0.544 
motor oilMotor oil or engine oil is an oil used for lubrication of various internal combustion engines. The main function is to lubricate moving parts; it also cleans, inhibits corrosion, improves sealing, and cools the engine by carrying heat away from moving parts.Motor oils are derived from... SAE 10 (20 °C) 
0.065 
65 
motor oilMotor oil or engine oil is an oil used for lubrication of various internal combustion engines. The main function is to lubricate moving parts; it also cleans, inhibits corrosion, improves sealing, and cools the engine by carrying heat away from moving parts.Motor oils are derived from... SAE 40 (20 °C) 
0.319 
319 
nitrobenzeneNitrobenzene is an organic compound with the chemical formula C6H5NO2. It is a waterinsoluble pale yellow oil with an almondlike odor. It freezes to give greenishyellow crystals. It is produced on a large scale as a precursor to aniline. Although occasionally used as a flavoring or perfume...

1.863 
1.863 
liquid nitrogen Liquid nitrogen is nitrogen in a liquid state at a very low temperature. It is produced industrially by fractional distillation of liquid air. Liquid nitrogen is a colourless clear liquid with density of 0.807 g/mL at its boiling point and a dielectric constant of 1.4... @ 77K 
1.58 
0.158 
propanol 1Propanol is a primary alcohol with the molecular formula of C3H8O, and a structural formula of CH3CH2CH2OH. It is also known as propan1ol, 1propyl alcohol, npropyl alcohol, npropanol, or simply propanol. It is an isomer of isopropanol . It is used as a solvent in the pharmaceutical...

1.945 
1.945 
olive oilOlive oil is an oil obtained from the olive , a traditional tree crop of the Mediterranean Basin. It is commonly used in cooking, cosmetics, pharmaceuticals, and soaps and as a fuel for traditional oil lamps...

.081 
81 
pitch Pitch is the name for any of a number of viscoelastic, solid polymers. Pitch can be made from petroleum products or plants. Petroleumderived pitch is also called bitumen. Pitch produced from plants is also known as resin. Products made from plant resin are also known as rosin.Pitch was...

2.3e8 
2.3e11 
sulfuric acidSulfuric acid is a strong mineral acid with the molecular formula . Its historical name is oil of vitriol. Pure sulfuric acid is a highly corrosive, colorless, viscous liquid. The salts of sulfuric acid are called sulfates...

2.42 
24.2 
waterWater is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often coexists on Earth with its solid state, ice, and gaseous state . Water also exists in a...

8.94 
0.894 
* These materials are highly
nonNewtonianA nonNewtonian fluid is a fluid whose flow properties differ in any way from those of Newtonian fluids. Most commonly the viscosity of nonNewtonian fluids is not independent of shear rate or shear rate history...
.
Viscosity of slurry
The term
slurryA slurry is, in general, a thick suspension of solids in a liquid.Examples of slurries:Examples of slurries include:* Lahars* A mixture of water and cement to form concrete* A mixture of water, gelling agent, and oxidizers used as an explosive...
designs 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:
where μ
_{s} and μ
_{l} are respectively the dynamic viscosity of the slurry and liquid (Pa·s), and μ
_{r} 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 ɸ of solid particles.
In the case of extremely low concentrations of fine particles, Einstein's equation may be used:
In the case of higher concentrations, a modified equation was proposed by Guth and Simha which takes into account interaction between the solid particles:
Further modification of this equation was proposed by Thomas from the fitting of empirical data:
where A = 0.00273 and B = 16.6.
In the case of very high concentrations, another empirical equation was proposed by Kitano et al.:
where A = 0.68 for smooth spherical particles.
Viscosity of solids
On the basis that all solids such as
graniteGranite is a common and widely occurring type of intrusive, felsic, igneous rock. Granite usually has a medium to coarsegrained texture. Occasionally some individual crystals are larger than the groundmass, in which case the texture is known as porphyritic. A granitic rock with a porphyritic...
flow in response to small
shear stressA shear stress, denoted \tau\, , is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section...
, some researchers have contended that substances known as
amorphous solidIn condensed matter physics, an amorphous or noncrystalline solid is a solid that lacks the longrange order characteristic of a crystal....
s, such as
glassGlass is an amorphous solid material. Glasses are typically brittle and optically transparent.The most familiar type of glass, used for centuries in windows and drinking vessels, is sodalime glass, composed of about 75% silica plus Na2O, CaO, and several minor additives...
and many polymers, may be considered to have viscosity. This has led some to the view that
solidSolid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...
s are simply "
liquidLiquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...
s" with a very high viscosity, typically greater than 10
^{12} Pa·s. This position is often adopted by supporters of the widely held misconception that glass flow can be observed in old buildings. This distortion is the result of the undeveloped glass making process of earlier eras, and not due to the viscosity of glass.
However, others argue that
solidSolid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...
s are, in general, elastic for small stresses while
fluidIn physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
s are not. Even if
solidSolid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...
s flow at higher stresses, they are characterized by their lowstress behavior. This distinction is muddled if measurements are continued over long time periods, such as the
Pitch drop experimentThe pitch drop experiment is a longterm experiment which measures the flow of a piece of pitch over many years. Pitch is the name for any of a number of highly viscous liquids which appear solid, most commonly bitumen...
. Viscosity may be an appropriate characteristic for
solidSolid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...
s in a
plasticIn physics and materials science, plasticity describes the deformation of a material undergoing nonreversible changes of shape in response to applied forces. For example, a solid piece of metal being bent or pounded into a new shape displays plasticity as permanent changes occur within the...
regime. The situation becomes somewhat confused as the term viscosity is sometimes used for solid materials, for example
Maxwell materialA Maxwell material is a viscoelastic material having the properties both of elasticity and viscosity. It is named for James Clerk Maxwell who proposed the model in 1867. It is also known as a Maxwell fluid. Definition :...
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
viscoelasticViscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like honey, resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain instantaneously when stretched and just...
. In
geologyGeology is the science comprising the study of solid Earth, the rocks of which it is composed, and the processes by which it evolves. Geology gives insight into the history of the Earth, as it provides the primary evidence for plate tectonics, the evolutionary history of life, and past climates...
, earth materials that exhibit viscous deformation at least three times greater than their elastic deformation are sometimes called
rheidIn geology, a rheid is a solid material that deforms by viscous flow. The term has the same Greek root as rheology, the science of viscoelasticity and nonlinear flow.Types of rheids:...
s.
Viscosity of amorphous materials
Viscous flow in
amorphous materialsIn condensed matter physics, an amorphous or noncrystalline solid is a solid that lacks the longrange order characteristic of a crystal....
(e.g. in
glassGlass is an amorphous solid material. Glasses are typically brittle and optically transparent.The most familiar type of glass, used for centuries in windows and drinking vessels, is sodalime glass, composed of about 75% silica plus Na2O, CaO, and several minor additives...
es 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
ArrheniustypeThe Arrhenius equation is a simple, but remarkably accurate, formula for the temperature dependence of the reaction rate constant, and therefore, rate of a chemical reaction. The equation was first proposed by the Dutch chemist J. H. van 't Hoff in 1884; five years later in 1889, the Swedish...
behavior: Q changes from a high value Q
_{H} at low temperatures (in the glassy state) to a low value Q
_{L} at high temperatures (in the liquid state). Depending on this change, amorphous materials are classified as either
 strong when: Q_{H} − Q_{L} < Q_{L} or
 fragile when: Q_{H} − Q_{L} ≥ Q_{L}.
The fragility of amorphous materials is numerically characterized by the Doremus’ fragility ratio:
and strong material have R
_{D} < 2 whereas fragile materials have R
_{D} ≥ 2.
The viscosity of amorphous materials is quite exactly described by a twoexponential equation:
with constants A
_{1}, A
_{2}, B, C and D related to thermodynamic parameters of joining bonds of an amorphous material.
Not very far from the glass transition temperature, T
_{g}, this equation can be approximated by a VogelFulcherTammann (VFT) equation.
If the temperature is significantly lower than the glass transition temperature, T < T
_{g}, then the twoexponential equation simplifies to an Arrhenius type equation:
with:
where H
_{d} is the enthalpy of formation of broken bonds (termed
configuron s) and H
_{m} is the
enthalpyEnthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...
of their motion. When the temperature is less than the glass transition temperature, T < T
_{g}, 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 > T
_{g}, the twoexponential equation also simplifies to an Arrhenius type equation:
with:
When the temperature is higher than the glass transition temperature, T > T
_{g}, the activation energy of viscosity is low because amorphous materials are melt and have most of their joining bonds broken which facilitates flow.
Eddy viscosity
In the study of
turbulenceIn fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time...
in
fluidIn physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
s, a common practical strategy for calculation is to ignore the smallscale vortices (or eddies) in the motion and to calculate a largescale motion with an eddy viscosity that characterizes the transport and dissipation of
energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
in the smallerscale flow (see
large eddy simulationLarge eddy simulation is a mathematical model for turbulence used in computational fluid dynamics. It was initially proposed in 1963 by Joseph Smagorinsky to simulate atmospheric air currents, and many of the issues unique to LES were first explored by Deardorff...
). Values of eddy viscosity used in modeling
oceanAn ocean is a major body of saline water, and a principal component of the hydrosphere. Approximately 71% of the Earth's surface is covered by ocean, a continuous body of water that is customarily divided into several principal oceans and smaller seas.More than half of this area is over 3,000...
circulation may be from 5x10
^{4} to 10
^{6} Pa·s depending upon the resolution of the numerical grid.
The linear viscous stress 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 seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
:
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
tensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multidimensional array of...
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 ∂
_{i} v
_{j} where ∂
_{i} is shorthand for ∂/∂x
_{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 notationIn mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae...
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 ω 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 rateofstrain tensor) can be broken down in a coordinateindependent (and therefore physically real) way is as the sum of a constant tensor (the rateofexpansion tensor) and a traceless symmetric tensor (the rateofshear tensor):
where δ
_{ij} is the
unit tensor. The most general linear relationship between the stress tensor σ and the rateofstrain tensor is then a linear combination of these two tensors:
where ς is the coefficient of bulk viscosity (or "second viscosity") and μ 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
pressurePressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure. Definition :...
. This pressure term (−p δ
_{ij}) must be added to the viscous stress tensor to obtain the total stress tensor for the fluid.
The infinitesimal force dF
_{i} on an infinitesimal area dA
_{i} is then given by the usual relationship:
See also
 Deborah number
The Deborah number is a dimensionless number, often used in rheology to characterize the fluidity of materials under specific flow conditions. It was originally proposed by Markus Reiner, a professor at Technion in Israel, inspired by a verse in the Bible, stating "The mountains flowed before the...
 Dilatant
A dilatant material is one in which viscosity increases with the rate of shear strain. Such a shear thickening fluid, also known by the acronym STF, is an example of a nonNewtonian fluid....
 Hyperviscosity syndrome
Hyperviscosity syndrome is a group of symptoms triggered by increase in the viscosity of the blood. Symptoms of high blood viscosity include spontaneous bleeding from mucous membranes, visual disturbances due to retinopathy, and neurologic symptoms ranging from headache and vertigo to seizures and...
 Inviscid flow
In fluid dynamics there are problems that are easily solved by using the simplifying assumption of an ideal fluid that has no viscosity. The flow of a fluid that is assumed to have no viscosity is called inviscid flow....
 Reyn
In fluid dynamics, the Reyn is a British unit of dynamic viscosity.Named in honour of Osbourne Reynolds, for whom Reynolds number is also named One Reyn is:Relation between Reyn and centipoise is approximately:...
 Reynolds number
 Trouton's ratio
 Viscoelasticity
Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like honey, resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain instantaneously when stretched and just...
 Viscosity index
Viscosity index is an arbitrary measure for the change of viscosity with temperature. It is used to characterize lubricating oil in the automotive industry....
 Joback method
The Joback method predicts eleven important and commonly used pure component thermodynamic properties from molecular structure only. Group Contribution Method :The Joback method is a group contribution method...
(estimation of the liquid viscosity from molecular structure)
 Microviscosity
Microviscosity, also known as microscopic viscosity, is the friction experienced by a single particle undergoing diffusion because of its interaction with its environment at the micrometer length scale. The concept of microviscosity is intimately related to the concept of single particle diffusion...
Further reading
 Hatschek, Emil (1928). The Viscosity of Liquids. New York: Van Nostrand. .
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