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The viscosity of a
fluid In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
is a measure of its
resistance Resistance may refer to: Arts, entertainment, and media Comics * Either of two similarly named but otherwise unrelated comic book series, both published by Wildstorm: ** ''Resistance'' (comics), based on the video game of the same title ** ''Th ...
to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example,
syrup In cooking Cooking, cookery, or culinary arts is the art, science, and craft of using heat In thermodynamics, heat is energy in transfer to or from a thermodynamic system, by mechanisms other than thermodynamic work or transfer o ...

syrup
has a higher viscosity than
water Water (chemical formula H2O) is an , transparent, tasteless, odorless, and , which is the main constituent of 's and the s of all known living organisms (in which it acts as a ). It is vital for all known forms of , even though it provide ...

water
. Viscosity can be conceptualized as quantifying the internal
frictional force Friction is the force In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), ...

frictional force
that arises between adjacent layers of fluid that are in relative motion. For instance, when a viscous fluid is forced through a tube, it flows more quickly near the tube's axis than near its walls. In such a case, experiments show that some stress (such as a
pressure Pressure (symbol: ''p'' or ''P'') is the force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...

pressure
difference between the two ends of the tube) is needed to sustain the flow through the tube. This is because a force is required to overcome the friction between the layers of the fluid which are in relative motion. So for a tube with a constant rate of flow, the strength of the compensating force is proportional to the fluid's viscosity. A fluid that has no resistance to
shear stress Shear stress, often denoted by (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

shear stress
is known as an ''ideal'' or ''inviscid'' fluid. Zero viscosity is observed only at
very low temperatures
very low temperatures
in
superfluids File:Liquid helium Rollin film.jpg, The liquid helium is in the superfluid phase. A thin invisible film creeps up the inside wall of the bowl and down on the outside. A drop forms. It will fall off into the liquid helium below. This will repeat unt ...
. Otherwise, the
second law of thermodynamics The second law of thermodynamics establishes the concept of entropy Entropy is a scientific concept, as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term an ...
requires all fluids to have positive viscosity; such fluids are technically said to be viscous or viscid. A fluid with a high viscosity, such as
pitch Pitch may refer to: Acoustic frequency * Pitch (music), the perceived frequency of sound including "definite pitch" and "indefinite pitch" ** Absolute pitch or "perfect pitch" ** Pitch class, a set of all pitches that are a whole number of octaves ...
, may appear to be a
solid Solid is one of the four fundamental states of matter (the others being liquid A liquid is a nearly incompressible fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied ...

solid
.


Etymology

The word "viscosity" is derived from the
Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an appa ...

Latin
' ("
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mistletoe
"). ''Viscum'' also referred to a viscous
glue adhesive dispensed from a tube Adhesive, also known as glue, cement, mucilage, or paste, is any non-metallic substance applied to one or both surfaces of two separate items that molecular binding , binds them together and resists their separati ...
derived from mistletoe berries.


Definition


Dynamic viscosity

In
materials science The interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic discipline An academic discipline or academic field is a subdivision of knowledge that is Education, taught and resea ...
and
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more specializ ...

engineering
, one is often interested in understanding the forces or stresses involved in the
deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (mechanics), such changes considered and analyzed as displacements of continuum bodies. * Defo ...
of a material. For instance, if the material were a simple spring, the answer would be given by
Hooke's law The balance wheel at the core of many mechanical clocks and watches depends on Hooke's law. Since the torque generated by the coiled spring is proportional to the angle turned by the wheel, its oscillations have a nearly constant period. Hooke ...
, which says that the force experienced by a spring is proportional to the distance displaced from equilibrium. Stresses which can be attributed to the deformation of a material from some rest state are called elastic stresses. In other materials, stresses are present which can be attributed to the rate of change of the deformation over time. These are called viscous stresses. For instance, in a fluid such as water the stresses which arise from shearing the fluid do not depend on the ''distance'' the fluid has been sheared; rather, they depend on how ''quickly'' the shearing occurs. Viscosity is the material property which relates the viscous stresses in a material to the rate of change of a deformation (the strain rate). Although it applies to general flows, it is easy to visualize and define in a simple shearing flow, such as a planar
Couette flowIn fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodynamics (the study of air and other gases in ...

Couette flow
. In the Couette flow, a fluid is trapped between two infinitely large plates, one fixed and one in parallel motion at constant speed u (see illustration to the right). If the speed of the top plate is low enough (to avoid turbulence), then in steady state the fluid particles move parallel to it, and their speed varies from 0 at the bottom to u at the top. Each layer of fluid moves faster than the one just below it, and friction between them gives rise to a
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...
resisting their relative motion. In particular, the fluid applies on the top plate a force in the direction opposite to its motion, and an equal but opposite force on the bottom plate. An external force is therefore required in order to keep the top plate moving at constant speed. In many fluids, the flow velocity is observed to vary linearly from zero at the bottom to u at the top. Moreover, the magnitude F of the force acting on the top plate is found to be proportional to the speed u and the area A of each plate, and inversely proportional to their separation y: : F=\mu A \frac. The proportionality factor is the ''dynamic viscosity'' of the fluid, often simply referred to as the ''viscosity''. It is denoted by the Greek letter mu (). The dynamic viscosity has the
dimensions thumb , 236px , The first four spatial dimensions, represented in a two-dimensional picture. In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature ...
\mathrm, therefore resulting in the
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system The metric system is a system of measurement A syste ...
and the derived units: : = \frac = \frac \cdot s = =
pressure Pressure (symbol: ''p'' or ''P'') is the force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...

pressure
multiplied by
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

time
. The afore mentioned ratio u/y is called the ''rate of shear deformation'' or ''
shear velocity Shear velocity, also called friction velocity, is a form by which a shear stress Shear stress, often denoted by ( Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of ...
'', and is the
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

derivative
of the fluid speed in the direction
perpendicular In elementary geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with propertie ...

perpendicular
to the plates (see illustrations to the right). If the velocity does not vary linearly with y, then the appropriate generalization is: :\tau=\mu \frac, where \tau = F / A, and \partial u / \partial y is the local shear velocity. This expression is referred to as Newton's law of viscosity. In shearing flows with planar symmetry, it is what ''defines'' \mu. It is a special case of the general definition of viscosity (see below), which can be expressed in coordinate-free form. Use of the Greek letter mu (\mu) for the dynamic viscosity (sometimes also called the ''absolute viscosity'') is common among
mechanical Mechanical may refer to: Machine * Mechanical system A machine is any physical system with ordered structural and functional properties. It may represent human-made or naturally occurring device molecular machine that uses Power (physics), p ...

mechanical
and
chemical engineer 200px, Chemical engineers design, construct and operate plants. In the field of engineering, a chemical engineer is a professional, equipped with the knowledge of chemical engineering upright=1.15, Chemical engineers design, construct and operate ...
s, as well as mathematicians and physicists. However, the Greek letter eta (\eta) is also used by chemists, physicists, and the
IUPAC The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering OrganizationsNational Adhering Organizations in chemistry are the organizations that work as the authoritative power over chemist ...
. The viscosity \mu is sometimes also called the ''shear viscosity''. However, at least one author discourages the use of this terminology, noting that \mu can appear in non-shearing flows in addition to shearing flows.


Kinematic viscosity

In fluid dynamics, it is sometimes more appropriate to work in terms of ''kinematic viscosity'' (sometimes also called the ''momentum diffusivity''), defined as the ratio of the dynamic viscosity () over the
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

density
of the fluid (). It is usually denoted by the Greek letter nu (): : \nu = \frac , and has the
dimensions thumb , 236px , The first four spatial dimensions, represented in a two-dimensional picture. In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature ...
\mathrm, therefore resulting in the
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system The metric system is a system of measurement A syste ...
and the derived units: : = \frac = \mathrm = \mathrm =
specific energy Specific energy or massic energy is energy In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, it ...

specific energy
multiplied by
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

time
.


General definition

In very general terms, the viscous stresses in a fluid are defined as those resulting from the relative velocity of different fluid particles. As such, the viscous stresses must depend on spatial gradients of the flow velocity. If the velocity gradients are small, then to a first approximation the viscous stresses depend only on the first derivatives of the velocity. (For Newtonian fluids, this is also a linear dependence.) In Cartesian coordinates, the general relationship can then be written as : \tau_ = \sum_k \sum_\ell \mu_ \frac, where \mu_ is a viscosity tensor that maps the velocity gradient tensor \partial v_k / \partial r_\ell onto the viscous stress tensor \tau_. Since the indices in this expression can vary from 1 to 3, there are 81 "viscosity coefficients" \mu_ in total. However, assuming that the viscosity rank-4 tensor is
isotropic Isotropy is uniformity in all orientations; it is derived from the Greek ''isos'' (ἴσος, "equal") and ''tropos'' (τρόπος, "way"). Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by t ...
reduces these 81 coefficients to three independent parameters \alpha, \beta, \gamma: : \mu_ = \alpha \delta_\delta_ + \beta \delta_\delta_ + \gamma \delta_\delta_, and furthermore, it is assumed that no viscous forces may arise when the fluid is undergoing simple rigid-body rotation, thus \beta = \gamma, leaving only two independent parameters. The most usual decomposition is in terms of the standard (scalar) viscosity \mu and the bulk viscosity \kappa such that \alpha = \kappa - \tfrac\mu and \beta = \gamma = \mu. In vector notation this appears as: : \boldsymbol = \mu \left nabla \mathbf + (\nabla \mathbf)^ \right- \left(\frac \mu - \kappa \right) (\nabla \cdot \mathbf) \mathbf, where \mathbf is the unit tensor, and the dagger \dagger denotes the
transpose In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces an ...

transpose
. This equation can be thought of as a generalized form of Newton's law of viscosity. The bulk viscosity (also called volume viscosity) expresses a type of internal friction that resists the shearless compression or expansion of a fluid. Knowledge of \kappa is frequently not necessary in fluid dynamics problems. For example, an incompressible fluid satisfies \nabla \cdot \mathbf = 0 and so the term containing \kappa drops out. Moreover, \kappa is often assumed to be negligible for gases since it is 0 in a
monatomic In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Phy ...
ideal gas An ideal gas is a theoretical gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion ...
. One situation in which \kappa can be important is the calculation of energy loss in
sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the b ...

sound
and
shock wave of an attached shock on a sharp-nosed supersonic F/A-18F Super Hornet in transonic flight Flight or flying is the process by which an object (physics), object motion (physics), moves through a space without contacting any planetary surfac ...
s, described by Stokes' law of sound attenuation, since these phenomena involve rapid expansions and compressions. It is worth emphasizing that the above expressions are not fundamental laws of nature, but rather definitions of viscosity. As such, their utility for any given material, as well as means for measuring or calculating the viscosity, must be established using separate means.


Momentum transport

Transport theory provides an alternative interpretation of viscosity in terms of momentum transport: viscosity is the material property which characterizes momentum transport within a fluid, just as
thermal conductivity The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa. Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...

thermal conductivity
characterizes
heat In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these ...

heat
transport, and (mass) diffusivity characterizes mass transport. To see this, note that in Newton's law of viscosity, \tau = \mu (\partial u / \partial y), the shear stress \tau has units equivalent to a momentum
flux of \mathbf(\mathbf) with the unit normal vector \mathbf(\mathbf) ''(blue arrows)'' at the point \mathbf multiplied by the area dS. The sum of \mathbf\cdot\mathbf dS for each patch on the surface is the flux through the surface Flux describes ...

flux
, i.e. momentum per unit time per unit area. Thus, \tau can be interpreted as specifying the flow of momentum in the y direction from one fluid layer to the next. Per Newton's law of viscosity, this momentum flow occurs across a velocity gradient, and the magnitude of the corresponding momentum flux is determined by the viscosity. The analogy with heat and mass transfer can be made explicit. Just as heat flows from high temperature to low temperature and mass flows from high density to low density, momentum flows from high velocity to low velocity. These behaviors are all described by compact expressions, called constitutive relations, whose one-dimensional forms are given here: : \begin \mathbf &= -D \frac & & \text \\\mathbf &= -k_t \frac & & \text \\\tau &= \mu \frac & & \text \end where \rho is the density, \mathbf and \mathbf are the mass and heat fluxes, and D and k_t are the mass diffusivity and thermal conductivity. The fact that mass, momentum, and energy (heat) transport are among the most relevant processes in continuum mechanics is not a coincidence: these are among the few physical quantities that are conserved at the microscopic level in interparticle collisions. Thus, rather than being dictated by the fast and complex microscopic interaction timescale, their dynamics occurs on macroscopic timescales, as described by the various equations of transport theory and hydrodynamics.


Newtonian and non-Newtonian fluids

Newton's law of viscosity is not a fundamental law of nature, but rather a
constitutive equation In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Sp ...
(like
Hooke's law The balance wheel at the core of many mechanical clocks and watches depends on Hooke's law. Since the torque generated by the coiled spring is proportional to the angle turned by the wheel, its oscillations have a nearly constant period. Hooke ...
,
Fick's law Fick's laws of diffusion describe diffusion File:DiffusionMicroMacro.gif, 250px, Diffusion from a microscopic and macroscopic point of view. Initially, there are solution, solute molecules on the left side of a barrier (purple line) and none ...
, and
Ohm's law Ohm's law states that the electric current, current through a Electrical conductor, conductor between two points is directly Proportionality (mathematics), proportional to the voltage across the two points. Introducing the constant of proporti ...

Ohm's law
) which serves to define the viscosity \mu. Its form is motivated by experiments which show that for a wide range of fluids, \mu is independent of strain rate. Such fluids are called
NewtonianNewtonian refers to the work of Isaac Newton, in particular: * Newtonian mechanics, i.e. classical mechanics * Newtonian telescope, a type of reflecting telescope * Newtonian cosmology * Newtonian dynamics * Newtonianism, the philosophical principle ...
.
Gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...

Gas
es,
water Water (chemical formula H2O) is an , transparent, tasteless, odorless, and , which is the main constituent of 's and the s of all known living organisms (in which it acts as a ). It is vital for all known forms of , even though it provide ...

water
, and many common liquids can be considered Newtonian in ordinary conditions and contexts. However, there are many
non-Newtonian fluid A non-Newtonian fluid is a fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external force. Fluids are a Phase (matter), phase of matter and include liquids, ...

non-Newtonian fluid
s that significantly deviate from this behavior. For example: * Shear-thickening (dilatant) liquids, whose viscosity increases with the rate of shear strain. * liquids, whose viscosity decreases with the rate of shear strain. *
Thixotropic Thixotropy is a time-dependent shear thinning In rheology, shear thinning is the non-Newtonian behavior of fluids whose viscosity The viscosity of a fluid is a measure of its drag (physics), resistance to deformation at a given rate. For liq ...
liquids, that become less viscous over time when shaken, agitated, or otherwise stressed. *
Rheopectic Rheopecty or rheopexy is the rare property of some non-Newtonian fluid A non-Newtonian fluid is a fluid that does not follow Viscosity#Newtonian and non-Newtonian fluids, Newton's law of viscosity, i.e., constant viscosity independent of stress. I ...
liquids, that become more viscous over time when shaken, agitated, or otherwise stressed. *
Bingham plastic A Bingham plastic is a viscoplastic material that behaves as a rigid body at low stresses but flows as a viscous The viscosity of a fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under ...
s that behave as a solid at low stresses but flow as a viscous fluid at high stresses. Trouton's ratio is the ratio of extensional viscosity to shear viscosity. For a Newtonian fluid, the Trouton ratio is 3. 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 fluid Compressible flow (or gas dynamics) is the branch of fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objec ...
s, 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 A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with ...

magnetic field
, possibly to the point of behaving like a solid.


In solids

The viscous forces that arise during fluid flow must not be confused with the elastic forces that arise in a solid in response to shear, compression or extension stresses. While in the latter the stress is proportional to the ''amount'' of shear deformation, in a fluid it is proportional to the ''rate'' of deformation over time. For this reason, 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 Granite () is a coarse-grained (phaneritic A phanerite is an igneous rock Igneous rock (derived from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Lat ...

granite
) will flow like liquids, albeit very slowly, even under arbitrarily small stress. Such materials are therefore best described as possessing both elasticity (reaction to deformation) and viscosity (reaction to rate of deformation); that is, being
viscoelastic In and , viscoelasticity is the property of that exhibit both and characteristics when undergoing . Viscous materials, like water, resist and linearly with time when a is applied. Elastic materials strain when stretched and immediately ret ...
. Viscoelastic solids may exhibit both shear viscosity and bulk viscosity. The extensional viscosity is a
linear combination In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 Geology (from the Ancient Greek Ancient Greek includes the forms of the Greek language Greek ( el, label=Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek ...

geology
, earth materials that exhibit viscous deformation at least three
orders of magnitude An order of magnitude is an approximation of the logarithm In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geome ...
greater than their elastic deformation are sometimes called rheids.


Measurement

Viscosity is measured with various types of
viscometer A viscometer (also called viscosimeter) is an instrument used to measure the viscosity of a fluid. For liquids with viscosities which vary with flow conditioning, flow conditions, an instrument called a rheometer is used. Thus, a rheometer can be co ...
s and
rheometer A 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 ...

rheometer
s. A rheometer is used for fluids that cannot be defined by a single value of viscosity and therefore require more parameters to be set and measured than is the case for a viscometer. Close temperature control of the fluid is essential to obtain accurate measurements, particularly in materials like lubricants, whose viscosity can double with a change of only 5 °C. For some fluids, the viscosity is constant over a wide range of shear rates (
Newtonian fluids A Newtonian fluid is a fluid in which the viscous stress tensor, viscous stresses arising from its Fluid dynamics, flow, at every point, are linearly correlated to the local strain rate—the derivative (mathematics), rate of change of its deformat ...
). The fluids without a constant viscosity (
non-Newtonian fluid A non-Newtonian fluid is a fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external force. Fluids are a Phase (matter), phase of matter and include liquids, ...

non-Newtonian fluid
s) cannot be described by a single number. Non-Newtonian fluids exhibit a variety of different correlations between shear stress and shear rate. One of the most common instruments for measuring kinematic viscosity is the glass capillary viscometer. In
coating A coating is a covering that is applied to the surface of an object, usually referred to as the substrate Substrate may refer to: Physical layers *Substrate (biology), the natural environment in which an organism lives, or the surface or medium ...
industries, viscosity may be measured with a cup in which the efflux time is measured. There are several sorts of cup – such as the
Zahn cup A 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 c ...
and the
Ford viscosity cup 150px, Ford viscosity cup 4 mm The Ford viscosity cup is a simple gravity Gravity (), or gravitation, is a list of natural phenomena, natural phenomenon by which all things with mass or energy—including planets, stars, galaxy, galaxies, an ...
– with the usage of each type varying mainly according to the industry. The efflux time can also be converted to kinematic viscosities (centistokes, cSt) through the conversion equations. Also used in coatings, a Stormer viscometer uses load-based rotation in order to determine viscosity. The viscosity is reported in Krebs units (KU), which are unique to Stormer viscometers. Vibrating viscometers can also be used to measure viscosity. Resonant, or vibrational viscometers work by creating shear waves within the liquid. In this method, the sensor is submerged in the fluid and is made to resonate at a specific frequency. As the surface of the sensor shears through the liquid, energy is lost due to its viscosity. This dissipated energy is then measured and converted into a viscosity reading. A higher viscosity causes a greater loss of energy. ''Extensional viscosity'' can be measured with various
rheometer A 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 ...

rheometer
s that apply extensional stress. Volume viscosity can be measured with an acoustic rheometer. is a calculation derived from tests performed on
drilling fluid Driller pouring anti-foaming agent down the drilling string on a drilling rig In geotechnical engineering#REDIRECT geotechnical engineering {{Redirect category shell, 1= {{R from other capitalisation ..., drilling fluid, also called drilli ...
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. Nanoviscosity (viscosity sensed by nanoprobes) can be measured by
Fluorescence correlation spectroscopy Fluorescence correlation spectroscopy (FCS) is a statistical analysis, via time correlation, of stationary fluctuations of the fluorescence intensity. Its theoretical underpinning originated from L. Onsager's regression hypothesis. The analysis ...
.


Units

The SI unit of dynamic viscosity is the
newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * Newton (film), ''Newton'' (film), a 2017 Indian fil ...
-second per square meter (N·s/m2), also frequently expressed in the equivalent forms
pascal Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, French ...
-
second The second (symbol: s, also abbreviated: sec) is the base unit of time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, th ...
(Pa·s) and
kilogram The kilogram (also kilogramme) is the base unit of mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of " ...
per meter per second (kg·m−1·s−1). The CGS unit is the poise (P, or g·cm−1·s−1 = 0.1 Pa·s), named after Jean Léonard Marie Poiseuille. It is commonly expressed, particularly in
ASTM ASTM International, formerly known as American Society for Testing and Materials, is an international standards organization A standards organization, standards body, standards developing organization (SDO), or standards setting organization ( ...
standards, as ''centipoise'' (cP), because it is more convenient (for instance the viscosity of water at 20 °C is about 1 cP), and one centipoise is equal to the SI millipascal second (mPa·s). The SI unit of kinematic viscosity is square meter per second (m2/s), whereas the CGS unit for kinematic viscosity is the stokes (St, or cm2·s−1 = 0.0001 m2·s−1), named after Sir
George Gabriel Stokes Sir George Gabriel Stokes, 1st Baronet, (; 13 August 1819 – 1 February 1903) was an Anglo-Irish physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in a ...

George Gabriel Stokes
. In U.S. usage, ''stoke'' is sometimes used as the singular form. The submultiple ''centistokes'' (cSt) is often used instead, 1 cSt = 1 mm2·s−1 = 10−6 m2·s−1. The kinematic viscosity of water at 20 °C is about 1 cSt. The most frequently used systems of US customary, or Imperial, units are the British Gravitational (BG) and English Engineering units, English Engineering (EE). In the BG system, dynamic viscosity has units of Pound (force), ''pound''-seconds per square foot (lb·s/ft2), and in the EE system it has units of pound (force), ''pound-force''-seconds per square foot (lbf·s/ft2). Note that the pound and pound-force are equivalent; the two systems differ only in how force and mass are defined. In the BG system the pound is a basic unit from which the unit of mass (the slug (unit), slug) is defined by Newton's Second Law, whereas in the EE system the units of force and mass (the pound-force and pound (mass), pound-mass respectively) are defined independently through the Second Law using the gc (engineering), proportionality constant ''gc''. Kinematic viscosity has units of square feet per second (ft2/s) in both the BG and EE systems. Nonstandard units include the reyn, a British unit of dynamic viscosity. In the automotive industry the viscosity index is used to describe the change of viscosity with temperature. The Multiplicative inverse, reciprocal of viscosity is ''fluidity'', usually symbolized by \phi = 1 / \mu or F = 1 / \mu, depending on the convention used, measured in ''reciprocal poise'' (P−1, or centimetre, cm·second, s·gram, g−1), sometimes called the ''rhe''. Fluidity is seldom used in
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more specializ ...

engineering
practice. At one time the petroleum industry relied on measuring kinematic viscosity by means of the Saybolt viscometer, and expressing kinematic viscosity in units of Saybolt universal seconds (SUS). Other abbreviations such as SSU (''Saybolt seconds universal'') or SUV (''Saybolt universal viscosity'') are sometimes used. Kinematic viscosity in centistokes can be converted from SUS according to the arithmetic and the reference table provided in
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D 2161.


Molecular origins

In general, the viscosity of a system depends in detail on how the molecules constituting the system interact. There are no simple but correct expressions for the viscosity of a fluid. The simplest exact expressions are the Green–Kubo relations for the linear shear viscosity or the ''transient time correlation function'' expressions derived by Evans and Morriss in 1988. Although these expressions are each exact, calculating the viscosity of a dense fluid using these relations currently requires the use of molecular dynamics computer simulations. On the other hand, much more progress can be made for a dilute gas. Even elementary assumptions about how gas molecules move and interact lead to a basic understanding of the molecular origins of viscosity. More sophisticated treatments can be constructed by systematically coarse-graining the equations of motion of the gas molecules. An example of such a treatment is Chapman–Enskog theory, which derives expressions for the viscosity of a dilute gas from the Boltzmann equation. Momentum transport in gases is generally mediated by discrete molecular collisions, and in liquids by attractive forces which bind molecules close together. Because of this, the dynamic viscosities of liquids are typically much larger than those of gases.


Pure gases

: Viscosity in gases arises principally from the molecular diffusion that transports momentum between layers of flow. An elementary calculation for a dilute gas at temperature T and density \rho gives :\mu = \alpha\rho\lambda\sqrt, where k_\text is the Boltzmann constant, m the molecular mass, and \alpha a numerical constant on the order of 1. The quantity \lambda, the mean free path, measures the average distance a molecule travels between collisions. Even without ''a priori'' knowledge of \alpha, this expression has interesting implications. In particular, since \lambda is typically inversely proportional to density and increases with temperature, \mu itself should increase with temperature and be ''independent'' of density at fixed temperature. In fact, both of these predictions persist in more sophisticated treatments, and accurately describe experimental observations. Note that this behavior runs counter to common intuition regarding liquids, for which viscosity typically ''decreases'' with temperature. For rigid elastic spheres of diameter \sigma, \lambda can be computed, giving : \mu = \frac \frac. In this case \lambda is independent of temperature, so \mu \propto T^. For more complicated molecular models, however, \lambda depends on temperature in a non-trivial way, and simple kinetic arguments as used here are inadequate. More fundamentally, the notion of a mean free path becomes imprecise for particles that interact over a finite range, which limits the usefulness of the concept for describing real-world gases.


Chapman–Enskog theory

A technique developed by Sydney Chapman (mathematician), Sydney Chapman and David Enskog in the early 1900s allows a more refined calculation of \mu. It is based on the Boltzmann equation, which provides a systematic statistical description of a dilute gas in terms of intermolecular interactions. As such, their technique allows accurate calculation of \mu for more realistic molecular models, such as those incorporating intermolecular attraction rather than just hard-core repulsion. It turns out that a more realistic modeling of interactions is essential for accurate prediction of the temperature dependence of \mu, which experiments show increases more rapidly than the T^ trend predicted for rigid elastic spheres. Indeed, the Chapman–Enskog analysis shows that the predicted temperature dependence can be tuned by varying the parameters in various molecular models. A simple example is the Sutherland model, which describes rigid elastic spheres with ''weak'' mutual attraction. In such a case, the attractive force can be treated Perturbation theory, perturbatively, which leads to a particularly simple expression for \mu: : \mu = \frac \left(\frac\right)^ \left(1 + \frac \right)^, where S is independent of temperature, being determined only by the parameters of the intermolecular attraction. To connect with experiment, it is convenient to rewrite as : \mu = \mu_0 \left(\frac\right)^ \frac, where \mu_0 is the viscosity at temperature T_0. If \mu is known from experiments at T = T_0 and at least one other temperature, then S can be calculated. It turns out that expressions for \mu obtained in this way are accurate for a number of gases over a sizable range of temperatures. On the other hand, argue that this success does not imply that molecules actually interact according to the Sutherland model. Rather, they interpret the prediction for \mu as a simple interpolation which is valid for some gases over fixed ranges of temperature, but otherwise does not provide a picture of intermolecular interactions which is fundamentally correct and general. Slightly more sophisticated models, such as the Lennard-Jones potential, may provide a better picture, but only at the cost of a more opaque dependence on temperature. In some systems the assumption of spherical symmetry must be abandoned as well, as is the case for vapors with highly polar molecules like Properties of water, H2O.


Bulk viscosity

In the kinetic-molecular picture, a non-zero bulk viscosity arises in gases whenever there are non-negligible relaxational timescales governing the exchange of energy between the translational energy of molecules and their internal energy, e.g. Rotational energy, rotational and vibrational. As such, the bulk viscosity is 0 for a monatomic ideal gas, in which the internal energy of molecules in negligible, but is nonzero for a gas like carbon dioxide, whose molecules possess both rotational and vibrational energy.


Pure liquids

In contrast with gases, there is no simple yet accurate picture for the molecular origins of viscosity in liquids. At the simplest level of description, the relative motion of adjacent layers in a liquid is opposed primarily by attractive molecular forces acting across the layer boundary. In this picture, one (correctly) expects viscosity to decrease with increasing temperature. This is because increasing temperature increases the random thermal motion of the molecules, which makes it easier for them to overcome their attractive interactions. Building on this visualization, a simple theory can be constructed in analogy with the discrete structure of a solid: groups of molecules in a liquid are visualized as forming "cages" which surround and enclose single molecules. These cages can be occupied or unoccupied, and stronger molecular attraction corresponds to stronger cages. Due to random thermal motion, a molecule "hops" between cages at a rate which varies inversely with the strength of molecular attractions. In Thermodynamic equilibrium, equilibrium these "hops" are not biased in any direction. On the other hand, in order for two adjacent layers to move relative to each other, the "hops" must be biased in the direction of the relative motion. The force required to sustain this directed motion can be estimated for a given shear rate, leading to where N_A is the Avogadro constant, h is the Planck constant, V is the volume of a mole (unit), mole of liquid, and T_b is the normal boiling point. This result has the same form as the widespread and accurate empirical relation where A and B are constants fit from data. On the other hand, several authors express caution with respect to this model. Errors as large as 30% can be encountered using equation (), compared with fitting equation () to experimental data. More fundamentally, the physical assumptions underlying equation () have been criticized. It has also been argued that the exponential dependence in equation () does not necessarily describe experimental observations more accurately than simpler, non-exponential expressions. In light of these shortcomings, the development of a less ad hoc model is a matter of practical interest. Foregoing simplicity in favor of precision, it is possible to write rigorous expressions for viscosity starting from the fundamental equations of motion for molecules. A classic example of this approach is Irving–Kirkwood theory. On the other hand, such expressions are given as averages over multiparticle Correlation function (statistical mechanics), correlation functions and are therefore difficult to apply in practice. In general, empirically derived expressions (based on existing viscosity measurements) appear to be the only consistently reliable means of calculating viscosity in liquids.


Mixtures and blends


Gaseous mixtures

The same molecular-kinetic picture of a single component gas can also be applied to a gaseous mixture. For instance, in the Chapman–Enskog approach the viscosity \mu_ of a binary mixture of gases can be written in terms of the individual component viscosities \mu_, their respective volume fractions, and the intermolecular interactions. As for the single-component gas, the dependence of \mu_ on the parameters of the intermolecular interactions enters through various collisional integrals which may not be expressible in terms of elementary functions. To obtain usable expressions for \mu_ which reasonably match experimental data, the collisional integrals typically must be evaluated using some combination of analytic calculation and empirical fitting. An example of such a procedure is the Sutherland approach for the single-component gas, discussed above.


Blends of liquids

As for pure liquids, the viscosity of a blend of liquids is difficult to predict from molecular principles. One method is to extend the molecular "cage" theory presented above for a pure liquid. This can be done with varying levels of sophistication. One useful expression resulting from such an analysis is the Lederer–Roegiers equation for a binary mixture: : \ln \mu_\text = \frac \ln \mu_1 + \frac \ln \mu_2, where \alpha is an empirical parameter, and x_ and \mu_ are the respective mole fractions and viscosities of the component liquids. Since blending is an important process in the lubricating and oil industries, a variety of empirical and propriety equations exist for predicting the viscosity of a blend, besides those stemming directly from molecular theory.


Solutions and suspensions


Aqeuous solutions

Depending on the solute and range of concentration, an aqueous electrolyte solution can have either a larger or smaller viscosity compared with pure water at the same temperature and pressure. For instance, a 20% saline (sodium chloride) solution has viscosity over 1.5 times that of pure water, whereas a 20% potassium iodide solution has viscosity about 0.91 times that of pure water. An idealized model of dilute electrolytic solutions leads to the following prediction for the viscosity \mu_s of a solution: : \frac = 1 + A \sqrt, where \mu_0 is the viscosity of the solvent, c is the concentration, and A is a positive constant which depends on both solvent and solute properties. However, this expression is only valid for very dilute solutions, having c less than 0.1 mol/L. For higher concentrations, additional terms are necessary which account for higher-order molecular correlations: : \frac = 1 + A \sqrt + B c + C c^2, where B and C are fit from data. In particular, a negative value of B is able to account for the decrease in viscosity observed in some solutions. Estimated values of these constants are shown below for sodium chloride and potassium iodide at temperature 25 °C (mol = Mole (unit), mole, L = liter).


Suspensions

In a suspension of solid particles (e.g. micron-size spheres suspended in oil), an effective viscosity \mu_ can be defined in terms of stress and strain components which are averaged over a volume large compared with the distance between the suspended particles, but small with respect to macroscopic dimensions. Such suspensions generally exhibit non-Newtonian behavior. However, for dilute systems in steady flows, the behavior is Newtonian and expressions for \mu_ can be derived directly from the particle dynamics. In a very dilute system, with volume fraction \phi \lesssim 0.02, interactions between the suspended particles can be ignored. In such a case one can explicitly calculate the flow field around each particle independently, and combine the results to obtain \mu_\text. For spheres, this results in the Einstein equation: : \mu_\text = \mu_0 \left(1 + \frac \phi \right), where \mu_0 is the viscosity of the suspending liquid. The linear dependence on \phi is a direct consequence of neglecting interparticle interactions; in general, one will have : \mu_\text = \mu_0 \left(1 + B \phi \right), where the coefficient B may depend on the particle shape (e.g. spheres, rods, disks). Experimental determination of the precise value of B is difficult, however: even the prediction B = 5/2 for spheres has not been conclusively validated, with various experiments finding values in the range 1.5 \lesssim B \lesssim 5. This deficiency has been attributed to difficulty in controlling experimental conditions. In denser suspensions, \mu_ acquires a nonlinear dependence on \phi, which indicates the importance of interparticle interactions. Various analytical and semi-empirical schemes exist for capturing this regime. At the most basic level, a term quadratic in \phi is added to \mu_\text: : \mu_\text = \mu_0 \left(1 + B \phi + B_1 \phi^2 \right), and the coefficient B_1 is fit from experimental data or approximated from the microscopic theory. In general, however, one should be cautious in applying such simple formulas since non-Newtonian behavior appears in dense suspensions (\phi \gtrsim 0.25 for spheres), or in suspensions of elongated or flexible particles. There is a distinction between a suspension of solid particles, described above, and an emulsion. The latter is a suspension of tiny droplets, which themselves may exhibit internal circulation. The presence of internal circulation can noticeably decrease the observed effective viscosity, and different theoretical or semi-empirical models must be used.


Amorphous materials

In the high and low temperature limits, viscous flow in Amorphous solid, amorphous materials (e.g. in glasses and melts) has the Arrhenius equation, Arrhenius form: :\mu = A e^, where is a relevant activation energy, given in terms of molecular parameters; is temperature; is the molar gas constant; and is approximately a constant. The activation energy takes a different value depending on whether the high or low temperature limit is being considered: it changes from a high value at low temperatures (in the glassy state) to a low value at high temperatures (in the liquid state). For intermediate temperatures, Q varies nontrivially with temperature and the simple Arrhenius form fails. On the other hand, the two-exponential equation :\mu = AT \exp\left(\frac\right) \left[ 1 + C \exp\left(\frac\right) \right], where A, B, C, D are all constants, provides a good fit to experimental data over the entire range of temperatures, while at the same time reducing to the correct Arrhenius form in the low and high temperature limits. Besides being a convenient fit to data, the expression can also be derived from various theoretical models of amorphous materials at the atomic level. A two-exponential equation for the viscosity can be derived within the Dyre shoving model of supercooled liquids, where the Arrhenius energy barrier is identified with the high-frequency shear modulus times a characteristic shoving volume. Upon specifying the temperature dependence of the shear modulus via thermal expansion and via the repulsive part of the intermolecular potential, another two-exponential equation is retrieved: : \mu = \exp where C_ denotes the high-frequency shear modulus of the material evaluated at a temperature equal to the glass transition temperature T_ , V_ is the so-called shoving volume, i.e. it is the characteristic volume of the group of atoms involved in the shoving event by which an atom/molecule escapes from the cage of nearest-neighbours, typically on the order of the volume occupied by few atoms. Furthermore, \alpha_ is the thermal expansion coefficient of the material, \lambda is a parameter which measures the steepness of the power-law rise of the ascending flank of the first peak of the radial distribution function, and is quantitatively related to the repulsive part of the interatomic potential. Finally, k_ denotes the Boltzmann constant.


Eddy viscosity

In the study of turbulence in
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s, a common practical strategy is to ignore the small-scale vortex, vortices (or eddy (fluid dynamics), eddies) in the motion and to calculate a large-scale motion with an ''effective'' viscosity, called the "eddy viscosity", which characterizes the transport and dissipation of energy in the smaller-scale flow (see large eddy simulation). In contrast to the viscosity of the fluid itself, which must be positive by the
second law of thermodynamics The second law of thermodynamics establishes the concept of entropy Entropy is a scientific concept, as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term an ...
, the eddy viscosity can be negative.


Selected substances

Observed values of viscosity vary over several orders of magnitude, even for common substances (see the order of magnitude table below). For instance, a 70% sucrose (sugar) solution has a viscosity over 400 times that of water, and 26000 times that of air. More dramatically, pitch has been estimated to have a viscosity 230 billion times that of water.


Water

The dynamic viscosity \mu of
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water
is about 0.89 mPa·s at room temperature (25 °C). As a function of temperature in kelvins, the viscosity can be estimated using the semi-empirical Vogel-Fulcher-Tammann equation: : \mu = A \exp\left( \frac \right) where ''A'' = 0.02939 mPa·s, ''B'' = 507.88 K, and ''C'' = 149.3 K. Experimentally determined values of the viscosity are also given in the table below. Note that at 20 °C the dynamic viscosity is about 1 cP and the kinematic viscosity is about 1 cSt.


Air

Under standard atmospheric conditions (25 °C and pressure of 1 bar), the dynamic viscosity of air is 18.5 μPa·s, roughly 50 times smaller than the viscosity of water at the same temperature. Except at very high pressure, the viscosity of air depends mostly on the temperature. Among the many possible approximate formulas for the temperature dependence (see Temperature dependence of viscosity), one is: :\eta_ = 2.791\cdot 10^ \cdot T^ which is accurate in the range -20 °C to 400 °C. For this formula to be valid, the temperature must be given in kelvins; \eta_ then corresponds to the viscosity in Pa·s.


Other common substances


Order of magnitude estimates

The following table illustrates the range of viscosity values observed in common substances. Unless otherwise noted, a temperature of 25 °C and a pressure of 1 atmosphere are assumed. Certain substances of variable composition or with non-Newtonian behavior are not assigned precise values, since in these cases viscosity depends on additional factors besides temperature and pressure.


See also


References


Footnotes


Citations


Sources

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External links


Fluid properties
– high accuracy calculation of viscosity for frequently encountered pure liquids and gases
Gas viscosity calculator as function of temperature

Air viscosity calculator as function of temperature and pressure


– a table of viscosities and vapor pressures for various fluids

– calculate coefficient of viscosity for mixtures of gases

– viscosity measurement, viscosity units and fixpoints, glass viscosity calculation

– conversion between kinematic and dynamic viscosity

– a table of water viscosity as a function of temperature
Vogel–Tammann–Fulcher Equation Parameters

Calculation of temperature-dependent dynamic viscosities for some common components


– United States Environmental Protection Agency
Artificial viscosity


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