The viscosity of a
fluid 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
** ''T ...
to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example,
syrup has a higher viscosity than
water
Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as ...
.
Viscosity quantifies the internal
frictional force 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. Experiments show that some
stress (such as a
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
difference between the two ends of the tube) is needed to sustain the flow. This is because a force is required to overcome the friction between the layers of the fluid which are in relative motion. For a tube with a constant rate of flow, the strength of the compensating force is proportional to the fluid's viscosity.
In general, viscosity depends on a fluid's state, such as its temperature, pressure, and rate of deformation. However, the dependence on some of these properties is negligible in certain cases. For example, the viscosity of a
Newtonian fluid does not vary significantly with the rate of deformation. Zero viscosity (no resistance to
shear stress) is observed only at
very low temperatures in
superfluids; otherwise, the
second law of thermodynamics
The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unles ...
requires all fluids to have positive viscosity. A fluid that has zero viscosity is called ''ideal'' or ''inviscid''.
Etymology
The word "viscosity" is derived from the
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power ...
' ("
mistletoe
Mistletoe is the common name for obligate hemiparasitic plants in the order Santalales. They are attached to their host tree or shrub by a structure called the haustorium, through which they extract water and nutrients from the host plant ...
"). ''Viscum'' also referred to a viscous
glue derived from mistletoe berries.
Definition
Dynamic viscosity

In
materials science and
engineering
Engineering is the use of scientific method, 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 rang ...
, one is often interested in understanding the forces or
stresses involved in the
deformation of a material. For instance, if the material were a simple spring, the answer would be given by
Hooke's law, 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 flow.
In the Couette flow, a fluid is trapped between two infinitely large plates, one fixed and one in parallel motion at constant speed
(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
at the bottom to
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 of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
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
at the top. Moreover, the magnitude of the force,
, acting on the top plate is found to be proportional to the speed
and the area
of each plate, and inversely proportional to their separation
:
:
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 , therefore resulting in the
SI units and the
derived units:
:
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
multiplied by
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, t ...
.
The aforementioned ratio
is called the ''rate of shear deformation'' or ''
shear velocity'', and is the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of the fluid speed in the direction
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
to the normal vector of the plates (see illustrations to the right). If the velocity does not vary linearly with
, then the appropriate generalization is:
:
where
, and
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''
. 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 (
) for the dynamic viscosity (sometimes also called the ''absolute viscosity'') is common among
mechanical and
chemical engineers, as well as mathematicians and physicists. However, the
Greek letter eta (
) is also used by chemists, physicists, and the
IUPAC. The viscosity
is sometimes also called the ''shear viscosity''. However, at least one author discourages the use of this terminology, noting that
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
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
of the fluid (). It is usually denoted by the
Greek letter nu ():
:
and has the
dimensions , therefore resulting in the
SI units and the
derived units:
:
specific energy multiplied by
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, t ...
.
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
:
where
is a viscosity tensor that maps the
velocity gradient tensor
onto the viscous stress tensor
. Since the indices in this expression can vary from 1 to 3, there are 81 "viscosity coefficients"
in total. However, assuming that the viscosity rank-4 tensor is
isotropic reduces these 81 coefficients to three independent parameters
,
,
:
:
and furthermore, it is assumed that no viscous forces may arise when the fluid is undergoing simple rigid-body rotation, thus
, leaving only two independent parameters. The most usual decomposition is in terms of the standard (scalar) viscosity
and the
bulk viscosity such that
and
. In vector notation this appears as:
:
where
is the unit tensor, and the dagger
denotes the
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
is frequently not necessary in fluid dynamics problems. For example, an incompressible fluid satisfies
and so the term containing
drops out. Moreover,
is often assumed to be negligible for gases since it is
in a
monatomic
In physics and chemistry, "monatomic" is a combination of the words "mono" and "atomic", and means "single atom". It is usually applied to gases: a monatomic gas is a gas in which atoms are not bound to each other. Examples at standard conditions ...
ideal gas. One situation in which
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 ...
and
shock wave
In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a me ...
s, described by
Stokes' law of sound attenuation Stokes's law of sound attenuation is a formula for the attenuation of sound in a Newtonian fluid, such as water or air, due to the fluid's viscosity. It states that the amplitude of a plane wave decreases exponentially with distance traveled, at a ...
, since these phenomena involve rapid expansions and compressions.
The defining equations for viscosity are not fundamental laws of nature, so their usefulness, as well as methods for measuring or calculating the viscosity, must be established using separate means. A potential issue is that viscosity depends, in principle, on the full microscopic state of the fluid, which encompasses the positions and momenta of every particle in the system. Such highly detailed information is typically not available in realistic systems. However, under certain conditions most of this information can be shown to be negligible. In particular, for Newtonian fluids near equilibrium and far from boundaries (bulk state), the viscosity depends only space- and time-dependent macroscopic fields (such as temperature and density) defining local equilibrium.
Nevertheless, viscosity may still carry a non-negligible dependence on several system properties, such as temperature, pressure, and the amplitude and frequency of any external forcing. Therefore, precision measurements of viscosity are only defined
with respect to a specific fluid state. To standardize comparisons among experiments and theoretical models, viscosity data is sometimes extrapolated to ideal limiting cases, such as the ''zero shear'' limit, or (for gases) the ''zero density'' limit.
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 ...
characterizes
heat
In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
transport, and (mass)
diffusivity characterizes mass transport. This perspective is implicit in Newton's law of viscosity,
, because the shear stress
has units equivalent to a momentum
flux
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ...
, i.e., momentum per unit time per unit area. Thus,
can be interpreted as specifying the flow of momentum in the
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:
:
where
is the density,
and
are the mass and heat fluxes, and
and
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 (like
Hooke's law,
Fick's law, and
Ohm's law) which serves to define the viscosity
. Its form is motivated by experiments which show that for a wide range of fluids,
is independent of strain rate. Such fluids are called
Newtonian.
Gases,
water
Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as ...
, and many common liquids can be considered Newtonian in ordinary conditions and contexts. However, there are many
non-Newtonian fluids that significantly deviate from this behavior. For example:
*
Shear-thickening (dilatant) liquids, whose viscosity increases with the rate of shear strain.
*
Shear-thinning liquids, whose viscosity decreases with the rate of shear strain.
*
Thixotropic liquids, that become less viscous over time when shaken, agitated, or otherwise stressed.
*
Rheopectic liquids, that become more viscous over time when shaken, agitated, or otherwise stressed.
*
Bingham plastics that behave as a solid at low stresses but flow as a viscous fluid at high stresses.
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 fluids, it depends on temperature and varies very slowly with pressure. The viscosity of some fluids may depend on other factors. A
magnetorheological fluid, for example, becomes thicker when subjected to a
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and t ...
, possibly to the point of behaving like a solid.
In solids
The viscous forces that arise during fluid flow are distinct from the
elastic forces that occur in a solid in response to shear, compression, or extension stresses. While in the latter the stress is proportional to the ''amount'' of shear deformation, in a fluid it is proportional to the ''rate'' of deformation over time. For this reason,
Maxwell
Maxwell may refer to:
People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage o ...
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) intrusive igneous rock composed mostly of quartz, alkali feldspar, and plagioclase. It forms from magma with a high content of silica and alkali metal oxides that slowly cools and solidifies und ...
) will flow like liquids, albeit very slowly, even under arbitrarily small stress. Such materials are best described as
viscoelastic—that is, possessing both elasticity (reaction to deformation) and viscosity (reaction to rate of deformation).
Viscoelastic solids may exhibit both shear viscosity and bulk viscosity. The
extensional viscosity is a
linear combination of the shear and bulk viscosities that describes the reaction of a solid elastic material to elongation. It is widely used for characterizing polymers.
In
geology
Geology () is a branch of natural science concerned with Earth and other astronomical objects, the features or rocks of which it is composed, and the processes by which they change over time. Modern geology significantly overlaps all other Ea ...
, earth materials that exhibit viscous deformation at least three
orders of magnitude greater than their elastic deformation are sometimes called
rheids.
Measurement
Viscosity is measured with various types of
viscometers and
rheometers. 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). The fluids without a constant viscosity (
non-Newtonian fluids) cannot be described by a single number. Non-Newtonian fluids exhibit a variety of different correlations between shear stress and shear rate.
One of the most common instruments for measuring kinematic viscosity is the glass capillary viscometer.
In
coating industries, viscosity may be measured with a cup in which the
efflux time is measured. There are several sorts of cup—such as the
Zahn cup and the
Ford viscosity cup—with the usage of each type varying mainly according to the industry.
Also used in coatings, a ''Stormer viscometer'' employs load-based rotation 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
rheometers that apply
extensional stress.
Volume viscosity can be measured with an
acoustic rheometer.
Apparent viscosity is a calculation derived from tests performed on
drilling fluid used in oil or gas well development. These calculations and tests help engineers develop and maintain the properties of the drilling fluid to the specifications required.
Nanoviscosity (viscosity sensed by nanoprobes) can be measured by
fluorescence correlation spectroscopy.
Units
The
SI unit of dynamic viscosity is the
newton-second per square meter (N·s/m
2), also frequently expressed in the equivalent forms
pascal-
second (Pa·s),
kilogram per meter per second (kg·m
−1·s
−1) and
poiseuille (Pl). The
CGS unit is the
poise
Poise may mean:
* Poise (unit), a measure of viscosity
* A concept similar to gracefulness
* Ferdinand Poise
image:Ferdinand Poise 1892.jpg,
Jean Alexandre Ferdinand Poise (3 June 1828 – 13 May 1892) was a French composer, mainly of opéra ...
(P, or g·cm
−1·s
−1 = 0.1 Pa·s), named after
Jean Léonard Marie Poiseuille. It is commonly expressed, particularly in
ASTM standards, as ''centipoise'' (cP). The centipoise is convenient because 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 (m
2/s), whereas the CGS unit for kinematic viscosity is the stokes (St, or cm
2·s
−1 = 0.0001 m
2·s
−1), named after Sir
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 mm
2·s
−1 = 10
−6 m
2·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 (EE). In the BG system, dynamic viscosity has units of
''pound''-seconds per square
foot (lb·s/ft
2), and in the EE system it has units of
''pound-force''-seconds per square foot (lbf·s/ft
2). 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) is defined by
Newton's Second Law, whereas in the EE system the units of force and mass (the pound-force and
pound-mass respectively) are defined independently through the Second Law using the
proportionality constant ''gc''.
Kinematic viscosity has units of square feet per second (ft
2/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
The viscosity index (VI) is an arbitrary, unit-less measure of a fluid's change in viscosity relative to temperature change. It is mostly used to characterize the viscosity-temperature behavior of lubricating oils. The lower the VI, the more the v ...
is used to describe the change of viscosity with temperature.
The
reciprocal of viscosity is ''fluidity'', usually symbolized by
or
, depending on the convention used, measured in ''reciprocal poise'' (P
−1, or
cm·
s·
g−1), sometimes called the ''rhe''. Fluidity is seldom used in
engineering
Engineering is the use of scientific method, 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 rang ...
practice.
At one time the petroleum industry relied on measuring kinematic viscosity by means of the
Saybolt viscometer Saybolt universal viscosity (SUV), and the related Saybolt FUROL viscosity (SFV), are specific standardised tests producing measures of kinematic viscosity. ''FUROL'' is an acronym for ''fuel and road oil''. Saybolt universal viscosity is specified ...
, 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
ASTM D 2161.
Molecular origins
Momentum transport in gases is mediated by discrete molecular collisions, and in liquids by attractive forces that bind molecules close together. Because of this, the dynamic viscosities of liquids are typically much larger than those of gases. In addition, viscosity tends to increase with temperature in gases and decrease with temperature in liquids.
Above the liquid-gas
critical point, the liquid and gas phases are replaced by a single
supercritical phase. In this regime, the mechanisms of momentum transport interpolate between liquid-like and gas-like behavior.
For example, along a supercritical
isobar (constant-pressure surface), the kinematic viscosity decreases at low temperature and increases at high temperature, with a minimum in between. A rough estimate for the value
at the minimum is
:
where
is the
Planck constant,
is the
electron mass, and
is the molecular mass.
In general, however, the viscosity of a system depends in detail on how the molecules constituting the system interact, and there are no simple but correct formulas for it. 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
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of th ...
computer simulations. Somewhat more progress can be made for a dilute gas, as 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
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Ler ...
.
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
and density
gives
:
where
is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas consta ...
,
the molecular mass, and
a numerical constant on the order of
. The quantity
, the
mean free path, measures the average distance a molecule travels between collisions. Even without ''a priori'' knowledge of
, this expression has nontrivial implications. In particular, since
is typically inversely proportional to density and increases with temperature,
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. By contrast, liquid viscosity typically decreases with temperature.
For rigid elastic spheres of diameter
,
can be computed, giving
:
In this case
is independent of temperature, so
. For more complicated molecular models, however,
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 Sydney Chapman may refer to:
*Sir Sydney Chapman (economist) (1871–1951), British economist and civil servant
* Sydney Chapman (mathematician) (1888–1970), FRS, British mathematician
*Sir Sydney Chapman (politician)
Sir Sydney Brookes Chapma ...
and
David Enskog 250px
David Enskog (22 April 1884, Västra Ämtervik, Sunne – 1 June 1947, Stockholm) was a Swedish mathematical physicist. Enskog helped develop the kinetic theory of gases by extending the Maxwell–Boltzmann equations.
Biography
Afte ...
in the early 1900s allows a more refined calculation of
. It is based on the
Boltzmann equation
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Ler ...
, which provides a statistical description of a dilute gas in terms of intermolecular interactions. The technique allows accurate calculation of
for molecular models that are more realistic than rigid elastic spheres, such as those incorporating intermolecular attractions. Doing so is necessary to reproduce the correct temperature dependence of
, which experiments show increases more rapidly than the
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
perturbatively, which leads to a simple expression for
:
:
where
is independent of temperature, being determined only by the parameters of the intermolecular attraction. To connect with experiment, it is convenient to rewrite as
:
where
is the viscosity at temperature
. If
is known from experiments at
and at least one other temperature, then
can be calculated. Expressions for
obtained in this way are qualitatively accurate for a number of simple gases. Slightly more sophisticated models, such as the
Lennard-Jones potential, may provide better agreement with experiments, but only at the cost of a more opaque dependence on temperature. In some systems, the assumption of
spherical symmetry
In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself.
Rotational circular symmetry is isomorphic with the circle group in the complex plane, or t ...
must be abandoned as well, as is the case for vapors with highly
polar molecules like
H2O. In these cases, the Chapman–Enskog analysis is significantly more complicated.
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 and
vibrational. As such, the bulk viscosity is
for a monatomic ideal gas, in which the internal energy of molecules in negligible, but is nonzero for a gas like
carbon dioxide
Carbon dioxide ( chemical formula ) is a chemical compound made up of molecules that each have one carbon atom covalently double bonded to two oxygen atoms. It is found in the gas state at room temperature. In the air, carbon dioxide is t ...
, 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
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
is the
Avogadro constant,
is the
Planck constant,
is the volume of a
mole of liquid, and
is the
normal boiling point. This result has the same form as the well-known empirical relation
where
and
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 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
of a binary mixture of gases can be written in terms of the individual component viscosities
, their respective volume fractions, and the intermolecular interactions. As for the single-component gas, the dependence of
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
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 expression resulting from such an analysis is the Lederer–Roegiers equation for a binary mixture:
:
where
is an empirical parameter, and
and
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.
Solutions and suspensions
Aqueous solutions
Depending on the
solute
In chemistry, a solution is a special type of homogeneous mixture composed of two or more substances. In such a mixture, a solute is a substance dissolved in another substance, known as a solvent. If the attractive forces between the solve ...
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
Sodium chloride , commonly known as salt (although sea salt also contains other chemical salts), is an ionic compound with the chemical formula NaCl, representing a 1:1 ratio of sodium and chloride ions. With molar masses of 22.99 and 35 ...
) 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
of a solution:
:
where
is the viscosity of the solvent,
is the concentration, and
is a positive constant which depends on both solvent and solute properties. However, this expression is only valid for very dilute solutions, having
less than 0.1 mol/L. For higher concentrations, additional terms are necessary which account for higher-order molecular correlations:
:
where
and
are fit from data. In particular, a negative value of
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, L =
liter).
Suspensions
In a suspension of solid particles (e.g.
micron-size spheres suspended in oil), an effective viscosity
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
can be derived directly from the particle dynamics. In a very dilute system, with volume fraction
, 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
. For spheres, this results in the Einstein equation:
:
where
is the viscosity of the suspending liquid. The linear dependence on
is a consequence of neglecting interparticle interactions. For dilute systems in general, one expects
to take the form
:
where the coefficient
may depend on the particle shape (e.g. spheres, rods, disks). Experimental determination of the precise value of
is difficult, however: even the prediction
for spheres has not been conclusively validated, with various experiments finding values in the range
. This deficiency has been attributed to difficulty in controlling experimental conditions.
In denser suspensions,
acquires a nonlinear dependence on
, 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
is added to
:
:
and the coefficient
is fit from experimental data or approximated from the microscopic theory. However, some authors advise caution in applying such simple formulas since non-Newtonian behavior appears in dense suspensions (
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 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 materials (e.g. in
glass
Glass is a non-Crystallinity, crystalline, often transparency and translucency, transparent, amorphous solid that has widespread practical, technological, and decorative use in, for example, window panes, tableware, and optics. Glass is most ...
es and melts) has the
Arrhenius form:
:
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,
varies nontrivially with temperature and the simple Arrhenius form fails. On the other hand, the two-exponential equation
:
where
,
,
,
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. This expression can be motivated 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
In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain:
:G \ \stack ...
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:
:
where
denotes the high-frequency
shear modulus
In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain:
:G \ \stack ...
of the material evaluated at a temperature equal to the
glass transition temperature
,
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,
is the
thermal expansion coefficient of the material,
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,
denotes the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas consta ...
.
Eddy viscosity
In the study of
turbulence in
fluids, a common practical strategy is to ignore the small-scale
vortices (or
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 physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
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 is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unles ...
, the eddy viscosity can be negative.
Prediction
Because viscosity depends continuously on temperature and pressure, it cannot be fully characterized by a finite number
of experimental measurements. Predictive formulas become necessary if experimental values are not available
at the temperatures and pressures of interest. This capability is important for thermophysical simulations,
in which the temperature and pressure of a fluid can vary continuously with space and time. A similar situation is encountered
for mixtures of pure fluids, where the viscosity depends continuously on the concentration ratios of the constituent fluids
For the simplest fluids, such as dilute monatomic gases and their mixtures, ''
ab initio''
quantum mechanical computations can accurately predict viscosity in terms
of fundamental atomic constants, i.e., without reference to existing viscosity measurements. For the special case of dilute helium,
uncertainties in the ''ab initio'' calculated viscosity are two order of magnitudes smaller than uncertainties in experimental values.
For most fluids, such high-accuracy, first-principles computations are not feasible. Rather, theoretical or empirical expressions
must be fit to existing viscosity measurements. If such an expression is fit to high-fidelity data over a large range of temperatures
and pressures, then it is called a "reference correlation" for that fluid. Reference correlations have been published for
many pure fluids; a few examples are
water
Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as ...
,
carbon dioxide
Carbon dioxide ( chemical formula ) is a chemical compound made up of molecules that each have one carbon atom covalently double bonded to two oxygen atoms. It is found in the gas state at room temperature. In the air, carbon dioxide is t ...
,
ammonia
Ammonia is an inorganic compound of nitrogen and hydrogen with the formula . A stable binary hydride, and the simplest pnictogen hydride, ammonia is a colourless gas with a distinct pungent smell. Biologically, it is a common nitrogeno ...
,
benzene
Benzene is an organic chemical compound with the molecular formula C6H6. The benzene molecule is composed of six carbon atoms joined in a planar ring with one hydrogen atom attached to each. Because it contains only carbon and hydrogen ato ...
, and
xenon. Many of these cover temperature and pressure ranges that encompass gas, liquid, and
supercritical phases.
Thermophysical modeling software often relies on reference correlations for predicting viscosity at user-specified temperature and pressure.
These correlations may be
proprietary. Examples are
REFPROP (proprietary) and
CoolProp
(open-source).
Viscosity can also be computed using formulas that express it in terms of the statistics of individual particle
trajectories. These formulas include the
Green–Kubo relations for the linear shear viscosity and the ''transient time correlation function''
expressions derived by Evans and Morriss in 1988.
The advantage of these expressions is that they are formally exact and valid for general systems. The disadvantage is that they require detailed
knowledge of particle trajectories, available only in computationally expensive simulations such as
molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of th ...
.
An accurate model for interparticle interactions is also required, which may be difficult to obtain for complex molecules.
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 of
water
Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as ...
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:
:
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. The values at 20 °C are a useful reference: there, 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:
:
which is accurate in the range −20 °C to 400 °C. For this formula to be valid, the temperature must be given in
kelvins;
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.
The values listed are representative estimates only, as they do not account for measurement uncertainties, variability in material definitions, or non-Newtonian behavior.
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 temperatureAir 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 ParametersCalculation of temperature-dependent dynamic viscosities for some common components–
United States Environmental Protection Agency
The Environmental Protection Agency (EPA) is an independent executive agency of the United States federal government tasked with environmental protection matters. President Richard Nixon proposed the establishment of EPA on July 9, 1970; it ...
Artificial viscosity
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