

Expressed in (SI unit):  Pa·s = kg/(s·m) 
Commonly used symbols:  μ 
Expressed in other quantities:  μ = G·t 
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or tensile stress. In everyday terms (and for fluids only), viscosity is "thickness". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity. Viscosity describes a fluid's internal resistance to flow and may be thought of as a measure of fluid friction. For example, highviscosity magma will create a tall, steep stratovolcano, because it cannot flow far before it cools, while lowviscosity lava will create a wide, shallowsloped shield volcano. Put simply, the less viscous something is, the greater its ease of movement (fluidity). ^{[1]} All real fluids (except superfluids) have some resistance to stress, but a fluid which has no resistance to shear stress is known as an ideal fluid or inviscid fluid. The study of viscosity is known as rheology.
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The word "viscosity" derives from the Latin word "viscum" for mistletoe. A viscous glue was made from mistletoe berries and used for limetwigs to catch birds.^{[2]}
Viscosity coefficients can be defined in two ways:
Viscosity is a tensorial quantity that can be decomposed in different ways into two independent components. The most usual decomposition yields the following viscosity coefficients:
Alternatively,
For example, at room temperature, water has a dynamic shear viscosity of about 1.0 × 10^{−3} Pa∙s and motor oil of about 250 × 10^{−3} Pa∙s.^{[3]}
In general, in any flow, layers move at different velocities and the fluid's viscosity arises from the shear stress between the layers that ultimately opposes any applied force.
Isaac Newton postulated that, for straight, parallel and uniform flow, the shear stress, τ, between layers is proportional to the velocity gradient, ∂u /∂y, in the direction perpendicular to the layers.
Here, the constant μ is known as the coefficient of viscosity, the viscosity, the dynamic viscosity, or the Newtonian viscosity.
This is a constitutive equation (like Hooke's law, Fick's law, Ohm's law). This means: it is not a fundamental law of nature, but a reasonable first approximation that holds in some materials and fails in others. Many fluids, such as water and most gases, satisfy Newton's criterion and are known as Newtonian fluids. NonNewtonian fluids exhibit a more complicated relationship between shear stress and velocity gradient than simple linearity.
The relationship between the shear stress and the velocity gradient can also be obtained by considering two plates closely spaced apart at a distance y, and separated by a homogeneous substance. Assuming that the plates are very large, with a large area A, such that edge effects may be ignored, and that the lower plate is fixed, let a force F be applied to the upper plate. If this force causes the substance between the plates to undergo shear flow (as opposed to just shearing elastically until the shear stress in the substance balances the applied force), the substance is called a fluid. The applied force is proportional to the area and velocity of the plate and inversely proportional to the distance between the plates. Combining these three relations results in the equation F = μ (Au/y), where μ is the proportionality factor called the dynamic viscosity (also called absolute viscosity, or simply viscosity). The equation can be expressed in terms of shear stress; τ = F/A = μ (u / y). The rate of shear deformation is u / y and can be also written as a shear velocity, du/dy. Hence, through this method, the relation between the shear stress and the velocity gradient can be obtained.
James Clerk Maxwell called viscosity fugitive elasticity because of the analogy that elastic deformation opposes shear stress in solids, while in viscous fluids, shear stress is opposed by rate of deformation.
Dynamic viscosity is measured with various types of rheometer. Close temperature control of the fluid is essential to accurate measurements, particularly in materials like lubricants, whose viscosity can double with a change of only 5 °C. For some fluids, it is a constant over a wide range of shear rates. These are Newtonian fluids.
The fluids without a constant viscosity are called nonNewtonian fluids. Their viscosity cannot be described by a single number. NonNewtonian fluids exhibit a variety of different correlations between shear stress and shear rate.
One of the most common instruments for measuring kinematic viscosity is the glass capillary viscometer.
In paint industries, viscosity is commonly measured with a Zahn cup, in which the efflux time is determined and given to customers. The efflux time can also be converted to kinematic viscosities (centistokes, cSt) through the conversion equations.
A Ford viscosity cup measures the rate of flow of a liquid. This, under ideal conditions, is proportional to the kinematic viscosity.
Also used in paint, a Stormer viscometer uses loadbased rotation in order to determine viscosity. The viscosity is reported in Krebs units (KU), which are unique to Stormer viscometers.
Vibrating viscometers can also be used to measure viscosity. These models such as the Dynatrol use vibration rather than rotation to measure viscosity.
Extensional viscosity can be measured with various rheometers that apply extensional stress.
Volume viscosity can be measured with acoustic rheometer.
The usual symbol for dynamic viscosity used by mechanical and chemical engineers — as well as fluid dynamicists — is the Greek letter mu (μ)^{[4]}^{[5]}^{[6]}^{[7]}. The symbol η is also used by chemists, physicists, and the IUPAC^{[8]}.
The SI physical unit of dynamic viscosity is the pascalsecond (Pa·s), which is identical to N·m^{−2}·s. If a fluid with a viscosity of one Pa·s is placed between two plates, and one plate is pushed sideways with a shear stress of one pascal, it moves a distance equal to the thickness of the layer between the plates in one second.
The cgs physical unit for dynamic viscosity is the poise^{[9]} (P), named after Jean Louis Marie Poiseuille. It is more commonly expressed, particularly in ASTM standards, as centipoise (cP). Water at 20 °C has a viscosity of 1.0020 cP or 0.001002 kilogram/meter second.
The relation to the SI unit is
In many situations, we are concerned with the ratio of the viscous force to the inertial force, the latter characterised by the fluid density ρ. This ratio is characterised by the kinematic viscosity (Greek letter nu, ν), defined as follows:
The SI unit of ν is m^{2}/s. The SI unit of ρ is kg/m^{3}.
The cgs physical unit for kinematic viscosity is the stokes (St), named after George Gabriel Stokes. It is sometimes expressed in terms of centistokes (cSt or ctsk). In U.S. usage, stoke is sometimes used as the singular form.
Water at 20 °C has a kinematic viscosity of about 1 cSt.
The kinematic viscosity is sometimes referred to as diffusivity of momentum, because it has the same unit as and is comparable to diffusivity of heat and diffusivity of mass. It is therefore used in dimensionless numbers which compare the ratio of the diffusivities.
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).^{[10]} Other abbreviations such as SSU (Saybolt Seconds Universal) or SUV (Saybolt Universal Viscosity) are sometimes used. Kinematic viscosity in centistoke can be converted from SUS according to the arithmetic and the reference table provided in ASTM D 2161.^{[11]}
The viscosity of a system is determined by how molecules constituting the system interact. There are no simple but correct expressions for the viscosity of a fluid. The simplest exact expressions are the Green–Kubo relations for the linear shear viscosity or the Transient Time Correlation Function expressions derived by Evans and Morriss in 1985. Although these expressions are each exact in order to calculate the viscosity of a dense fluid, using these relations requires the use of molecular dynamics computer simulations.
Viscosity in gases arises principally from the molecular diffusion that transports momentum between layers of flow. The kinetic theory of gases allows accurate prediction of the behavior of gaseous viscosity.
Within the regime where the theory is applicable:
James Clerk Maxwell published a famous paper in 1866 using the kinetic theory of gases to study gaseous viscosity.^{[14]} To understand why the viscosity is independent of pressure consider two adjacent boundary layers (A and B) moving with respect to each other. The internal friction (the viscosity) of the gas is determined by the probability a particle of layer A enters layer B with a corresponding transfer of momentum. Maxwell's calculations showed him that the viscosity coefficient is proportional to both the density, the mean free path and the mean velocity of the atoms. On the other hand, the mean free path is inversely proportional to the density. So an increase of pressure doesn't result in any change of the viscosity.
In relation to diffusion, the kinematic viscosity provides a better understanding of the behavior of mass transport of a dilute species. Viscosity is related to shear stress and the rate of shear in a fluid, which illustrates its dependence on the mean free path, λ, of the diffusing particles.
From fluid mechanics, shear stress, τ, on a unit area moving parallel to itself, is found to be proportional to the rate of change of velocity with distance perpendicular to the unit area:
for a unit area parallel to the xz plane, moving along the x axis. We will derive this formula and show how μ is related to λ.
Interpreting shear stress as the time rate of change of momentum, p, per unit area A (rate of momentum flux) of an arbitrary control surface gives
where is the average velocity along x of fluid molecules hitting the unit area, with respect to the unit area.
Further manipulation will show^{[15]}
where
The ChapmanEnskog equation^{[16]} may be used to estimate viscosity for a dilute gas. This equation is based on a semitheoretical assumption by Chapman and Enskog. The equation requires three empirically determined parameters: the collision diameter (σ), the maximum energy of attraction divided by the Boltzmann constant (є/к) and the collision integral (ω(T^{*})).
with
In liquids, the additional forces between molecules become important. This leads to an additional contribution to the shear stress though the exact mechanics of this are still controversial.^{[citation needed]} Thus, in liquids:
The dynamic viscosities of liquids are typically several orders of magnitude higher than dynamic viscosities of gases.
The viscosity of the blend of two or more liquids can be estimated using the Refutas equation^{[17]}^{[18]}. The calculation is carried out in three steps.
The first step is to calculate the Viscosity Blending Number (VBN) (also called the Viscosity Blending Index) of each component of the blend:
where v is the kinematic viscosity in centistokes (cSt). It is important that the kinematic viscosity of each component of the blend be obtained at the same temperature.
The next step is to calculate the VBN of the blend, using this equation:
where x_{X} is the mass fraction of each component of the blend.
Once the viscosity blending number of a blend has been calculated using equation (2), the final step is to determine the kinematic viscosity of the blend by solving equation (1) for v:
where VBN_{Blend} is the viscosity blending number of the blend.
The viscosity of air and water are by far the two most important materials for aviation aerodynamics and shipping fluid dynamics. Temperature plays the main role in determining viscosity.
The viscosity of air depends mostly on the temperature. At 15.0 °C, the viscosity of air is 1.78 × 10^{−5} kg/(m·s), 17.8 μPa.s or 1.78 × 10^{−4} P. One can get the viscosity of air as a function of temperature from the Gas Viscosity Calculator
The dynamic viscosity of water is 8.90 × 10^{−4} Pa·s or 8.90 × 10^{−3} dyn·s/cm^{2} or 0.890 cP at about 25 °C.
Water has a viscosity of 0.0091 poise at 25 °C, or 1 centipoise at 20 °C.
As a function of temperature T (K): μ(Pa·s) = A × 10^{B/(T−C)}
where A=2.414 × 10^{−5} Pa·s ; B = 247.8 K ; and C = 140 K^{[citation needed]}.
Viscosity of liquid water at different temperatures up to the normal boiling point is listed below.
Temperature
[°C] 
Viscosity
[mPa·s] 

10  1.308 
20  1.002 
30  0.7978 
40  0.6531 
50  0.5471 
60  0.4668 
70  0.4044 
80  0.3550 
90  0.3150 
100  0.2822 
Some dynamic viscosities of Newtonian fluids are listed below:
Gases (at 0 °C):  viscosity
[μPa·s] 

hydrogen  8.4 
air  17.4 
xenon  21.2 
Liquids (at 25 °C):  viscosity
[Pa·s] 
viscosity
[cP=mPa.s] 

acetone^{[19]}  3.06e4  0.306 
benzene^{[19]}  6.04e4  0.604 
blood (37 °C)^{[20]}  3e3 to 4e3  3–4 
castor oil^{[19]}  0.985  985 
corn syrup^{[19]}  1.3806  1380.6 
ethanol^{[19]}  1.074e3  1.074 
ethylene glycol  1.61e2  16.1 
glycerol^{[21]}  1.49 (at 20 °C)  1490 
HFO380  2.022  2022 
mercury^{[19]}  1.526e3  1.526 
methanol^{[19]}  5.44e4  0.544 
nitrobenzene^{[19]}  1.863e3  1.863 
liquid nitrogen @ 77K  1.58e4  0.158 
propanol^{[19]}  1.945e3  1.945 
olive oil  .081  81 
pitch  2.3e8  2.3e11 
sulfuric acid^{[19]}  2.42e2  24.2 
water  8.94e4  0.894 
Fluids with variable compositions  viscosity
[Pa·s] 
viscosity
[cP] 

honey  2–10  2,000–10,000 
molasses  5–10  5,000–10,000 
molten glass  10–1,000  10,000–1,000,000 
chocolate syrup  10–25  10,000–25,000 
molten chocolate^{*}  45–130 ^{[22]}  45,000–130,000 
ketchup^{*}  50–100  50,000–100,000 
peanut butter^{*}  ~250  ~250,000 
shortening^{*}  ~250  250,000 
* These materials are highly nonNewtonian.
On the basis that all solids such as granite^{[23]} flow to a small extent in response to small shear stress, some researchers^{[24]} have contended that substances known as amorphous solids, such as glass and many polymers, may be considered to have viscosity. This has led some to the view that solids are simply liquids with a very high viscosity, typically greater than 10^{12} Pa·s. This position is often adopted by supporters of the widely held misconception that glass flow can be observed in old buildings. This distortion is more likely the result of the glass making process rather than the viscosity of glass.^{[25]}
However, others argue that solids are, in general, elastic for small stresses while fluids are not.^{[26]} Even if solids flow at higher stresses, they are characterized by their lowstress behavior. This distinction can become muddled if measurements are continued over long time periods, such as the Pitch drop experiment. Viscosity may be an appropriate characteristic for solids in a plastic regime. The situation becomes somewhat confused as the term viscosity is sometimes used for solid materials, for example Maxwell materials, to describe the relationship between stress and the rate of change of strain, rather than rate of shear.
These distinctions may be largely resolved by considering the constitutive equations of the material in question, which take into account both its viscous and elastic behaviors. Materials for which both their viscosity and their elasticity are important in a particular range of deformation and deformation rate are called viscoelastic. In geology, earth materials that exhibit viscous deformation at least three times greater than their elastic deformation are sometimes called rheids.
Viscous flow in amorphous materials (e.g. in glasses and melts)^{[28]}^{[29]}^{[30]} is a thermally activated process:
where Q is activation energy, T is temperature, R is the molar gas constant and A is approximately a constant.
The viscous flow in amorphous materials is characterized by a deviation from the Arrheniustype behavior: Q changes from a high value Q_{H} at low temperatures (in the glassy state) to a low value Q_{L} at high temperatures (in the liquid state). Depending on this change, amorphous materials are classified as either
The fragility of amorphous materials is numerically characterized by the Doremus’ fragility ratio:
and strong material have R_{D} < 2 whereas fragile materials have R_{D} ≥ 2.
The viscosity of amorphous materials is quite exactly described by a twoexponential equation:
with constants A_{1}, A_{2}, B, C and D related to thermodynamic parameters of joining bonds of an amorphous material.
Not very far from the glass transition temperature, T_{g}, this equation can be approximated by a VogelFulcherTammann (VFT) equation.
If the temperature is significantly lower than the glass transition temperature, T < T_{g}, then the twoexponential equation simplifies to an Arrhenius type equation:
with:
where H_{d} is the enthalpy of formation of broken bonds (termed configuron s) and H_{m} is the enthalpy of their motion. When the temperature is less than the glass transition temperature, T < T_{g}, the activation energy of viscosity is high because the amorphous materials are in the glassy state and most of their joining bonds are intact.
If the temperature is highly above the glass transition temperature, T > T_{g}, the twoexponential equation also simplifies to an Arrhenius type equation:
with:
When the temperature is higher than the glass transition temperature, T > T_{g}, the activation energy of viscosity is low because amorphous materials are melt and have most of their joining bonds broken which facilitates flow.
The negativeonethird of the trace of the stress tensor is often identified with the thermodynamic pressure,
which only depends upon the equilibrium state potentials like temperature and density (equation of state). In general, the trace of the stress tensor is the sum of thermodynamic pressure contribution plus another contribution which is proportional to the divergence of the velocity field. This constant of proportionality is called the volume viscosity.
In the study of turbulence in fluids, a common practical strategy for calculation is to ignore the smallscale vortices (or eddies) in the motion and to calculate a largescale motion with an eddy viscosity that characterizes the transport and dissipation of energy in the smallerscale flow (see large eddy simulation). Values of eddy viscosity used in modeling ocean circulation may be from 5x10^{4} to 10^{6} Pa·s depending upon the resolution of the numerical grid.
The reciprocal of viscosity is fluidity, usually symbolized by φ = 1 / μ or F = 1 / μ, depending on the convention used, measured in reciprocal poise (cm·s·g^{−1}), sometimes called the rhe. Fluidity is seldom used in engineering practice.
The concept of fluidity can be used to determine the viscosity of an ideal solution. For two components a and b, the fluidity when a and b are mixed is
which is only slightly simpler than the equivalent equation in terms of viscosity:
where χ_{a} and χ_{b} is the mole fraction of component a and b respectively, and μ_{a} and μ_{b} are the components pure viscosities.
Viscous forces in a fluid are a function of the rate at which the fluid velocity is changing over distance. The velocity at any point r is specified by the velocity field v(r). The velocity at a small distance dr from point r may be written as a Taylor series:
where dv / dr is shorthand for the dyadic product of the del operator and the velocity:
This is just the Jacobian of the velocity field.
Viscous forces are the result of relative motion between elements of the fluid, and so are expressible as a function of the velocity field. In other words, the forces at r are a function of v(r) and all derivatives of v(r) at that point. In the case of linear viscosity, the viscous force will be a function of the Jacobian tensor alone. For almost all practical situations, the linear approximation is sufficient.
If we represent x, y, and z by indices 1, 2, and 3 respectively, the i,j component of the Jacobian may be written as ∂_{i} v_{j} where ∂_{i} is shorthand for ∂/∂x_{i}. Note that when the first and higher derivative terms are zero, the velocity of all fluid elements is parallel, and there are no viscous forces.
Any matrix may be written as the sum of an antisymmetric matrix and a symmetric matrix, and this decomposition is independent of coordinate system, and so has physical significance. The velocity field may be approximated as:
where Einstein notation is now being used in which repeated indices in a product are implicitly summed. The second term from the right is the asymmetric part of the first derivative term, and it represents a rigid rotation of the fluid about r with angular velocity ω where:
For such a rigid rotation, there is no change in the relative positions of the fluid elements, and so there is no viscous force associated with this term. The remaining symmetric term is responsible for the viscous forces in the fluid. Assuming the fluid is isotropic (i.e. its properties are the same in all directions), then the most general way that the symmetric term (the rateofstrain tensor) can be broken down in a coordinateindependent (and therefore physically real) way is as the sum of a constant tensor (the rateofexpansion tensor) and a traceless symmetric tensor (the rateofshear tensor):
where δ_{ij} is the unit tensor. The most general linear relationship between the stress tensor σ and the rateofstrain tensor is then a linear combination of these two tensors:^{[31]}
where ς is the coefficient of bulk viscosity (or "second viscosity") and μ is the coefficient of (shear) viscosity.
The forces in the fluid are due to the velocities of the individual molecules. The velocity of a molecule may be thought of as the sum of the fluid velocity and the thermal velocity. The viscous stress tensor described above gives the force due to the fluid velocity only. The force on an area element in the fluid due to the thermal velocities of the molecules is just the hydrostatic pressure. This pressure term (−p δ_{ij}) must be added to the viscous stress tensor to obtain the total stress tensor for the fluid.
The infinitesimal force dF_{i} on an infinitesimal area dA_{i} is then given by the usual relationship:

