The viscosity of a fluid is a measure of its resistance 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.
Viscosity can be conceptualized as quantifying the friction that arises between adjacent layers of fluid that are in relative motion. For instance, when a 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 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: the strength of this force is proportional to the viscosity.
A fluid that has no resistance to shear stress is known as an ideal or inviscid fluid. Zero viscosity is observed only at cryogenics in Superfluidity. Otherwise, the second law of thermodynamics requires all fluids to have positive viscosity; such fluids are technically said to be viscous or viscid. A fluid with a relatively high viscosity, such as pitch, may appear to be a solid.
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 $u$ (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 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 $y$:
Use of the Greek letter mu ($\backslash mu$) for the viscosity is common among mechanical and chemical engineers, as well as physicists.
However, the Greek letter eta ($\backslash eta$) is also used by chemists, physicists, and the IUPAC.
where $\backslash mu\_\{ijkl\}$ is a viscosity tensor that maps the strain rate tensor $\backslash partial\; v\_k\; /\; \backslash partial\; r\_l$ onto the viscous stress tensor $\backslash tau\_\{ij\}$.Bird, Steward, & Lightfoot, p. 18 (Note that this source uses a alternate sign convention, which has been reversed here.) Since the indices in this expression can vary from 1 to 3, there are 81 "viscosity coefficients" $\backslash mu\_\{ijkl\}$ in total. However, due to spatial symmetries these coefficients are not all independent. For instance, for isotropic Newtonian fluids, the 81 coefficients can be reduced to 2 independent parameters. The most usual decomposition yields the standard (scalar) viscosity $\backslash mu$ and the bulk viscosity $\backslash kappa$:
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 $\backslash kappa$ is frequently not necessary in fluid dynamics problems. For example, an incompressible fluid satisfies $\backslash nabla\; \backslash cdot\; \backslash mathbf\{v\}\; =\; 0$ and so the term containing $\backslash kappa$ drops out. Moreover, $\backslash kappa$ is often assumed to be negligible for gases since it is $0$ in a monoatomic ideal gas. One situation in which $\backslash kappa$ can be important is the calculation of energy loss in sound and , 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.
Consistent with this nomenclature, the viscosity $\backslash mu$ is frequently called the dynamic viscosity or absolute viscosity, and has units force × time/area.
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 onedimensional forms are given here:
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.
The Trouton ratio or Trouton's ratio is the ratio of extensional viscosity to shear viscosity. For a Newtonian fluid, the Trouton ratio is 3.
Shearthinning liquids are very commonly, but misleadingly, described as thixotropic.
Even for a Newtonian fluid, the viscosity usually depends on its composition and temperature. For gases and other compressible fluids, it depends on temperature and varies very slowly with pressure. The viscosity of some fluids may depend on other factors. A magnetorheological fluid, for example, becomes thicker when subjected to a magnetic field, possibly to the point of behaving like a solid.
However, many liquids (including water) will briefly react like elastic solids when subjected to sudden stress. Conversely, many "solids" (even granite) will flow like liquids, albeit very slowly, even under arbitrarily small stress. Such materials are therefore best described as possessing both elasticity (reaction to deformation) and viscosity (reaction to rate of deformation); that is, being viscoelasticity.
Indeed, some authors have claimed that , such as glass and many polymers, are actually liquids with a very high viscosity (greater than 10^{12} Pa·s). However, other authors dispute this hypothesis, claiming instead that there is some threshold for the stress, below which most solids will not flow at all, and that alleged instances of glass flow in window panes of old buildings are due to the crude manufacturing process of older eras rather than to the viscosity of glass.
Viscoelastic solids may exhibit both shear viscosity and bulk viscosity. The extensional viscosity is a linear combination of the shear and bulk viscosities that describes the reaction of a solid elastic material to elongation. It is widely used for characterizing polymers.
In geology, earth materials that exhibit viscous deformation at least three orders of magnitude greater than their elastic deformation are sometimes called .
For some fluids, the viscosity is constant over a wide range of shear rates (Newtonian fluids). The fluids without a constant viscosity (nonNewtonian fluids) 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 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. The efflux time can also be converted to kinematic viscosities (centistokes, cSt) through the conversion equations.
Also used in coatings, 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. 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 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.
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), 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.
The reciprocal of viscosity is fluidity, usually symbolized by $\backslash phi\; =\; 1\; /\; \backslash mu$ or $F\; =\; 1\; /\; \backslash mu$, depending on the convention used, measured in reciprocal poise (P^{−1}, or centimetre·second·gram^{−1}), sometimes called the rhe. Fluidity is seldom used in engineering practice.
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.
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).ASTM D 2161 (2005) "Standard Practice for Conversion of Kinematic Viscosity to Saybolt Universal Viscosity or to Saybolt Furol Viscosity", p. 1 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.
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.
! Elementary calculation of viscosity for a dilute gas 
Consider a dilute gas moving parallel to the $x$axis with velocity $u(y)$ that depends only on the $y$ coordinate. To simplify the discussion, the gas is assumed to have uniform temperature and density.
Under these assumptions, the $x$ velocity of a molecule passing through $y\; =\; 0$ is equal to whatever velocity that molecule had when its mean free path $\backslash lambda$ began. Because $\backslash lambda$ is typically small compared with macroscopic scales, the average $x$ velocity of such a molecule has the form
where $\backslash alpha$ is a numerical constant on the order of $1$. (Some authors estimate $\backslash alpha\; =\; 2/3$; on the other hand, a more careful calculation for rigid elastic spheres gives $\backslash alpha\; \backslash simeq\; 0.998$.) Now, because half the molecules on either side are moving towards $y=0$, and doing so on average with half the average moleculer speed $(8\; k\_\backslash text\{B\}\; T/\backslash pi\; m)^\{1/2\}$, the momentum flux from either side is
According to the definition of viscosity, this momentum flux should be equal to $\backslash mu\; \backslash frac\{d\; u\}\{d\; y\}(0)$, which leads to

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 $\backslash rho$ gives
where $k\_\backslash text\{B\}$ is the Boltzmann constant, $m$ the molecular mass, and $\backslash alpha$ a numerical constant on the order of $1$. The quantity $\backslash lambda$, the mean free path, measures the average distance a molecule travels between collisions. Even without a priori knowledge of $\backslash alpha$, this expression has interesting implications. In particular, since $\backslash lambda$ is typically inversely proportional to density and increases with temperature, $\backslash mu$ 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 $\backslash sigma$, $\backslash lambda$ can be computed, giving
In this case $\backslash lambda$ is independent of temperature, so $\backslash mu\; \backslash propto\; T^\{1/2\}$. For more complicated molecular models, however, $\backslash lambda$ depends on temperature in a nontrivial 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 realworld gases.Chapman & Cowling, p. 103
It turns out that a more realistic modeling of interactions is essential for accurate prediction of the temperature dependence of $\backslash mu$, which experiments show increases more rapidly than the $T^\{1/2\}$ 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,The discussion which follows draws from Chapman & Cowling, pp. 232237. which describes rigid elastic spheres with weak mutual attraction. In such a case, the attractive force can be treated perturbatively, which leads to a particularly simple expression for $\backslash mu$:
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.Bird, Steward, & Lightfoot, pp. 2931 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 $N\_A$ is the Avogadro constant, $h$ is the Planck constant, $V$ is the volume of a 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.Reid & Sherwood, pp. 203204 One 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 extensively criticized. It has also been argued that the exponential dependence in equation () does not necessarily describe experimental observations more accurately than simpler, nonexponential expressions.Hildebrand p. 37
In light of these shortcomings, the development of a less adhoc 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 IrvingKirkwood 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.Reid & Sherwood, pp. 206209
where $\backslash alpha$ is an empirical parameter, and $x\_\{1,2\}$ and $\backslash mu\_\{1,2\}$ 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.
$\backslash mu\_\{\backslash text\{eff\}\}\; =\; \backslash mu\_0\; \backslash left(1\; +\; \backslash frac\{5\}\{2\}\; \backslash phi\; \backslash right),$
where $\backslash mu\_0$ is the viscosity of the suspending liquid. The linear dependence on $\backslash phi$ is a direct consequence of neglecting interparticle interactions; in general, one will have:
$\backslash mu\_\{\backslash text\{eff\}\}\; =\; \backslash mu\_0\; \backslash left(1\; +\; B\; \backslash phi\; \backslash right),$
where the coefficient $B$ may depend on the particle shape (e.g. spheres, rods, disks).Bird, Steward, & Lightfoot p. 32 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 $1.5\; \backslash lesssim\; B\; \backslash lesssim\; 5$. This deficiency has been attributed to difficulty in controlling experimental conditions.
In denser suspensions, $\backslash mu\_\{\backslash text\{eff\}\}$ acquires a nonlinear dependence on $\backslash phi$, which indicates the importance of interparticle interactions. Various analytical and semiempirical schemes exist for capturing this regime. At the most basic level, a term quadratic in $\backslash phi$ is added to $\backslash mu\_\{\backslash text\{eff\}\}$:
$\backslash mu\_\{\backslash text\{eff\}\}\; =\; \backslash mu\_0\; \backslash left(1\; +\; B\; \backslash phi\; +\; B\_1\; \backslash phi^2\; \backslash right),$
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 nonNewtonian behavior appears in dense suspensions ($\backslash phi\; \backslash gtrsim\; 0.25$ 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 semiempirical models must be used.Bird, Steward, & Lightfoot p. 33
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 twoexponential equation
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.
where = , = 247.8 K, and = 140 K.
Experimentally determined values of the viscosity at various temperatures are given below.
+Viscosity of water at various temperatures  
10  1.3059 
20  1.0016 
30  0.79722 
50  0.54652 
70  0.40355 
90  0.31417 
Benzene  0.604  25 
Water  1.0016  20 
Mercury  1.526  25 
Whole milk  2.12  20 
Olive oil  56.2  26 
Honey  $\backslash approx$ 200010000  20 
Ketchup  $\backslash approx$ 500020000  25 
Peanut butter  $\backslash approx$ 10^{4}10^{6}  
Pitch  1030 (variable) 

