Viscosity

Illustration of a planar Couette flow. Since the shearing flow is opposed by friction between adjacent layers of fluid (which are in relative motion), a force is required to sustain the motion of the upper plate. The relative strength of this force is a measure of the fluid's viscosity.

In a general parallel flow, the shear stress is proportional to the gradient of the velocity.

In materials science and engineering, 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 ${\displaystyle u}$ (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 ${\displaystyle 0}$ at the bottom to ${\displaystyle u}$ at the top.[5] 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 ${\displaystyle u}$ at the top. Moreover, the magnitude ${\displaystyle F}$ of the force acting on the top plate is found to be proportional to the speed ${\displaystyle u}$ and the area ${\displaystyle A}$ of each plate, and inversely proportional to their separation ${\displaystyle y}$:

${\displaystyle F=\mu A{\frac {u}{y}}.}$

The proportionality factor ${\displaystyle \mu }$ is the dynamic viscosity of the fluid, with units of ${\displaystyle {\text{Pa}}\cdot {\text{s}}}$ (pascal-second), often simply referred to as the viscosity. The ratio ${\displaystyle u/y}$ is called the rate of shear deformation or shear velocity, and is the derivative of the fluid speed in the direction perpendicular to the plates (see illustrations to the right). If the velocity does not vary linearly with ${\displaystyle y}$, then the appropriate generalization is:

${\displaystyle \tau =\mu {\frac {\partial u}{\partial y}},}$

where ${\displaystyle \tau =F/A}$, and ${\displaystyle \partial u/\partial y}$ is the local shear velocity. This expression is referred to as Newton's law of viscosity. In shearing flows with planar symmetry, it is what defines ${\displaystyle \mu }$. It is a special case of the general definition of viscosity (see below), which can be expressed in coordinate-free form.

Use of the Greek letter mu (${\displaystyle \mu }$) for the dynamic viscosity (sometimes also called the absolute viscosity) is common among mechanical and chemical engineers, as well as mathematicians and physicists.[6][7][8] However, the Greek letter eta (${\displaystyle \eta }$) is also used by chemists, physicists, and the IUPAC.[9] The viscosity ${\displaystyle \mu }$ is sometimes also called the shear viscosity. However, at least one author discourages the use of this terminology, noting that ${\displaystyle \mu }$ can appear in nonshearing flows in addition to shearing flows.[10]

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.[11] (For Newtonian fluids, this is also a linear dependence.) In Cartesian coordinates, the general relationship can then be written as

${\displaystyle \tau _{ij}=\sum _{k}\sum _{\ell }\mu _{ijk\ell }{\frac {\partial v_{k}}{\partial r_{\ell }}},}$

where ${\displaystyle \mu _{ijk\ell }}$ is a viscosity tensor that maps the velocity gradient tensor ${\displaystyle \partial v_{k}/\partial r_{\ell }}$ onto the viscous stress tensor ${\displaystyle \tau _{ij}}$.[12] Since the indices in this expression can vary from 1 to 3, there are 81 "viscosity coefficients" ${\displaystyle \mu _{ijkl}}$ in total. However, assuming that the viscosity rank-4 tensor is isotropic reduces these 81 coefficients to three independent parameters ${\displaystyle \alpha }$, ${\displaystyle \beta }$, ${\displaystyle \gamma }$:

${\displaystyle \mu _{ijk\ell }=\alpha \delta _{ij}\delta _{k\ell }+\beta \delta _{ik}\delta _{j\ell }+\gamma \delta _{i\ell }\delta _{jk},}$

and furthermore, it is assumed that no viscous forces may arise when the fluid is undergoing simple rigid-body rotation, thus ${\displaystyle \beta =\gamma }$, leaving only two independent parameters.[11] The most usual decomposition is in terms of the standard (scalar) viscosity ${\displaystyle \mu }$ and the bulk viscosity ${\displaystyle \kappa }$ such that ${\displaystyle \alpha =\kappa -{\tfrac {2}{3}}\mu }$ and ${\displaystyle \beta =\gamma =\mu }$. In vector notation this appears as:

${\displaystyle \mathbf {\tau } =\mu \left[\nabla \mathbf {v} +(\nabla \mathbf {v} )^{\dagger }\right]-\left({\frac {2}{3}}\mu -\kappa \right)(\nabla \cdot \mathbf {v} )\mathbf {\delta } ,}$

where ${\displaystyle \mathbf {\delta } }$ is the unit tensor, and the dagger ${\displaystyle \dagger }$ denotes the transpose.[10][13] 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 ${\displaystyle \kappa }$ is frequently not necessary in fluid dynamics problems. For example, an incompressible fluid satisfies ${\displaystyle \nabla \cdot \mathbf {v} =0}$ and so the term containing ${\displaystyle \kappa }$ drops out. Moreover, ${\displaystyle \kappa }$ is often assumed to be negligible for gases since it is ${\displaystyle 0}$ in a monoatomic ideal gas.[10] One situation in which ${\displaystyle \kappa }$ can be important is the calculation of energy loss in sound and shock waves, 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.

In fluid dynamics, it is sometimes more convenient to work in terms of the kinematic viscosity (sometimes also called the momentum diffusivity), defined as the ratio of the viscosity μ to the density of the fluid ρ. It is usually denoted by the Greek letter nu (ν) and has dimension ${\displaystyle \mathrm {(length)^{2}/time} }$:

${\displaystyle \nu ={\frac {\mu }{\rho }},}$

in order to distinguish it from the dynamic viscosity ${\displaystyle \mu }$, which has dimension ${\displaystyle \mathrm {force\times time/area} }$.

Elementary calculation of viscosity for a dilute gas

Consider a dilute gas moving parallel to the ${\displaystyle x}$-axis with velocity ${\displaystyle u(y)}$ that depends only on the ${\displaystyle y}$ coordinate. To simplify the discussion, the gas is assumed to have uniform temperature and density. Under these assumptions, the ${\displaystyle x}$ velocity of a molecule passing through ${\displaystyle y=0}$ is equal to whatever velocity that molecule had when its mean free path ${\displaystyle \lambda }$ began. Because ${\displaystyle \lambda }$ is typically small compared with macroscopic scales, the average ${\displaystyle x}$ velocity of such a molecule has the form ${\displaystyle u(0)\pm \alpha \lambda {\frac {du}{dy}}(0),}$ where ${\displaystyle \alpha }$ is a numerical constant on the order of ${\displaystyle 1}$. (Some authors estimate ${\displaystyle \alpha =2/3}$;[14][28] on the other hand, a more careful calculation for rigid elastic spheres gives ${\displaystyle \alpha \simeq 0.998}$.) Now, because half the molecules on either side are moving towards ${\displaystyle y=0}$, and doing so on average with half the average molecular speed ${\displaystyle (8k_{\text{B}}T/\pi m)^{1/2}}$, the momentum flux from either side is ${\displaystyle {\frac {1}{4}}\rho \cdot {\sqrt {\frac {8k_{\text{B}}T}{\pi m}}}\cdot \left(u(0)\pm \alpha \lambda {\frac {du}{dy}}(0)\right).}$ The net momentum flux at ${\displaystyle y=0}$ is the difference of the two: ${\displaystyle -{\frac {1}{2}}\rho \cdot {\sqrt {\frac {8k_{\text{B}}T}{\pi m}}}\cdot \alpha \lambda {\frac {du}{dy}}(0).}$ According to the definition of viscosity, this momentum flux should be equal to ${\displaystyle -\mu {\frac {du}{dy}}(0)}$, which leads to ${\displaystyle \mu =\alpha \rho \lambda {\sqrt {\frac {2k_{\text{B}}T}{\pi m}}}.}$

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 ${\displaystyle T}$ and density ${\displaystyle \rho }$ gives

${\displaystyle \mu =\alpha \rho \lambda {\sqrt {\frac {2k_{\text{B}}T}{\pi m}}},}$

where ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant, ${\displaystyle m}$ the molecular mass, and ${\displaystyle \alpha }$ a numerical constant on the order of ${\displaystyle 1}$. The quantity ${\displaystyle \lambda }$, the mean free path, measures the average distance a molecule travels between collisions. Even without a priori knowledge of ${\displaystyle \alpha }$, this expression has interesting implications. In particular, since ${\displaystyle \lambda }$ is typically inversely proportional to density and increases with temperature, ${\displaystyle \mu }$ 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.[14][28]

For rigid elastic spheres of diameter ${\displaystyle \sigma }$, ${\displaystyle \lambda }$ can be computed, giving

${\displaystyle \mu ={\frac {\alpha }{\pi ^{3/2}}}{\frac {\sqrt {k_{\text{B}}mT}}{\sigma ^{2}}}.}$

In this case ${\displaystyle \lambda }$ is independent of temperature, so ${\displaystyle \mu \propto T^{1/2}}$. For more complicated molecular models, however, ${\displaystyle \lambda }$ depends on temperature in a non-trivial way, and simple kinetic arguments as used here are inadequate. More fundamentally, the notion of a mean free path becomes imprecise for particles that interact over a finite range, which limits the usefulness of the concept for describing real-world gases.[29]

A technique developed by Sydney Chapman and David Enskog in the early 1900s allows a more refined calculation of ${\displaystyle \mu }$.[27] It is based on the Boltzmann equation, which provides a systematic statistical description of a dilute gas in terms of intermolecular interactions.[30] As such, their technique allows accurate calculation of ${\displaystyle \mu }$ for more realistic molecular models, such as those incorporating intermolecular attraction rather than just hard-core repulsion.

It turns out that a more realistic modeling of interactions is essential for accurate prediction of the temperature dependence of ${\displaystyle \mu }$, which experiments show increases more rapidly than the ${\displaystyle T^{1/2}}$ trend predicted for rigid elastic spheres.[14] 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,[lower-alpha 1] 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 ${\displaystyle \mu }$:

${\displaystyle \mu ={\frac {5}{16\sigma ^{2}}}\left({\frac {k_{\text{B}}mT}{\pi }}\right)^{1/2}\left(1+{\frac {S}{T}}\right)^{-1},}$

where ${\displaystyle S}$ is independent of temperature, being determined only by the parameters of the intermolecular attraction. To connect with experiment, it is convenient to rewrite as

${\displaystyle \mu =\mu _{0}\left({\frac {T}{T_{0}}}\right)^{3/2}{\frac {T_{0}+S}{T+S}},}$

where ${\displaystyle \mu _{0}}$ is the viscosity at temperature ${\displaystyle T_{0}}$.[31] If ${\displaystyle \mu }$ is known from experiments at ${\displaystyle T=T_{0}}$ and at least one other temperature, then ${\displaystyle S}$ can be calculated. It turns out that expressions for ${\displaystyle \mu }$ obtained in this way are accurate for a number of gases over a sizable range of temperatures. On the other hand, Chapman & Cowling 1970 argue that this success does not imply that molecules actually interact according to the Sutherland model. Rather, they interpret the prediction for ${\displaystyle \mu }$ as a simple interpolation which is valid for some gases over fixed ranges of temperature, but otherwise does not provide a picture of intermolecular interactions which is fundamentally correct and general. Slightly more sophisticated models, such as the Lennard-Jones potential, may provide a better picture, but only at the cost of a more opaque dependence on temperature. In some systems the assumption of spherical symmetry must be abandoned as well, as is the case for vapors with highly polar molecules like H₂O.[32][33]

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 ${\displaystyle 0}$ for a monatomic ideal gas, in which the internal energy of molecules in negligible, but is nonzero for a gas like carbon dioxide, whose molecules possess both rotational and vibrational energy.[34][35]

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.[36]

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.[37] 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

${\displaystyle \mu \approx {\frac {N_{A}h}{V}}\operatorname {exp} \left(3.8{\frac {T_{b}}{T}}\right),}$

(1)

where ${\displaystyle N_{A}}$ is the Avogadro constant, ${\displaystyle h}$ is the Planck constant, ${\displaystyle V}$ is the volume of a mole of liquid, and ${\displaystyle T_{b}}$ is the normal boiling point. This result has the same form as the widespread and accurate empirical relation

${\displaystyle \mu =Ae^{B/T},}$

(2)

where ${\displaystyle A}$ and ${\displaystyle B}$ are constants fit from data.[37][38] On the other hand, several authors express caution with respect to this model. Errors as large as 30% can be encountered using equation (1), compared with fitting equation (2) to experimental data.[37] More fundamentally, the physical assumptions underlying equation (1) have been criticized.[39] It has also been argued that the exponential dependence in equation (1) does not necessarily describe experimental observations more accurately than simpler, non-exponential expressions.[40][41]

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.[42] 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.[43]

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 ${\displaystyle \mu _{\text{mix}}}$ of a binary mixture of gases can be written in terms of the individual component viscosities ${\displaystyle \mu _{1,2}}$, their respective volume fractions, and the intermolecular interactions.[27] As for the single-component gas, the dependence of ${\displaystyle \mu _{\text{mix}}}$ 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 ${\displaystyle \mu _{\text{mix}}}$ 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.

As for pure liquids, the viscosity of a blend of liquids is difficult to predict from molecular principles. One method is to extend the molecular "cage" theory presented above for a pure liquid. This can be done with varying levels of sophistication. One useful expression resulting from such an analysis is the Lederer–Roegiers equation for a binary mixture:

${\displaystyle \ln \mu _{\text{blend}}={\frac {x_{1}}{x_{1}+\alpha x_{2}}}\ln \mu _{1}+{\frac {\alpha x_{2}}{x_{1}+\alpha x_{2}}}\ln \mu _{2},}$

where ${\displaystyle \alpha }$ is an empirical parameter, and ${\displaystyle x_{1,2}}$ and ${\displaystyle \mu _{1,2}}$ are the respective mole fractions and viscosities of the component liquids.[44]

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.[44]

Depending on the solute and range of concentration, an aqueous electrolyte solution can have either a larger or smaller viscosity compared with pure water at the same temperature and pressure. For instance, a 20% saline (sodium chloride) solution has viscosity over 1.5 times that of pure water, whereas a 20% potassium iodide solution has viscosity about 0.91 times that of pure water.

An idealized model of dilute electrolytic solutions leads to the following prediction for the viscosity ${\displaystyle \mu _{s}}$ of a solution:[45]

${\displaystyle {\frac {\mu _{s}}{\mu _{0}}}=1+A{\sqrt {c}},}$

where ${\displaystyle \mu _{0}}$ is the viscosity of the solvent, ${\displaystyle c}$ is the concentration, and ${\displaystyle A}$ is a positive constant which depends on both solvent and solute properties. However, this expression is only valid for very dilute solutions, having ${\displaystyle c}$ less than 0.1 mol/L.[46] For higher concentrations, additional terms are necessary which account for higher-order molecular correlations:

${\displaystyle {\frac {\mu _{s}}{\mu _{0}}}=1+A{\sqrt {c}}+Bc+Cc^{2},}$

where ${\displaystyle B}$ and ${\displaystyle C}$ are fit from data. In particular, a negative value of ${\displaystyle B}$ is able to account for the decrease in viscosity observed in some solutions. Estimated values of these constants are shown below for sodium chloride and potassium iodide at temperature 25 °C (mol = mole, L = liter).[45]

Solute	${\displaystyle A}$ (mol−1/2 L1/2)	${\displaystyle B}$ (mol−1 L)	${\displaystyle C}$ (mol−2 L2)
Sodium chloride (NaCl)	0.0062	0.0793	0.0080
Potassium iodide (KI)	0.0047	−0.0755	0.0000

In a suspension of solid particles (e.g. micron-size spheres suspended in oil), an effective viscosity ${\displaystyle \mu _{\text{eff}}}$ 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.[47] Such suspensions generally exhibit non-Newtonian behavior. However, for dilute systems in steady flows, the behavior is Newtonian and expressions for ${\displaystyle \mu _{\text{eff}}}$ can be derived directly from the particle dynamics. In a very dilute system, with volume fraction ${\displaystyle \phi \lesssim 0.02}$, interactions between the suspended particles can be ignored. In such a case one can explicitly calculate the flow field around each particle independently, and combine the results to obtain ${\displaystyle \mu _{\text{eff}}}$. For spheres, this results in the Einstein equation:

${\displaystyle \mu _{\text{eff}}=\mu _{0}\left(1+{\frac {5}{2}}\phi \right),}$

where ${\displaystyle \mu _{0}}$ is the viscosity of the suspending liquid. The linear dependence on ${\displaystyle \phi }$ is a direct consequence of neglecting interparticle interactions; in general, one will have

${\displaystyle \mu _{\text{eff}}=\mu _{0}\left(1+B\phi \right),}$

where the coefficient ${\displaystyle B}$ may depend on the particle shape (e.g. spheres, rods, disks).[48] Experimental determination of the precise value of ${\displaystyle B}$ is difficult, however: even the prediction ${\displaystyle B=5/2}$ for spheres has not been conclusively validated, with various experiments finding values in the range ${\displaystyle 1.5\lesssim B\lesssim 5}$. This deficiency has been attributed to difficulty in controlling experimental conditions.[49]

In denser suspensions, ${\displaystyle \mu _{\text{eff}}}$ acquires a nonlinear dependence on ${\displaystyle \phi }$, which indicates the importance of interparticle interactions. Various analytical and semi-empirical schemes exist for capturing this regime. At the most basic level, a term quadratic in ${\displaystyle \phi }$ is added to ${\displaystyle \mu _{\text{eff}}}$:

${\displaystyle \mu _{\text{eff}}=\mu _{0}\left(1+B\phi +B_{1}\phi ^{2}\right),}$

and the coefficient ${\displaystyle B_{1}}$ is fit from experimental data or approximated from the microscopic theory. In general, however, one should be cautious in applying such simple formulas since non-Newtonian behavior appears in dense suspensions (${\displaystyle \phi \gtrsim 0.25}$ for spheres),[49] or in suspensions of elongated or flexible particles.[47]

There is a distinction between a suspension of solid particles, described above, and an emulsion. The latter is a suspension of tiny droplets, which themselves may exhibit internal circulation. The presence of internal circulation can noticeably decrease the observed effective viscosity, and different theoretical or semi-empirical models must be used.[50]

Common glass viscosity curves[51]

In the high and low temperature limits, viscous flow in amorphous materials (e.g. in glasses and melts)[52][53][54] has the Arrhenius form:

${\displaystyle \mu =Ae^{Q/(RT)},}$

where Q is a relevant activation energy, given in terms of molecular parameters; T is temperature; R is the molar gas constant; and A is approximately a constant. The activation energy Q takes a different value depending on whether the high or low temperature limit is being considered: it 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).

Common logarithm of viscosity against temperature for B₂O₃, showing two regimes

For intermediate temperatures, ${\displaystyle Q}$ varies nontrivially with temperature and the simple Arrhenius form fails. On the other hand, the two-exponential equation

${\displaystyle \mu =AT\exp \left({\frac {B}{RT}}\right)\left[1+C\exp \left({\frac {D}{RT}}\right)\right],}$

where ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$, ${\displaystyle D}$ 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.[53]

A two-exponential equation for the viscosity can be derived within the Dyre shoving model of supercooled liquids, where the Arrhenius energy barrier is identified with the high-frequency shear modulus times a characteristic shoving volume.[55] 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:[56]

${\displaystyle \mu =\exp {\left\{{\frac {V_{c}C_{G}}{k_{B}T}}\exp {\left[(2+\lambda )\alpha _{T}T_{g}\left(1-{\frac {T}{T_{g}}}\right)\right]}\right\}}}$

where ${\displaystyle C_{G}}$ denotes the high-frequency shear modulus of the material evaluated at a temperature equal to the glass transition temperature ${\displaystyle T_{g}}$, ${\displaystyle V_{c}}$ 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, ${\displaystyle \alpha _{T}}$ is the thermal expansion coefficient of the material, ${\displaystyle \lambda }$ is a parameter which measures the steepness of the power-law rise of the ascending flank of the first peak of the radial distribution function, and is quantitatively related to the repulsive part of the interatomic potential.[56] Finally, ${\displaystyle k_{B}}$ denotes the Boltzmann constant.

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 the smaller-scale flow (see large eddy simulation).[57][58] In contrast to the viscosity of the fluid itself, which must be positive by the second law of thermodynamics, the eddy viscosity can be negative.[59][60]

The dynamic viscosity ${\displaystyle \mu }$ of water is about 0.89 mPa·s at room temperature (25 °C). As a function of temperature in kelvins, the viscosity can be estimated using the semi-empirical Vogel-Fulcher-Tammann equation:

${\displaystyle \mu =A\exp \left({\frac {B}{T-C}}\right)}$

where A = 0.02939 mPa·s, B = 507.88 K, and C = 149.3 K.[63] Experimentally determined values of the viscosity are also given in the table below. Note that at 20 °C the dynamic viscosity is about 1 cP and the kinematic viscosity is about 1 cSt.

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:[64]

${\displaystyle \eta _{\text{air}}=2.791\cdot 10^{-7}\cdot T^{0.7355}}$

which is accurate in the range -20 °C to 400 °C. For this formula to be valid, the temperature must be given in kelvins; ${\displaystyle \eta _{\text{air}}}$ then corresponds to the viscosity in Pa·s.

Honey being drizzled

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