Viscosity | |
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A simulation of liquids with different viscosities. The liquid on the right has higher viscosity than the liquid on the left. | |

Common symbols | η , μ |

Derivations from other quantities | μ = G · t |

Part of a series on | ||||

Continuum mechanics | ||||
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Laws
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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.^{ [1] }

- Etymology
- Definition
- Simple definition
- General definition
- Dynamic and kinematic viscosity
- Momentum transport
- Newtonian and non-Newtonian fluids
- In solids
- Measurement
- Units
- Molecular origins
- Pure gases
- Pure liquids
- Mixtures and blends
- Solutions and suspensions
- Amorphous materials
- Eddy viscosity
- Selected substances
- Water
- Air
- Other common substances
- Order of magnitude estimates
- See also
- References
- Footnotes
- Citations
- Sources
- External links

Viscosity can be conceptualized as quantifying the internal frictional force 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 very low temperatures in superfluids. Otherwise, the second law of thermodynamics requires all fluids to have positive viscosity;^{ [2] }^{ [3] } such fluids are technically said to be viscous or viscid. A fluid with a high viscosity, such as pitch, may appear to be a solid.

The word "viscosity" is derived from the Latin * viscum* ("mistletoe").

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 (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.^{ [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 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 viscosity of the fluid, with units of (pascal-second). The ratio 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 , 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 viscosity is common among mechanical and chemical engineers, as well as physicists.^{ [6] }^{ [7] }^{ [8] } However, the Greek letter eta () is also used by chemists, physicists, and the IUPAC.^{ [9] } The viscosity is sometimes also referred to as the *shear viscosity*. However, at least one author discourages the use of this terminology, noting that 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

where is a viscosity tensor that maps the velocity gradient tensor onto the viscous stress tensor .^{ [12] } 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.^{ [11] } 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.^{ [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 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 monoatomic ideal gas.^{ [10] } One situation in which 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 common to work in terms of the *kinematic viscosity* (also called "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 :

Consistent with this nomenclature, the viscosity is frequently called the *dynamic viscosity* or *absolute viscosity*, and has units force × time/area.

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 characterizes heat transport, and (mass) diffusivity characterizes mass transport.^{ [14] } To see this, note that in Newton's law of viscosity, , the shear stress has units equivalent to a momentum flux, 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.^{ [15] } 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.

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, 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 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 (dilatant) 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.^{ [16] }^{ [17] } Shear-thinning liquids are very commonly, but misleadingly, described as thixotropic.^{ [18] }

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.

The viscous forces that arise during fluid flow must not be confused with the elastic forces that arise in a solid in response to shear, compression or extension stresses. While in the latter the stress is proportional to the *amount* of shear deformation, in a fluid it is proportional to the *rate* of deformation over time. (For this reason, Maxwell 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) will flow like liquids, albeit very slowly, even under arbitrarily small stress.^{ [19] } Such materials are therefore best described as possessing both elasticity (reaction to deformation) and viscosity (reaction to rate of deformation); that is, being viscoelastic.

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 rheids.^{ [20] }

Viscosity is measured with various types of viscometers and rheometers. A rheometer is used for those 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 acquire accurate measurements, particularly in materials like lubricants, whose viscosity can double with a change of only 5 °C.^{ [21] }

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. The efflux time can also be converted to kinematic viscosities (centistokes, cSt) through the conversion equations.

Also used in coatings, a Stormer viscometer uses load-based rotation in order to determine viscosity. The viscosity is reported in Krebs units (KU), which are unique to Stormer viscometers.

Vibrating viscometers can also be used to measure viscosity. Resonant, or vibrational viscometers work by creating shear waves within the liquid. In this method, the sensor is submerged in the fluid and is made to resonate at a specific frequency. As the surface of the sensor shears through the liquid, energy is lost due to its viscosity. This dissipated energy is then measured and converted into a viscosity reading. A higher viscosity causes a greater loss of energy.^{[ citation needed ]}

*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.

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) and kilogram per meter per second (kg·m^{−1}·s^{−1}). The CGS unit is the poise (P, or g·cm^{−1}·s^{−1} = 0.1 Pa·s),^{ [22] } named after Jean Léonard Marie Poiseuille. It is commonly expressed, particularly in ASTM standards, as *centipoise* (cP), because centipoise is equal to the SI millipascal seconds (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.^{ [23] } 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 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}). Note that the pound and pound-force are equivalent; the two systems differ only in how force and mass are defined. In the BG system the pound is a basic unit from which the unit of mass (the slug) 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 [[gc (engineering) proportionality constant ''g<sub>c</sub>'']].

Kinematic viscosity has units of square feet per second (ft^{2}) in both the BG and EE systems.

Nonstandard units include the reyn, a British unit of dynamic viscosity.^{[ citation needed ]} In the automotive industry the viscosity index 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 practice.

At one time the petroleum industry relied on measuring kinematic viscosity by means of the Saybolt viscometer, and expressing kinematic viscosity in units of Saybolt universal seconds (SUS).^{ [24] } 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.

In general, the viscosity of a system depends in detail on how the molecules constituting the system interact. There are no simple but correct expressions for the viscosity of a fluid. The simplest exact expressions are the Green–Kubo relations for the linear shear viscosity or the *transient time correlation function* expressions derived by Evans and Morriss in 1988.^{ [25] } Although these expressions are each exact, calculating the viscosity of a dense fluid using these relations currently requires the use of molecular dynamics computer simulations. On the other hand, much more progress can be made for a dilute gas. Even elementary assumptions about how gas molecules move and interact lead to a basic understanding of the molecular origins of viscosity. More sophisticated treatments can be constructed by systematically coarse-graining the equations of motion of the gas molecules. An example of such a treatment is Chapman–Enskog theory, which derives expressions for the viscosity of a dilute gas from the Boltzmann equation.^{ [26] }

Momentum transport in gases is generally mediated by discrete molecular collisions, and in liquids by attractive forces which bind molecules close together.^{ [14] } 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 -axis with velocity that depends only on the coordinate. To simplify the discussion, the gas is assumed to have uniform temperature and density.

Under these assumptions, the velocity of a molecule passing through is equal to whatever velocity that molecule had when its mean free path began. Because is typically small compared with macroscopic scales, the average velocity of such a molecule has the form

where is a numerical constant on the order of . (Some authors estimate ;

^{ [14] }^{ [27] }on the other hand, a more careful calculation for rigid elastic spheres gives .) Now, because*half*the molecules on either side are moving towards , and doing so on average with*half*the average moleculer speed , the momentum flux from either side isThe

*net*momentum flux at is the difference of the two:According to the definition of viscosity, this momentum flux should be equal to , 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 and density gives

where is the Boltzmann constant, 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 interesting 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. Note that this behavior runs counter to common intuition regarding liquids, for which viscosity typically *decreases* with temperature.^{ [14] }^{ [27] }

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.^{ [28] }

A technique developed by Sydney Chapman and David Enskog in the early 1900s allows a more refined calculation of .^{ [26] } It is based on the Boltzmann equation, which provides a systematic statistical description of a dilute gas in terms of intermolecular interactions.^{ [29] } As such, their technique allows accurate calculation of 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 , which experiments show increases more rapidly than the 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 :

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 .^{ [30] } If is known from experiments at and at least one other temperature, then can be calculated. It turns out that expressions for 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 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 H_{2}O.^{ [31] }^{ [32] }

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, whose molecules possess both rotational and vibrational energy.^{ [33] }^{ [34] }

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.^{ [35] }

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.^{ [36] } 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

**(1)**

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 widespread and accurate empirical relation

**(2)**

where and are constants fit from data.^{ [36] }^{ [37] } 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.^{ [36] } More fundamentally, the physical assumptions underlying equation (** 1 **) have been criticized.^{ [38] } 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.^{ [39] }^{ [40] }

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.^{ [41] } 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.^{ [42] }

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.^{ [26] } 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.

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:

where is an empirical parameter, and and are the respective mole fractions and viscosities of the component liquids.^{ [43] }

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.^{ [43] }

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 of a solution:^{ [44] }

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.^{ [45] } 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).^{ [44] }

Solute | (mol^{−1/2} L^{1/2}) | (mol^{−1} L) | (mol^{−2} L^{2}) |
---|---|---|---|

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 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.^{ [46] } 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 direct consequence of neglecting interparticle interactions; in general, one will have

where the coefficient may depend on the particle shape (e.g. spheres, rods, disks).^{ [47] } 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.^{ [48] }

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. In general, however, one should be cautious in applying such simple formulas since non-Newtonian behavior appears in dense suspensions ( for spheres),^{ [48] } or in suspensions of elongated or flexible particles.^{ [46] }

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.^{ [49] }

In the high and low temperature limits, viscous flow in amorphous materials (e.g. in glasses and melts)^{ [51] }^{ [52] }^{ [53] } has the Arrhenius form:

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).

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. Besides being a convenient fit to data, the expression can also be derived from various theoretical models of amorphous materials at the atomic level.^{ [52] }

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.^{ [54] } Upon specifying the temperature dependence of the shear modulus via thermal expansion and via the repulsive part of the intermolecular potential, a two-exponential equation is retrieved as in the Krausser-Samwer-Zaccone (KSZ) equation (developed by Alessio Zaccone, J. Krausser and Konrad Samwer). The parameters in the KSZ equation are expressed in terms of physical quantities such as the repulsive part of the interatomic or intermolecular interaction, the thermal expansion coefficient and the glass transition temperature of the system.^{ [55] } The KSZ equation reads as:

where denotes the high-frequency shear modulus 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.^{ [55] } Finally, 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).^{ [56] }^{ [57] } 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.^{ [58] }^{ [59] }

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.^{ [61] } More dramatically, pitch has been estimated to have a viscosity 230 billion times that of water.^{ [60] }

The dynamic viscosity 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:

where *A* = 0.02939 mPa·s, *B* = 507.88 K, and *C* = 149.3 K.^{ [62] } Experimentally determined values of the viscosity are also given in the table below.

Temperature (°C) | Viscosity (mPa·s) |
---|---|

10 | 1.3059 |

20 | 1.0016 |

30 | 0.79722 |

50 | 0.54652 |

70 | 0.40355 |

90 | 0.31417 |

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:^{ [63] }

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.

Substance | Viscosity (mPa·s) | Temperature (°C) | Ref. |
---|---|---|---|

Benzene | 0.604 | 25 | ^{ [61] } |

Water | 1.0016 | 20 | |

Mercury | 1.526 | 25 | |

Whole milk | 2.12 | 20 | ^{ [64] } |

Dark beer | 2.53 | 20 | ^{ [65] } |

Olive oil | 56.2 | 26 | ^{ [64] } |

Honey | 2000–10000 | 20 | ^{ [66] } |

Ketchup ^{ [lower-alpha 2] } | 5000–20000 | 25 | ^{ [67] } |

Peanut butter ^{ [lower-alpha 2] } | 10^{4}–10^{6} | ^{ [68] } | |

Pitch | 2.3×10^{11} | 10–30 (variable) | ^{ [60] } |

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.

Factor (Pa·s) | Description | Examples | Values (Pa·s) | Ref. |
---|---|---|---|---|

10^{−6} | Lower range of gaseous viscosity | Butane | 7.49 × 10^{−6} | ^{ [69] } |

Hydrogen | 8.8 × 10^{−6} | ^{ [70] } | ||

10^{−5} | Upper range of gaseous viscosity | Krypton | 2.538 × 10^{−5} | ^{ [71] } |

Neon | 3.175 × 10^{−5} | |||

10^{−4} | Lower range of liquid viscosity | Pentane | 2.24 × 10^{−4} | ^{ [61] } |

Gasoline | 6 × 10^{−4} | |||

Water | 8.90 × 10^{−4} | ^{ [61] } | ||

10^{−3} | Typical range for small-molecule Newtonian liquids | Ethanol | 1.074 × 10^{−3} | |

Mercury | 1.526 × 10^{−3} | |||

Whole milk (20 °C) | 2.12 × 10^{−3} | ^{ [64] } | ||

Blood | 4 × 10^{−3} | |||

10^{−2} – 10^{0} | Oils and long-chain hydrocarbons | Linseed oil | 0.028 | |

Olive oil | 0.084 | ^{ [64] } | ||

SAE 10 Motor oil | 0.085 to 0.14 | |||

Castor oil | 0.1 | |||

SAE 20 Motor oil | 0.14 to 0.42 | |||

SAE 30 Motor oil | 0.42 to 0.65 | |||

SAE 40 Motor oil | 0.65 to 0.90 | |||

Glycerine | 1.5 | |||

Pancake syrup | 2.5 | |||

10^{1} – 10^{3} | Pastes, gels, and other semisolids (generally non-Newtonian) | Ketchup | ≈ 10^{1} | ^{ [67] } |

Mustard | ||||

Sour cream | ≈ 10^{2} | |||

Peanut butter | ^{ [68] } | |||

Lard | ≈ 10^{3} | |||

≈10^{8} | Viscoelastic polymers | Pitch | ^{ [60] } | |

≈10^{21} | Certain solids under a viscoelastic description | Mantle (geology) |

- Dashpot
- Deborah number
- Dilatant
- Herschel–Bulkley fluid
- Hyperviscosity syndrome
- Intrinsic viscosity
- Inviscid flow
- Joback method (estimation of liquid viscosity from molecular structure)
- Kaye effect
- Microviscosity
- Morton number
- Quasi-solid
- Rheology
- Stokes flow
- Superfluid helium-4
- Viscoplasticity
- Viscosity models for mixtures

A **viscometer** is an instrument used to measure the viscosity of a fluid. For liquids with viscosities which vary with flow conditions, an instrument called a rheometer is used. Thus, a rheometer can be considered as a special type of viscometer. Viscometers only measure under one flow condition.

In physics, the **Navier–Stokes equations** are a set of partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes.

The **Grashof number** (**Gr**) is a dimensionless number in fluid dynamics and heat transfer which approximates the ratio of the buoyancy to viscous force acting on a fluid. It frequently arises in the study of situations involving natural convection and is analogous to the Reynolds number. It's believed to be named after Franz Grashof. Though this grouping of terms had already been in use, it wasn't named until around 1921, 28 years after Franz Grashof's death. It's not very clear why the grouping was named after him.

In physics and fluid mechanics, a **boundary layer** is the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant.

A **Newtonian fluid** is a fluid in which the viscous stresses arising from its flow, at every point, are linearly correlated to the local strain rate—the rate of change of its deformation over time. That is equivalent to saying those forces are proportional to the rates of change of the fluid's velocity vector as one moves away from the point in question in various directions.

**Shear stress**, often denoted by **τ**, is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. Normal stress, on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts.

A **power-law fluid**, or the **Ostwald–de Waele relationship**, is a type of generalized Newtonian fluid for which the shear stress, *τ*, is given by

A **Bingham plastic** is a viscoplastic material that behaves as a rigid body at low stresses but flows as a viscous fluid at high stress. It is named after Eugene C. Bingham who proposed its mathematical form.

**Hemorheology**, also spelled **haemorheology**, or **blood rheology**, is the study of flow properties of blood and its elements of plasma and cells. Proper tissue perfusion can occur only when blood's rheological properties are within certain levels. Alterations of these properties play significant roles in disease processes. Blood viscosity is determined by plasma viscosity, hematocrit and mechanical properties of red blood cells. Red blood cells have unique mechanical behavior, which can be discussed under the terms erythrocyte deformability and erythrocyte aggregation. Because of that, blood behaves as a non-Newtonian fluid. As such, the viscosity of blood varies with shear rate. Blood becomes less viscous at high shear rates like those experienced with increased flow such as during exercise or in peak-systole. Therefore, blood is a shear-thinning fluid. Contrarily, blood viscosity increases when shear rate goes down with increased vessel diameters or with low flow, such as downstream from an obstruction or in diastole. Blood viscosity also increases with increases in red cell aggregability.

**Darcy's law** is an equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on results of experiments on the flow of water through beds of sand, forming the basis of hydrogeology, a branch of earth sciences.

The **Marangoni effect** is the mass transfer along an interface between two fluids due to a gradient of the surface tension. In the case of temperature dependence, this phenomenon may be called **thermo-capillary convection**.

In fluid dynamics, **Couette flow** is the flow of a viscous fluid in the space between two surfaces, one of which is moving tangentially relative to the other. The configuration often takes the form of two parallel plates or the gap between two concentric cylinders. The flow is driven by virtue of viscous drag force acting on the fluid, but may additionally be motivated by an applied pressure gradient in the flow direction. The Couette configuration models certain practical problems, like flows in the Earth's mantle and the atmosphere, flow in lightly loaded journal bearings, and is often employed in viscometry and to demonstrate approximations of reversibility. This type of flow is named in honor of Maurice Couette, a Professor of Physics at the French University of Angers in the late 19th century.

Viscosity depends strongly on temperature. In liquids it usually decreases with increasing temperature, whereas in gases viscosity increases with increasing temperature. This article discusses several models of this dependence, ranging from rigorous first-principles calculations for monatomic gases, to empirical correlations for liquids.

**Fluid mechanics** is the branch of physics concerned with the mechanics of fluids and the forces on them. It has applications in a wide range of disciplines, including mechanical, civil, chemical and biomedical engineering, geophysics, oceanography, meteorology, astrophysics, and biology.

The **Brinkman number** (**Br**) is a dimensionless number related to heat conduction from a wall to a flowing viscous fluid, commonly used in polymer processing. It is named after the Dutch mathematician and physicist Henri Brinkman. There are several definitions; one is

The intent of this article is to highlight the important points of the **derivation of the Navier–Stokes equations** as well as its application and formulation for different families of fluids.

**Volume viscosity** is a material property relevant for characterizing fluid flow. Common symbols are or . It has dimensions, and the corresponding SI unit is the pascal-second (Pa·s).

The **Herschel–Bulkley fluid** is a generalized model of a non-Newtonian fluid, in which the strain experienced by the fluid is related to the stress in a complicated, non-linear way. Three parameters characterize this relationship: the consistency *k*, the flow index *n*, and the yield shear stress . The consistency is a simple constant of proportionality, while the flow index measures the degree to which the fluid is shear-thinning or shear-thickening. Ordinary paint is one example of a shear-thinning fluid, while oobleck provides one realization of a shear-thickening fluid. Finally, the yield stress quantifies the amount of stress that the fluid may experience before it yields and begins to flow.

The **Reynolds number** helps predict flow patterns in different fluid flow situations. At low Reynolds numbers, flows tend to be dominated by laminar (sheet-like) flow, while at high Reynolds numbers flows tend to be turbulent. The turbulence results from differences in the fluid's speed and direction, which may sometimes intersect or even move counter to the overall direction of the flow. These eddy currents begin to churn the flow, using up energy in the process, which for liquids increases the chances of cavitation. Reynolds numbers are an important dimensionless quantity in fluid mechanics.

The **viscous stress tensor** is a tensor used in continuum mechanics to model the part of the stress at a point within some material that can be attributed to the strain rate, the rate at which it is deforming around that point.

- ↑ The discussion which follows draws from Chapman & Cowling 1970, pp. 232–237
- 1 2 These materials are highly non-Newtonian.

- ↑ Symon 1971.
- ↑ Balescu 1975, pp. 428–429.
- ↑ Landau & Lifshitz 1987.
- ↑ Harper, Douglas (n.d.). "viscous (adj.)".
*Online Etymology Dictionary*. Retrieved 19 September 2019. - ↑ Mewis & Wagner 2012, p. 19.
- ↑ Streeter, Wylie & Bedford 1998.
- ↑ Holman 2002.
- ↑ Incropera et al. 2007.
- ↑ Nič et al. 1997.
- 1 2 3 Bird, Stewart & Lightfoot 2007, p. 19.
- 1 2 Landau & Lifshitz 1987, pp. 44–45.
- ↑ Bird, Stewart & Lightfoot 2007, p. 18: Note that this source uses an alternative sign convention, which has been reversed here.
- ↑ Landau & Lifshitz 1987, p. 45.
- 1 2 3 4 5 Bird, Stewart & Lightfoot 2007.
- ↑ Schroeder 1999.
- ↑ Różańska et al. 2014, pp. 47–55.
- ↑ Trouton 1906, pp. 426–440.
- ↑ Mewis & Wagner 2012, pp. 228–230.
- ↑ Kumagai, Sasajima & Ito 1978, pp. 157–161.
- ↑ Scherer, Pardenek & Swiatek 1988, p. 14.
- ↑ Hannan, Henry (2007).
*Technician's Formulation Handbook for Industrial and Household Cleaning Products*. Waukesha, Wisconsin: Kyral LLC. p. 7. ISBN 978-0-6151-5601-9. - ↑ McNaught & Wilkinson 1997, poise.
- ↑ Gyllenbok 2018, p. 213.
- ↑
*ASTM D2161 : Standard Practice for Conversion of Kinematic Viscosity to Saybolt Universal Viscosity or to Saybolt Furol Viscosity*, ASTM, 2005, p. 1 - ↑ Evans & Morriss 1988, pp. 4142–4148.
- 1 2 3 Chapman & Cowling 1970.
- 1 2 Bellac, Mortessagne & Batrouni 2004.
- ↑ Chapman & Cowling 1970, p. 103.
- ↑ Cercignani 1975.
- ↑ Sutherland 1893, pp. 507–531.
- ↑ Bird, Stewart & Lightfoot 2007, pp. 25–27.
- ↑ Chapman & Cowling 1970, pp. 235–237.
- ↑ Chapman & Cowling 1970, pp. 197, 214–216.
- ↑ Cramer 2012, p. 066102-2.
- ↑ Reid & Sherwood 1958, p. 202.
- 1 2 3 Bird, Stewart & Lightfoot 2007, pp. 29–31.
- ↑ Reid & Sherwood 1958, pp. 203–204.
- ↑ Hildebrand 1958.
- ↑ Hildebrand 1958, p. 37.
- ↑ Egelstaff 1992, p. 264.
- ↑ Irving & Kirkwood 1949, pp. 817–829.
- ↑ Reid & Sherwood 1958, pp. 206–209.
- 1 2 Zhmud 2014, p. 22.
- 1 2 Viswanath et al. 2007.
- ↑ Abdulagatov, Zeinalova & Azizov 2006, pp. 75–88.
- 1 2 Bird, Stewart & Lightfoot 2007, pp. 31–33.
- ↑ Bird, Stewart & Lightfoot 2007, p. 32.
- 1 2 Mueller, Llewellin & Mader 2009, pp. 1201–1228.
- ↑ Bird, Stewart & Lightfoot 2007, p. 33.
- ↑ Fluegel 2007.
- ↑ Doremus 2002, pp. 7619–7629.
- 1 2 Ojovan, Travis & Hand 2007, p. 415107.
- ↑ Ojovan & Lee 2004, pp. 3803–3810.
- ↑ Dyre, Olsen & Christensen 1996, p. 2171.
- 1 2 Krausser, Samwer & Zaccone 2015, p. 13762.
- ↑ Bird, Stewart & Lightfoot 2007, p. 163.
- ↑ Lesieur 2012, pp. 2–.
- ↑ Sivashinsky & Yakhot 1985, p. 1040.
- ↑ Xie & Levchenko 2019, p. 045434.
- 1 2 3 4 Edgeworth, Dalton & Parnell 1984, pp. 198–200.
- 1 2 3 4 5 Rumble 2018.
- ↑ Viswanath & Natarajan 1989, pp. 714–715.
- ↑ tec-science (2020-03-25). "Viscosity of liquids and gases".
*tec-science*. Retrieved 2020-05-07. - 1 2 3 4 Fellows 2009.
- ↑ Severa & Los 2008.
- ↑ Yanniotis, Skaltsi & Karaburnioti 2006, pp. 372–377.
- 1 2 Koocheki et al. 2009, pp. 596–602.
- 1 2 Citerne, Carreau & Moan 2001, pp. 86–96.
- ↑ Kestin, Khalifa & Wakeham 1977.
- ↑ Assael et al. 2018.
- ↑ Kestin, Ro & Wakeham 1972.

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